**Spacecraft Navigation**

**Chapter 3**

Provisional chapter

**Fractal Pyramid: A New Math Tool to Reorient and**

DOI: 10.5772/intechopen.71751

Fractal Pyramid: A New Math Tool to Reorient and

An original mathematical instrument matching two different operational procedures aimed to change orientation and velocity of a spacecraft is suggested and described in detail. The tool's basements, quaternion algebra with its square-root (pregeometric) image, and fractal surface are represented in a parenthetical but in a sufficient format, indicating their principle properties providing solution to the operational task. A supplementary notion of vector-quaternion version of relativity theory is introduced since the spacecraft-observer mechanical system appears congenitally relativistic. The new tool is shown to have a simple pregeometric image of a fractal pyramid whose tilt and distortion evoke needed changes in the spacecraft's motion parameters, and the respective math procedures proved to be simplified compared with the traditionally used

In classical mechanics, rotation of a rigid body (in particular, a spacecraft) and its translational motion are normally regarded as drastically different actions leading to changes in its position and are respectively described by different groups. Relativistic mechanics, in its turn, deals with these two types of motions "more homogeneously" since rotation and linear motion are described in this case by 4 � 4 matrices from the Lorentz group SOð Þ 1; 3 . However, it is well known that the special relativity limits itself by inertial motions of the involved frames of reference while use of general relativity comprising any types of motion but demanding math methods of tensor calculus seems unapproved sophisticated. Happily, there exists a simpler vector version of the relativity theory admitting arbitrary accelerated motion of the frames. A brief formulation of the theory is made with the help of quaternion vector units, each set of the

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

Keywords: spacecraft motion, operation, quaternion, fractal surface

**Accelerate a Spacecraft**

Accelerate a Spacecraft

http://dx.doi.org/10.5772/intechopen.71751

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Alexander P. Yefremov

Alexander P. Yefremov

Abstract

math methods.

1. Introduction

Provisional chapter

#### **Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft** Fractal Pyramid: A New Math Tool to Reorient and

DOI: 10.5772/intechopen.71751

Alexander P. Yefremov

Additional information is available at the end of the chapter Alexander P. Yefremov

http://dx.doi.org/10.5772/intechopen.71751 Additional information is available at the end of the chapter

Accelerate a Spacecraft

#### Abstract

An original mathematical instrument matching two different operational procedures aimed to change orientation and velocity of a spacecraft is suggested and described in detail. The tool's basements, quaternion algebra with its square-root (pregeometric) image, and fractal surface are represented in a parenthetical but in a sufficient format, indicating their principle properties providing solution to the operational task. A supplementary notion of vector-quaternion version of relativity theory is introduced since the spacecraft-observer mechanical system appears congenitally relativistic. The new tool is shown to have a simple pregeometric image of a fractal pyramid whose tilt and distortion evoke needed changes in the spacecraft's motion parameters, and the respective math procedures proved to be simplified compared with the traditionally used math methods.

Keywords: spacecraft motion, operation, quaternion, fractal surface

#### 1. Introduction

In classical mechanics, rotation of a rigid body (in particular, a spacecraft) and its translational motion are normally regarded as drastically different actions leading to changes in its position and are respectively described by different groups. Relativistic mechanics, in its turn, deals with these two types of motions "more homogeneously" since rotation and linear motion are described in this case by 4 � 4 matrices from the Lorentz group SOð Þ 1; 3 . However, it is well known that the special relativity limits itself by inertial motions of the involved frames of reference while use of general relativity comprising any types of motion but demanding math methods of tensor calculus seems unapproved sophisticated. Happily, there exists a simpler vector version of the relativity theory admitting arbitrary accelerated motion of the frames. A brief formulation of the theory is made with the help of quaternion vector units, each set of the

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

units representing a Cartesian-type frame of reference. In this case, the rotation-and-translation operator is given by 3 � 3 matrix belonging to the group SOð Þ 3;С known to be 1:1 isomorphic to the group SOð Þ 1; 3 . However, the calculations of the body's complex motions even within the framework of the vector-quaternion relativity remain prolonged and cumbersome, a simpler method is desired. Such a method is found due to existence of 1:2 isomorphism of the groups SOð Þ 3; С and SLð Þ 2;С , the last being a spinor group operating in fractal two-dimensional complex-number valued space (a fractal surface). It is necessary to mention that the subgroup of SLð Þ 2;С , rotational group SUð Þ2 , is normally used in space-flight practice, providing comparatively simple mathematical computations for a spacecraft reorientation tasks [1, 2]. This method is based upon similarity-type transformations of the initial quaternion triad, in fact assuming nontrivial multiplication of at least three different quaternions, though it straightforwardly gives the data describing the axis of single rotation and value of the respective angle. However, this method provides no translational motion.

12 <sup>¼</sup> <sup>1</sup>,<sup>i</sup>

scalar ð Þa and vector bkq<sup>k</sup>

Chivita symbols (see e.g., [4]).

square root from the norm) j j <sup>q</sup> � ffiffiffiffiffi

� � � � 2

j j q

ð Þ ac � b1d<sup>1</sup> � b2d<sup>2</sup> � b3d<sup>3</sup>

� �, <sup>B</sup> <sup>¼</sup> d e

þð Þ ad<sup>3</sup> þ cb<sup>3</sup> þ b1d<sup>2</sup> � b2d<sup>1</sup>

then from definition of the norm one finds

<sup>2</sup> <sup>¼</sup> <sup>q</sup>1q<sup>2</sup> � � � �

multiplication in this algebra is no more associative).

f �d

right ¼ q1q2= q<sup>2</sup>

has the form

q1=q<sup>2</sup> � �

<sup>A</sup> <sup>¼</sup> a b c �a <sup>2</sup> <sup>¼</sup> <sup>j</sup>

<sup>q</sup> � <sup>a</sup> <sup>þ</sup> bkq<sup>k</sup> has its conjugate <sup>q</sup> � <sup>a</sup> � bkqk, the norm <sup>q</sup><sup>2</sup> �

<sup>2</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> ¼ �1, <sup>1</sup><sup>i</sup> <sup>¼</sup> <sup>i</sup><sup>1</sup> <sup>¼</sup> <sup>i</sup>, <sup>1</sup><sup>j</sup> <sup>¼</sup> <sup>j</sup><sup>1</sup> <sup>¼</sup> <sup>j</sup>, <sup>1</sup><sup>k</sup> <sup>¼</sup> <sup>k</sup><sup>1</sup> <sup>¼</sup> k, ij ¼ �ji <sup>¼</sup> <sup>k</sup>, jk ¼ �kj <sup>¼</sup> <sup>i</sup>, ki ¼ �ik <sup>¼</sup> j, : (1)

� � parts <sup>q</sup> � <sup>a</sup> <sup>þ</sup> bkqk, where a, bk <sup>∈</sup> <sup>R</sup>, and the multiplication table (1)

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft

� �

� � <sup>¼</sup> <sup>q</sup>1q2q2q<sup>1</sup> <sup>¼</sup> <sup>q</sup>1q1q2q<sup>2</sup> <sup>¼</sup> <sup>q</sup><sup>1</sup>

<sup>p</sup> . Inverse number is <sup>q</sup>�<sup>1</sup> <sup>¼</sup> <sup>q</sup>=j j <sup>q</sup>

. If q is a product of two multipliers q<sup>1</sup> ¼ a þ bkq<sup>k</sup> and q<sup>2</sup> ¼ c þ dnqn,

� �, traceless: TrA <sup>¼</sup> TrB <sup>¼</sup> 0, and not degenerate: det<sup>A</sup> 6¼ 0,

, (2)

http://dx.doi.org/10.5772/intechopen.71751

37

� � qq ¼ qq, and the modulus (positive

� �

� � � � 2 q2 � � � � 2

<sup>2</sup> <sup>þ</sup> <sup>d</sup><sup>2</sup> 3 � �: (4)

<sup>1</sup> <sup>þ</sup> <sup>d</sup><sup>2</sup>

<sup>2</sup> <sup>þ</sup> ð Þ ad<sup>2</sup> <sup>þ</sup> cb<sup>2</sup> <sup>þ</sup> <sup>b</sup>3d<sup>1</sup> � <sup>b</sup>1d<sup>3</sup>

2

left ¼ q2q1= q<sup>2</sup> � � � � <sup>2</sup> and

; so, for two

: (3)

2 þ

Q-numbers and the multiplication law (1) can be more compactly rewritten in the vector (and tensor) notations i, j, k ! q1, q2, q<sup>3</sup> ! qk, j, k, l, m, n… ¼ 1, 2, 3; then, a quaternion is a sum of

1q<sup>k</sup> ¼ qk1 ¼ qk, q<sup>k</sup> q<sup>l</sup> ¼ �δkl þ εklj q<sup>j</sup>

Summation in repeated indices is implied, and δkl and εklj are the 3D Kronecker and Levi-

Quaternions admit the same operations as real and complex numbers. Comparison of Q-numbers is reduced to their equality: two Q-numbers are equal if coefficients at respective units are equal. Commutative addition (subtraction) of Q-numbers is made by components. Q-numbers are multiplied as polynomials; the rules (1, 2) state that multiplication is noncommutative (left and right products are defined), but still associative. A quaternion

> qq <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a<sup>2</sup> þ bkbk

quaternions q<sup>1</sup> and q2, division (left and right) is defined as q1=q<sup>2</sup>

� � <sup>q</sup>1q<sup>2</sup>

Written in components, Eq. (3) becomes the famous identity of four squares

<sup>2</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup>

detB 6¼ 0. We use the matrices to build two different imaginary units as

<sup>2</sup> <sup>þ</sup> ð Þ ad<sup>1</sup> <sup>þ</sup> cb<sup>1</sup> <sup>þ</sup> <sup>b</sup>2d<sup>2</sup> � <sup>b</sup>3d<sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> 3 � � <sup>c</sup><sup>2</sup> <sup>þ</sup> <sup>d</sup><sup>2</sup>

Identities of the type (4) exist only in four algebras: of real numbers (trivial identity), of complex numbers (two squares), of quaternions (four squares), and of octonions (the last exclusive algebra with one real and seven imaginary units admits identity of eight squares;

Geometrically, the imaginary Q-units are associated with three unit vectors initiating a Cartesian coordinate system (Q-triad, Q-frame). This image, in particular, follows from the fact that, according to Eq. (2), each imaginary unit appears as ordered product of the two others: q<sup>1</sup> ¼ q2q3, q<sup>2</sup> ¼ q3q1, q<sup>3</sup> ¼ q1q<sup>2</sup> (vector products in Gibbs-Heaviside algebra). One can easily construct a set of such units. To demonstrate this, we consider a couple of 2 � 2-matrices,

<sup>2</sup> <sup>¼</sup> <sup>q</sup>1q<sup>2</sup>

In this study, we suggest an essential development of the last (single rotation) method leading, first, to noticeable simplification of computations, and second, to possibility of introduction of additional parameters responsible for the spacecraft acceleration. This development is fully based on fundamental properties of subgeometric dyad forming the fractal space in a way underlying the 3D physical space. Moreover, we suggest subgeometric images (fractal joystick and fractal pyramid) of the math tools realizing the spacecraft's reorientation and acceleration tasks. As well, we give a brief comparative analysis of simplicity (or complexity) of conventional and new methods.

The study is composed as following. In Sections 2–4 we offer a detailed mathematical introduction. In Section 2, we renew our knowledge of quaternion algebra giving traditional (Hamiltonian) and more compact (tensor) notions and correlations. In Section 3, we briefly reproduce the quaternion version of the relativity theory. In Section 4, we consider main notions and properties of the 2D fractal space and show how to build a 3D frame out of a dyad element.

Sections 5–7 are devoted to new math methods making operations of a spacecraft simpler and more functional. Section 5 is devoted to presentation of three methods to reorient a spacecraft with accent on convenience of the single rotation method involving a fractal joystick model. In Section 6, we suggest a very simple way to introduce (apart from space rotation) an acceleration of the spacecraft and demonstrate a subgeometric image of the respective math tool having a shape of fractal pyramid. Finally, in Section 7, we give a sketch of a technological map previewing necessary steps to simultaneously reorient and accelerate the spacecraft followed by a series of relevant pictures.

#### 2. Basic notions and relations of quaternion algebra

Quaternion (Q-) numbers were discovered by Hamilton in 1843 [3]. A quaternion is a math object of the type q ¼ a1 þ bi þ cj þ dk (in Hamilton's notation), where a, b, c, d are real coefficients at the real unit 1 (the symbol is normally omitted in the number) and at three imaginary units i, j, k forming the postulated multiplication table (16 equalities).

$$\begin{aligned} \mathbf{1}^2 = 1, \mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = -1, \mathbf{i}\mathbf{i} = \mathbf{i}\mathbf{1} = \mathbf{i}, \mathbf{1}\mathbf{j} = \mathbf{j}\mathbf{1} = \mathbf{j}, \mathbf{1}\mathbf{k} = \mathbf{k}\mathbf{1} = \mathbf{k}\mathbf{\\ \mathbf{i}}\mathbf{j} = -\mathbf{j}\mathbf{i} = \mathbf{k}, \mathbf{j}\mathbf{k} = -\mathbf{k}\mathbf{j} = \mathbf{i}, \mathbf{k}\mathbf{i} = -\mathbf{i}\mathbf{k} = \mathbf{j}\mathbf{}\end{aligned} \tag{1}$$

Q-numbers and the multiplication law (1) can be more compactly rewritten in the vector (and tensor) notations i, j, k ! q1, q2, q<sup>3</sup> ! qk, j, k, l, m, n… ¼ 1, 2, 3; then, a quaternion is a sum of scalar ð Þa and vector bkq<sup>k</sup> � � parts <sup>q</sup> � <sup>a</sup> <sup>þ</sup> bkqk, where a, bk <sup>∈</sup> <sup>R</sup>, and the multiplication table (1) has the form

units representing a Cartesian-type frame of reference. In this case, the rotation-and-translation operator is given by 3 � 3 matrix belonging to the group SOð Þ 3;С known to be 1:1 isomorphic to the group SOð Þ 1; 3 . However, the calculations of the body's complex motions even within the framework of the vector-quaternion relativity remain prolonged and cumbersome, a simpler method is desired. Such a method is found due to existence of 1:2 isomorphism of the groups SOð Þ 3; С and SLð Þ 2;С , the last being a spinor group operating in fractal two-dimensional complex-number valued space (a fractal surface). It is necessary to mention that the subgroup of SLð Þ 2;С , rotational group SUð Þ2 , is normally used in space-flight practice, providing comparatively simple mathematical computations for a spacecraft reorientation tasks [1, 2]. This method is based upon similarity-type transformations of the initial quaternion triad, in fact assuming nontrivial multiplication of at least three different quaternions, though it straightforwardly gives the data describing the axis of single rotation and value of the respec-

In this study, we suggest an essential development of the last (single rotation) method leading, first, to noticeable simplification of computations, and second, to possibility of introduction of additional parameters responsible for the spacecraft acceleration. This development is fully based on fundamental properties of subgeometric dyad forming the fractal space in a way underlying the 3D physical space. Moreover, we suggest subgeometric images (fractal joystick and fractal pyramid) of the math tools realizing the spacecraft's reorientation and acceleration tasks. As well, we give a brief comparative analysis of simplicity (or complexity) of conven-

The study is composed as following. In Sections 2–4 we offer a detailed mathematical introduction. In Section 2, we renew our knowledge of quaternion algebra giving traditional (Hamiltonian) and more compact (tensor) notions and correlations. In Section 3, we briefly reproduce the quaternion version of the relativity theory. In Section 4, we consider main notions and properties of the 2D fractal space and show how to build a 3D frame out of a dyad element.

Sections 5–7 are devoted to new math methods making operations of a spacecraft simpler and more functional. Section 5 is devoted to presentation of three methods to reorient a spacecraft with accent on convenience of the single rotation method involving a fractal joystick model. In Section 6, we suggest a very simple way to introduce (apart from space rotation) an acceleration of the spacecraft and demonstrate a subgeometric image of the respective math tool having a shape of fractal pyramid. Finally, in Section 7, we give a sketch of a technological map previewing necessary steps to simultaneously reorient and accelerate the spacecraft

Quaternion (Q-) numbers were discovered by Hamilton in 1843 [3]. A quaternion is a math object of the type q ¼ a1 þ bi þ cj þ dk (in Hamilton's notation), where a, b, c, d are real coefficients at the real unit 1 (the symbol is normally omitted in the number) and at three imaginary

tive angle. However, this method provides no translational motion.

tional and new methods.

36 Space Flight

followed by a series of relevant pictures.

2. Basic notions and relations of quaternion algebra

units i, j, k forming the postulated multiplication table (16 equalities).

$$\mathbf{1}\mathbf{q}\_{k} = \mathbf{q}\_{k}\mathbf{1} = \mathbf{q}\_{k'} \quad \mathbf{q}\_{k} \ \mathbf{q}\_{l} = -\delta\_{kl} + \varepsilon\_{kl\dot{\jmath}} \ \mathbf{q}\_{\dot{\jmath}'} \tag{2}$$

Summation in repeated indices is implied, and δkl and εklj are the 3D Kronecker and Levi-Chivita symbols (see e.g., [4]).

Quaternions admit the same operations as real and complex numbers. Comparison of Q-numbers is reduced to their equality: two Q-numbers are equal if coefficients at respective units are equal. Commutative addition (subtraction) of Q-numbers is made by components. Q-numbers are multiplied as polynomials; the rules (1, 2) state that multiplication is noncommutative (left and right products are defined), but still associative. A quaternion <sup>q</sup> � <sup>a</sup> <sup>þ</sup> bkq<sup>k</sup> has its conjugate <sup>q</sup> � <sup>a</sup> � bkqk, the norm <sup>q</sup><sup>2</sup> � � � � � qq ¼ qq, and the modulus (positive square root from the norm) j j <sup>q</sup> � ffiffiffiffiffi qq <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a<sup>2</sup> þ bkbk <sup>p</sup> . Inverse number is <sup>q</sup>�<sup>1</sup> <sup>¼</sup> <sup>q</sup>=j j <sup>q</sup> 2 ; so, for two quaternions q<sup>1</sup> and q2, division (left and right) is defined as q1=q<sup>2</sup> � � left ¼ q2q1= q<sup>2</sup> � � � � <sup>2</sup> and q1=q<sup>2</sup> � � right ¼ q1q2= q<sup>2</sup> � � � � 2 . If q is a product of two multipliers q<sup>1</sup> ¼ a þ bkq<sup>k</sup> and q<sup>2</sup> ¼ c þ dnqn, then from definition of the norm one finds

$$\left|\langle q\rangle\right|^2 = \left|q\_1q\_2\right|^2 = \left(q\_1q\_2\right)\left(\overline{q\_1q\_2}\right) = q\_1q\_2\overline{q}\_2\overline{q}\_1 = q\_1\overline{q}\_1q\_2\overline{q}\_2 = \left|q\_1\right|^2\left|q\_2\right|^2. \tag{3}$$

Written in components, Eq. (3) becomes the famous identity of four squares

$$\begin{aligned} \left(ac - b\_1d\_1 - b\_2d\_2 - b\_3d\_3\right)^2 &+ \left(ad\_1 + cb\_1 + b\_2d\_2 - b\_3d\_2\right)^2 + \left(ad\_2 + cb\_2 + b\_3d\_1 - b\_1d\_3\right)^2 + \\ \left(+ ad\_3 + cb\_3 + b\_1d\_2 - b\_2d\_1\right)^2 &= \left(a^2 + b\_1^2 + b\_2^2 + b\_3^2\right)\left(c^2 + d\_1^2 + d\_2^2 + d\_3^2\right). \end{aligned} \tag{4}$$

Identities of the type (4) exist only in four algebras: of real numbers (trivial identity), of complex numbers (two squares), of quaternions (four squares), and of octonions (the last exclusive algebra with one real and seven imaginary units admits identity of eight squares; multiplication in this algebra is no more associative).

Geometrically, the imaginary Q-units are associated with three unit vectors initiating a Cartesian coordinate system (Q-triad, Q-frame). This image, in particular, follows from the fact that, according to Eq. (2), each imaginary unit appears as ordered product of the two others: q<sup>1</sup> ¼ q2q3, q<sup>2</sup> ¼ q3q1, q<sup>3</sup> ¼ q1q<sup>2</sup> (vector products in Gibbs-Heaviside algebra). One can easily construct a set of such units. To demonstrate this, we consider a couple of 2 � 2-matrices, <sup>A</sup> <sup>¼</sup> a b c �a � �, <sup>B</sup> <sup>¼</sup> d e f �d � �, traceless: TrA <sup>¼</sup> TrB <sup>¼</sup> 0, and not degenerate: det<sup>A</sup> 6¼ 0, detB 6¼ 0. We use the matrices to build two different imaginary units as

$$\mathbf{q}\_1 = \frac{A}{\sqrt{\det A}}, \quad \mathbf{q}\_2 = \frac{B}{\sqrt{\det B}}.\tag{5}$$

Y N

nN ! Rk 0

Product of multiple hyperbolic rotations is physically sensible if accompanied by real rotations in the framework of vector version of theory of relativity (see Section 3); so in general, the

> <sup>m</sup><sup>1</sup> ::R<sup>α</sup><sup>N</sup> nN ::H<sup>η</sup><sup>s</sup>

The second type of transformations is performed by an operator U and its inverse U�<sup>1</sup> is

<sup>0</sup> <sup>¼</sup> <sup>U</sup>qkU�<sup>1</sup>

It is evident that the transformation (12) keeps the form of the basic law (2). The operators U are known to form the (spinor) group U ∈SLð Þ 2;C of special linear 2D transformations over field of complex numbers; this group is 2:1 isomorphic to SOð Þ 3;C and similarly to the Lorentz group. A special case of the transformation (12) is a real rotation made by means of the subgroup SUð Þ2 ∈ SLð Þ 2;C , and this spinor subgroup is 2:1 isomorphic to vector group SOð Þ 3;R . It is necessary to note that the transformation of the type (12) with U ∈SUð Þ2 is most frequently used for solution of a spacecraft orientation problem (see

As well, in formulation of quaternion relativity (see Section 3), we shall need notion of a biquaternion (BQ-) number. Such a number has the form b ¼ x þ ykqk, where x, yk ∈ C while 1, q<sup>k</sup> are Q-units. BQ-numbers admit addition, multiplication, and conjugation b ¼ x � ykqk. But the norm is not well defined since the product bb <sup>¼</sup> x2 <sup>þ</sup> ykyk in general is not a real (and positive) number. A real number "norm" exists in the subset of vector biquaternions

There are evidently zero dividers in Eq. (14), hence division is not well defined, but the subset (13 and 14) comprises basic formulas describing relative motion of arbitrary accelerated frames

ms ! Ok<sup>0</sup>

mSOð Þ 3;R : (10)

http://dx.doi.org/10.5772/intechopen.71751

39

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft

<sup>m</sup> ∈SOð Þ 3;C (11)

: (12)

b ¼ ð Þ wk þ i zk q<sup>k</sup> (13)

wkzk <sup>¼</sup> <sup>0</sup> ! k k<sup>b</sup> <sup>2</sup> <sup>¼</sup> bb <sup>¼</sup> wkwk � zkzk: (14)

j¼1 Rαj nj <sup>¼</sup> <sup>R</sup><sup>α</sup><sup>1</sup> n1 ⋯R<sup>α</sup><sup>N</sup>

ms <sup>¼</sup> <sup>R</sup><sup>α</sup><sup>1</sup>

<sup>n</sup><sup>1</sup> ::H<sup>η</sup><sup>1</sup>

qk

Y N

Y M

s¼1 Rαj njH<sup>η</sup><sup>s</sup>

whose real and imaginary parts are mutually orthogonal

j¼1

matrices of the type

are used in applications.

given as

Section 5.2).

of reference.

We form the product of the two units and demand that its trace vanishes that is given as

$$\mathbf{q}\_1 \mathbf{q}\_2 = \frac{AB}{\sqrt{\det A \det B}'} \quad \text{Tr}(AB) = 0;\tag{6}$$

then Eq. (6) gives expression for the third imaginary Q-unit q1q<sup>2</sup> ¼ q3, and as a whole, we get the Q-triad <sup>q</sup>k, the real unit always remaining the unit matrix 1 � 1 0 0 1 � �. One readily checks up that the triad given by Eqs. (5) and (6) identically satisfies the multiplication law (2). Built in a similar way, the simplest representation of Q-units q~k is given by the Pauli matrices p~k with factor –i: <sup>q</sup>~<sup>k</sup> ¼ �ip~<sup>k</sup>

$$\mathbf{1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \ \mathbf{q}\_{\bar{\mathbf{1}}} = -i \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \ \mathbf{q}\_{\bar{\mathbf{2}}} = -i \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \ \mathbf{q}\_{\bar{\mathbf{3}}} = -i \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \tag{7}$$

and the imaginary Q-triad given as Eq. (7) describes a constant Q-vector frame.

However, a Q-frame may be variable, rotating, and moving. There are two types of transformations changing the frame but retaining the form of the multiplication law (2). The first is rotational-type transformation

$$\mathbf{q}\_{k'} = O\_{k'n} \mathbf{q}\_n \tag{8}$$

where Ok<sup>0</sup> <sup>n</sup> is a 3 � 3-matrix (its components are in general complex numbers) having orthogonal properties Ok 0 nOm<sup>0</sup> <sup>n</sup> ¼ δkm, hence this matrix belongs to the special orthogonal group of 3D rotations over field of complex numbers Ok<sup>0</sup> <sup>n</sup> ∈SOð Þ 3; C . The matrix On<sup>0</sup> <sup>k</sup> can be always represented as a product of plane (or simple) rotations, irreducible representations of SOð Þ <sup>3</sup>;<sup>C</sup> . For such matrices, a special notation will be used, e.g., <sup>O</sup><sup>Θ</sup> <sup>n</sup> , where the lower index indicates the rotation axis (the frame's unit vector) and upper index shows the rotation angle. Depending on the math nature of the angle Θ, we distinguish two types of simple rotations. If <sup>Θ</sup> <sup>¼</sup> <sup>α</sup><sup>∈</sup> <sup>R</sup>, then we have a real simple rotation <sup>O</sup><sup>Θ</sup> <sup>n</sup> ! <sup>R</sup><sup>α</sup> <sup>n</sup>; if the angle is imaginary Θ ¼ η ∈i R, then we have a simple hyperbolic rotation O<sup>Θ</sup> <sup>n</sup> ! <sup>H</sup><sup>η</sup> <sup>n</sup>; for example Eq. (9)

$$R\_3^a \equiv \begin{pmatrix} \cos a & \sin a & 0 \\ -\sin a & \cos a & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad H\_3^\eta \equiv \begin{pmatrix} \cos h\eta & -i\sin h\eta & 0 \\ i\sin h\eta & \cos h\eta & 0 \\ 0 & 0 & 1 \end{pmatrix}. \tag{9}$$

Superposition of any number (N) of real rotations (product of relevant matrices) gives a (nonplane) real rotation

$$\prod\_{j=1}^{N} \mathbb{R}\_{n\_j}^{a\_j} = \mathbb{R}\_{n\_1}^{a\_1} \cdots \mathbb{R}\_{n\_N}^{a\_N} \to \mathbb{R}\_{k'm} \text{SO}(3, \mathbb{R}).\tag{10}$$

Product of multiple hyperbolic rotations is physically sensible if accompanied by real rotations in the framework of vector version of theory of relativity (see Section 3); so in general, the matrices of the type

$$\prod\_{j=1}^{N} \prod\_{s=1}^{M} R\_{n\_j}^{a\_j} H\_{m\_s}^{\eta\_s} = R\_{n\_1}^{a\_1} \dots H\_{m\_1}^{\eta\_1} \dots R\_{n\_N}^{a\_N} \dots H\_{m\_s}^{\eta\_s} \to \mathcal{O}\_{k'm} \in \mathcal{SO}(3, \mathbb{C}) \tag{11}$$

are used in applications.

<sup>q</sup><sup>1</sup> <sup>¼</sup> <sup>A</sup>

<sup>q</sup>1q<sup>2</sup> <sup>¼</sup> AB ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

the Q-triad <sup>q</sup>k, the real unit always remaining the unit matrix 1 � 1 0

0 1 1 0 � �

and the imaginary Q-triad given as Eq. (7) describes a constant Q-vector frame.

SOð Þ <sup>3</sup>;<sup>C</sup> . For such matrices, a special notation will be used, e.g., <sup>O</sup><sup>Θ</sup>

cos α sin α 0 � sin α cos α 0 0 01

, q<sup>~</sup><sup>1</sup> ¼ �i

factor –i: <sup>q</sup>~<sup>k</sup> ¼ �ip~<sup>k</sup>

38 Space Flight

where Ok<sup>0</sup>

onal properties Ok

<sup>1</sup> <sup>¼</sup> 1 0 0 1 � �

rotational-type transformation

0 nOm<sup>0</sup>

Rα <sup>3</sup> �

(nonplane) real rotation

rotations over field of complex numbers Ok<sup>0</sup>

<sup>Θ</sup> <sup>¼</sup> <sup>α</sup><sup>∈</sup> <sup>R</sup>, then we have a real simple rotation <sup>O</sup><sup>Θ</sup>

then we have a simple hyperbolic rotation O<sup>Θ</sup>

0

BB@

ffiffiffiffiffiffiffiffiffiffiffi

We form the product of the two units and demand that its trace vanishes that is given as

det<sup>A</sup> <sup>p</sup> , <sup>q</sup><sup>2</sup> <sup>¼</sup> <sup>B</sup>

then Eq. (6) gives expression for the third imaginary Q-unit q1q<sup>2</sup> ¼ q3, and as a whole, we get

up that the triad given by Eqs. (5) and (6) identically satisfies the multiplication law (2). Built in a similar way, the simplest representation of Q-units q~k is given by the Pauli matrices p~k with

, q<sup>~</sup><sup>2</sup> ¼ �i

However, a Q-frame may be variable, rotating, and moving. There are two types of transformations changing the frame but retaining the form of the multiplication law (2). The first is

q<sup>k</sup><sup>0</sup> ¼ Ok<sup>0</sup>

represented as a product of plane (or simple) rotations, irreducible representations of

indicates the rotation axis (the frame's unit vector) and upper index shows the rotation angle. Depending on the math nature of the angle Θ, we distinguish two types of simple rotations. If

<sup>n</sup> ! <sup>H</sup><sup>η</sup>

Superposition of any number (N) of real rotations (product of relevant matrices) gives a

1

CCA , H<sup>η</sup> <sup>3</sup> �

<sup>n</sup> is a 3 � 3-matrix (its components are in general complex numbers) having orthog-

<sup>n</sup> ! <sup>R</sup><sup>α</sup>

0

BB@

<sup>n</sup> ¼ δkm, hence this matrix belongs to the special orthogonal group of 3D

<sup>n</sup> ∈SOð Þ 3; C . The matrix On<sup>0</sup>

<sup>n</sup>; for example Eq. (9)

cos hη �i sin hη 0 i sin hη cos hη 0 0 01

ffiffiffiffiffiffiffiffiffiffi

0 �i i 0 � �

det<sup>B</sup> <sup>p</sup> : (5)

detAdet<sup>B</sup> <sup>p</sup> , Tr AB ð Þ¼ <sup>0</sup>; (6)

0 1 � �

, q<sup>~</sup><sup>3</sup> ¼ �i

<sup>n</sup>q<sup>n</sup> (8)

1 0 0 1 � �

. One readily checks

, (7)

<sup>k</sup> can be always

CCA: (9)

<sup>n</sup> , where the lower index

<sup>n</sup>; if the angle is imaginary Θ ¼ η ∈i R,

1

The second type of transformations is performed by an operator U and its inverse U�<sup>1</sup> is given as

$$\mathbf{q}\_{k'} = \mathcal{U}\mathbf{q}\_k\mathcal{U}^{-1}.\tag{12}$$

It is evident that the transformation (12) keeps the form of the basic law (2). The operators U are known to form the (spinor) group U ∈SLð Þ 2;C of special linear 2D transformations over field of complex numbers; this group is 2:1 isomorphic to SOð Þ 3;C and similarly to the Lorentz group. A special case of the transformation (12) is a real rotation made by means of the subgroup SUð Þ2 ∈ SLð Þ 2;C , and this spinor subgroup is 2:1 isomorphic to vector group SOð Þ 3;R . It is necessary to note that the transformation of the type (12) with U ∈SUð Þ2 is most frequently used for solution of a spacecraft orientation problem (see Section 5.2).

As well, in formulation of quaternion relativity (see Section 3), we shall need notion of a biquaternion (BQ-) number. Such a number has the form b ¼ x þ ykqk, where x, yk ∈ C while 1, q<sup>k</sup> are Q-units. BQ-numbers admit addition, multiplication, and conjugation b ¼ x � ykqk. But the norm is not well defined since the product bb <sup>¼</sup> x2 <sup>þ</sup> ykyk in general is not a real (and positive) number. A real number "norm" exists in the subset of vector biquaternions

$$b = \begin{pmatrix} \mathbf{w}\_k + \mathbf{i} \ \mathbf{z}\_k \end{pmatrix} \ \mathbf{q}\_k \tag{13}$$

whose real and imaginary parts are mutually orthogonal

$$\mathfrak{w}\_k \mathfrak{z}\_k = \mathbf{0} \to \|\mathfrak{b}\|^2 = b\overline{\mathfrak{b}} = \mathfrak{w}\_k \mathfrak{w}\_k - \mathfrak{z}\_k \mathfrak{z}\_k. \tag{14}$$

There are evidently zero dividers in Eq. (14), hence division is not well defined, but the subset (13 and 14) comprises basic formulas describing relative motion of arbitrary accelerated frames of reference.

#### 3. Vector-quaternion version of the relativity theory

According to Eqs. (13) and (14), the interval of Einstein's relativity theory<sup>1</sup>

$$d\mathbf{s}^2 = d\mathbf{x}\_0^2 - d\mathbf{x}\_k d\mathbf{x}\_k = dt^2 - dr^2 \tag{15}$$

vectors q<sup>2</sup> and q3. It is easily checked up that all matrices O of the type (19) constitute a subgroup

Main idea of Q-version of relativity is to replace line element of Einstein's relativity (15) and its invariance under Lorentz group by adequate BQ-vector (16) invariant under rotational group represented by matrices O ∈SOð Þ 1; 2 . Then, instead of quadratic form of four-dimensional coordinates, an observer has at his disposal a movable Q-triad with time and distances measured along its unit vectors and dealt with the vector basement as with the Newtonian mechanics or general relativity in tetrad formulation. However, on this way, an essential peculiarity arises. Eq. (16) implies that the constructed space-time model has six dimensions, and it is a symmetric sum of two three-dimensional (3D) spaces Q<sup>6</sup> ¼ R<sup>3</sup> ⊕ T3, where R<sup>3</sup> is the usual 3D space where coordinate and velocity change, whereas T<sup>3</sup> is also a 3D space but imaginary with respect to R3. In this model, the observer works only with some sections of the 6D space; but since the objects of the observations are found in real 3D space, and

Physical measurements in the Q-model are made with the help of three spatial rulers q<sup>k</sup> and built-in geometric clock represented by "imaginary time rulers" (Pauli-type matrices) p<sup>k</sup> � iqk,

Now, the principal statement of the Q-version of relativity follows: all physically sustainable

The sustainability means form-invariance of BQ-vector (16) or (20) under transformations (21). Kinematic effects of special relativity are straightforwardly found in the Q-version; here, we demonstrate only one effect important for fractal pyramid technology accelerating a spacecraft

Boost. Σ-observer always can align one of his spatial vectors (e.g., q2) with velocity of moving

<sup>Σ</sup><sup>0</sup> <sup>¼</sup> <sup>H</sup><sup>η</sup>

<sup>3</sup> from Eq. (9b) (rotation about q<sup>3</sup> by angle η). This simple rotation, physi-

with an observer in the

ds ¼ ekdtp<sup>k</sup> þ dxk qk: (20)

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft

http://dx.doi.org/10.5772/intechopen.71751

41

Σ<sup>0</sup> ¼ OΣ, O ∈ SOð Þ 1; 2 : (21)

ds ¼ dtp<sup>1</sup> þ dr q2: (22)

<sup>3</sup> Σ, (23)

imaginary time axis is distinguished, an illusion of four dimensions emerges.

initial point represents full physical frame of reference, Eq. (16) can be rewritten as

the two triads being obviously co-aligned. The tool-set Σ � pk; q<sup>k</sup>

frames of reference are interconnected by "rotational equations"

Let the frame Σ<sup>0</sup> be a result of a hyperbolic rotation of a constant frame Σ

cally a boost, obviously keeping BQ-vector (20) form-invariant

body, so basic BQ-vector can be written in the form

(see Section 6).

with the matrix H<sup>η</sup>

SOð Þ 1; 2 ⊂ SOð Þ 3;C of the ordered rotations of Q-triads.

admits a BQ-square root

$$d\mathbf{s} = (\mathrm{i}e\_k dt + d\mathbf{x}\_k) \cdot \mathbf{q}\_{k'} \tag{16}$$

where displacement of observed object dxk is orthogonal to a unit vector ek directing change in time dt : ekdxk <sup>¼</sup> 0. Under these conditions, square of Eq. (16) yields Eq. (15) <sup>d</sup>sd<sup>s</sup> <sup>¼</sup> ds<sup>2</sup> . It is convenient to explicitly relate displacement dxk to a plane orthogonal to time-directing vector ek with the help of metric-projector bkn � δkn � eken dxk ¼ dxnbnk,

then the orthogonality condition is fulfilled automatically ekdxk ¼ ekdxnbkn ¼ 0:

The interval (15) is invariant under Lorentz transformations of coordinate system dx<sup>α</sup><sup>0</sup> <sup>¼</sup> <sup>L</sup><sup>α</sup><sup>0</sup> <sup>λ</sup> dx<sup>λ</sup>, Lα0 <sup>λ</sup> ∈ SOð Þ 1; 3 , while the Q-frame can be subject to SOð Þ 3;C rotations q<sup>k</sup><sup>0</sup> ¼ Ok<sup>0</sup> lql ; simultaneous application of the transformations, together with demand that the BQ-vector (16) form be conserved, leads to correlation between components of matrices Ok<sup>0</sup> <sup>l</sup> and L<sup>α</sup><sup>0</sup> λ <sup>2</sup> [5, 6]

$$\text{ie}\_k \mathbf{O}\_{s'k} = \text{ie}\_{s'} \mathbf{L}\_{0'0} + \mathbf{L}\_{m'0} \mathbf{b}\_{m's'} \tag{17}$$

$$
\hbar\_{nk}O\_{s'k} = -\mathrm{i}\varepsilon\_{s'}L\_{0'm} - L\_{m'k}b\_{m's'}.\tag{18}
$$

Eqs. (17) and (18) in particular mean that within the groupSOð Þ 3;C a set of ordered simple rotations of the type (11) are distinguished, real and hyperbolic, each performed about one-unit vector of Q-triad. If for instance, direction No. 1 of L<sup>α</sup><sup>0</sup> <sup>λ</sup> is not involved in the transformation (ek ¼ ek<sup>0</sup> ¼ δ1k), then Eqs. (17) and (18) represent the matrix O as function of components of Lorentz matrix L

$$\mathbf{O}\_{k'm} = \begin{pmatrix} L^{0'}\_{~0} & -iL^{0'}\_{~2} & -iL^{0'}\_{~3} \\ iL^{2'}\_{~0} & L^{2'}\_{~2} & L^{2'}\_{~3} \\ iL^{3'}\_{~0} & L^{3'}\_{~2} & L^{3'}\_{~3} \end{pmatrix}. \tag{19}$$

The matrix (19) may describe a series of simple rotations, but real rotations should be always performed about vector q<sup>1</sup> (initial or transformed), while hyperbolic rotations are allowed about

$$\eta\_{\alpha\beta} = \text{diag}(1, -1, -1, -1)$$

<sup>1</sup> Standard interval of special relativity is regarded for simplicity; similarly, interval of general relativity can be considered in tangent space ds<sup>2</sup> <sup>¼</sup> <sup>θ</sup><sup>2</sup> <sup>0</sup> � <sup>θ</sup>kθ<sup>k</sup> with <sup>θ</sup>ð Þ <sup>α</sup> <sup>¼</sup> <sup>g</sup>ð Þ <sup>α</sup> <sup>λ</sup>dy<sup>λ</sup> being basic one-form and Greek indices in brackets enumerating tangent space tetrad, and those without brackets are related to curved manifold holonomic coordinates

<sup>2</sup> ds<sup>2</sup> <sup>¼</sup> <sup>θ</sup><sup>2</sup> <sup>0</sup> � <sup>θ</sup>kθkθð Þ <sup>α</sup> <sup>¼</sup> <sup>g</sup>ð Þ <sup>α</sup> <sup>λ</sup>dy<sup>λ</sup>: four-dimensional indices are raised and lowered by Minkowski metric ηαβ <sup>¼</sup> diag 1ð Þ ; �1; �1; �1 .

vectors q<sup>2</sup> and q3. It is easily checked up that all matrices O of the type (19) constitute a subgroup SOð Þ 1; 2 ⊂ SOð Þ 3;C of the ordered rotations of Q-triads.

3. Vector-quaternion version of the relativity theory

ek with the help of metric-projector bkn � δkn � eken dxk ¼ dxnbnk,

conserved, leads to correlation between components of matrices Ok<sup>0</sup>

Ok<sup>0</sup> <sup>m</sup> ¼

Q-triad. If for instance, direction No. 1 of L<sup>α</sup><sup>0</sup>

iekOs<sup>0</sup>

bnkOs<sup>0</sup>

then the orthogonality condition is fulfilled automatically ekdxk ¼ ekdxnbkn ¼ 0:

<sup>λ</sup> ∈ SOð Þ 1; 3 , while the Q-frame can be subject to SOð Þ 3;C rotations q<sup>k</sup><sup>0</sup> ¼ Ok<sup>0</sup>

The interval (15) is invariant under Lorentz transformations of coordinate system dx<sup>α</sup><sup>0</sup>

<sup>k</sup> ¼ ies0L<sup>0</sup><sup>0</sup>

<sup>k</sup> ¼ �ies0L<sup>0</sup><sup>0</sup>

then Eqs. (17) and (18) represent the matrix O as function of components of Lorentz matrix L

L00

0

BB@

tangent space tetrad, and those without brackets are related to curved manifold holonomic coordinates

iL<sup>2</sup><sup>0</sup>

iL<sup>3</sup><sup>0</sup>

admits a BQ-square root

Lα0

40 Space Flight

1

2 ds<sup>2</sup> <sup>¼</sup> <sup>θ</sup><sup>2</sup>

in tangent space ds<sup>2</sup> <sup>¼</sup> <sup>θ</sup><sup>2</sup>

diag 1ð Þ ; �1; �1; �1 .

According to Eqs. (13) and (14), the interval of Einstein's relativity theory<sup>1</sup>

ds<sup>2</sup> <sup>¼</sup> dx<sup>2</sup>

where displacement of observed object dxk is orthogonal to a unit vector ek directing change in time dt : ekdxk <sup>¼</sup> 0. Under these conditions, square of Eq. (16) yields Eq. (15) <sup>d</sup>sd<sup>s</sup> <sup>¼</sup> ds<sup>2</sup>

convenient to explicitly relate displacement dxk to a plane orthogonal to time-directing vector

application of the transformations, together with demand that the BQ-vector (16) form be

Eqs. (17) and (18) in particular mean that within the groupSOð Þ 3;C a set of ordered simple rotations of the type (11) are distinguished, real and hyperbolic, each performed about one-unit vector of

<sup>0</sup> �iL<sup>0</sup><sup>0</sup>

<sup>0</sup> L<sup>2</sup><sup>0</sup>

<sup>0</sup> L<sup>3</sup><sup>0</sup>

The matrix (19) may describe a series of simple rotations, but real rotations should be always performed about vector q<sup>1</sup> (initial or transformed), while hyperbolic rotations are allowed about

Standard interval of special relativity is regarded for simplicity; similarly, interval of general relativity can be considered

ηαβ ¼ diag 1ð Þ ; �1; �1; �1

<sup>0</sup> � <sup>θ</sup>kθkθð Þ <sup>α</sup> <sup>¼</sup> <sup>g</sup>ð Þ <sup>α</sup> <sup>λ</sup>dy<sup>λ</sup>: four-dimensional indices are raised and lowered by Minkowski metric ηαβ <sup>¼</sup>

<sup>0</sup> þ Lm<sup>0</sup>

<sup>m</sup> � Lm<sup>0</sup>

<sup>0</sup>bm<sup>0</sup>

<sup>2</sup> �iL<sup>0</sup><sup>0</sup> 3

3

<sup>0</sup> � <sup>θ</sup>kθ<sup>k</sup> with <sup>θ</sup>ð Þ <sup>α</sup> <sup>¼</sup> <sup>g</sup>ð Þ <sup>α</sup> <sup>λ</sup>dy<sup>λ</sup> being basic one-form and Greek indices in brackets enumerating

1

<sup>2</sup> L<sup>2</sup><sup>0</sup> 3

<sup>2</sup> L<sup>3</sup><sup>0</sup>

kbm<sup>0</sup>

<sup>0</sup> � dxkdxk <sup>¼</sup> dt<sup>2</sup> � dr<sup>2</sup> (15)

ds ¼ ð Þ iekdt þ dxk qk, (16)

. It is

<sup>¼</sup> <sup>L</sup><sup>α</sup><sup>0</sup> <sup>λ</sup> dx<sup>λ</sup>,

; simultaneous

lql

<sup>s</sup><sup>0</sup> , (17)

<sup>s</sup><sup>0</sup> : (18)

CCA: (19)

<sup>l</sup> and L<sup>α</sup><sup>0</sup> λ <sup>2</sup> [5, 6]

<sup>λ</sup> is not involved in the transformation (ek ¼ ek<sup>0</sup> ¼ δ1k),

Main idea of Q-version of relativity is to replace line element of Einstein's relativity (15) and its invariance under Lorentz group by adequate BQ-vector (16) invariant under rotational group represented by matrices O ∈SOð Þ 1; 2 . Then, instead of quadratic form of four-dimensional coordinates, an observer has at his disposal a movable Q-triad with time and distances measured along its unit vectors and dealt with the vector basement as with the Newtonian mechanics or general relativity in tetrad formulation. However, on this way, an essential peculiarity arises. Eq. (16) implies that the constructed space-time model has six dimensions, and it is a symmetric sum of two three-dimensional (3D) spaces Q<sup>6</sup> ¼ R<sup>3</sup> ⊕ T3, where R<sup>3</sup> is the usual 3D space where coordinate and velocity change, whereas T<sup>3</sup> is also a 3D space but imaginary with respect to R3. In this model, the observer works only with some sections of the 6D space; but since the objects of the observations are found in real 3D space, and imaginary time axis is distinguished, an illusion of four dimensions emerges.

Physical measurements in the Q-model are made with the help of three spatial rulers q<sup>k</sup> and built-in geometric clock represented by "imaginary time rulers" (Pauli-type matrices) p<sup>k</sup> � iqk, the two triads being obviously co-aligned. The tool-set Σ � pk; q<sup>k</sup> with an observer in the initial point represents full physical frame of reference, Eq. (16) can be rewritten as

$$d\mathbf{s} = e\_k dt \mathbf{p}\_k + d\mathbf{x}\_k \ \mathbf{q}\_k. \tag{20}$$

Now, the principal statement of the Q-version of relativity follows: all physically sustainable frames of reference are interconnected by "rotational equations"

$$
\Sigma' = O\Sigma, \quad O \in SO(1,2). \tag{21}
$$

The sustainability means form-invariance of BQ-vector (16) or (20) under transformations (21). Kinematic effects of special relativity are straightforwardly found in the Q-version; here, we demonstrate only one effect important for fractal pyramid technology accelerating a spacecraft (see Section 6).

Boost. Σ-observer always can align one of his spatial vectors (e.g., q2) with velocity of moving body, so basic BQ-vector can be written in the form

$$d\mathbf{s} = d\mathbf{t}\mathbf{p}\_1 \, \, +dr \, \, \mathbf{q}\_2. \tag{2}$$

Let the frame Σ<sup>0</sup> be a result of a hyperbolic rotation of a constant frame Σ

$$
\Sigma' = H\_3^\eta \,\,\,\Sigma \,\,\,\tag{23}
$$

with the matrix H<sup>η</sup> <sup>3</sup> from Eq. (9b) (rotation about q<sup>3</sup> by angle η). This simple rotation, physically a boost, obviously keeping BQ-vector (20) form-invariant

$$dt\mathbf{p}\_1 + dr\mathbf{q}\_2 = dt'\mathbf{p}\_{1'} + dr'\mathbf{q}\_{2'} \tag{24}$$

G<sup>A</sup>

and two nilpotent matrices

<sup>B</sup> � aAaB, HA

Next, we built sum and difference of the idempotent matrices

<sup>B</sup> � GA

<sup>B</sup> � GA

(31d), then we obtain the basis of quaternion (and biquaternion) numbers

basis. The unit q<sup>3</sup> defined in Eqs. (32), (31b) is the characteristic example

<sup>q</sup><sup>k</sup><sup>0</sup> <sup>¼</sup> <sup>U</sup>qkU�<sup>1</sup>

q3 B

respectively; the eigenvalues are <sup>þ</sup><sup>i</sup> (for <sup>a</sup>) and �<sup>i</sup> (for <sup>b</sup>), and GA

As mentioned above, the similarity transformation of the units

from the viewpoint of the 3D space described by the triad vectors qk.

<sup>B</sup> <sup>þ</sup> <sup>H</sup><sup>A</sup>

<sup>B</sup> � <sup>H</sup><sup>A</sup>

<sup>B</sup> <sup>¼</sup> <sup>D</sup>AB <sup>þ</sup> FAB <sup>¼</sup> aAbB <sup>þ</sup> <sup>b</sup>AaB, <sup>~</sup><sup>I</sup>

If the units Eqs. (31b) and (31c) are slightly corrected so that their product is the third unit

Now, we recall the spectral theorem (of the matrix theory) stating that any invertible matrix with distinct eigenvalues can be represented as a sum of idempotent projectors with the eigenvalues as coefficients, the projectors being direct products of vectors of a biorthogonal

<sup>A</sup> <sup>¼</sup> iaAaB � ibAbB <sup>¼</sup> iG<sup>A</sup>

Right and left eigenfunctions of <sup>q</sup><sup>3</sup> are vectors aA, bB and covectors aA, bB of the dyad,

preserves the form of algebras' multiplication law (2). Therefore, vector units from Eq. (32) can be obtained from a single unit, say, q<sup>3</sup> by a transformation (34). Then, all vector units have same eigenvalues �i, and the eigenfunctions of the derived units are linear combinations of the eigenfunctions of the initial unit [9]. This also means that the mapping (34) is a secondary one, but the primary one is SLð Þ 2;C transformation of dyad vectors, thus forming a set of spinors

<sup>E</sup> � <sup>E</sup><sup>A</sup>

<sup>K</sup><sup>~</sup> � <sup>K</sup><sup>~</sup> <sup>A</sup>

<sup>~</sup><sup>I</sup> � <sup>~</sup><sup>I</sup> A

J � J A

and sum and difference of the nilpotent matrices

<sup>B</sup> � <sup>b</sup>AbB ! <sup>G</sup><sup>A</sup>

BGB <sup>C</sup> <sup>¼</sup> GA

C, H<sup>A</sup> BH<sup>B</sup> <sup>C</sup> <sup>¼</sup> <sup>H</sup><sup>A</sup>

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft

<sup>B</sup> <sup>¼</sup> <sup>D</sup>AB � <sup>F</sup>AB <sup>¼</sup> <sup>a</sup>AbB � <sup>b</sup>AaB, J<sup>2</sup> ¼ �<sup>E</sup> : (31d)

<sup>1</sup> � E, <sup>q</sup><sup>1</sup> ¼ �<sup>i</sup> <sup>~</sup>I, <sup>q</sup><sup>2</sup> ¼ �i J, <sup>q</sup><sup>3</sup> ¼ �<sup>i</sup> <sup>K</sup><sup>~</sup> : (32)

<sup>B</sup> � iH<sup>A</sup>

<sup>B</sup> , H<sup>A</sup>

, U ∈ SLð Þ 2; C (34)

<sup>B</sup> <sup>¼</sup> aAaB <sup>þ</sup> <sup>b</sup>AbB, E<sup>2</sup> <sup>¼</sup> E, (31a)

<sup>B</sup> <sup>¼</sup> aAaB � <sup>b</sup>AbB, <sup>K</sup><sup>~</sup> <sup>2</sup> <sup>¼</sup> E, (31b)

<sup>2</sup> <sup>¼</sup> E, (31c)

<sup>B</sup> : (33)

<sup>B</sup> are the projectors.

<sup>D</sup>AB � aAbB, FAB � bAaB ! <sup>D</sup>ABDBC <sup>¼</sup> <sup>0</sup>, FAB <sup>F</sup>BC <sup>¼</sup> <sup>0</sup>: (30b)

<sup>C</sup> , (30a)

43

http://dx.doi.org/10.5772/intechopen.71751

yields familiar coordinate transformations

dt<sup>0</sup> ¼ dtcosh η þ dr sinh η, dr<sup>0</sup> ¼ dt sinh η þ drcosh η (25)

with respective effects of length and time segments contraction. If observed particle is the body of reference of the frame Σ<sup>0</sup> , then dr<sup>0</sup> ¼ 0, and one finds that the frame Σ<sup>0</sup> is moving with the velocity

$$V = \frac{dr}{dt} = \tanh\psi.\tag{26}$$

Specific features of the Q-vector version of relativity will be effectively used below in the fractalpyramid math method to operate a spacecraft. Now, we turn to notions of a fractal space.

#### 4. Fractal space underlying physical space

In this section, we show that a 3D space (e.g., physical space) may be endowed with a pregeometry [7] mathematically described by a complex-numbered surface, a 2D fractal space, each its vector having dimensionality half compared to that of the 3D space. We start with 2D space and construct out of its basic elements a basis of 3D space.

Let there exist a smooth 2D space (surface) endowed with a metric gAB (and inverse: gBCgBC ! gABgBC <sup>¼</sup> <sup>δ</sup><sup>C</sup> <sup>A</sup>) and with a system of coordinates xA <sup>¼</sup> <sup>x</sup><sup>1</sup>; <sup>x</sup><sup>2</sup> ; here A, B, C <sup>¼</sup> <sup>1</sup>, 2, δC <sup>A</sup> is a 2D Kronecker symbol, summation in repeated indices is also implied. The line element of the surface is

$$d\mathbf{s}^2 = \mathbf{g}\_{AB} d\mathbf{x}^A d\mathbf{x}^B;\tag{27}$$

the surface may be curved, so covariant and contravariant metric components differ. In a point, we choose a couple of unit orthogonal vectors aA, b<sup>B</sup> (a dyad)

$$
\mathcal{g}\_{AB}a^A a^B = 1,\tag{28}
$$

$$
\mathcal{g}\_{AB}a^A b^B = 0.\tag{29}
$$

A domain of the surface in vicinity of the dyad's initial point (together with respective part of tangent plane having the metric <sup>δ</sup>MN <sup>¼</sup> <sup>δ</sup>MN <sup>¼</sup> <sup>δ</sup><sup>N</sup> <sup>m</sup>) will be called a "2D-cell."

Considering direct (tensor) products of the dyad vectors with mixed components [8], we can construct only four such products (2 � 2 matrices): two idempotent matrices

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft http://dx.doi.org/10.5772/intechopen.71751 43

$$\mathbf{G}\_{\mathcal{B}}^{A} \equiv \boldsymbol{\sigma}^{A} \boldsymbol{a}\_{\mathcal{B}}, \quad \boldsymbol{H}\_{\mathcal{B}}^{A} \equiv \boldsymbol{b}^{A} \boldsymbol{b}\_{\mathcal{B}} \to \mathbf{G}\_{\mathcal{B}}^{A} \mathbf{G}\_{\mathcal{C}}^{B} = \mathbf{G}\_{\mathcal{C}}^{A} \quad \boldsymbol{H}\_{\mathcal{B}}^{A} \boldsymbol{H}\_{\mathcal{C}}^{B} = \boldsymbol{H}\_{\mathcal{C}}^{A} \tag{30a}$$

and two nilpotent matrices

dtp<sup>1</sup> þ drq<sup>2</sup> ¼ dt<sup>0</sup>

<sup>V</sup> <sup>¼</sup> dr

with respective effects of length and time segments contraction. If observed particle is the body

Specific features of the Q-vector version of relativity will be effectively used below in the fractalpyramid math method to operate a spacecraft. Now, we turn to notions of a fractal space.

In this section, we show that a 3D space (e.g., physical space) may be endowed with a pregeometry [7] mathematically described by a complex-numbered surface, a 2D fractal space, each its vector having dimensionality half compared to that of the 3D space. We start with 2D space

Let there exist a smooth 2D space (surface) endowed with a metric gAB (and inverse:

<sup>A</sup> is a 2D Kronecker symbol, summation in repeated indices is also implied. The line element

the surface may be curved, so covariant and contravariant metric components differ. In a point,

A domain of the surface in vicinity of the dyad's initial point (together with respective part of

Considering direct (tensor) products of the dyad vectors with mixed components [8], we can

construct only four such products (2 � 2 matrices): two idempotent matrices

<sup>A</sup>) and with a system of coordinates xA <sup>¼</sup> <sup>x</sup><sup>1</sup>; <sup>x</sup><sup>2</sup> ; here A, B, C <sup>¼</sup> <sup>1</sup>, 2,

ds<sup>2</sup> <sup>¼</sup> gABdxAdxB; (27)

gABaAaB <sup>¼</sup> <sup>1</sup>, (28)

gABaAbB <sup>¼</sup> <sup>0</sup>: (29)

<sup>m</sup>) will be called a "2D-cell."

yields familiar coordinate transformations

4. Fractal space underlying physical space

and construct out of its basic elements a basis of 3D space.

we choose a couple of unit orthogonal vectors aA, b<sup>B</sup> (a dyad)

tangent plane having the metric <sup>δ</sup>MN <sup>¼</sup> <sup>δ</sup>MN <sup>¼</sup> <sup>δ</sup><sup>N</sup>

of reference of the frame Σ<sup>0</sup>

gBCgBC ! gABgBC <sup>¼</sup> <sup>δ</sup><sup>C</sup>

of the surface is

δC

velocity

42 Space Flight

p<sup>1</sup><sup>0</sup> þ dr<sup>0</sup>

dt<sup>0</sup> ¼ dtcosh η þ dr sinh η, dr<sup>0</sup> ¼ dt sinh η þ drcosh η (25)

, then dr<sup>0</sup> ¼ 0, and one finds that the frame Σ<sup>0</sup> is moving with the

dt <sup>¼</sup> tanhψ: (26)

q<sup>2</sup><sup>0</sup> (24)

$$D^{A\_{\mathcal{B}}} \equiv a^A b\_{\mathcal{B}} \quad F^{A\_{\mathcal{B}}} \equiv b^A a\_{\mathcal{B}} \to D^{A\_{\mathcal{B}}} D^{B\_{\mathcal{C}}} = 0, \quad F^{A\_{\mathcal{B}}} F^{B\_{\mathcal{C}}} = 0. \tag{30b}$$

Next, we built sum and difference of the idempotent matrices

$$E \equiv E\_B^A \equiv G\_B^A + H\_B^A = a^A a\_B + b^A b\_{B'} \quad E^2 = E,\tag{31a}$$

$$
\tilde{\mathbf{K}} \equiv \tilde{\mathbf{K}}\_{\mathcal{B}}^{A} \equiv \mathbf{G}\_{\mathcal{B}}^{A} - H\_{\mathcal{B}}^{A} = a^{A}a\_{\mathcal{B}} - b^{A}b\_{\mathcal{B}\prime} \quad \tilde{\mathbf{K}}^{2} = \mathbf{E},\tag{31b}
$$

and sum and difference of the nilpotent matrices

$$
\tilde{I} \equiv \tilde{I}\_B^A = D^{A\_B} + F^{A\_B} = a^A b\_B + b^A a\_{B\prime} \quad \tilde{I}^2 = E,\tag{31c}
$$

$$J \equiv I\_{\mathcal{B}}^{A} = D^{A\_{\mathcal{B}}} - F^{A\_{\mathcal{B}}} = a^{A}b\_{\mathcal{B}} - b^{A}a\_{\mathcal{B}}, \quad J^{2} = -E \,. \tag{31d}$$

If the units Eqs. (31b) and (31c) are slightly corrected so that their product is the third unit (31d), then we obtain the basis of quaternion (and biquaternion) numbers

$$1 \equiv E, \quad \mathbf{q}\_1 = -i \ \mathbf{\tilde{l}}, \mathbf{q}\_2 = -i \ \mathbf{j}, \mathbf{q}\_3 = -i \ \mathbf{\tilde{K}}.\tag{32}$$

Now, we recall the spectral theorem (of the matrix theory) stating that any invertible matrix with distinct eigenvalues can be represented as a sum of idempotent projectors with the eigenvalues as coefficients, the projectors being direct products of vectors of a biorthogonal basis. The unit q<sup>3</sup> defined in Eqs. (32), (31b) is the characteristic example

$$\left|\mathbf{q}\_{\mathbf{3}}\right|\_{A}^{B} = \mathrm{i}\boldsymbol{\sigma}^{A}\boldsymbol{a}\_{\mathbf{B}} - \mathrm{i}\boldsymbol{b}^{A}\boldsymbol{b}\_{\mathbf{B}} = \mathrm{i}\mathbb{G}\_{\mathrm{B}}^{A} - \mathrm{i}\mathrm{H}\_{\mathrm{B}}^{A}.\tag{33}$$

Right and left eigenfunctions of <sup>q</sup><sup>3</sup> are vectors aA, bB and covectors aA, bB of the dyad, respectively; the eigenvalues are <sup>þ</sup><sup>i</sup> (for <sup>a</sup>) and �<sup>i</sup> (for <sup>b</sup>), and GA <sup>B</sup> , H<sup>A</sup> <sup>B</sup> are the projectors.

As mentioned above, the similarity transformation of the units

$$\mathfrak{q}\_{\mathbb{K}} = \mathcal{U}\mathfrak{q}\_{\mathbb{k}}\mathcal{U}^{-1}, \quad \mathcal{U} \in \mathcal{SL}(2, \mathbb{C}) \tag{34}$$

preserves the form of algebras' multiplication law (2). Therefore, vector units from Eq. (32) can be obtained from a single unit, say, q<sup>3</sup> by a transformation (34). Then, all vector units have same eigenvalues �i, and the eigenfunctions of the derived units are linear combinations of the eigenfunctions of the initial unit [9]. This also means that the mapping (34) is a secondary one, but the primary one is SLð Þ 2;C transformation of dyad vectors, thus forming a set of spinors from the viewpoint of the 3D space described by the triad vectors qk.

Hereinafter, we introduce shorter 2D-index-free matrix notations for the dyad: a vector is a column, a co-vector is a row, and a parity indicator þ or � marks the sign of the eigenvalue �i

$$a^A \to \psi^+, \quad a\_A \to \phi^+, \quad b^A \to \psi^-, \quad b\_A \to \phi^-; \tag{35}$$

this helps to rewrite the above expressions more compactly. The dyad orthonormality conditions (28, 29) acquire the form

$$
\varphi^{\pm}\psi^{\pm} = 1, \quad \varphi^{\mp}\psi^{\pm} = \varphi^{\pm}\psi^{\mp} = 0. \tag{36}
$$

the idempotent projectors are denoted as C<sup>þ</sup> � G ¼ ψþφþ, C� � H ¼ ψ�φ� ,

and the units (32) are expressed through the single dyad vectors (co-vectors) as

$$\mathbf{1} = \boldsymbol{\psi}^{+} \boldsymbol{\phi}^{+} + \boldsymbol{\psi}^{-} \boldsymbol{\phi}^{-},\tag{37a}$$

(i) a series of subsequent several angles rotation and (ii) a one-angle rotation about an instant axis. Mixed variants exist, but are less productive, and they are not normally considered.

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft

http://dx.doi.org/10.5772/intechopen.71751

45

If magnitudes involved in calculations are generically measured in real numbers, then both techniques (i) and (ii) are based on the vector rotation group SOð Þ 3; R . Math content of the technique (i) implies a multiple set of plane rotations [of type of Eq. (9a)] by Euler (or Krylov, or others) angles about selected axes. The technique (ii) in its turn represents a nontrivial problem

Quaternions are widely known to fit better than real numbers for the orientation tasks mostly due to the fact that three vector units represent models of three mutually orthogonal gyroscope axes. As well, use of the Q-algebra formalism essentially simplifies calculations, especially for the technique (ii), since both the vector rotation group SOð Þ 3;R and its spinor "equivalent" SUð Þ2 reflection group can be used whatever enigmatic were formulas describing spinor rotations. However, the quaternion algebra reveals its unique property to split axial 3D vectors into dyad sets belonging to a fractal subspace as in Eq. (37), see also the basic work [10]. The above-described fractalization procedure, mathematically nontrivial and much less known, on the one hand clarifies "mysterious" two-side SUð Þ2 quaternion vector multiplication and on the other hand endows all algebraic objects and actions with distinct geometric sense; moreover, the calculations become most primitive. Solution of a spacecraft reorientation task as transformation of a fractal dyad represents the third math method (iii) suggested here. However, all three math methods are described

5.1. Quaternion SO(3,R) approach to the reorientation problem: Technique (i)

q<sup>n</sup><sup>0</sup> ¼ Rn<sup>0</sup>

with the above parameters of the probe's orientations are given by the matrices

Outlined above technique (i) demands that the matrix Rn<sup>0</sup>

Orientation of a spacecraft in 3D space is determined by three angles between axes of some global coordinate system and unit vectors of a frame attached to the moving body taking into account its physical symmetry. The global coordinates, e.g., are represented by a spherical system, and its local initiating vectors pointing: q<sup>1</sup> to the north along the Earth's meridian), q<sup>2</sup> along a parallel, and q<sup>3</sup> to zenith direction. The directing vectors q<sup>k</sup> are considered constant. Then, the orientation of a spacecraft bearing a frame q<sup>k</sup><sup>0</sup> , (with q<sup>1</sup><sup>0</sup> along the body, q<sup>2</sup><sup>0</sup> a transverse one, and q<sup>3</sup><sup>0</sup> along gravity) is determined by three angles: "yaw" ψ, the angle between q<sup>1</sup> and q<sup>1</sup><sup>0</sup> (rotation about q3); "roll" φ, angle q<sup>2</sup> � q<sup>2</sup><sup>0</sup> (rotation about q1); and "pitch" θ, angle q<sup>3</sup> � q<sup>3</sup><sup>0</sup> (rotation about q2). Within these notations, the spacecraft's orientation in the space is described by the matrix

simple rotations, irreducible representations of SOð Þ 3;R [a special notation for such matrix is

<sup>n</sup>, see Section 2, Eqs. (9, 10)], each performed about a frame's unit vector. Simple rotations

<sup>k</sup>qk, R∈ SOð Þ 3;R : (39)

<sup>k</sup> be represented as a product of

of determining the instant axis of a single rotation.

in detail in this section.

equation

Rα

$$\mathbf{q}\_1 = -i \ (\psi^+ \varphi^- + \psi^- \varphi^-), \tag{37b}$$

$$\mathbf{q}\_2 = \psi^+ \varphi^- - \psi^- \varphi^-,\tag{37c}$$

$$\mathbf{q}\_3 = i \ (\psi^+ \phi^+ - \psi^- \phi^-). \tag{37d}$$

Eq. (37) obviously demonstrates that the dyad elements are in a way "square roots" from 3D vector units. So, if we put dimensionality of any 3D line to be a unity, then dimensionality of a line on the 2D space (e.g., dimensionality of a dyad vector) must be ½; hence from the viewpoint of the 3D space, the surface determined by a dyad is fractal. The next important observation concerns transformations. The transformation (34) clearly results from the SLð Þ 2; C transformations of the dyad vectors (covectors)

$$
\psi'^{\pm} = \mathcal{U}\psi^{\pm}, \quad \varphi' = \varphi'^{\mp}\mathcal{U}^{-1}.\tag{38}
$$

So, apart from vector-type (8) and spinor-type (12) transformations of a Q-triad (an element of 3D space), there exists a possibility to deal with more fundamental math elements, vectors, and covectors describing "pregeometric" 2D cell of a fractal surface. These simpler math objects are subject to evidently simpler mapping (38); moreover, in the following sections, we will show that the operators of the transformations, being themselves BQ-numbers, suggest simpler and less numerous equations to solve, thus reducing degree of math load and probability of mistakes.

#### 5. Three methods to reorient a spacecraft and fractal joystick

The orientation tasks are relevant with computations over 3D flat space modeling a local domain of the physical space. Two types of the orientation problem solutions are traditional: (i) a series of subsequent several angles rotation and (ii) a one-angle rotation about an instant axis. Mixed variants exist, but are less productive, and they are not normally considered.

Hereinafter, we introduce shorter 2D-index-free matrix notations for the dyad: a vector is a column, a co-vector is a row, and a parity indicator þ or � marks the sign of the eigenvalue �i

this helps to rewrite the above expressions more compactly. The dyad orthonormality condi-

Eq. (37) obviously demonstrates that the dyad elements are in a way "square roots" from 3D vector units. So, if we put dimensionality of any 3D line to be a unity, then dimensionality of a line on the 2D space (e.g., dimensionality of a dyad vector) must be ½; hence from the viewpoint of the 3D space, the surface determined by a dyad is fractal. The next important observation concerns transformations. The transformation (34) clearly results from the SLð Þ 2; C

<sup>ψ</sup>0� <sup>¼</sup> <sup>U</sup>ψ�, <sup>φ</sup><sup>0</sup> <sup>¼</sup> <sup>φ</sup>0�U�<sup>1</sup>

So, apart from vector-type (8) and spinor-type (12) transformations of a Q-triad (an element of 3D space), there exists a possibility to deal with more fundamental math elements, vectors, and covectors describing "pregeometric" 2D cell of a fractal surface. These simpler math objects are subject to evidently simpler mapping (38); moreover, in the following sections, we will show that the operators of the transformations, being themselves BQ-numbers, suggest simpler and less numerous equations to solve, thus reducing degree of math load and probability of mistakes.

The orientation tasks are relevant with computations over 3D flat space modeling a local domain of the physical space. Two types of the orientation problem solutions are traditional:

5. Three methods to reorient a spacecraft and fractal joystick

the idempotent projectors are denoted as C<sup>þ</sup> � G ¼ ψþφþ, C� � H ¼ ψ�φ� , and the units (32) are expressed through the single dyad vectors (co-vectors) as

tions (28, 29) acquire the form

44 Space Flight

transformations of the dyad vectors (covectors)

<sup>a</sup><sup>A</sup> ! <sup>ψ</sup>þ, aA ! <sup>φ</sup>þ, b<sup>A</sup> ! <sup>ψ</sup>�, bA ! <sup>φ</sup>�; (35)

<sup>φ</sup>�ψ� <sup>¼</sup> <sup>1</sup>, <sup>φ</sup><sup>∓</sup>ψ� <sup>¼</sup> <sup>φ</sup>�ψ<sup>∓</sup> <sup>¼</sup> <sup>0</sup> , (36)

1 ¼ ψþφ<sup>þ</sup> þ ψ�φ�, (37a)

q<sup>2</sup> ¼ ψþφ� � ψ�φ�, (37c)

q<sup>3</sup> ¼ i ψþφ<sup>þ</sup> � ψ�φ� ð Þ: (37d)

: (38)

q<sup>1</sup> ¼ �i ψþφ� þ ψ�φ� ð Þ, (37b)

If magnitudes involved in calculations are generically measured in real numbers, then both techniques (i) and (ii) are based on the vector rotation group SOð Þ 3; R . Math content of the technique (i) implies a multiple set of plane rotations [of type of Eq. (9a)] by Euler (or Krylov, or others) angles about selected axes. The technique (ii) in its turn represents a nontrivial problem of determining the instant axis of a single rotation.

Quaternions are widely known to fit better than real numbers for the orientation tasks mostly due to the fact that three vector units represent models of three mutually orthogonal gyroscope axes. As well, use of the Q-algebra formalism essentially simplifies calculations, especially for the technique (ii), since both the vector rotation group SOð Þ 3;R and its spinor "equivalent" SUð Þ2 reflection group can be used whatever enigmatic were formulas describing spinor rotations. However, the quaternion algebra reveals its unique property to split axial 3D vectors into dyad sets belonging to a fractal subspace as in Eq. (37), see also the basic work [10]. The above-described fractalization procedure, mathematically nontrivial and much less known, on the one hand clarifies "mysterious" two-side SUð Þ2 quaternion vector multiplication and on the other hand endows all algebraic objects and actions with distinct geometric sense; moreover, the calculations become most primitive. Solution of a spacecraft reorientation task as transformation of a fractal dyad represents the third math method (iii) suggested here. However, all three math methods are described in detail in this section.

#### 5.1. Quaternion SO(3,R) approach to the reorientation problem: Technique (i)

Orientation of a spacecraft in 3D space is determined by three angles between axes of some global coordinate system and unit vectors of a frame attached to the moving body taking into account its physical symmetry. The global coordinates, e.g., are represented by a spherical system, and its local initiating vectors pointing: q<sup>1</sup> to the north along the Earth's meridian), q<sup>2</sup> along a parallel, and q<sup>3</sup> to zenith direction. The directing vectors q<sup>k</sup> are considered constant. Then, the orientation of a spacecraft bearing a frame q<sup>k</sup><sup>0</sup> , (with q<sup>1</sup><sup>0</sup> along the body, q<sup>2</sup><sup>0</sup> a transverse one, and q<sup>3</sup><sup>0</sup> along gravity) is determined by three angles: "yaw" ψ, the angle between q<sup>1</sup> and q<sup>1</sup><sup>0</sup> (rotation about q3); "roll" φ, angle q<sup>2</sup> � q<sup>2</sup><sup>0</sup> (rotation about q1); and "pitch" θ, angle q<sup>3</sup> � q<sup>3</sup><sup>0</sup> (rotation about q2). Within these notations, the spacecraft's orientation in the space is described by the matrix equation

$$\mathbf{q}\_{\mathbf{r}\prime} = \mathcal{R}\_{\mathbf{r}\prime k} \mathbf{q}\_{\mathbf{k}\prime} R \in SO(3, R). \tag{39}$$

Outlined above technique (i) demands that the matrix Rn<sup>0</sup> <sup>k</sup> be represented as a product of simple rotations, irreducible representations of SOð Þ 3;R [a special notation for such matrix is Rα <sup>n</sup>, see Section 2, Eqs. (9, 10)], each performed about a frame's unit vector. Simple rotations with the above parameters of the probe's orientations are given by the matrices

$$R\_3^\psi \equiv \begin{pmatrix} \cos\psi & \sin\psi & 0\\ -\sin\psi & \cos\psi & 0\\ 0 & 0 & 1 \end{pmatrix},\\ R\_2^\psi \equiv \begin{pmatrix} \cos\varphi & 0 & -\sin\varphi\\ 0 & 1 & 0\\ \sin\varphi & 0 & \cos\varphi \end{pmatrix},\\ R\_1^\theta \equiv \begin{pmatrix} 1 & 0 & 0\\ 0 & \cos\theta & -\sin\theta\\ 0 & \sin\theta & \cos\theta \end{pmatrix}.\tag{40}$$

<sup>U</sup> <sup>¼</sup> x z

where

and q is a Q-vector unit

the matrix U is unimodular if

ing geometric analysis.

that vectors of the transformed frame q<sup>k</sup>

therefore,

w y � � <sup>¼</sup> <sup>x</sup> <sup>þ</sup> <sup>y</sup>

<sup>a</sup> <sup>¼</sup> <sup>x</sup> <sup>þ</sup> <sup>y</sup>

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>a</sup><sup>2</sup> <sup>p</sup>

<sup>q</sup><sup>k</sup><sup>0</sup> � <sup>U</sup>qkU�<sup>1</sup> <sup>¼</sup> cos <sup>α</sup> <sup>þ</sup> sin <sup>α</sup>lnq<sup>n</sup>

<sup>q</sup> <sup>¼</sup> <sup>1</sup>

2 þ

<sup>2</sup> , b <sup>¼</sup>

x � y <sup>2</sup> <sup>z</sup> <sup>w</sup> � <sup>x</sup> � <sup>y</sup> 2

<sup>a</sup> � cos <sup>α</sup>, b <sup>¼</sup> ffiffiffiffiffiffiffiffi

with lk representing cosines of angles between Q-vectors q<sup>k</sup> and the direction determined by q. With the help of Eqs. (46) and (2), we reproduce the transformation (42) in the developed form

Eq. (47) in fact interlinks the SOð Þ 3; R rotation matrix components and the parameters of SUð Þ2 transformations of a Q-frame [compare with (39)]. As well, Eq. (47) helps to make the follow-

Multiplied by lk (with summation in index k), Eq. (47) yields the equality lkq<sup>k</sup><sup>0</sup> ¼ lnqn, meaning

frame qk; i.e., the transformation may be represented as a conical rotation about lk, Φ � 2α, which is angle of the rotation in the orthogonal plane with the metric pkn ¼ δkn � lkln [see the

<sup>¼</sup> 2 sin <sup>2</sup>αlkln <sup>q</sup><sup>n</sup> <sup>þ</sup> cos 2αq<sup>k</sup> <sup>þ</sup> sin 2αlnεnkmq<sup>m</sup> <sup>¼</sup> ¼ lkln þ cos 2α δkn � lk ½ � ð Þþ ln sin 2αlmεmkn qn:

bkbk

<sup>U</sup> <sup>¼</sup> cos <sup>α</sup> <sup>þ</sup> ð Þ sin <sup>α</sup> lnqn, U�<sup>1</sup> <sup>¼</sup> cos <sup>α</sup> � ð Þ sin <sup>α</sup> lnq<sup>n</sup> (46)

� �q<sup>k</sup> cos <sup>α</sup> � sin <sup>α</sup>lmq<sup>m</sup>

The unit vector (44) represented through the constant basis (7) has the form; q ¼ lkq<sup>k</sup> ¼ ð Þ bk=b q<sup>k</sup> where lk ¼ bk=b are components of a unit vector pointing in 3D space a vector with components bk, then the condition det<sup>U</sup> <sup>¼</sup> xy � wz <sup>¼</sup> 1 takes the form <sup>a</sup><sup>2</sup> <sup>þ</sup> b2 <sup>¼</sup> 1, <sup>b</sup><sup>2</sup> <sup>¼</sup> bkbk. This general biquaternion case will be used in subsequent studies when combined rotation-plus-translational motion is regarded (see Section 6). In this section, we consider only quaternion case: a, bk ∈ R, so

0 B@

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>x</sup> <sup>þ</sup> <sup>y</sup> 2 � �<sup>2</sup> <sup>r</sup>

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>x</sup> <sup>þ</sup> <sup>y</sup> 2 � �<sup>2</sup> <sup>r</sup>

1

q � a þ bq (42)

47

http://dx.doi.org/10.5772/intechopen.71751

, (43)

CA, <sup>q</sup><sup>2</sup> ¼ �1: (44)

<sup>p</sup> � sin <sup>α</sup> (45)

� � <sup>¼</sup>

<sup>0</sup> have the same projections onto vector lk as the initial

(47)

Direct reorientation problem, i.e., reaching object's assigned orientation, can be solved by a sequence of plane rotations mathematically described by a sequent multiplication of matrices [see Eq. (10)]. This problem has no unique solution since the group SOð Þ 3;R is not commutative; i.e., different multiplication order of the matrices (40) with the same parameters (angles) generally gives different result; e.g., the products <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>ψ</sup> <sup>3</sup> <sup>R</sup><sup>φ</sup> <sup>2</sup> R<sup>θ</sup> <sup>1</sup> and <sup>R</sup><sup>0</sup> <sup>¼</sup> <sup>R</sup><sup>θ</sup> 1R<sup>φ</sup> <sup>2</sup> <sup>R</sup><sup>ψ</sup> <sup>3</sup> are, in general, different R 6¼ R<sup>0</sup> . Vice versa, different orders of the matrix product with other parameters may yield the same result, e.g., products <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>ψ</sup> <sup>3</sup> <sup>R</sup><sup>φ</sup> <sup>2</sup> R<sup>θ</sup> <sup>1</sup> and <sup>R</sup><sup>0</sup> <sup>¼</sup> <sup>R</sup><sup>θ</sup><sup>0</sup> <sup>1</sup> <sup>R</sup><sup>φ</sup><sup>0</sup> <sup>2</sup> <sup>R</sup><sup>ψ</sup><sup>0</sup> <sup>3</sup> may represent equivalent rotational result R ¼ R<sup>0</sup> . The possibility to represent an arbitrary SOð Þ 3;R matrix as a product of its irreducible representations given in different order in particular entails uncertainty in solution of the inverse problem when one has to determine values of angles securing an assigned reorientation of the spacecraft. Therefore, the technique (i) does not provide single-valued results.

Even with more difficult, we meet trying to use matrices from the group SOð Þ 3; R in the technique (ii). As is known from the theory of matrices (see e.g., [11]) in this case, we have to solve the characteristic equation RX ¼ X searching for the matrix operator R an eigenvector X with unit eigenvalue, the vector X pointing direction of the instant rotation axis. This tough algebraic task then followed by sophisticated calculations aimed to find the instant rotation angle. The use of hypercomplex numbers essentially helped to avoid these math troubles, and about half of a century ago, quaternion algebra became a common tool serving for engineering goals of navigation and orientation. Indeed, the similarity transformation UqU�<sup>1</sup> of a quaternion q performed with the help of auxiliary quaternion U � a þ bq geometrically leads to conical rotation of the vector part of q about an axis whose direction is determined by the unit Q-vector q (e.g., [2]); the value of the instant rotation angle is computed as 2 arctanð Þ b=a . Below, we suggest a detailed analysis of this type of description of rotations.

#### 5.2. Reorientation by a single rotation of the quaternion frame: Technique (ii)

Consider a 2 � 2matrix (with complex-number components) <sup>U</sup> � x z w y � �, belonging to a special linear group U ∈ SLð Þ 2, C , detU ¼ xy � wz ¼ 1. The multiplication law (6) is obviously form invariant under the similarity-type transformation

$$\mathbf{q}\_{\mathbf{r}\prime} = \mathcal{U}\mathbf{q}\_n\mathcal{U}^{-1}.\tag{41}$$

One readily demonstrates that the matrix U is a biquaternion with the definable norm; indeed,

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft http://dx.doi.org/10.5772/intechopen.71751 47

$$U = \begin{pmatrix} x & z \\ w & y \end{pmatrix} = \frac{x+y}{2} + \sqrt{1 - \left(\frac{x+y}{2}\right)^2} \mathbf{q} \equiv a + b\mathbf{q} \tag{42}$$

where

Rψ <sup>3</sup> �

46 Space Flight

0 B@

eral, different R 6¼ R<sup>0</sup>

equivalent rotational result R ¼ R<sup>0</sup>

provide single-valued results.

cosψ sinψ 0 � sinψ cosψ 0 0 01 1 CA, R<sup>φ</sup> <sup>2</sup> �

generally gives different result; e.g., the products <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>ψ</sup>

may yield the same result, e.g., products <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>ψ</sup>

0 B@

cosφ 0 � sinφ 01 0 sinφ 0 cosφ

Direct reorientation problem, i.e., reaching object's assigned orientation, can be solved by a sequence of plane rotations mathematically described by a sequent multiplication of matrices [see Eq. (10)]. This problem has no unique solution since the group SOð Þ 3;R is not commutative; i.e., different multiplication order of the matrices (40) with the same parameters (angles)

as a product of its irreducible representations given in different order in particular entails uncertainty in solution of the inverse problem when one has to determine values of angles securing an assigned reorientation of the spacecraft. Therefore, the technique (i) does not

Even with more difficult, we meet trying to use matrices from the group SOð Þ 3; R in the technique (ii). As is known from the theory of matrices (see e.g., [11]) in this case, we have to solve the characteristic equation RX ¼ X searching for the matrix operator R an eigenvector X with unit eigenvalue, the vector X pointing direction of the instant rotation axis. This tough algebraic task then followed by sophisticated calculations aimed to find the instant rotation angle. The use of hypercomplex numbers essentially helped to avoid these math troubles, and about half of a century ago, quaternion algebra became a common tool serving for engineering goals of navigation and orientation. Indeed, the similarity transformation UqU�<sup>1</sup> of a quaternion q performed with the help of auxiliary quaternion U � a þ bq geometrically leads to conical rotation of the vector part of q about an axis whose direction is determined by the unit Q-vector q (e.g., [2]); the value of the instant rotation angle is computed as 2 arctanð Þ b=a .

Below, we suggest a detailed analysis of this type of description of rotations.

Consider a 2 � 2matrix (with complex-number components) <sup>U</sup> � x z

form invariant under the similarity-type transformation

5.2. Reorientation by a single rotation of the quaternion frame: Technique (ii)

special linear group U ∈ SLð Þ 2, C , detU ¼ xy � wz ¼ 1. The multiplication law (6) is obviously

<sup>q</sup><sup>n</sup><sup>0</sup> <sup>¼</sup> <sup>U</sup>qnU�<sup>1</sup>

One readily demonstrates that the matrix U is a biquaternion with the definable norm; indeed,

1

<sup>3</sup> <sup>R</sup><sup>φ</sup> <sup>2</sup> R<sup>θ</sup>

. Vice versa, different orders of the matrix product with other parameters

<sup>3</sup> <sup>R</sup><sup>φ</sup> <sup>2</sup> R<sup>θ</sup> CA, R<sup>θ</sup> <sup>1</sup> � 0 B@

<sup>1</sup> and <sup>R</sup><sup>0</sup> <sup>¼</sup> <sup>R</sup><sup>θ</sup>

<sup>1</sup> <sup>R</sup><sup>φ</sup><sup>0</sup> <sup>2</sup> <sup>R</sup><sup>ψ</sup><sup>0</sup>

w y � �

: (41)

, belonging to a

<sup>1</sup> and <sup>R</sup><sup>0</sup> <sup>¼</sup> <sup>R</sup><sup>θ</sup><sup>0</sup>

. The possibility to represent an arbitrary SOð Þ 3;R matrix

10 0 0 cos θ � sin θ 0 sin θ cos θ

> 1R<sup>φ</sup> <sup>2</sup> <sup>R</sup><sup>ψ</sup>

1 CA:

<sup>3</sup> are, in gen-

<sup>3</sup> may represent

(40)

$$a = \frac{x+y}{2}, b = \sqrt{1 - \left(\frac{x+y}{2}\right)^2},\tag{43}$$

and q is a Q-vector unit

$$\mathbf{q} = \frac{1}{\sqrt{1 - a^2}} \begin{pmatrix} \frac{\mathbf{x} - \mathbf{y}}{2} & \mathbf{z} \\ \mathbf{w} & -\frac{\mathbf{x} - \mathbf{y}}{2} \end{pmatrix} \mathbf{q}^2 = -1. \tag{44}$$

The unit vector (44) represented through the constant basis (7) has the form; q ¼ lkq<sup>k</sup> ¼ ð Þ bk=b q<sup>k</sup> where lk ¼ bk=b are components of a unit vector pointing in 3D space a vector with components bk, then the condition det<sup>U</sup> <sup>¼</sup> xy � wz <sup>¼</sup> 1 takes the form <sup>a</sup><sup>2</sup> <sup>þ</sup> b2 <sup>¼</sup> 1, <sup>b</sup><sup>2</sup> <sup>¼</sup> bkbk. This general biquaternion case will be used in subsequent studies when combined rotation-plus-translational motion is regarded (see Section 6). In this section, we consider only quaternion case: a, bk ∈ R, so the matrix U is unimodular if

$$a \equiv \cos \alpha, \ b = \sqrt{b\_k b\_k} \equiv \sin \alpha \tag{45}$$

therefore,

$$\boldsymbol{U} = \cos a + (\sin a) \; l\_n \mathbf{q}\_{n'} \; \mathcal{U}^{-1} = \cos a - (\sin a) \; l\_n \mathbf{q}\_n \tag{46}$$

with lk representing cosines of angles between Q-vectors q<sup>k</sup> and the direction determined by q. With the help of Eqs. (46) and (2), we reproduce the transformation (42) in the developed form

$$\begin{array}{rcl} \mathbf{q}\_{k'} & \equiv & \mathcal{U}\mathbf{q}\_k \mathcal{U}^{-1} = \left(\cos\alpha + \sin a \, l\_n \mathbf{q}\_n\right) \mathbf{q}\_k \left(\cos\alpha - \sin a \, l\_n \mathbf{q}\_m\right) =\\ & & = & 2\sin^2 a l\_k l\_n \mathbf{q}\_n + \cos 2a \mathbf{q}\_k + \sin 2a l\_n \varepsilon\_{nkm} \mathbf{q}\_m =\\ & & = & \left[l\_k l\_n + \cos 2a(\delta\_{kn} - l\_k l\_n) + \sin 2a l\_n \varepsilon\_{mkn}\right] \mathbf{q}\_n. \end{array} \tag{47}$$

Eq. (47) in fact interlinks the SOð Þ 3; R rotation matrix components and the parameters of SUð Þ2 transformations of a Q-frame [compare with (39)]. As well, Eq. (47) helps to make the following geometric analysis.

Multiplied by lk (with summation in index k), Eq. (47) yields the equality lkq<sup>k</sup><sup>0</sup> ¼ lnqn, meaning that vectors of the transformed frame q<sup>k</sup> <sup>0</sup> have the same projections onto vector lk as the initial frame qk; i.e., the transformation may be represented as a conical rotation about lk, Φ � 2α, which is angle of the rotation in the orthogonal plane with the metric pkn ¼ δkn � lkln [see the second term in Eq. (47)]. Let two unit vectors ek, nk form this plane pkn ¼ eken þ nknn, then lmεmkn ¼ eknn � ennk, and the SOð Þ 3;R -matrix comprised in Eq. (47) acquires the form

$$R\_{k'n} = l\_k l\_n + \cos\Phi \ (e\_k e\_n + n\_k n\_n) + \sin\Phi \ (e\_k n\_n - e\_k n\_n). \tag{48}$$

Introducing now two artificial unit vectors with complex number components sk � ð Þ ek <sup>þ</sup> ink <sup>=</sup> ffiffiffi 2 <sup>p</sup> and <sup>s</sup><sup>∗</sup> <sup>k</sup> � ð Þ ek � ink <sup>=</sup> ffiffiffi 2 <sup>p</sup> , we get the final (canonical) expression

$$R\_{kn} = l\_k l\_n + e^{i\Phi} \left. s\_k s\_n^\* + e^{-i\Phi} \right| s\_k^\* s\_n. \tag{49}$$

domain of complex-number valued 2D fractal space [see Eqs. (37)]. Then, rotation (reorientation) of the frame q<sup>k</sup> by the technique (ii) on the base of the transformation (42) induces

<sup>ψ</sup>0� <sup>¼</sup> <sup>U</sup>ψ�,φ0� <sup>¼</sup> <sup>φ</sup>�U�<sup>1</sup>

Further on, we use for the dyad the eigenvectors ψ� [and eigencovectors as Hermitian conju-

Normalization and orthogonality conditions are identically satisfied. The matrix U, as a qua-

<sup>¼</sup> cos <sup>α</sup> þ �l1<sup>i</sup> <sup>ψ</sup>þφ� <sup>þ</sup> <sup>ψ</sup>�φ<sup>þ</sup> <sup>ð</sup> Þ þ <sup>l</sup><sup>2</sup> <sup>ψ</sup>þφ� � <sup>ψ</sup>�φ<sup>þ</sup> <sup>ð</sup> Þ þ <sup>l</sup>3<sup>i</sup> <sup>ψ</sup>þφ<sup>þ</sup> � <sup>ψ</sup>�φ� ½ � ð Þ sin <sup>α</sup>, (55a)

<sup>U</sup>�<sup>1</sup> <sup>¼</sup> cos <sup>α</sup> � �l1<sup>i</sup> <sup>ψ</sup>þφ� <sup>þ</sup> <sup>ψ</sup>�φ<sup>þ</sup> <sup>ð</sup> Þ þ <sup>l</sup><sup>2</sup> <sup>ψ</sup>þφ� � <sup>ψ</sup>�φ<sup>þ</sup> <sup>ð</sup> Þ þ <sup>l</sup>3<sup>i</sup> <sup>ψ</sup>þφ<sup>þ</sup> � <sup>ψ</sup>�φ� ½ � ð Þ sin <sup>α</sup>: (55b)

Eq. (56) shows that the nonlinear problem formulated within the technique (ii), on the fractal

To get technological formulas convenient for fast numerical computation, we denote the final

The second vector ψ0� and the co-vectors are simply expressed through the factors (57) and

: (53)

http://dx.doi.org/10.5772/intechopen.71751

49

,φ� <sup>¼</sup> ð Þ 1 0 : (54)

] of q<sup>3</sup> of any Q-triad, where respective eigenvalues being �i.

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft

0

ψ0þ ¼ ð Þ cos α þ il<sup>3</sup> sin α ψ<sup>þ</sup> � sin αð Þ il<sup>1</sup> þ l<sup>2</sup> ψ�, (56a)

ψ0� ¼ sin αð Þ �il<sup>1</sup> þ l<sup>2</sup> ψ<sup>þ</sup> þ ð Þ cos α � il<sup>3</sup> sin α ψ�, (56b)

φ0þ ¼ ð Þ cos α þ il<sup>3</sup> sin α φ<sup>þ</sup> � sin αð Þ il<sup>1</sup> þ l<sup>2</sup> φ�, (56c)

φ0� ¼ sin αð Þ il<sup>1</sup> þ l<sup>2</sup> φ<sup>þ</sup> þ ð Þ cos α þ il<sup>3</sup> sin α φ�: (56d)

A � cos α þ il<sup>3</sup> sin α, B � sin αð Þ il<sup>1</sup> þ l<sup>2</sup> : (57)

ψ0þ � Aψ<sup>þ</sup> � Bψ�: (58a)

specific type of the "interior" rotation on the fractal surface level [see Eq. (38)]

,φ<sup>þ</sup> <sup>¼</sup> ð Þ 0 1 ,ψ� <sup>¼</sup> <sup>1</sup>

In the simplest case of q<sup>3</sup> from Eq. (7), the constant dyad is

<sup>ψ</sup><sup>þ</sup> <sup>¼</sup> <sup>0</sup> 1

ternion (46), is expressible in terms of the fractal basis

surface level, is reduced to a linear task of the 2D basis rotation.

Then, we notice that only one new dyad vector is to be computed,

gation of the vectors <sup>φ</sup>� <sup>¼</sup> <sup>ψ</sup>� <sup>T</sup>

U ¼ cos α þ lnq<sup>n</sup> sin α ¼

Therefore, Eq. (53) takes the form

values of the new 2D basis as

their complex conjugation

Eq. (49) is just an explicit formulation of the spectral theorem applied on a 3D orthogonal matrix. Since its determinant differs from zero, this matrix is nonsingular, all its eigenvalues λð Þ<sup>i</sup> are different, so it is simple; therefore, it can be expanded into a series of projectors Cð Þ<sup>i</sup> with λð Þ<sup>i</sup> as coefficients

$$\mathcal{R} = \sum\_{i=1}^{3} \lambda\_{(i)} \mathbb{C}\_{(i)}.\tag{50}$$

Here, <sup>λ</sup>ð Þ<sup>1</sup> <sup>¼</sup> 1, <sup>λ</sup>ð Þ<sup>2</sup> <sup>¼</sup> <sup>e</sup><sup>i</sup><sup>Φ</sup>, <sup>λ</sup>ð Þ<sup>3</sup> <sup>¼</sup> <sup>e</sup>�i<sup>Φ</sup>, <sup>C</sup>ð Þ<sup>1</sup> kn <sup>¼</sup> lkln, <sup>C</sup>ð2Þkn <sup>¼</sup> sks� <sup>n</sup>, <sup>C</sup>ð Þ<sup>3</sup> kn <sup>¼</sup> sks<sup>∗</sup> <sup>n</sup>; the projectors are idempotents CN ð Þ<sup>i</sup> <sup>¼</sup> <sup>C</sup>ð Þ<sup>i</sup> , <sup>N</sup> being a natural number, TrCð Þ<sup>i</sup> <sup>¼</sup> 1, detCð Þ<sup>i</sup> <sup>¼</sup> 0. It is important to note that the decomposition of a matrix R into the series (49, 50) necessarily leads to appearance of the complex-numbered 2D basis sk s<sup>∗</sup> <sup>k</sup> ; we will indicate similar features in the fractal technique (iii) below.

The value of the single rotation angle follows from computation of the trace of the matrix (49)

$$\Phi = 2\alpha = \arccos\left(\frac{O\_{k'k} - 1}{2}\right);\tag{51}$$

antisymmetric part of the matrix yields the components of unit vector directing the rotation axis

$$l\_{\circ} = \mathrm{is}\_{k} \mathrm{s}\_{m}^{\*} \varepsilon\_{kn\circ} = \frac{O\_{k'm} \varepsilon\_{kn\circ}}{\sqrt{(\mathfrak{B} - O\_{n'n})(1 + O\_{n'n})}} \cdot \tag{52}$$

Eqs. (51) and (52) represent parameters of the single rotations, the angle Φ. and components lj of the vector pointing the rotation axis, as functions of an arbitrary SOð Þ 3; C rotation angles, e.g., yaw, roll, and pitch f g ψ;φ; θ , and parameters of an equivalent single rotation, the value of the angle Φ and components (in the initial frame) lj of the vector pointing the rotation axis.

#### 5.3. Reorientation as transformation of a fractal surface, technique (iii)

In Section 4, we demonstrated that each vector of any Q-triad q<sup>k</sup> is a linear combination of vector-covector direct products of its proper biorthogonal basis ψ�; φ� � � belonging to a domain of complex-number valued 2D fractal space [see Eqs. (37)]. Then, rotation (reorientation) of the frame q<sup>k</sup> by the technique (ii) on the base of the transformation (42) induces specific type of the "interior" rotation on the fractal surface level [see Eq. (38)]

$$
\psi'^{\pm} = \mathcal{U}\psi^{\pm}, \mathcal{q}'^{\pm} = \mathcal{q}^{\pm}\mathcal{U}^{-1}.\tag{53}
$$

Further on, we use for the dyad the eigenvectors ψ� [and eigencovectors as Hermitian conjugation of the vectors <sup>φ</sup>� <sup>¼</sup> <sup>ψ</sup>� <sup>T</sup> ] of q<sup>3</sup> of any Q-triad, where respective eigenvalues being �i. In the simplest case of q<sup>3</sup> from Eq. (7), the constant dyad is

$$
\psi^+ = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \varphi^+ = \begin{pmatrix} 0 & 1 \end{pmatrix}, \psi^- = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \varphi^- = \begin{pmatrix} 1 & 0 \end{pmatrix}. \tag{54}
$$

Normalization and orthogonality conditions are identically satisfied. The matrix U, as a quaternion (46), is expressible in terms of the fractal basis

$$\begin{aligned} \mathbf{U} &= \cos a + l\_n \mathbf{q}\_n \sin a = \\ \mathbf{f} &= \cos a + \left[ -l\_1 \mathbf{i} (\psi^+ \boldsymbol{\varphi}^- + \psi^- \boldsymbol{\varphi}^+) + l\_2 (\psi^+ \boldsymbol{\varphi}^- - \psi^- \boldsymbol{\varphi}^+) + l\_3 \mathbf{i} (\psi^+ \boldsymbol{\varphi}^+ - \psi^- \boldsymbol{\varphi}^-) \right] \sin a, \end{aligned} \tag{55a}$$

$$\Delta U^{-1} = \cos a - [-l\_1 i(\psi^+ \varphi^- + \psi^- \varphi^+) + l\_2(\psi^+ \varphi^- - \psi^- \varphi^+) + l\_3 i(\psi^+ \varphi^+ - \psi^- \varphi^-)] \sin a. \tag{55b}$$

Therefore, Eq. (53) takes the form

second term in Eq. (47)]. Let two unit vectors ek, nk form this plane pkn ¼ eken þ nknn, then

Introducing now two artificial unit vectors with complex number components sk � ð Þ ek <sup>þ</sup> ink <sup>=</sup> ffiffiffi

<sup>i</sup><sup>Φ</sup> sks ∗ <sup>n</sup> þ e

Eq. (49) is just an explicit formulation of the spectral theorem applied on a 3D orthogonal matrix. Since its determinant differs from zero, this matrix is nonsingular, all its eigenvalues λð Þ<sup>i</sup> are different, so it is simple; therefore, it can be expanded into a series of projectors Cð Þ<sup>i</sup> with

<sup>p</sup> , we get the final (canonical) expression

<sup>R</sup> <sup>¼</sup> <sup>X</sup> 3

i¼1

note that the decomposition of a matrix R into the series (49, 50) necessarily leads to appear-

The value of the single rotation angle follows from computation of the trace of the matrix (49)

antisymmetric part of the matrix yields the components of unit vector directing the rotation axis

Eqs. (51) and (52) represent parameters of the single rotations, the angle Φ. and components lj of the vector pointing the rotation axis, as functions of an arbitrary SOð Þ 3; C rotation angles, e.g., yaw, roll, and pitch f g ψ;φ; θ , and parameters of an equivalent single rotation, the value of the angle Φ and components (in the initial frame) lj of the vector pointing the rotation axis.

In Section 4, we demonstrated that each vector of any Q-triad q<sup>k</sup> is a linear combination of vector-covector direct products of its proper biorthogonal basis ψ�; φ� � � belonging to a

<sup>m</sup>εknj <sup>¼</sup> Ok<sup>0</sup>

Φ ¼ 2α ¼ arccos

Rkn ¼ lkln þ e

Here, <sup>λ</sup>ð Þ<sup>1</sup> <sup>¼</sup> 1, <sup>λ</sup>ð Þ<sup>2</sup> <sup>¼</sup> <sup>e</sup><sup>i</sup><sup>Φ</sup>, <sup>λ</sup>ð Þ<sup>3</sup> <sup>¼</sup> <sup>e</sup>�i<sup>Φ</sup>, <sup>C</sup>ð Þ<sup>1</sup> kn <sup>¼</sup> lkln, <sup>C</sup>ð2Þkn <sup>¼</sup> sks�

lj ¼ isks ∗

5.3. Reorientation as transformation of a fractal surface, technique (iii)

ance of the complex-numbered 2D basis sk s<sup>∗</sup>

<sup>n</sup> ¼ lkln þ cosΦ ð Þþ eken þ nknn sinΦ ð Þ eknn � eknn : (48)

�i<sup>Φ</sup> s ∗

ð Þ<sup>i</sup> <sup>¼</sup> <sup>C</sup>ð Þ<sup>i</sup> , <sup>N</sup> being a natural number, TrCð Þ<sup>i</sup> <sup>¼</sup> 1, detCð Þ<sup>i</sup> <sup>¼</sup> 0. It is important to

Ok<sup>0</sup> <sup>k</sup> � 1 2 � �

<sup>m</sup>εkmj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 � On<sup>0</sup> ð Þ<sup>n</sup> 1 þ On<sup>0</sup> ð Þ<sup>n</sup>

<sup>k</sup> sn: (49)

<sup>n</sup>; the projectors are

λð Þ<sup>i</sup> Cð Þ<sup>i</sup> : (50)

<sup>n</sup>, <sup>C</sup>ð Þ<sup>3</sup> kn <sup>¼</sup> sks<sup>∗</sup>

<sup>k</sup> ; we will indicate similar features in the fractal

<sup>p</sup> : (52)

; (51)

lmεmkn ¼ eknn � ennk, and the SOð Þ 3;R -matrix comprised in Eq. (47) acquires the form

Rk 0

2

<sup>k</sup> � ð Þ ek � ink <sup>=</sup> ffiffiffi

2 <sup>p</sup> and <sup>s</sup><sup>∗</sup>

48 Space Flight

λð Þ<sup>i</sup> as coefficients

idempotents CN

technique (iii) below.

$$
\psi'^+ = (\cos a + il\_3 \sin a)\psi^+ - \sin a (il\_1 + l\_2)\psi^-,\tag{56a}
$$

$$
\psi^{\prime -} = \sin a (-il\_1 + l\_2) \psi^+ + (\cos a - il\_3 \sin a) \psi^-, \tag{56b}
$$

$$
\phi'^+ = (\cos a + il\_3 \sin a)\phi^+ - \sin a (il\_1 + l\_2)\phi^-,\tag{56c}
$$

$$
\phi'^- = \sin a (\text{il}\_1 + l\_2) \phi^+ + (\cos a + \text{il}\_3 \sin a) \phi^-. \tag{56d}
$$

Eq. (56) shows that the nonlinear problem formulated within the technique (ii), on the fractal surface level, is reduced to a linear task of the 2D basis rotation.

To get technological formulas convenient for fast numerical computation, we denote the final values of the new 2D basis as

$$A \equiv \cos a + il\_3 \sin a, \ B \equiv \sin a (il\_1 + l\_2). \tag{57}$$

Then, we notice that only one new dyad vector is to be computed,

$$
\psi^{\prime +} \equiv A \psi^+ - B \psi^- \,. \tag{58a}
$$

The second vector ψ0� and the co-vectors are simply expressed through the factors (57) and their complex conjugation

Figure 1. Fractal "joystick tool".

$$
\psi^{\prime -} = B^\* \psi^+ + A^\* \psi^-, \\
\varphi^{\prime +} \equiv A^\* \varphi^+ - B^\* \varphi^-, \\
\varphi^{\prime -} = B \varphi^+ + A \varphi^-. \tag{58b}
$$

This helps to represent the 3D reorientation processes "subgeometrically", on the 2D fractal level, as a displacement of a "joystick" tool (see [12] and Figure 1).

2D complex-numbered space can be imaged as a pyramid (with no base) consisting of one real, one imaginary, and two mixed real-imaginary joined surfaces. The joystick has one of its end matched with the pyramid's top by a hinge; a certain shift of the stick gives components of a new dyad vectors and co-vectors. From these fractal elements, a new Q-frame providing the assigned reorientation of the spacecraft is straightforwardly built.

All reorientation parameters providing operations in the fractal space are in fact the components of the matrix U ∈ SUð Þ2 ; therefore, the unit vector directing the axis of instant rotation is given by Eq. (52); the fractal rotation angle is

$$\alpha = \arccos\left(\frac{\sqrt{1 + O\_{k'k}}}{2}\right). \tag{59}$$

• The angle of fractal rotation is computed [Eq. (59)].

operational systems realizing the reorientation.

Eq. (49)].

number surface.

lation of the technique (iii).

O<sup>i</sup><sup>η</sup> <sup>3</sup> ¼ 0 B@

about one axis

• The dyad and resulting Q-triad are computed [Eqs. (56), (37), much simpler than in

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51

• If the computed and assigned frames match, then the rotation parameters are sent to the

The study suggested in Section 5 gives detailed analysis of math mechanisms linking two different approaches to solution of an object's reorientation task, a consequent 3D rotations described by matrices and a single rotation about an instant axis described by matrices. We like to emphasize importance (and original form) of Eqs. (48) and (49) explicitly demonstrating the projector-eigenvalue decomposition of any SOð Þ 3; R matrix, so immediately giving technological values of the single rotation. Another novel math feature of the problem is its connection with subgeometric properties of a fractal complex

However, thorough analysis of the Q-math reveals its additional, and important, option quite helpful in operational tasks. Namely, extension of the groups SOð Þ 3;R and SUð Þ2 to the rotations with complex parameters, SOð Þ 3;С and SLð Þ 2; C , respectively, with the vectorquaternion version of relativity theory taken into account, may open a possibility not only reorient but as well simultaneously endow a spacecraft with velocity assigned in value and direction. Apparently, this math tool matching rotations and accelerations, if possible in 3D space, should exist as fractal mechanism. Designing of such original (and exotic) operational

In this section, we essentially extend the methods briefly described above. The crucial point of the extension is introduction of an imaginary parameter of rotation, thus involving hyperbolic functions. We assume that this action will result in possibility to control not only orientation, but as well dynamics of the spacecraft. We will prove the assumption within extended formu-

But at first, to make the picture more clear, we show it in framework of 3D serial rotations [technique (i)], and for simplicity, we implement just one supplement plane hyperbolic rotation

1

so that hyperbolic functions are introduced. Then, complete rotational operator is

CA <sup>¼</sup>

0 B@

cosh η �i sinh η 0 i sinh η cosh η 0 0 01

1

CA � <sup>H</sup><sup>η</sup>

<sup>3</sup>, (60)

instrument is a challenging task; it is in detail analyzed in the next section.

6. Hyperbolic rotations and a fractal pyramid

cos ðÞ � iη sin ð Þ iη 0 sin ð Þ iη cos ð Þ iη 0 0 01

Eqs. (59), (52), (56), and (37) suggest a very simple algorithm for computation of all parameters of a single rotation and resulting matrices of a reoriented Q-triad describing new orientation of a spacecraft.

The technological scheme of the reorientation procedure can be briefly outlined as the following steps:


The study suggested in Section 5 gives detailed analysis of math mechanisms linking two different approaches to solution of an object's reorientation task, a consequent 3D rotations described by matrices and a single rotation about an instant axis described by matrices. We like to emphasize importance (and original form) of Eqs. (48) and (49) explicitly demonstrating the projector-eigenvalue decomposition of any SOð Þ 3; R matrix, so immediately giving technological values of the single rotation. Another novel math feature of the problem is its connection with subgeometric properties of a fractal complex number surface.

However, thorough analysis of the Q-math reveals its additional, and important, option quite helpful in operational tasks. Namely, extension of the groups SOð Þ 3;R and SUð Þ2 to the rotations with complex parameters, SOð Þ 3;С and SLð Þ 2; C , respectively, with the vectorquaternion version of relativity theory taken into account, may open a possibility not only reorient but as well simultaneously endow a spacecraft with velocity assigned in value and direction. Apparently, this math tool matching rotations and accelerations, if possible in 3D space, should exist as fractal mechanism. Designing of such original (and exotic) operational instrument is a challenging task; it is in detail analyzed in the next section.

#### 6. Hyperbolic rotations and a fractal pyramid

<sup>ψ</sup>0� <sup>¼</sup> <sup>B</sup><sup>∗</sup>

Figure 1. Fractal "joystick tool".

50 Space Flight

given by Eq. (52); the fractal rotation angle is

a spacecraft.

ing steps:

<sup>ψ</sup><sup>þ</sup> <sup>þ</sup> <sup>A</sup><sup>∗</sup>

level, as a displacement of a "joystick" tool (see [12] and Figure 1).

assigned reorientation of the spacecraft is straightforwardly built.

<sup>ψ</sup>�,φ0þ � <sup>A</sup><sup>∗</sup>

α ¼ arccos

• A spacecraft reorientation is assigned by a series of simple rotations [Eq. (40)].

• Components of the rotation axis vector are computed [Eq. (52)].

This helps to represent the 3D reorientation processes "subgeometrically", on the 2D fractal

2D complex-numbered space can be imaged as a pyramid (with no base) consisting of one real, one imaginary, and two mixed real-imaginary joined surfaces. The joystick has one of its end matched with the pyramid's top by a hinge; a certain shift of the stick gives components of a new dyad vectors and co-vectors. From these fractal elements, a new Q-frame providing the

All reorientation parameters providing operations in the fractal space are in fact the components of the matrix U ∈ SUð Þ2 ; therefore, the unit vector directing the axis of instant rotation is

p

Eqs. (59), (52), (56), and (37) suggest a very simple algorithm for computation of all parameters of a single rotation and resulting matrices of a reoriented Q-triad describing new orientation of

The technological scheme of the reorientation procedure can be briefly outlined as the follow-

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ Ok<sup>0</sup> k

2 !

<sup>φ</sup><sup>þ</sup> � <sup>B</sup><sup>∗</sup>φ�,φ0� <sup>¼</sup> <sup>B</sup>φ<sup>þ</sup> <sup>þ</sup> <sup>A</sup>φ�: (58b)

: (59)

In this section, we essentially extend the methods briefly described above. The crucial point of the extension is introduction of an imaginary parameter of rotation, thus involving hyperbolic functions. We assume that this action will result in possibility to control not only orientation, but as well dynamics of the spacecraft. We will prove the assumption within extended formulation of the technique (iii).

But at first, to make the picture more clear, we show it in framework of 3D serial rotations [technique (i)], and for simplicity, we implement just one supplement plane hyperbolic rotation about one axis

$$O\_3^{i\eta} = \begin{pmatrix} \cos\left(i\eta\right) & -\sin\left(i\eta\right) & 0\\ \sin\left(i\eta\right) & \cos\left(i\eta\right) & 0\\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} \cosh\eta & -i\sinh\eta & 0\\ i\sinh\eta & \cosh\eta & 0\\ 0 & 0 & 1 \end{pmatrix} \equiv H\_{3'}^{\eta} \tag{60}$$

so that hyperbolic functions are introduced. Then, complete rotational operator is

$$O = H\_3^\eta R \tag{61}$$

is made in the fractal surface format, and then returned into 3D space. Despite seeming

So, following the ideology of geometrization of the algebraic actions, we plunge into the fractal medium, and we consider the technique (iii). We rewrite fractal mapping with the operator

<sup>2</sup> � <sup>i</sup> sinh <sup>η</sup>

<sup>2</sup> lkq<sup>k</sup>

q3ψ<sup>þ</sup> ¼ þiψþ, q3ψ� ¼ �iψþ, φþq<sup>3</sup> ¼ þiφþ, φ�q<sup>3</sup> ¼ �iφ�; (71)

ψ0�, φ<sup>þ</sup> ¼ e

<sup>2</sup> <sup>i</sup>q<sup>1</sup> � <sup>q</sup><sup>2</sup>

2 q3

<sup>ψ</sup>0� (69)

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<sup>ψ</sup><sup>~</sup> �: (70)

�η=2

φ0þ, φ� ¼ e

�ηψ�φ<sup>þ</sup> ð Þ: (73)

; (74)

: (75a)

η=2

φ0�: (72)

complexity of the given expressions, the final calculation is shown to be very simple.

where the intermediate dyad is a result of the real rotation (similar with the covectors)

Φ <sup>2</sup> <sup>þ</sup> sin <sup>Φ</sup>

We also stress that all dyad elements used in the computations are always the eigenvectors

hence, Eq. (69) produces a new fractal basis simply multiplying the intermediate dyad by an

By other words, one dyad vector and one co-vector (here ψþand φ�) become longer, and the others (ψ� and φþ) become shorter, all of them though preserving unit length, i.e., rescaled.

This primitive mapping has clear physical sense concerning kinematic of a spacecraft. To reveal it, we, using Eq. (75), build an "imaginary constituent" of the 3D frame vector q<sup>1</sup><sup>0</sup> as in

, <sup>ψ</sup>�φ<sup>þ</sup> <sup>¼</sup> <sup>1</sup>

iq<sup>1</sup><sup>0</sup> ¼ cosh η iq<sup>1</sup> þ tanhη q<sup>2</sup>

Eq. (75a) rewritten in terms of the Pauli-type matrices [as in Eqs. (20), (22)] p � iq has the form

�η=2

ψ0þ, ψ� ¼ e

<sup>q</sup><sup>1</sup><sup>0</sup> ¼ �<sup>i</sup> <sup>ψ</sup>0þφ0� <sup>þ</sup> <sup>ψ</sup>0�φ0þ ¼ �i eηψþφ� <sup>þ</sup> <sup>e</sup>

<sup>2</sup> <sup>i</sup>q<sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup>

<sup>ψ</sup>� <sup>¼</sup> <sup>U</sup>ψ0� <sup>¼</sup> cosh <sup>η</sup>

ψ0� ¼ cos

η=2

<sup>ψ</sup>þφ� <sup>¼</sup> <sup>1</sup>

(eigencovectors) of the quaternion unit q<sup>3</sup>

<sup>2</sup> <sup>þ</sup> sin <sup>η</sup> 2 <sup>ψ</sup>0þ <sup>¼</sup> <sup>e</sup>

However from Eqs. (37b, c), we find

substitution of the Eq. (74) into Eq. (73) yields

(62) in the form

exponent

Eq. (37b).

<sup>ψ</sup><sup>þ</sup> <sup>¼</sup> cosh <sup>η</sup>

We rewrite the operator (61) in the spinor-type form where the tilde denotes some initial basis

$$
\Delta U = \left(\cosh\frac{\eta}{2} - i\sinh\frac{\eta}{2}\mathbf{q}\_3\right)\left(\cos\frac{\Phi}{2} + \sin\frac{\Phi}{2}l\_{\bar{k}}\mathbf{q}\_{\bar{k}}\right),
\tag{62}
$$

and the components of the instant rotation axis vector given by Eq. (52). It is important to note that in the computation procedure, we have to deal with vectors belonging to the same frame. Therefore, we express q<sup>3</sup><sup>0</sup> ¼ R3nq<sup>n</sup> and make multiplication in Eq. (62) to obtain

$$\mathcal{U} = \cosh\frac{\eta}{2}\cos\frac{\Phi}{2} - i\sinh\frac{\eta}{2}\mathcal{R}\_{3\dot{n}}l\_{\dot{n}} + \left[\cosh\frac{\eta}{2}\sin\frac{\Phi}{2}l\_{\dot{n}} - i\sinh\frac{\eta}{2}\left(\cos\frac{\Phi}{2}\mathcal{R}\_{3\dot{n}} + \sin\frac{\Phi}{2}\mathcal{R}\_{3\dot{\beta}}l\_{\dot{n}}\varepsilon\_{\dot{j}\dot{m}}\right)\right]\mathbf{q}\_{\dot{n}}.\tag{63}$$

This expression is again a quaternion and we denote it as

$$\mathcal{U} = \cos\Theta + (\sin\Theta)\mathbf{q}\_\prime \mathcal{U}^{-1} = \cos\Theta - (\sin\Theta)\mathbf{q} \tag{64}$$

where

$$\cos\Theta \equiv \cos\frac{\eta}{2}\cos\frac{\Phi}{2} - i\sinh\eta\sin\frac{\Phi}{2}R\_{3\bar{\eta}}l\_{\bar{\eta}\nu} \tag{65}$$

$$\sin\Theta \mathbf{q} \equiv \left[ \cosh\frac{\eta}{2} \sin\frac{\Phi}{2} l\_{\hat{n}} - i \sinh\frac{\eta}{2} \left( \cos\frac{\Phi}{2} R\_{3\hat{n}} + \sin\frac{\Phi}{2} R\_{3\hat{n}} l \|r\_{j\text{inv}}\right) \right] \mathbf{q}\_{\text{n}\prime} \tag{66}$$

parameter Θ being a complex number. One straightforwardly verifies fulfilling the identity

$$\begin{split} \left(\cos^{2}\Theta + \left(\sin^{2}\Theta\right)\right)\mathbf{q}^{2} &= \left(\cosh\frac{\eta}{2}\cos\frac{\Phi}{2} - i\sinh\eta\sin\frac{\Phi}{2}R\_{3i}l\bar{\eta}\right)\left(\cosh\frac{\eta}{2}\cos\frac{\Phi}{2} - i\sinh\eta\sin\frac{\Phi}{2}R\_{3\bar{\eta}}l\bar{\eta}\right) + \\ &+ \left[\cosh\frac{\eta}{2}\sin\frac{\Phi}{2}l\bar{\eta} - i\sinh\frac{\eta}{2}\left(\cos\frac{\Phi}{2}R\_{3i\bar{\eta}} + \sin\frac{\Phi}{2}R\_{3j}l\bar{\eta}\varepsilon\_{j\bar{\eta}\bar{\eta}}\right)\right] \times \\ &\times \left[\cosh\frac{\eta}{2}\sin\frac{\Phi}{2}l\bar{\eta} - i\sinh\frac{\eta}{2}\left(\cosh\frac{\Phi}{2}R\_{3i\bar{\eta}} + \sin\frac{\Phi}{2}R\_{3j}l\bar{\eta}\varepsilon\_{j\bar{\eta}\bar{\eta}}\right)\right] \mathbf{q}\bar{\eta}\,\bar{\mathbf{q}}\bar{\eta} = 1. \end{split} \tag{67}$$

Expression for the vector-directing axis of the single rotation is found from Eqs. (65) and (66)

$$l\_{n} = \frac{\cosh\frac{\eta}{2}, \sin\frac{\Phi}{2}l\_{\hat{n}}, -i, \sinh\frac{\eta}{2}, \left(\cos\frac{\Phi}{2}R\_{3\hat{n}} + \sin\frac{\Phi}{2}R\_{\hat{n}\hat{j}}l\|\varepsilon\_{j\text{nn}}\right)}{\sqrt{1 - \left(\cosh\frac{\eta}{2}\cos\frac{\Phi}{2} - i\sinh\eta\sin\frac{\Phi}{2}R\_{3\hat{n}}l\_{\hat{n}}\right)\left(\cosh\frac{\eta}{2}\cos\frac{\Phi}{2} - i\sinh\eta\sin\frac{\Phi}{2}R\_{3\hat{p}}l\_{\hat{p}}\right)}}\tag{68}$$

Eq. (61) represents an operator performing the serial rotation, and Eqs. (65), (68) give parameters of a single rotation. Physical content of this rotation is easily revealed when the mapping is made in the fractal surface format, and then returned into 3D space. Despite seeming complexity of the given expressions, the final calculation is shown to be very simple.

So, following the ideology of geometrization of the algebraic actions, we plunge into the fractal medium, and we consider the technique (iii). We rewrite fractal mapping with the operator (62) in the form

$$
\psi^{\pm} = \mathcal{U}\psi'^{\pm} = \left(\cosh\frac{\eta}{2} - i\sinh\frac{\eta}{2}\mathbf{q}\_3\right)\psi'^{\pm} \tag{69}
$$

where the intermediate dyad is a result of the real rotation (similar with the covectors)

$$
\psi^{\prime \pm} = \left(\cos\frac{\Phi}{2} + \sin\frac{\Phi}{2}l\_k \mathbf{q}\_k\right) \tilde{\psi}^{\pm}.\tag{70}
$$

We also stress that all dyad elements used in the computations are always the eigenvectors (eigencovectors) of the quaternion unit q<sup>3</sup>

$$\mathbf{q}\_3 \psi^+ = +i\psi^+, \quad \mathbf{q}\_3 \psi^- = -i\psi^+, \quad \wp^+ \mathbf{q}\_3 = +i\wp^+, \quad \wp^- \mathbf{q}\_3 = -i\wp^-; \tag{71}$$

hence, Eq. (69) produces a new fractal basis simply multiplying the intermediate dyad by an exponent

$$\psi^{+} = \left(\cosh\frac{\eta}{2} + \sin\frac{\eta}{2}\right)\psi^{\prime +} = \varepsilon^{\eta/2}\psi^{\prime +}, \ \psi^{-} = \varepsilon^{-\eta/2}\psi^{\prime -}, \ \phi^{+} = \varepsilon^{-\eta/2}\phi^{\prime +}, \ \phi^{-} = \varepsilon^{\eta/2}\phi^{\prime -}. \tag{72}$$

By other words, one dyad vector and one co-vector (here ψþand φ�) become longer, and the others (ψ� and φþ) become shorter, all of them though preserving unit length, i.e., rescaled.

This primitive mapping has clear physical sense concerning kinematic of a spacecraft. To reveal it, we, using Eq. (75), build an "imaginary constituent" of the 3D frame vector q<sup>1</sup><sup>0</sup> as in Eq. (37b).

$$\mathbf{q}\_{1'} = -i \left( \psi^{'+} \phi^{'-} + \psi^{'-} \phi^{'+} \right) = -i (\varepsilon^{\eta} \psi^{+} \phi^{-} + \varepsilon^{-\eta} \psi^{-} \phi^{+}).\tag{73}$$

However from Eqs. (37b, c), we find

<sup>O</sup> <sup>¼</sup> <sup>H</sup><sup>η</sup>

We rewrite the operator (61) in the spinor-type form where the tilde denotes some initial basis

and the components of the instant rotation axis vector given by Eq. (52). It is important to note that in the computation procedure, we have to deal with vectors belonging to the same frame.

cos Φ <sup>2</sup> <sup>þ</sup> sin <sup>Φ</sup>

<sup>2</sup> ln<sup>~</sup> � <sup>i</sup> sinh <sup>η</sup>

<sup>2</sup> � <sup>i</sup> sinh <sup>η</sup> sin <sup>Φ</sup>

Φ

<sup>2</sup> <sup>R</sup>3n<sup>~</sup> ln<sup>~</sup>

<sup>2</sup> <sup>R</sup>3n<sup>~</sup> <sup>þ</sup> sin <sup>Φ</sup>

<sup>2</sup> <sup>R</sup>3n<sup>~</sup> <sup>þ</sup> sin <sup>Φ</sup>

� �

� �

<sup>2</sup> <sup>R</sup>3n<sup>~</sup> <sup>þ</sup> sin <sup>Φ</sup>

<sup>2</sup> cos <sup>Φ</sup>

<sup>2</sup> cos

parameter Θ being a complex number. One straightforwardly verifies fulfilling the identity

<sup>2</sup> cos <sup>Φ</sup>

<sup>2</sup> cosh <sup>Φ</sup>

Expression for the vector-directing axis of the single rotation is found from Eqs. (65) and (66)

<sup>2</sup> ; cos <sup>Φ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>R</sup>3n<sup>~</sup> ln<sup>~</sup> <sup>Þ</sup> cosh <sup>η</sup>

�

Eq. (61) represents an operator performing the serial rotation, and Eqs. (65), (68) give parameters of a single rotation. Physical content of this rotation is easily revealed when the mapping

<sup>2</sup> � <sup>i</sup> sinh <sup>η</sup> sin <sup>Φ</sup>

<sup>2</sup> cos <sup>Φ</sup> 2

<sup>U</sup> <sup>¼</sup> cos <sup>Θ</sup> <sup>þ</sup> ð Þ sin <sup>Θ</sup> <sup>q</sup>, U�<sup>1</sup> <sup>¼</sup> cos <sup>Θ</sup> � ð Þ sin <sup>Θ</sup> <sup>q</sup> (64)

<sup>2</sup> <sup>R</sup>3n<sup>~</sup> <sup>þ</sup> sin <sup>Φ</sup>

cosh <sup>η</sup>

<sup>2</sup> <sup>R</sup>3~<sup>j</sup>

<sup>2</sup> <sup>R</sup>3~<sup>l</sup>

<sup>2</sup> R3~<sup>j</sup>

<sup>2</sup> cos <sup>Φ</sup>

lm~ εjmnÞ

lp~εlpsÞ

lm~ εjmn

<sup>2</sup> � <sup>i</sup> sinh <sup>η</sup> sin <sup>Φ</sup>

� �

�

2 q3

<sup>2</sup> � <sup>i</sup> sinh <sup>η</sup>

Therefore, we express q<sup>3</sup><sup>0</sup> ¼ R3nq<sup>n</sup> and make multiplication in Eq. (62) to obtain

<sup>2</sup> sin <sup>Φ</sup>

<sup>2</sup> cos Φ

<sup>2</sup> ln<sup>~</sup> � <sup>i</sup> sinh <sup>η</sup>

� �

<sup>2</sup> ln<sup>~</sup> � <sup>i</sup> sinh <sup>η</sup>

<sup>2</sup> ln<sup>~</sup> � <sup>i</sup> sinh <sup>η</sup>

<sup>2</sup> ln<sup>~</sup> ; �i; sinh <sup>η</sup>

<sup>2</sup> � <sup>i</sup> sinh <sup>η</sup> sin <sup>Φ</sup>

<sup>R</sup>3n<sup>~</sup> ln<sup>~</sup> <sup>þ</sup> cosh <sup>η</sup>

cos <sup>Θ</sup> � cos <sup>η</sup>

<sup>2</sup> sin <sup>Φ</sup>

<sup>2</sup> cos <sup>Φ</sup>

<sup>2</sup> sin <sup>Φ</sup>

<sup>2</sup> sin <sup>Φ</sup>

<sup>2</sup> ; sin <sup>Φ</sup>

q � �

�

� �

<sup>U</sup> <sup>¼</sup> cosh <sup>η</sup>

This expression is again a quaternion and we denote it as

<sup>U</sup> <sup>¼</sup> cosh <sup>η</sup>

52 Space Flight

where

<sup>2</sup> cos <sup>Φ</sup>

<sup>2</sup> � <sup>i</sup> sinh <sup>η</sup> 2

sin <sup>Θ</sup><sup>q</sup> � cosh <sup>η</sup>

cos <sup>2</sup><sup>Θ</sup> <sup>þ</sup> sin <sup>2</sup><sup>Θ</sup> � �q<sup>2</sup> <sup>¼</sup> cosh <sup>η</sup>

ln ¼

�

<sup>þ</sup> cosh <sup>η</sup>

� cosh <sup>η</sup>

cosh <sup>η</sup>

<sup>2</sup> cos <sup>Φ</sup>

<sup>1</sup> � cosh <sup>η</sup>

�

�

<sup>3</sup>R (61)

(62)

(63)

(66)

þ

(67)

(68)

<sup>2</sup> <sup>R</sup>3~<sup>p</sup> lp<sup>~</sup>

qn:

<sup>2</sup> <sup>l</sup><sup>~</sup>kq<sup>~</sup><sup>k</sup> <sup>Þ</sup>,

<sup>R</sup>3n<sup>~</sup> <sup>þ</sup> sin <sup>Φ</sup>

� �

2 R3~<sup>j</sup> lm<sup>~</sup> εjmnÞ

<sup>2</sup> <sup>R</sup>3n<sup>~</sup> ln<sup>~</sup> , (65)

<sup>2</sup> � <sup>i</sup> sinh <sup>η</sup> sin <sup>Φ</sup>

qn~q~s ¼ 1:

<sup>2</sup> R3~<sup>p</sup> l<sup>~</sup><sup>p</sup> Þ:

� �

�

qn,

<sup>2</sup> <sup>R</sup>3n<sup>~</sup> lm<sup>~</sup> <sup>ε</sup>jmn;<sup>Þ</sup>

$$
\psi^{+}\varphi^{-} = \frac{1}{2} \left( i\mathbf{q}\_{1} + \mathbf{q}\_{2} \right) , \ \psi^{-}\varphi^{+} = \frac{1}{2} \left( i\mathbf{q}\_{1} - \mathbf{q}\_{2} \right) ; \tag{74}
$$

substitution of the Eq. (74) into Eq. (73) yields

$$i\mathbf{q}\_{1'} = \cosh\eta \begin{pmatrix} i\mathbf{q}\_1 + \tanh\eta \ \mathbf{q}\_2 \end{pmatrix}.\tag{75a}$$

Eq. (75a) rewritten in terms of the Pauli-type matrices [as in Eqs. (20), (22)] p � iq has the form

$$\mathbf{p}\_{1'} = \cosh \eta (\mathbf{p}\_1 + \tanh \eta \ \mathbf{q}\_2). \tag{75b}$$

Using results of Section 3, we associate the hyperbolic functions with the time ratio

$$
\cosh \eta = \frac{dt}{dt'}\tag{76}
$$

simpler model of the joystick, and moreover, to make the picture symmetric, we show positive

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55

Computations providing the spacecraft's reorientation and acceleration are performed on the

and negative directions of the pyramid (see Figure 2).

fractal level by Eq. (58) with the functions A, B generalized as

(linking time dt of an immobile frame and proper time dt<sup>0</sup> of moving spacecraft) and with the relative velocity ratio (c is speed of light).

$$
\tanh \eta = V/c.\tag{77}
$$

Then, Eq. (75b) takes the form of "vector interval" of quaternion version of relativity theory (23)

$$dt'\ \mathbf{p}\_{1'} = dt\left(\mathbf{p}\_1 + \frac{V}{c}\ \ \mathbf{q}\_2\right);\tag{78}$$

when squared, it gives the spacecraft's special relativistic space-time interval linked with the frame at rest by the Lorentz (hyperbolic) transformation

$$dt'^2 = dt^2 \,\,\left(1 - V^2/c^2\right) \tag{79}$$

describing kinematics of a frame moving along q<sup>2</sup> with velocity V, while the vectors p1(or p<sup>1</sup>0) play the role of direction of time in the immobile (or moving) spacecraft. It is always possible to choose the direction q<sup>2</sup> as pointing the "yaw" of a spacecraft. In particular, the velocity can be small sufficiently to reduce the calculations into classical format

$$V/c = \tanh \eta \approx \eta \tag{80}$$

besides, the velocity modulus may be variable in time; hence, the spacecraft is accelerated.

So, introducing imaginary rotation angles, we obtain a possibility to control an arbitrary space reorientation of a spacecraft with variation of its velocity in the direction that can be as well changing with time (In this sample, the vector q<sup>2</sup> is in fact permanently rotating.)

This math tool has two important properties. First, a spacecraft endowed by the tool with a velocity is initially described as a relativistic system; one comes to the classical mechanics considering the hyperbolic parameter small. Second, the tool accelerates the spacecraft always in the direction of the frame vector appointed to indicate "yaw"; if this vector rotates, changing the yaw, the acceleration arrow changes with it; i.e., the spacecraft is accelerated along a curve line. These properties can be useful in real motion control.

On the 2D fractal level, the spacecraft's more complex 3D motion comprising reorientation and acceleration is accompanied by respective rotation and deformation of the mentioned above fractal pyramid. Here, this subgeometric image of the math instrument necessarily enriches a simpler model of the joystick, and moreover, to make the picture symmetric, we show positive and negative directions of the pyramid (see Figure 2).

p<sup>1</sup><sup>0</sup> ¼ cosh η p<sup>1</sup> þ tanhη q<sup>2</sup>

cosh <sup>η</sup> <sup>¼</sup> dt

(linking time dt of an immobile frame and proper time dt<sup>0</sup> of moving spacecraft) and with the

Then, Eq. (75b) takes the form of "vector interval" of quaternion version of relativity theory

when squared, it gives the spacecraft's special relativistic space-time interval linked with the

describing kinematics of a frame moving along q<sup>2</sup> with velocity V, while the vectors p1(or p<sup>1</sup>0) play the role of direction of time in the immobile (or moving) spacecraft. It is always possible to choose the direction q<sup>2</sup> as pointing the "yaw" of a spacecraft. In particular, the velocity can be

dt0<sup>2</sup> <sup>¼</sup> dt<sup>2</sup> <sup>1</sup> � <sup>V</sup><sup>2</sup>

besides, the velocity modulus may be variable in time; hence, the spacecraft is accelerated.

changing with time (In this sample, the vector q<sup>2</sup> is in fact permanently rotating.)

So, introducing imaginary rotation angles, we obtain a possibility to control an arbitrary space reorientation of a spacecraft with variation of its velocity in the direction that can be as well

This math tool has two important properties. First, a spacecraft endowed by the tool with a velocity is initially described as a relativistic system; one comes to the classical mechanics considering the hyperbolic parameter small. Second, the tool accelerates the spacecraft always in the direction of the frame vector appointed to indicate "yaw"; if this vector rotates, changing the yaw, the acceleration arrow changes with it; i.e., the spacecraft is accelerated along a curve

On the 2D fractal level, the spacecraft's more complex 3D motion comprising reorientation and acceleration is accompanied by respective rotation and deformation of the mentioned above fractal pyramid. Here, this subgeometric image of the math instrument necessarily enriches a

V <sup>c</sup> <sup>q</sup><sup>2</sup> 

=c

dt<sup>0</sup> p<sup>1</sup><sup>0</sup> ¼ dt p<sup>1</sup> þ

Using results of Section 3, we associate the hyperbolic functions with the time ratio

relative velocity ratio (c is speed of light).

frame at rest by the Lorentz (hyperbolic) transformation

small sufficiently to reduce the calculations into classical format

line. These properties can be useful in real motion control.

(23)

54 Space Flight

: (75b)

dt<sup>0</sup> (76)

; (78)

tanhη ¼ V=c: (77)

<sup>2</sup> (79)

V=c ¼ tanhη ≈ η (80)

Computations providing the spacecraft's reorientation and acceleration are performed on the fractal level by Eq. (58) with the functions A, B generalized as

Figure 2. Case (a): The spacecraft performs a 3D rotation, the pyramid is tilted by respective halfangle. Rotations and displacements of a spacecraft (Pioneer-10) accompanied by respective 2D rotations and deformations of the fractal pyramid. Case (b): The reoriented spacecraft rectilinearly moves with some velocity, and the tilted pyramid is distorted: Two its edges become shorter, and the other two edges become longer. Case (c): The spacecraft ("frees-framed") is reoriented by another angle, and the distorted pyramid as tilted by respective halfangle. Case (d): The spacecraft moves along a curve trajectory with changing velocity (accelerated), and the pyramid is subject to permanent respective tilt and distortion.

$$A \equiv \left(\cos\frac{\Phi}{2} + il\_3 \sin\frac{\Phi}{2}\right) e^{\eta/2}, \ B \equiv \sin\frac{\Phi}{2} \text{ ( $i$  } l\_1 + l\_2) e^{-\eta/2},\tag{81a}$$

with hyperbolic conjugation ( <sup>⊕</sup> : <sup>e</sup>�η=<sup>2</sup> ! <sup>e</sup> <sup>∓</sup>η=<sup>2</sup> ), similar to the complex conjugation, introduced, e.g.,

$$A^{\Theta} \equiv \left(\cos\frac{\Phi}{2} + il\_3 \sin\frac{\Phi}{2}\right) e^{-\eta/2},\ B^{\*\Theta} \equiv \sin\frac{\Phi}{2} \ (-i\ l\_1 + l\_2) e^{\eta/2},\tag{81b}$$

where vector lk directs axis of the single space rotation by angle Φ. Then (as in Section 5), only one equation is to be solved, e.g., that determining the dyad vector

$$
\psi'^+ = A \ \psi^+ - B \ \psi^-,\tag{82a}
$$

7. Technological scheme and concluding remarks

• Parameters as functions of time must be determined and input.

and velocity until the assigned values are achieved. And we emphasize two most important results of this study.

simple algorithms, other approaches having no such advantages.

(Approved for public release; distribution unlimited)

Institute of Gravitation and Cosmology, RUDN University, Moscow, Russia

Address all correspondence to: a.yefremov@rudn.ru

above approach.

literature.

Author details

References

Alexander P. Yefremov

TT-F-15414)

A sketch of technological scheme aimed to realize mixed rotation-acceleration maneuver of a spacecraft can be suggested as the following consequence of actions fit for any mentioned

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft

http://dx.doi.org/10.5772/intechopen.71751

57

• The initial and final parameters of reorientation and acceleration are assigned and memorized.

• Process of computation of quantum steps starts resulting in obtaining of a series of related parameter values describing the orientation and velocity of the spacecraft's frame.

• The data of each step are transmitted to the systems changing the spacecraft orientation

First, we succeeded to show that an extrarotation by an imaginary angle entails endowing a spacecraft with a (relativistic) velocity, hence in addition to reorientation, to accelerate it. This math observation seems to be a novel one since no similar information is met in related

Second, we show that the most mathematically economical way to compute operational parameters needed for realization of the maneuver is to utilize the "fractal pyramid" technique (definitely a new tool) comprising minimal number of math actions, where major of them are

[1] Halijak Ch.A. Quaternions Applied to Missle Systems, US Army Missle Research and Development Command, AD A 052232, Redstone Arsenal, Alabama 35809, Apr, 18, 1978

[2] Branets VN, Shmyglevskiy IP. Application of Quaternions to Rigid Body Rotation Problems, (translated from the Russian in 1974). Washington D.C.: Scientific Translation Service, National Aeronautics and Space Administrationp. 352, (1973. Report No.NASA-

• Time intervals are divided into standard steps (quantized), the standard input.

and rest of the dyad elements is found by primitive math actions

$$\psi^{\prime}A^\*\psi^+ + B^\*\psi^- , \ \phi^{\prime \prime} = A \ast^{\Theta} \ \ \phi^+ - B \ast^{\Theta} \ \ \ \phi^- \ \ \ \ \ \ \phi^{\prime \prime} = A^{\Theta} \ \phi^+ + B^{\Theta} \ \phi^- . \tag{82b}$$

Eqs. (82), (37) immediately give expressions of all spacecraft's frame vectors, thus solving the reorientation and acceleration problem in explicit form.

One straightforwardly finds that use of the fractal technique (iii) essentially simplifies computation procedures. In paper [13], we compare math difficulty of the discussed three techniques in solution of the simple problem of the spacecraft's one-plane space rotation and acceleration. It is demonstrated there that the techniques (i) and (ii) demand solution of at least seven equations, among them are matrix equations, while the fractal technique (iii) suggests solution of only four relatively simple algebraic equations.

### 7. Technological scheme and concluding remarks

A sketch of technological scheme aimed to realize mixed rotation-acceleration maneuver of a spacecraft can be suggested as the following consequence of actions fit for any mentioned above approach.


And we emphasize two most important results of this study.

First, we succeeded to show that an extrarotation by an imaginary angle entails endowing a spacecraft with a (relativistic) velocity, hence in addition to reorientation, to accelerate it. This math observation seems to be a novel one since no similar information is met in related literature.

Second, we show that the most mathematically economical way to compute operational parameters needed for realization of the maneuver is to utilize the "fractal pyramid" technique (definitely a new tool) comprising minimal number of math actions, where major of them are simple algorithms, other approaches having no such advantages.

#### Author details

A � cos

<sup>A</sup> <sup>⊕</sup> � cos

ψ0�A<sup>∗</sup>

duced, e.g.,

56 Space Flight

Φ

Φ

<sup>2</sup> <sup>þ</sup> il<sup>3</sup> sin <sup>Φ</sup>

<sup>2</sup> <sup>þ</sup> il<sup>3</sup> sin <sup>Φ</sup>

one equation is to be solved, e.g., that determining the dyad vector

and rest of the dyad elements is found by primitive math actions

reorientation and acceleration problem in explicit form.

of only four relatively simple algebraic equations.

2

with changing velocity (accelerated), and the pyramid is subject to permanent respective tilt and distortion.

2

e �η=2

e η=2

with hyperbolic conjugation ( <sup>⊕</sup> : <sup>e</sup>�η=<sup>2</sup> ! <sup>e</sup> <sup>∓</sup>η=<sup>2</sup> ), similar to the complex conjugation, intro-

Figure 2. Case (a): The spacecraft performs a 3D rotation, the pyramid is tilted by respective halfangle. Rotations and displacements of a spacecraft (Pioneer-10) accompanied by respective 2D rotations and deformations of the fractal pyramid. Case (b): The reoriented spacecraft rectilinearly moves with some velocity, and the tilted pyramid is distorted: Two its edges become shorter, and the other two edges become longer. Case (c): The spacecraft ("frees-framed") is reoriented by another angle, and the distorted pyramid as tilted by respective halfangle. Case (d): The spacecraft moves along a curve trajectory

where vector lk directs axis of the single space rotation by angle Φ. Then (as in Section 5), only

Eqs. (82), (37) immediately give expressions of all spacecraft's frame vectors, thus solving the

One straightforwardly finds that use of the fractal technique (iii) essentially simplifies computation procedures. In paper [13], we compare math difficulty of the discussed three techniques in solution of the simple problem of the spacecraft's one-plane space rotation and acceleration. It is demonstrated there that the techniques (i) and (ii) demand solution of at least seven equations, among them are matrix equations, while the fractal technique (iii) suggests solution

, B � sin <sup>Φ</sup>

, B∗ ⊕ � sin <sup>Φ</sup>

<sup>ψ</sup><sup>þ</sup> <sup>þ</sup> <sup>B</sup><sup>∗</sup>ψ�, <sup>φ</sup>0þ <sup>¼</sup> <sup>A</sup><sup>∗</sup> <sup>⊕</sup> <sup>φ</sup><sup>þ</sup> � <sup>B</sup><sup>∗</sup> <sup>⊕</sup> <sup>φ</sup>�, <sup>φ</sup>0� <sup>¼</sup> <sup>A</sup> <sup>⊕</sup> <sup>φ</sup><sup>þ</sup> <sup>þ</sup> <sup>B</sup> <sup>⊕</sup> <sup>φ</sup>�: (82b)

<sup>2</sup> ð Þ i l<sup>1</sup> <sup>þ</sup> <sup>l</sup><sup>2</sup> <sup>e</sup>

<sup>2</sup> ð Þ �i l<sup>1</sup> <sup>þ</sup> <sup>l</sup><sup>2</sup> <sup>e</sup>

ψ0þ ¼ A ψ<sup>þ</sup> � B ψ�, (82a)

�η=2

η=2

, (81a)

, (81b)

Alexander P. Yefremov

Address all correspondence to: a.yefremov@rudn.ru

Institute of Gravitation and Cosmology, RUDN University, Moscow, Russia

#### References


[3] Hamilton WR The Mathematical Papers of William Rowan Hamilton. Vol. 3. Cambridge: Cambridge University Press; 1967

**Chapter 4**

Provisional chapter

**Code Optimization for Strapdown Inertial Navigation**

DOI: 10.5772/intechopen.71732

Code Optimization for Strapdown Inertial Navigation

Inertial navigation systems are in common use for decades due to its advantages. Since INS outputs are usually used for inputs in different control algorithms (depending on applications), INS will induce certain errors and limitations. This chapter deals with optimization of the inertial navigation algorithm against limitations due to the accuracy and stability of signals from the sensors and constraints resulting from the integration step and processor speed used for embedded applications. Inertial navigation considered here is "strapdown" inertial navigation system (SINS) which assumes a fixed inertial measurement unit (IMU). In this chapter, fundamentals of strapdown inertial navigation will be presented as well as three different algorithms which will be analyzed in regard to numerical stability, time consumption and processor load criteria.

Keywords: strapdown inertial navigation system, quaternions, forward Euler

INS is inertial navigation system, the system that determines the position based on the output of the motion sensors: accelerometers and gyroscopes. The first INS was based on accelerometers mounted on gimbal platform, to ensure measurement of acceleration in navigational frame. Nowadays "strapdown" inertial navigation system (SINS) is in common use, due to its mechanical simplicity, reduced size and price compered to platform INS. Strapdown inertial navigation system implies a fixed inertial measurement unit (IMU), whereby the analytical picture of the navigation system is obtained from the integration of the gyroscope rates.

The main problem that arises when SINS is used is the exact determination of the orientation based on the gyroscopes outputs. Every error made in this stage will affect the error of projection of the gravitational acceleration. Accelerations are integrated twice in order to

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

**System Algorithm**

System Algorithm

Abstract

1. Introduction

Ivana Todić and Vladimir Kuzmanović

Ivana Todić and Vladimir Kuzmanović

http://dx.doi.org/10.5772/intechopen.71732

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

integration, code optimization, code analyses


Provisional chapter

#### **Code Optimization for Strapdown Inertial Navigation System Algorithm** Code Optimization for Strapdown Inertial Navigation System Algorithm

DOI: 10.5772/intechopen.71732

Ivana Todić and Vladimir Kuzmanović

Additional information is available at the end of the chapter Ivana Todić and Vladimir Kuzmanović

http://dx.doi.org/10.5772/intechopen.71732 Additional information is available at the end of the chapter

#### Abstract

[3] Hamilton WR The Mathematical Papers of William Rowan Hamilton. Vol. 3. Cambridge:

[4] Yefremov AP. Quaternions and biquaternions: Algebra, geometry and physical theories. Hypercomplex Numbers in Geometry and Physics. 2004;1:104-119, arXiv: math-ph/050

[5] Yefremov AP. Bi-quaternion square roots, rotational relativity, and dual space-time inter-

[6] Yefremov AP. Quaternion model of relativity: Solutions for non-inertial motions and new

[7] Wheeler JA. Pregeometry: Motivations and prospects. In: Marlov AR, editor. Quantum

[8] Yefremov AP. Splitting of 3D quaternion dimensions into 2D-cells and a "world screen

[9] Yefremov AP. Fundamental properties of quaternion Spinors. Gravitation and Cosmol-

[10] Yefremov AP. The conic-gearing image of a complex number and a spinor-born surface

[11] Lancaster P, Tismenetsky M. The Theory of Matrices with Applications. 2nd ed. San

[12] Yefremov AP. Fractal Surface as the Simplest Tool to Control Orientation of a Spacecraft.

[13] Yefremov AP. New fractal math tool providing simultaneous reorientation and accelera-

Theory and Gravitation. New York: Academic Press; 1080. pp. 1-11

technology". Advanced Science Letters. 2012;5(1):288-293

Cambridge University Press; 1967

ogy. 2010;16(2):137-139

vals. Gravitation & Cosmology. 2007;133(51):178-184

effects. Advanced Science Letters. 2008;1:179-186

geometry. Gravitation and Cosmology. 2011;17(1)

tion of spacecraft. Acta Astronautica. 2017;139:481-485

Diego, London: Academic Press; 1985. p. 154

Acta Astronautica. 2016;129:174

1055

58 Space Flight

Inertial navigation systems are in common use for decades due to its advantages. Since INS outputs are usually used for inputs in different control algorithms (depending on applications), INS will induce certain errors and limitations. This chapter deals with optimization of the inertial navigation algorithm against limitations due to the accuracy and stability of signals from the sensors and constraints resulting from the integration step and processor speed used for embedded applications. Inertial navigation considered here is "strapdown" inertial navigation system (SINS) which assumes a fixed inertial measurement unit (IMU). In this chapter, fundamentals of strapdown inertial navigation will be presented as well as three different algorithms which will be analyzed in regard to numerical stability, time consumption and processor load criteria.

Keywords: strapdown inertial navigation system, quaternions, forward Euler integration, code optimization, code analyses

#### 1. Introduction

INS is inertial navigation system, the system that determines the position based on the output of the motion sensors: accelerometers and gyroscopes. The first INS was based on accelerometers mounted on gimbal platform, to ensure measurement of acceleration in navigational frame. Nowadays "strapdown" inertial navigation system (SINS) is in common use, due to its mechanical simplicity, reduced size and price compered to platform INS. Strapdown inertial navigation system implies a fixed inertial measurement unit (IMU), whereby the analytical picture of the navigation system is obtained from the integration of the gyroscope rates.

The main problem that arises when SINS is used is the exact determination of the orientation based on the gyroscopes outputs. Every error made in this stage will affect the error of projection of the gravitational acceleration. Accelerations are integrated twice in order to

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

determine the position, so any errors made when determining the orientation will cause the error in position determination to increase exponentially with integration time.

Errors when determining orientation are caused by the gyroscope performance and precision, as well as signal processing methods used for processing gyroscope outputs. Besides hardware limitations of the gyroscopes, algorithms used for orientation calculation also cause errors. This chapter focuses only on errors caused by applied algorithms and on optimization of these algorithms in terms of time consumption and processor load.

#### 2. Fundamentals of inertial navigation

The basic idea of inertial navigation is based on the integration of acceleration measured by the accelerometers; see [1]. The accelerometers measure the specific force that can be represented as:

$$\mathbf{f} = \mathbf{a} - \mathbf{g} \tag{1}$$

<sup>ω</sup><sup>E</sup> ¼ � VN

<sup>ω</sup><sup>N</sup> <sup>¼</sup> VE R<sup>λ</sup> þ h

<sup>ω</sup>up <sup>¼</sup> VE

reference ellipsoid in the north-south and east-west directions, respectively.

where Re is the equatorial radius of the Earth, <sup>e</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>b</sup><sup>2</sup>

navigation frame, using information from the gyroscopes.

2.1. Determination of angular increments and transformation matrix

where ωxb, yb, zb is the gyroscope signals in the body coordinate frame.

αxb, yb, zb ¼

ellipsoid.

this example is ts = 2 ms:

R<sup>ϕ</sup> þ h

<sup>R</sup><sup>λ</sup> <sup>þ</sup> <sup>h</sup> tan <sup>ϕ</sup>

<sup>1</sup> � <sup>e</sup><sup>2</sup> sin <sup>2</sup><sup>ϕ</sup> � �<sup>3</sup>

<sup>1</sup> � <sup>e</sup><sup>2</sup> sin <sup>2</sup><sup>ϕ</sup> � �<sup>1</sup>

As the accelerometers measure acceleration in the coordinate frame related to the object, it is necessary to determine the transformation matrix from the body frame (see [4]) into the

The navigation algorithm adopted here can be divided into two parts. The first part that works with higher frequency plays the role of determining velocity and angle increments, while the other part of the algorithm that works eight times slower provides information on the position and the speed in the navigation coordinate frame (usually required by the guidance law in the case of the missile application). Such algorithm is advantageous from the point of optimization of the calculation time in the control computer, which can be divided into eight different steps. Also, this SINS algorithm proved to be mathematically more stable in relation to others, in determining the quaternion position at the same sampling time. Namely, when integrating angular velocities in order to obtain the angular position, depending on the size of the integration step, the quaternion error increases over time, and in addition to renormalization, it also affects the overall error in position and velocity. This error does not occur with this algorithm.

The first step in determining the transformation matrix is the determination of angular inclusions, and as explained above, this process is repeated with the basic integration step which in

> ðtkþts tk

2

Code Optimization for Strapdown Inertial Navigation System Algorithm

http://dx.doi.org/10.5772/intechopen.71732

2

where h is the height above the reference ellipsoid, Rϕ, R<sup>λ</sup> is the radius of the curvature of the

<sup>R</sup><sup>ϕ</sup> <sup>¼</sup> Re <sup>1</sup> � <sup>e</sup><sup>2</sup> � �

<sup>R</sup><sup>λ</sup> <sup>¼</sup> Re

(4)

61

(5)

<sup>a</sup><sup>2</sup> is the eccentricity of the reference

ωxb, yb, zbdt (6)

where a is the absolute acceleration, acceleration in relation to the inertial coordinate frame, g is the gravitational acceleration.

In this chapter, the effect of the rotation of the Earth (which can simply be introduced into equations for the needs of systems operating in a longer time interval) is neglected.

In accordance with the previous assumption, the following relationship between acceleration and velocity in relation to the inertial coordinate frame is:

$$\begin{aligned} \mathbf{a} &= \frac{d\mathbf{V}}{dt}\Big|\_{I} \\ \frac{d\mathbf{V}}{dt}\Big|\_{I} &= \frac{d\mathbf{V}}{dt}\Big|\_{N} + \mathbf{a}\_{N} \times \mathbf{V} \end{aligned} \tag{2}$$

where <sup>d</sup><sup>V</sup> dt <sup>N</sup> is the speed derivative relative to the navigation coordinate frame, ω<sup>N</sup> is the absolute angular velocity of the navigation coordinate frame.

In the inertial navigation algorithm, for the navigation coordinate frame, the ENUp coordinate frame has been adopted; see [2]. This choice is made due to the desire to have the height coordinate positive and on the other hand in order to more accurately determine the azimuth numerically.

In accordance with the ENUp coordinate frame, the following relations apply:

$$\begin{aligned} f\_E &= \frac{dV\_E}{dt} + \omega\_N V\_{up} - \omega\_{up} V\_N\\ f\_N &= \frac{dV\_N}{dt} - \omega\_E V\_{up} + \omega\_{up} V\_E\\ f\_{up} &= \frac{dV\_{up}}{dt} + \omega\_E V\_N - \omega\_N V\_E + \mathbf{g} \end{aligned} \tag{3}$$

As a result of the WGS84 standard for the Earth shape (see [3]), projection of angular speeds of the ENUp coordinate frame has been adopted in the following form:

Code Optimization for Strapdown Inertial Navigation System Algorithm http://dx.doi.org/10.5772/intechopen.71732 61

$$\begin{aligned} \omega\_E &= -\frac{V\_N}{R\_\phi + h} \\ \omega\_N &= \frac{V\_E}{R\_\lambda + h} \\ \omega\_{up} &= \frac{V\_E}{R\_\lambda + h} \tan \phi \end{aligned} \tag{4}$$

where h is the height above the reference ellipsoid, Rϕ, R<sup>λ</sup> is the radius of the curvature of the reference ellipsoid in the north-south and east-west directions, respectively.

determine the position, so any errors made when determining the orientation will cause the

Errors when determining orientation are caused by the gyroscope performance and precision, as well as signal processing methods used for processing gyroscope outputs. Besides hardware limitations of the gyroscopes, algorithms used for orientation calculation also cause errors. This chapter focuses only on errors caused by applied algorithms and on optimization of these

The basic idea of inertial navigation is based on the integration of acceleration measured by the accelerometers; see [1]. The accelerometers measure the specific force that can be represented as:

where a is the absolute acceleration, acceleration in relation to the inertial coordinate frame, g is

In this chapter, the effect of the rotation of the Earth (which can simply be introduced into

In accordance with the previous assumption, the following relationship between acceleration

In the inertial navigation algorithm, for the navigation coordinate frame, the ENUp coordinate frame has been adopted; see [2]. This choice is made due to the desire to have the height coordinate positive and on the other hand in order to more accurately determine the azimuth numerically.

dt <sup>þ</sup> <sup>ω</sup>NVup � <sup>ω</sup>upVN

dt � <sup>ω</sup>EVup <sup>þ</sup> <sup>ω</sup>upVE

As a result of the WGS84 standard for the Earth shape (see [3]), projection of angular speeds of

dt <sup>þ</sup> <sup>ω</sup>EVN � <sup>ω</sup>NVE <sup>þ</sup> <sup>g</sup>

<sup>N</sup> is the speed derivative relative to the navigation coordinate frame, ω<sup>N</sup> is the

equations for the needs of systems operating in a longer time interval) is neglected.

<sup>a</sup> <sup>¼</sup> <sup>d</sup><sup>V</sup> dt I

In accordance with the ENUp coordinate frame, the following relations apply:

<sup>f</sup> <sup>E</sup> <sup>¼</sup> dVE

<sup>f</sup> <sup>N</sup> <sup>¼</sup> dVN

<sup>f</sup> up <sup>¼</sup> dVup

the ENUp coordinate frame has been adopted in the following form:

dV dt <sup>I</sup> <sup>¼</sup> <sup>d</sup><sup>V</sup> dt 

f ¼ a � g (1)

<sup>N</sup> <sup>þ</sup> <sup>ω</sup><sup>N</sup> � <sup>V</sup> (2)

(3)

error in position determination to increase exponentially with integration time.

algorithms in terms of time consumption and processor load.

and velocity in relation to the inertial coordinate frame is:

absolute angular velocity of the navigation coordinate frame.

2. Fundamentals of inertial navigation

the gravitational acceleration.

where <sup>d</sup><sup>V</sup> dt 

60 Space Flight

$$\begin{aligned} R\_{\phi} &= \frac{R\_{\epsilon} \left(1 - e^{2}\right)}{\left(1 - e^{2} \sin^{2} \phi\right)^{\frac{3}{2}}} \\\\ R\_{\Lambda} &= \frac{R\_{\epsilon}}{\left(1 - e^{2} \sin^{2} \phi\right)^{\frac{1}{2}}} \end{aligned} \tag{5}$$

where Re is the equatorial radius of the Earth, <sup>e</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>b</sup><sup>2</sup> <sup>a</sup><sup>2</sup> is the eccentricity of the reference ellipsoid.

As the accelerometers measure acceleration in the coordinate frame related to the object, it is necessary to determine the transformation matrix from the body frame (see [4]) into the navigation frame, using information from the gyroscopes.

The navigation algorithm adopted here can be divided into two parts. The first part that works with higher frequency plays the role of determining velocity and angle increments, while the other part of the algorithm that works eight times slower provides information on the position and the speed in the navigation coordinate frame (usually required by the guidance law in the case of the missile application). Such algorithm is advantageous from the point of optimization of the calculation time in the control computer, which can be divided into eight different steps. Also, this SINS algorithm proved to be mathematically more stable in relation to others, in determining the quaternion position at the same sampling time. Namely, when integrating angular velocities in order to obtain the angular position, depending on the size of the integration step, the quaternion error increases over time, and in addition to renormalization, it also affects the overall error in position and velocity. This error does not occur with this algorithm.

#### 2.1. Determination of angular increments and transformation matrix

The first step in determining the transformation matrix is the determination of angular inclusions, and as explained above, this process is repeated with the basic integration step which in this example is ts = 2 ms:

$$\alpha\_{\mathbf{x}\_{b},\mathbf{y}\_{b},z\_{b}} = \int\_{t\_{k}}^{t\_{k}+t\_{s}} \alpha\_{\mathbf{x}\_{b},\mathbf{y}\_{b},z\_{b}} dt \tag{6}$$

where ωxb, yb, zb is the gyroscope signals in the body coordinate frame.

The process of calculating the position quaternion or the transformation matrix is also divided into two parts.

The first part is the calculation of the quaternion between the navigation coordinate frame and the body frame, assuming that the navigation coordinate frame can be considered inert during one step of integration.

The second part is used for the quaternion correction due to the rotation of the navigation coordinate frame.

If we compare these two transformations, we can conclude that the first transformation is the rotation of "fast" motion. One of the reasons why this algorithm proved to be numerically more stable is the separation of the integration of the "fast" rotation from the integration of the "slow" rotation.

If we compare the angular rates of those two motions, we can conclude that the "slow" rotation rates are four or more times lower than the "fast" rotation rates which leads to numerical integral errors when these two rotations are combined.

In accordance with the above, the following relations apply:

$$\begin{aligned} \mathbf{q}\_{n+1}^{l} &= \mathbf{q}\_{n} \Delta \mathbf{q}\_{f} \\ \mathbf{q}\_{n+1} &= \Delta \mathbf{q}\_{s} \mathbf{q}\_{n+1}^{l} \end{aligned} \tag{7}$$

ΔΦ ¼

þ 1 2

where

ΔΦxb ΔΦyb ΔΦzb

X 4

αxbð Þj

þ 2 <sup>3</sup> <sup>P</sup><sup>1</sup>

αxbð Þ4 αybð Þ4 αzbð Þ4 αxbð Þ2 αybð Þ2 αzbð Þ2

<sup>30</sup> ð Þ <sup>P</sup><sup>1</sup> � <sup>P</sup><sup>2</sup>

ð Þþ ts <sup>α</sup><sup>k</sup>�<sup>1</sup>

0 �αzbð Þj αybð Þj αzbð Þj 0 �αxbð Þj �αybð Þj αxbð Þj 0

> cos Ωtm 2

<sup>Ω</sup> sin <sup>Ω</sup>tm 2

<sup>Ω</sup> sin <sup>Ω</sup>tm 2

<sup>Ω</sup> sin <sup>Ω</sup>tm 2

� �

� �

� <sup>Ω</sup><sup>x</sup>

� <sup>Ω</sup><sup>y</sup>

� <sup>Ω</sup><sup>z</sup>

where. Ωx, Ωy, Ω<sup>z</sup> is the projections of the absolute angular velocity of the navigation coordi-

αxbð Þ4 αybð Þ4 αzbð Þ4

Code Optimization for Strapdown Inertial Navigation System Algorithm

αxbð Þ3 αybð Þ3 αzbð Þ3

3 7 5

0

BB@

ð Þ ts

1

http://dx.doi.org/10.5772/intechopen.71732

CCA

αxbð Þ4 αybð Þ4 αzbð Þ4

1

(10)

63

(11)

(12)

CCA

0

BB@

<sup>α</sup>ðÞ¼ <sup>j</sup> <sup>α</sup><sup>k</sup>

If we return to the quaternion of slow rotation, the following relationship is valid:

Δq<sup>s</sup> ¼

1

CCA þ 1

αybð Þj

αzbð Þj

P<sup>j</sup> ¼

If we neglect the rotation of the Earth, the following applies:

<sup>Ω</sup><sup>x</sup> ¼ � Vy Ry � Vx a e 2 b13b<sup>23</sup>

<sup>Ω</sup><sup>y</sup> <sup>¼</sup> Vx Rx þ Vy a e 2 b13b<sup>23</sup>

Ω<sup>z</sup> ¼ 0 1 Rx ¼ 1 a

1 Ry ¼ 1 a 1 � e <sup>2</sup> b<sup>33</sup> 2 <sup>2</sup> <sup>þ</sup> <sup>e</sup> 2 b<sup>13</sup> <sup>2</sup> � <sup>h</sup> a

1 � e <sup>2</sup> b<sup>33</sup> 2 <sup>2</sup> <sup>þ</sup> <sup>e</sup> 2 b<sup>23</sup> <sup>2</sup> � <sup>h</sup> a

2 6 4

j¼1

X 4

j¼1

X 4

j¼1

αxbð Þ3 αybð Þ3 αzbð Þ3

0

BB@

ð Þ P<sup>1</sup> þ P<sup>2</sup>

nate frame on its axes.

where q<sup>I</sup> is the quaternion of rotation from the body to the inertial coordinate frame, q is the quaternion of rotation from the body to the navigational coordinate frame, Δq<sup>f</sup> is the quaternion of fast rotation increment, Δq<sup>s</sup> is the quaternion of slow rotation increment.

The quaternion of fast rotation can be represented in the form of a rotary vector as follows:

$$
\Delta \mathbf{q}\_f = \begin{bmatrix}
\Delta q\_{f0} \\
\Delta q\_{f1} \\
\Delta q\_{f2} \\
\Delta q\_{f3} \\
\Delta q\_{f3}
\end{bmatrix} = \begin{bmatrix}
\cos\frac{\Delta \Phi}{2} \\
\frac{\Delta \Phi\_{xb}}{\Delta \Phi} \sin\frac{\Delta \Phi}{2} \\
\frac{\Delta \Phi\_{yb}}{\Delta \Phi} \sin\frac{\Delta \Phi}{2} \\
\frac{\Delta \Phi\_{zb}}{\Delta \Phi} \sin\frac{\Delta \Phi}{2}
\end{bmatrix} \tag{8}
$$

The following relationship holds for small angles:

$$
\Delta \Phi = \int\_{t\_n}^{t\_n + t\_n} \omega dt + \frac{1}{2} \int\_{t\_n}^{t\_n + t\_n} (\Phi \times \omega) dt \tag{9}
$$

where tm = 8ts is the slow integration step.

To solve the previous equation, a four-step algorithm will be used (Conning correction [5–7]):

Code Optimization for Strapdown Inertial Navigation System Algorithm http://dx.doi.org/10.5772/intechopen.71732 63

$$\begin{aligned} \Delta \mathbf{O} &= \begin{bmatrix} \Delta \Phi\_{xb} \\ \Delta \Phi\_{yb} \\ \Delta \Phi\_{zb} \end{bmatrix} = \begin{bmatrix} \sum\_{j=1}^{4} a\_{xb}(j) \\ \sum\_{j=1}^{4} a\_{yb}(j) \\ \sum\_{j=1}^{4} a\_{zb}(j) \\ \sum\_{j=1}^{4} a\_{zb}(j) \end{bmatrix} + \frac{2}{3} \left( \mathbf{P}\_{1} \begin{bmatrix} a\_{xb}(2) \\ a\_{yb}(2) \\ a\_{zb}(2) \end{bmatrix} + \mathbf{P}\_{3} \begin{bmatrix} a\_{xb}(4) \\ a\_{yb}(4) \\ a\_{zb}(4) \end{bmatrix} \right) \\ + \frac{1}{2} (\mathbf{P}\_{1} + \mathbf{P}\_{2}) \left( \begin{bmatrix} a\_{xb}(3) \\ a\_{yb}(3) \\ a\_{zb}(3) \end{bmatrix} + \begin{bmatrix} a\_{xb}(4) \\ a\_{yb}(4) \\ a\_{zb}(4) \end{bmatrix} \right) + \frac{1}{30} (\mathbf{P}\_{1} - \mathbf{P}\_{2}) \left( \begin{bmatrix} a\_{xb}(3) \\ a\_{yb}(3) \\ a\_{zb}(3) \end{bmatrix} - \begin{bmatrix} a\_{xb}(4) \\ a\_{yb}(4) \\ a\_{zb}(4) \end{bmatrix} \right) \end{aligned} \tag{10}$$

where

(7)

(8)

The process of calculating the position quaternion or the transformation matrix is also divided

The first part is the calculation of the quaternion between the navigation coordinate frame and the body frame, assuming that the navigation coordinate frame can be considered inert during

The second part is used for the quaternion correction due to the rotation of the navigation

If we compare these two transformations, we can conclude that the first transformation is the rotation of "fast" motion. One of the reasons why this algorithm proved to be numerically more stable is the separation of the integration of the "fast" rotation from the integration of the

If we compare the angular rates of those two motions, we can conclude that the "slow" rotation rates are four or more times lower than the "fast" rotation rates which leads to

> <sup>n</sup>þ<sup>1</sup> <sup>¼</sup> <sup>q</sup>nΔq<sup>f</sup> <sup>q</sup><sup>n</sup>þ<sup>1</sup> <sup>¼</sup> <sup>Δ</sup>qsq<sup>I</sup>

where q<sup>I</sup> is the quaternion of rotation from the body to the inertial coordinate frame, q is the quaternion of rotation from the body to the navigational coordinate frame, Δq<sup>f</sup> is the quater-

The quaternion of fast rotation can be represented in the form of a rotary vector as follows:

nþ1

cos ΔΦ 2

ð Þ Φ � ω dt (9)

ΔΦxb <sup>Δ</sup><sup>Φ</sup> sin <sup>Δ</sup><sup>Φ</sup> 2

ΔΦyb <sup>Δ</sup><sup>Φ</sup> sin <sup>Δ</sup><sup>Φ</sup> 2

ΔΦzb <sup>Δ</sup><sup>Φ</sup> sin <sup>Δ</sup><sup>Φ</sup> 2

ðtnþtm tn

numerical integral errors when these two rotations are combined.

Δq<sup>f</sup> ¼

ΔΦ ¼

ðtnþtm tn

ωdt þ 1 2

To solve the previous equation, a four-step algorithm will be used (Conning correction [5–7]):

The following relationship holds for small angles:

where tm = 8ts is the slow integration step.

qI

nion of fast rotation increment, Δq<sup>s</sup> is the quaternion of slow rotation increment.

Δqf <sup>0</sup> Δqf <sup>1</sup> Δqf <sup>2</sup> Δqf <sup>3</sup>

In accordance with the above, the following relations apply:

into two parts.

62 Space Flight

one step of integration.

coordinate frame.

"slow" rotation.

$$\alpha(j) = \alpha^k(t\_s) + \alpha^{k-1}(t\_s)$$

$$\mathbf{P}\_j = \begin{bmatrix} 0 & -\alpha\_{zb}(j) & \alpha\_{yb}(j) \\ \alpha\_{zb}(j) & 0 & -\alpha\_{xb}(j) \\ -\alpha\_{yb}(j) & \alpha\_{xb}(j) & 0 \end{bmatrix}$$

If we return to the quaternion of slow rotation, the following relationship is valid:

$$
\Delta \mathbf{q}\_s = \begin{bmatrix}
\cos \frac{\Omega t\_m}{2} \\
\end{bmatrix} \tag{11}
$$

where. Ωx, Ωy, Ω<sup>z</sup> is the projections of the absolute angular velocity of the navigation coordinate frame on its axes.

If we neglect the rotation of the Earth, the following applies:

$$\begin{aligned} \Omega\_x &= -\frac{V\_y}{R\_y} - \frac{V\_x}{a} e^2 b\_{13} b\_{23} \\ \Omega\_y &= \frac{V\_x}{R\_x} + \frac{V\_y}{a} e^2 b\_{13} b\_{23} \\ \Omega\_z &= 0 \\ \frac{1}{R\_x} &= \frac{1}{a} \left( 1 - e^2 \frac{b\_{33}}{2} + e^2 b\_{13} ^2 - \frac{h}{a} \right) \\ \frac{1}{R\_y} &= \frac{1}{a} \left( 1 - e^2 \frac{b\_{33}}{2} ^2 + e^2 b\_{23} ^2 - \frac{h}{a} \right) \end{aligned} \tag{12}$$

where bij are members of the transformation matrix from the Earth-coordinate frame (ECEF) into the navigation coordinate frame B<sup>n</sup> ECEF.

The Poisson equation for the transformation matrix from the coordinate frame related to the Earth (ECEF) in the navigation coordinate frame can be written in the following form:

$$\begin{aligned} \mathbf{B}\_{n}^{ECEF} &= \mathbf{B}\_{n}^{ECEF} \Delta \mathbf{o}\_{n-ECEF} \\ \mathbf{B}\_{ECEF}^{n} &= \left(\mathbf{B}\_{n}^{ECEF}\right)^{T} \\ \Delta \mathbf{o}\_{n-ECEF} &= \begin{bmatrix} 0 & 0 & \Omega\_{y} \\ 0 & 0 & -\Omega\_{x} \\ -\Omega\_{y} & \Omega\_{x} & 0 \end{bmatrix} \end{aligned} \tag{13}$$

The absolute acceleration can be written in the following form:

where <sup>d</sup><sup>V</sup> dt � �

following:

defined:

velocity of the body coordinate frame.

and the previous equation can be written like

Ðtkþtm tk

Ðtkþtm tk

Ðtkþtm tk

dV~ xb dt dt <sup>¼</sup>

dV<sup>~</sup> yb dt dt <sup>¼</sup>

dV~ zb dt dt <sup>¼</sup> dV dt � � � � I <sup>¼</sup> <sup>d</sup><sup>V</sup> dt � � � � b

dV dt � � � � b <sup>¼</sup> <sup>d</sup><sup>V</sup> dt � � � � I

ðtkþtm tk

ðtkþtm tk

ðtkþtm tk

[5–7]) from which the step of slow integration was adopted as tm = 8ts:

dVxb dt dt <sup>þ</sup>

dVyb dt dt <sup>þ</sup>

dVzb dt dt <sup>þ</sup>

The recursive solution of the previous equations is done in eight steps (sculling correction; see

Wxb, <sup>k</sup> ¼ Wxb, <sup>k</sup>�<sup>1</sup> þ Wyb, <sup>k</sup>�<sup>1</sup>αzb, <sup>k</sup> � Wzb, <sup>k</sup>�<sup>1</sup>αyb, <sup>k</sup> þ ΔWxb, <sup>k</sup> Wyb, <sup>k</sup> ¼ Wyb, <sup>k</sup>�<sup>1</sup> þ Wzb , <sup>k</sup>�<sup>1</sup>αxb, <sup>k</sup> � Wxb, <sup>k</sup>�<sup>1</sup>αzb, <sup>k</sup> þ ΔWyb, <sup>k</sup> Wzb, <sup>k</sup> ¼ Wzb, <sup>k</sup>�<sup>1</sup> þ Wxb, <sup>k</sup>�<sup>1</sup>αyb, <sup>k</sup> � Wyb, <sup>k</sup>�<sup>1</sup>αxb, <sup>k</sup> þ ΔWzb, <sup>k</sup>

Wzb, <sup>k</sup> ¼ Wzb, <sup>k</sup>�<sup>1</sup> þ Wxb, <sup>k</sup>αyb, <sup>k</sup> � Wyb, <sup>k</sup>αxb, <sup>k</sup> þ ΔWzb, <sup>k</sup> Wyb, <sup>k</sup> ¼ Wyb, <sup>k</sup>�<sup>1</sup> þ Wzb , <sup>k</sup>αxb , <sup>k</sup> � Wxb, <sup>k</sup>αzb, <sup>k</sup> þ ΔWyb, <sup>k</sup> Wxb, <sup>k</sup> ¼ Wxb, <sup>k</sup>�<sup>1</sup> þ Wyb, <sup>k</sup>αzb , <sup>k</sup> � Wzb, <sup>k</sup>αyb, <sup>k</sup> þ ΔWxb, <sup>k</sup>

After calculating the velocity increments in the body coordinate frame, it is possible to determine the increment of the velocities in the navigation coordinate frame, since the matrix of transformation between the body and the navigational coordinate frame has already been

The initial values in each new step of slow integration are Wxb = Wyb = Wzb = 0.

<sup>b</sup> is the total speed derivatives with respect to the body coordinate frame, <sup>d</sup><sup>V</sup>

total speed derivatives with respect to the inertial coordinate frame, ω<sup>b</sup> is the absolute angular

The specific force projections acting in the body coordinate frame are obtained from the accelerometer. Accordingly, the integration will be performed in the body coordinate frame,

If we apply integration with the slow integration step to the previous equation, we obtain the

ðtkþtm tk

ðtkþtm tk

ðtkþtm tk

ωyb

ωzbVyb � ωyb

� �dt

ωxbVzb � ωzbVxb ð Þdt

Vxb � ωxbVyb � �dt

þ ω<sup>b</sup> � V (17)

http://dx.doi.org/10.5772/intechopen.71732

Code Optimization for Strapdown Inertial Navigation System Algorithm

� ω<sup>b</sup> � V (18)

Vzb

dt � � <sup>I</sup> is the 65

(19)

(20)

The recursive solution of the Poisson equation can be represented in the following way:

$$\begin{aligned} b\_{12}(N) &= b\_{12}(N-1) - \Omega\_y b\_{32}(N-1)t\_m \\ b\_{22}(N) &= b\_{22}(N-1) + \Omega\_x b\_{32}(N-1)t\_m \\ b\_{32}(N) &= b\_{32}(N-1) + \left(\Omega\_y b\_{12}(N-1) - \Omega\_x b\_{22}(N-1)\right)t\_m \\ b\_{13}(N) &= b\_{13}(N-1) - \Omega\_y b\_{33}(N-1)t\_m \\ b\_{23}(N) &= b\_{23}(N-1) + \Omega\_x b\_{33}(N-1)t\_m \\ b\_{33}(N) &= b\_{33}(N-1) + \left(\Omega\_y b\_{13}(N-1) - \Omega\_x b\_{23}(N-1)\right)t\_m \\ b\_{31}(N) &= b\_{12}(N)b\_{23}(N) - b\_{22}(N)b\_{13}(N) \end{aligned} \tag{14}$$

With the quaternion of fast and the quaternion of slow rotations defined above, on the basis of Eq. (7), the quaternion of total rotation can be determined and with its direct cosine matrix representing the transformation from the body to the navigation coordinate frame. This matrix will be updated with the time step of the slow integration:

$$\mathbf{C}\_{b}^{n} = \begin{bmatrix} 1 - 2\left(q\_{2}^{-2} + q\_{3}^{-2}\right) & 2\left(q\_{1}q\_{2} - q\_{0}q\_{3}\right) & 2\left(q\_{0}q\_{2} + q\_{1}q\_{3}\right) \\ 2\left(q\_{1}q\_{2} + q\_{0}q\_{3}\right) & 1 - 2\left(q\_{1}^{-2} + q\_{3}^{-2}\right) & 2\left(q\_{2}q\_{3} - q\_{0}q\_{1}\right) \\ 2\left(q\_{1}q\_{3} - q\_{0}q\_{2}\right) & 2\left(q\_{0}q\_{1} + q\_{2}q\_{3}\right) & 1 - 2\left(q\_{1}^{-2} + q\_{2}^{-2}\right) \end{bmatrix} \tag{15}$$

#### 2.2. Determination of speed and position in space

Previously defined method used for determining the angle increments based on measured gyroscope signals can now be used in the same way to define the speed increments based on signals from the accelerometer. These increments are also determined by the fast integration step ts:

$$
\Delta W\_{x\_b, y\_b, z\_b} = \int\_{t\_k}^{t\_k + t\_s} a\_{x\_b, y\_b, z\_b} dt \tag{16}
$$

where axb, yb, zb is the signals from the accelerometer in the body coordinate frame.

The absolute acceleration can be written in the following form:

where bij are members of the transformation matrix from the Earth-coordinate frame (ECEF)

The Poisson equation for the transformation matrix from the coordinate frame related to the

<sup>n</sup> Δω<sup>n</sup>�ECEF

0 0 Ω<sup>y</sup>

(13)

(14)

(15)

0 0 �Ω<sup>x</sup>

�Ω<sup>y</sup> Ω<sup>x</sup> 0

ECEF.

B\_ ECEF

Bn

<sup>n</sup> <sup>¼</sup> <sup>B</sup>ECEF

ECEF <sup>¼</sup> <sup>B</sup>ECEF n � �<sup>T</sup>

Δω<sup>n</sup>�ECEF ¼

b12ð Þ¼ N b12ð Þ� N � 1 Ωyb32ð Þ N � 1 tm b22ð Þ¼ N b22ð Þþ N � 1 Ωxb32ð Þ N � 1 tm

b13ð Þ¼ N b13ð Þ� N � 1 Ωyb33ð Þ N � 1 tm b23ð Þ¼ N b23ð Þþ N � 1 Ωxb33ð Þ N � 1 tm

b31ð Þ¼ N b12ð Þ N b23ð Þ� N b22ð Þ N b13ð Þ N

will be updated with the time step of the slow integration:

1 � 2 q<sup>2</sup>

2.2. Determination of speed and position in space

2 q1q<sup>2</sup> þ q0q<sup>3</sup>

2 q1q<sup>3</sup> � q0q<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>q</sup><sup>3</sup>

� � <sup>1</sup> � <sup>2</sup> <sup>q</sup><sup>1</sup>

� � <sup>2</sup> <sup>q</sup>0q<sup>1</sup> <sup>þ</sup> <sup>q</sup>2q<sup>3</sup>

ΔWxb,yb, zb ¼

where axb, yb, zb is the signals from the accelerometer in the body coordinate frame.

<sup>2</sup> � � <sup>2</sup> <sup>q</sup>1q<sup>2</sup> � <sup>q</sup>0q<sup>3</sup>

Cn <sup>b</sup> ¼

Earth (ECEF) in the navigation coordinate frame can be written in the following form:

The recursive solution of the Poisson equation can be represented in the following way:

<sup>b</sup>32ð Þ¼ <sup>N</sup> <sup>b</sup>32ð Þþ <sup>N</sup> � <sup>1</sup> <sup>Ω</sup>yb12ð Þ� <sup>N</sup> � <sup>1</sup> <sup>Ω</sup>xb22ð Þ <sup>N</sup> � <sup>1</sup> � �tm

<sup>b</sup>33ð Þ¼ <sup>N</sup> <sup>b</sup>33ð Þþ <sup>N</sup> � <sup>1</sup> <sup>Ω</sup>yb13ð Þ� <sup>N</sup> � <sup>1</sup> <sup>Ω</sup>xb23ð Þ <sup>N</sup> � <sup>1</sup> � �tm

With the quaternion of fast and the quaternion of slow rotations defined above, on the basis of Eq. (7), the quaternion of total rotation can be determined and with its direct cosine matrix representing the transformation from the body to the navigation coordinate frame. This matrix

Previously defined method used for determining the angle increments based on measured gyroscope signals can now be used in the same way to define the speed increments based on signals from the accelerometer. These increments are also determined by the fast integration step ts:

> ðtkþts tk

� � <sup>2</sup> <sup>q</sup>0q<sup>2</sup> <sup>þ</sup> <sup>q</sup>1q<sup>3</sup>

<sup>2</sup> � � <sup>2</sup> <sup>q</sup>2q<sup>3</sup> � <sup>q</sup>0q<sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>q</sup><sup>3</sup>

� � <sup>1</sup> � <sup>2</sup> <sup>q</sup><sup>1</sup>

� �

� �

<sup>2</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> <sup>2</sup> � �

axb, yb, zbdt (16)

into the navigation coordinate frame B<sup>n</sup>

64 Space Flight

$$\left.\frac{d\mathbf{V}}{dt}\right|\_{\mathbf{I}} = \frac{d\mathbf{V}}{dt}\bigg|\_{\mathbf{b}} + \mathbf{a}\_{\mathbf{b}} \times \mathbf{V} \tag{17}$$

where <sup>d</sup><sup>V</sup> dt � � <sup>b</sup> is the total speed derivatives with respect to the body coordinate frame, <sup>d</sup><sup>V</sup> dt � � <sup>I</sup> is the total speed derivatives with respect to the inertial coordinate frame, ω<sup>b</sup> is the absolute angular velocity of the body coordinate frame.

The specific force projections acting in the body coordinate frame are obtained from the accelerometer. Accordingly, the integration will be performed in the body coordinate frame, and the previous equation can be written like

$$
\left.\frac{d\mathbf{V}}{dt}\right|\_b = \frac{d\mathbf{V}}{dt}\Big|\_l - \boldsymbol{\omega}\_b \times \mathbf{V} \tag{18}
$$

If we apply integration with the slow integration step to the previous equation, we obtain the following:

$$\int\_{t\_k}^{t\_k + t\_n} \frac{d\bar{V}\_{x\_0}}{dt} dt = \int\_{t\_k}^{t\_k + t\_n} \frac{d\bar{V}\_{x\_0}}{dt} dt + \int\_{t\_k}^{t\_k + t\_n} \left(\omega\_{z\_k} V\_{y\_b} - \omega\_{y\_b} V\_{z\_b}\right) dt$$

$$\int\_{t\_k}^{t\_k + t\_n} \frac{d\bar{V}\_{y\_b}}{dt} dt = \int\_{t\_k}^{t\_k + t\_n} \frac{d\bar{V}\_{y\_b}}{dt} dt + \int\_{t\_k}^{t\_k + t\_n} (\omega\_{x\_b} V\_{z\_b} - \omega\_{z\_b} V\_{x\_b}) dt \tag{19}$$

$$\int\_{t\_k}^{t\_k + t\_n} \frac{d\bar{V}\_{z\_b}}{dt} dt = \int\_{t\_k}^{t\_k + t\_n} \frac{d\bar{V}\_{z\_b}}{dt} dt + \int\_{t\_k}^{t\_k + t\_n} (\omega\_{y\_b} V\_{x\_b} - \omega\_{x\_b} V\_{y\_b}) dt$$

The recursive solution of the previous equations is done in eight steps (sculling correction; see [5–7]) from which the step of slow integration was adopted as tm = 8ts:

$$\begin{aligned} W\_{3,k} &= W\_{3,k-1} + W\_{y\_\nu,k-1} \alpha\_{z\_\nu,k} - W\_{z\_\nu,k-1} \alpha\_{y\_\nu,k} + \Delta W\_{x\_\nu,k} \\ W\_{y\_\nu,k} &= W\_{y\_\nu,k-1} + W\_{z\_\nu,k-1} \alpha\_{x\_\nu,k} - W\_{x\_\nu,k-1} \alpha\_{z\_\nu,k} + \Delta W\_{y\_\nu,k} \\ W\_{z\_\nu,k} &= W\_{z\_\nu,k-1} + W\_{x\_\nu,k-1} \alpha\_{y\_\nu,k} - W\_{y\_\nu,k-1} \alpha\_{x\_\nu,k} + \Delta W\_{z\_\nu,k} \\ W\_{z\_\nu,k} &= W\_{z\_\nu,k-1} + W\_{x\_\nu,k} \alpha\_{y\_\nu,k} - W\_{y\_\nu,k} \alpha\_{x\_\nu,k} + \Delta W\_{z\_\nu,k} \\ W\_{y\_\nu,k} &= W\_{y\_\nu,k-1} + W\_{z\_\nu,k} \alpha\_{x\_\nu,k} - W\_{x\_\nu,k} \alpha\_{z\_\nu,k} + \Delta W\_{y\_\nu,k} \\ W\_{x\_\nu,k} &= W\_{x\_\nu,k-1} + W\_{y\_\nu,k} \alpha\_{z\_\nu,k} - W\_{z\_\nu,k} \alpha\_{y\_\nu,k} + \Delta W\_{x\_\nu,k} \end{aligned} \tag{20}$$

The initial values in each new step of slow integration are Wxb = Wyb = Wzb = 0.

After calculating the velocity increments in the body coordinate frame, it is possible to determine the increment of the velocities in the navigation coordinate frame, since the matrix of transformation between the body and the navigational coordinate frame has already been defined:

$$
\begin{bmatrix}
\Delta W\_x \\
\Delta W\_y \\
\Delta W\_z
\end{bmatrix} = \mathbf{C}\_b^n \begin{bmatrix}
W\_{x\_b} \\
W\_{y\_b} \\
W\_{z\_b}
\end{bmatrix} \tag{21}
$$

<sup>E</sup> <sup>¼</sup> <sup>180</sup>

<sup>N</sup> <sup>¼</sup> <sup>180</sup>

<sup>h</sup> <sup>¼</sup> <sup>Ð</sup><sup>t</sup>

frame, we can get to the relations for angular positions:

3. Strapdown INS (SINS) algorithms

Figure 1. Forward Euler SINS algorithm block diagram.

<sup>π</sup> ð Þ <sup>λ</sup> � <sup>λ</sup><sup>0</sup> cos <sup>φ</sup><sup>0</sup>

<sup>π</sup> <sup>φ</sup> � <sup>φ</sup><sup>0</sup> � �a

Similarly, using the matrix definition from the navigation coordinate frame and the body

Cb <sup>n</sup>ð Þ <sup>1</sup>; <sup>1</sup>

Cb <sup>n</sup>ð Þ <sup>2</sup>; <sup>1</sup> � �

> Cb <sup>n</sup>ð Þ <sup>3</sup>; <sup>2</sup>

Cb <sup>n</sup>ð Þ <sup>3</sup>; <sup>3</sup> � �

Three SINS algorithms based on previously defined mathematical model will be presented here. The basic solution of SINS is forward Euler method applied to the main equations for rotation and translation. Block diagram of this method is presented in Figure 1. In this algorithm there

<sup>n</sup> ð Þ ð Þ <sup>3</sup>; <sup>1</sup>

<sup>t</sup><sup>0</sup> Vzdt

ψ ¼ arctan

φ ¼ arctan

θ ¼ arcsin Cb

is no division to the fast and the slow rotation, and all calculation is done in each step.

� �a

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(26)

67

(27)

The speed of the object relative to the Earth in the navigation coordinate frame can now be represented by the following relations, with the remark that the Earth's rotation that is neglected:

$$\begin{aligned} V\_x &= W\_x - \int\_{t\_0}^t V\_z \Omega\_y dt \\\\ V\_y &= W\_y + \int\_{t\_0}^t V\_z \Omega\_x dt \\\\ V\_z &= W\_z - \int\_{t\_0}^t \left( V\_y \Omega\_x - V\_x \Omega\_y + g \right) dt \end{aligned} \tag{22}$$

where Ωx, Ωy, Ω<sup>z</sup> is the projections of the absolute angular velocity of the navigation coordinate frame on its axes, Wx, Wy, Wz is the sums of projections of velocity increments in the navigational frame.

The determination of the position in the navigation coordinate frame can be solved in two ways: by integration of the velocities, which is the case in determining the height, or by the relationship between the matrix defined by Poisson's equation and its definitions:

$$\mathbf{B}\_{ECF}^{H} = \begin{bmatrix} -\sin\varrho\cos\lambda\sin\varepsilon - \sin\lambda\cos\varepsilon & \sin\varrho\sin\lambda\sin\varepsilon + \cos\lambda\cos\varepsilon & \cos\varrho\sin\varepsilon \\ -\sin\varrho\cos\lambda\cos\varepsilon + \sin\lambda\sin\varepsilon & -\sin\varrho\sin\lambda\cos\varepsilon - \cos\lambda\sin\varepsilon & \cos\varrho\cos\varepsilon \\ \cos\varrho\cos\lambda & \cos\varrho\sin\lambda & \sin\varrho \end{bmatrix} \tag{23}$$

where φ is the latitude, λ is the longitude, ε is the azimuth.

Geographical navigation parameters can be determined from the relation of the preceding equation and Eq. (14):

$$\begin{aligned} \phi &= \arctan \frac{b\_{33}}{b\_0} \quad [-90, +90] \\\\ \lambda &= \arctan \frac{b\_{32}}{b\_{31}} \quad [-180, 180] \\\\ \varepsilon &= \arctan \frac{b\_{13}}{b\_{23}} \quad [0, 360] \\\\ b\_0 &= \sqrt{{b\_{13}}^2 + {b\_{23}}^2} \end{aligned} \tag{24}$$

As the azimuth is now defined, projections of speed in the ENUp coordinate frame can be determined:

$$\begin{aligned} V\_N &= V\_y \cos \varepsilon + V\_x \sin \varepsilon \\ V\_E &= -V\_y \sin \varepsilon + V\_x \cos \varepsilon \end{aligned} \tag{25}$$

The position in the ENUp frame can be determined as

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$$\begin{aligned} E &= \frac{180}{\pi} (\lambda - \lambda\_0) \cos \left( q\_0 \right) a \\ N &= \frac{180}{\pi} \left( q - q\_0 \right) a \\ h &= \int\_{t\_0}^{t} V\_z dt \end{aligned} \tag{26}$$

Similarly, using the matrix definition from the navigation coordinate frame and the body frame, we can get to the relations for angular positions:

$$\begin{aligned} \psi &= \arctan\left(\frac{\mathbb{C}\_{b}"{(1,1)}}{\mathbb{C}\_{b}"{(2,1)}}\right) \\ \varphi &= \arctan\left(\frac{\mathbb{C}\_{b}"{(3,2)}}{\mathbb{C}\_{b}"{(3,3)}}\right) \\ \theta &= \arcsin(\mathbb{C}\_{b}"{(3,1)}) \end{aligned} \tag{27}$$

#### 3. Strapdown INS (SINS) algorithms

ΔWx ΔWy ΔWz 3 7 7 <sup>5</sup> <sup>¼</sup> <sup>C</sup><sup>n</sup> b

The speed of the object relative to the Earth in the navigation coordinate frame can now be represented by the following relations, with the remark that the Earth's rotation that is neglected:

<sup>t</sup><sup>0</sup> VzΩydt

<sup>t</sup><sup>0</sup> VzΩxdt

where Ωx, Ωy, Ω<sup>z</sup> is the projections of the absolute angular velocity of the navigation coordinate frame on its axes, Wx, Wy, Wz is the sums of projections of velocity increments in the naviga-

The determination of the position in the navigation coordinate frame can be solved in two ways: by integration of the velocities, which is the case in determining the height, or by the

> � sinφcos λ sin ε � sin λ cos ε sinφsin λ sin ε þ cos λ cos ε cosφsin ε � sinφcos λ cos ε þ sin λ sin ε � sinφsin λ cos ε � cos λ sin ε cosφcos ε cosφcos λ cosφsin λ sinφ

Geographical navigation parameters can be determined from the relation of the preceding

b<sup>33</sup> b0

b<sup>32</sup> b<sup>31</sup>

b<sup>13</sup> b<sup>23</sup>

As the azimuth is now defined, projections of speed in the ENUp coordinate frame can be

VN ¼ Vy cos ε þ Vx sin ε VE ¼ �Vy sin ε þ Vx cos ε

½ � �90; þ90

½ � �180; 180

½ � 0; 360

relationship between the matrix defined by Poisson's equation and its definitions:

ϕ ¼ arctan

λ ¼ arctan

ε ¼ arctan

<sup>b</sup><sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b<sup>13</sup> <sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>23</sup> <sup>2</sup> p Wxb Wyb Wzb

<sup>t</sup><sup>0</sup> VyΩ<sup>x</sup> � VxΩ<sup>y</sup> <sup>þ</sup> <sup>g</sup> � �dt

(21)

(22)

(24)

(25)

Vx <sup>¼</sup> Wx � <sup>Ð</sup><sup>t</sup>

Vy <sup>¼</sup> Wy <sup>þ</sup> <sup>Ð</sup><sup>t</sup>

Vz <sup>¼</sup> Wz � <sup>Ð</sup><sup>t</sup>

where φ is the latitude, λ is the longitude, ε is the azimuth.

The position in the ENUp frame can be determined as

tional frame.

66 Space Flight

equation and Eq. (14):

determined:

Bn ECEF ¼ Three SINS algorithms based on previously defined mathematical model will be presented here.

The basic solution of SINS is forward Euler method applied to the main equations for rotation and translation. Block diagram of this method is presented in Figure 1. In this algorithm there is no division to the fast and the slow rotation, and all calculation is done in each step.

Figure 1. Forward Euler SINS algorithm block diagram.

The other solution of SINS algorithm—the regular SINS—based on mathematical model previously defined is presented in Figure 2 as block diagram. The regular SINS algorithm calculates the velocity and angle increments eight times, and in the last step, Conning and Sculling corrections are implemented including all the other equations in Figure 2.

Figure 3. Divided SINS algorithm main flowchart.

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Figure 2. Regular SINS algorithm block diagram.

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Figure 3. Divided SINS algorithm main flowchart.

The other solution of SINS algorithm—the regular SINS—based on mathematical model previously defined is presented in Figure 2 as block diagram. The regular SINS algorithm calculates the velocity and angle increments eight times, and in the last step, Conning and Sculling

corrections are implemented including all the other equations in Figure 2.

68 Space Flight

Figure 2. Regular SINS algorithm block diagram.

Figure 4. IMU procedure flowchart—part one.

The last solution that is considered is SINS algorithm divided in eight steps. This eight-step algorithm naturally arose as a consequence of Conning equation, and it is presented in the following flowcharts.

Similar to the IMU algorithm, the navigation procedure is also divided into several steps

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From Table 1, it can be seen that forward Euler algorithm provides all SINS output values in every step unlike regular and divided SINS which will provide outputs eight times slower.

Availability of output data calculated by all three SINS algorithms is presented in Table 1.

shown in Figure 7.

Figure 6. IMU procedure flowchart—part three.

Figure 5. IMU procedure flowchart—part two.

From Figure 3, it can be seen that in each step, the main algorithm will call IMU and navigation procedures. This means that in each step, some part of calculation will be completed.

From Figure 4, it can be seen that sculling correction will be calculated in each step (Figures 5 and 6).

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Figure 5. IMU procedure flowchart—part two.

Figure 6. IMU procedure flowchart—part three.

The last solution that is considered is SINS algorithm divided in eight steps. This eight-step algorithm naturally arose as a consequence of Conning equation, and it is presented in the

From Figure 3, it can be seen that in each step, the main algorithm will call IMU and navigation procedures. This means that in each step, some part of calculation will be completed.

From Figure 4, it can be seen that sculling correction will be calculated in each step (Figures 5 and 6).

following flowcharts.

70 Space Flight

Figure 4. IMU procedure flowchart—part one.

Similar to the IMU algorithm, the navigation procedure is also divided into several steps shown in Figure 7.

Availability of output data calculated by all three SINS algorithms is presented in Table 1.

From Table 1, it can be seen that forward Euler algorithm provides all SINS output values in every step unlike regular and divided SINS which will provide outputs eight times slower.

case of the divided SINS algorithm, the entire mission algorithm can be optimized in these

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The regular and the divided SINS algorithms are based on the same numerical integration, and the results of those two algorithms are equal in time. On the other side, we can compare quaternion stability of forward Euler integration and regular SINS algorithm in time. Quaternion norm which needs to be equal to one for quaternion of rotation is sensitive to the integration step for forward Euler integration. Both algorithms were implemented in MATLAB Simulink. Norm of quaternion is presented in Figure 8 for the same integration step of 2 ms and for the same input data of gyroscopes presented in Figure 9. From Figure 8 it can be seen that the quaternion norm will be affected whenever there is significant move-

Quaternion norm error will further affect all outputs of SINS algorithm, and that will lead to error accumulation over time. Figure 10 represents angle errors for the same simulation.

eight steps.

ment of the object.

Figure 8. Quaternion norm error comparison.

Figure 7. Navigation procedure flowchart.


Table 1. Comparison of available data in each step for different SINS algorithms.

Generally, guidance and autopilot algorithms do not require inputs with such high frequency, and both regular and divided SINS will usually satisfy requirements; see [8]. On the other hand, if we compare the regular and the divided SINS algorithm, we can see that in the case of the divided SINS algorithm, the entire mission algorithm can be optimized in these eight steps.

The regular and the divided SINS algorithms are based on the same numerical integration, and the results of those two algorithms are equal in time. On the other side, we can compare quaternion stability of forward Euler integration and regular SINS algorithm in time. Quaternion norm which needs to be equal to one for quaternion of rotation is sensitive to the integration step for forward Euler integration. Both algorithms were implemented in MATLAB Simulink. Norm of quaternion is presented in Figure 8 for the same integration step of 2 ms and for the same input data of gyroscopes presented in Figure 9. From Figure 8 it can be seen that the quaternion norm will be affected whenever there is significant movement of the object.

Quaternion norm error will further affect all outputs of SINS algorithm, and that will lead to error accumulation over time. Figure 10 represents angle errors for the same simulation.

Figure 8. Quaternion norm error comparison.

Generally, guidance and autopilot algorithms do not require inputs with such high frequency, and both regular and divided SINS will usually satisfy requirements; see [8]. On the other hand, if we compare the regular and the divided SINS algorithm, we can see that in the

Algorithm step 0 1234567

Vxyz, Bn ECEF, H, lat, lon, Cn b , ϕ, θ,ψ

Regular SINS — —————— Vxyz,

Bn ECEF, lat, lon Vxyz, Bn ECEF, H, lat, lon, Cn b , ϕ, θ,ψ

Vxyz, Bn ECEF, H, lat, lon, Cn b , ϕ, θ,ψ

Vxyz, Bn ECEF, H, lat, lon, Cn b , ϕ, θ,ψ

—————

Vxyz, Bn ECEF, H, lat, lon, Cn b , ϕ, θ, ψ

Vxyz, Bn ECEF, H, lat, lon, Cn b , ϕ, θ, ψ

Bn ECEF, H, lat, lon, Cn b , ϕ, θ, ψ

Vxyz, Bn ECEF, H, lat, lon, Cn b , ϕ, θ,ψ

Cn b , ϕ, θ,ψ

Table 1. Comparison of available data in each step for different SINS algorithms.

Figure 7. Navigation procedure flowchart.

Bn ECEF, H, lat, lon, Cn b , ϕ, θ,ψ

H

Forward Euler method Vxyz,

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Divided SINS Vx, Vy, Vz,

4. Time consumption and processor load comparison of the regular

Forward Euler algorithm, regular SINS algorithm and SINS algorithm divided into eight different steps presented here were compared in terms of processor load and time it takes for all necessary calculations to complete. PC with Intel Core 2 Duo P8600 processor and 4 GB of RAM was used as a testbed for comparison of the three mentioned algorithms. Ubuntu 16.04 LTS operating system in real-time mode was used for time measurements and result generation.

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Instead of using real sensors to feed the data to the algorithm, the data were read from the files that contained recorded sensor outputs from INS tests previously performed. All the data were memory mapped to avoid any loss of time due to IO operations, thus making the algorithm exclusively CPU bound. Real-time interval timer set to 2 ms was used as the time frame generator for the INS algorithm in order to mimic real-life operation. Every 2 ms, an interrupt would occur causing the next piece of data to be fed to the algorithm, and the next step of the algorithm would be performed. In the case of the regular SINS algorithm, the entire quaternion calculation will be performed in every eighth step. In the case of the divided SINS algorithm, a piece of that calculation will be calculated in all of those seven middle steps as well as in the final eighth step, thus optimizing processor load and dividing calculation time across all steps

Statistics that are compared after the completion of the two SINS algorithms are the total time spent in every eight steps of the algorithm, average amount of time spent in every step and average processor load in each of the steps of the algorithm. Total time spent in every step of the algorithm depends on the number of steps and as such is not important as a performance measure. Average time and average processor load in each step of the algorithm are used for performance comparison. In Linux, there are three distinct time measures of process execution.

Wall clock time is the amount of calendar time that elapsed from starting the process or the stopwatch until moment "now". Thus, wall clock time includes the time the process has spent waiting for its turn on the CPU besides the time it actually spent running on the CPU. User time is the time the process spent executing on the CPU in user mode, while system time is the time the process spent executing on the CPU in system or kernel mode. User and system time measure the actual time the process spent using the CPU, and total amount of time spent on

All of these considered, wall clock represents the time that would be measured using a stopwatch. Although wall clock time heavily relies on the operating system load, on the scheduling policy used by the operating system and on the number of cores the CPU has, it can be used as a measure of time since all versions of the algorithm are subjected to the same conditions during the testing procedure. Even though the wall clock time is measured, it is not actually used in time comparison of the two mentioned algorithms. Instead, user and system time are used for comparison, because they rely only on the performance of the CPU, and the

actual time it takes for calculations in the algorithm is the sum of these two times.

SINS, the divided SINS and the forward Euler algorithms

in the algorithm evenly.

Those are wall clock time, user time and system time.

the CPU is calculated as the sum of these two time measurements.

Figure 9. Input data from gyroscopes used for simulation.

Figure 10. Angle error accumulation in time.

#### 4. Time consumption and processor load comparison of the regular SINS, the divided SINS and the forward Euler algorithms

Forward Euler algorithm, regular SINS algorithm and SINS algorithm divided into eight different steps presented here were compared in terms of processor load and time it takes for all necessary calculations to complete. PC with Intel Core 2 Duo P8600 processor and 4 GB of RAM was used as a testbed for comparison of the three mentioned algorithms. Ubuntu 16.04 LTS operating system in real-time mode was used for time measurements and result generation.

Instead of using real sensors to feed the data to the algorithm, the data were read from the files that contained recorded sensor outputs from INS tests previously performed. All the data were memory mapped to avoid any loss of time due to IO operations, thus making the algorithm exclusively CPU bound. Real-time interval timer set to 2 ms was used as the time frame generator for the INS algorithm in order to mimic real-life operation. Every 2 ms, an interrupt would occur causing the next piece of data to be fed to the algorithm, and the next step of the algorithm would be performed. In the case of the regular SINS algorithm, the entire quaternion calculation will be performed in every eighth step. In the case of the divided SINS algorithm, a piece of that calculation will be calculated in all of those seven middle steps as well as in the final eighth step, thus optimizing processor load and dividing calculation time across all steps in the algorithm evenly.

Statistics that are compared after the completion of the two SINS algorithms are the total time spent in every eight steps of the algorithm, average amount of time spent in every step and average processor load in each of the steps of the algorithm. Total time spent in every step of the algorithm depends on the number of steps and as such is not important as a performance measure. Average time and average processor load in each step of the algorithm are used for performance comparison. In Linux, there are three distinct time measures of process execution. Those are wall clock time, user time and system time.

Figure 9. Input data from gyroscopes used for simulation.

74 Space Flight

Figure 10. Angle error accumulation in time.

Wall clock time is the amount of calendar time that elapsed from starting the process or the stopwatch until moment "now". Thus, wall clock time includes the time the process has spent waiting for its turn on the CPU besides the time it actually spent running on the CPU. User time is the time the process spent executing on the CPU in user mode, while system time is the time the process spent executing on the CPU in system or kernel mode. User and system time measure the actual time the process spent using the CPU, and total amount of time spent on the CPU is calculated as the sum of these two time measurements.

All of these considered, wall clock represents the time that would be measured using a stopwatch. Although wall clock time heavily relies on the operating system load, on the scheduling policy used by the operating system and on the number of cores the CPU has, it can be used as a measure of time since all versions of the algorithm are subjected to the same conditions during the testing procedure. Even though the wall clock time is measured, it is not actually used in time comparison of the two mentioned algorithms. Instead, user and system time are used for comparison, because they rely only on the performance of the CPU, and the actual time it takes for calculations in the algorithm is the sum of these two times.

All times mentioned are given in microseconds. Total amount of time spent in the step of the algorithm is calculated as the sum of user and system time. Average processor load is calculated as

$$PL = \frac{\overline{u} + \overline{s}}{t} \tag{28}$$

the regular SINS algorithm and even more so compared to the divided SINS algorithm. Processor load and time spent calculating in each step of the regular and the divided SINS algorithms vary from step to step, whereas the time it takes for the forward Euler algorithm to do its calculations can be considered as constant. Relative time gain (T) and processor load gain (PL) of the divided SINS (DSINS) and the regular SINS (RSINS) algorithm over the

Algorithm step 0 1 234567

Table 5. Regular and divided SINS processor load and time gains over the forward Euler algorithm.

RSINS T (%) �66.33 �64.64 �49.47 �55.86 �66.87 �67.37 �66.43 �21.65 RSINS PL (%) �66.23 �64.61 �49.35 �55.84 �66.88 �67.35 �66.56 �21.75 DSINS T (%) �66.23 �64.36 �49.70 �50.01 �66.37 �65.80 �63.14 �61.84 DSINS PL (%) �66.23 �64.61 �50.00 �50.32 �66.56 �65.91 �63.31 �62.66

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In this chapter, navigation algorithm based on strapdown inertial navigation system algorithm optimized for coding in eight steps is presented. This algorithm proved to be a good option in situations where time and processor speed are limiting factors. Average time necessary for the regular SINS algorithm to complete all the steps and perform one full calculation is 21.02957 μs, whereas the divided SINS algorithm needs 19.24816 μs to perform the same operation, which scales to 8.47% improvement in time consumption. Even more important than time consumption improvement is the processor load in each timer interval, which is more uniformly distributed across all the steps in the divided SINS algorithm. Uniformly distributed processor load allows for easier design and development of multithreaded applications, as well as more free resources for the control computer to gather information about its surroundings and to issue commands to other devices in the control chain accordingly.

Also this algorithm proved to be mathematically more stable in term of quaternion norm, which mean that there is less error in angle computation and cumulatively in trajectory

2

2 Faculty of Mathematics, University of Belgrade, Belgrade, Republic of Serbia

1 Faculty of Mechanical Engineering, University of Belgrade, Belgrade, Republic of Serbia

forward Euler algorithm are presented in Table 5.

5. Conclusion

calculation.

Ivana Todić

Author details

1

\* and Vladimir Kuzmanović

\*Address all correspondence to: itodic@mas.bg.ac.rs

where u is the average user time, s is the average system time, t is the time sample duration.

The results obtained after time measurements of the regular SINS algorithm are presented in Table 2.

The results obtained after time measurements of the divided SINS algorithm are presented in Table 3.

Results presented in the tables are not comparable to the execution times on faster or slower processors. Even though exact times are not comparable when a switch to a different CPU is made, their ratio will still hold. Relative time gain and processor load gain of the divided SINS over the regular SINS are presented in Table 4.

For the sake of completeness, forward Euler version of the SINS algorithm that performs all calculations in each timer interrupt was also taken into consideration. Every 2 ms both quaternions are calculated as well as navigation parameters. Basically, this approach has no notable steps, so the previous method of time measurement is not applicable here. Instead, the average time necessary for the calculation of both quaternions and navigation parameters is taken as the performance measure.

On average, it takes 6.16023 μs for the forward Euler algorithm to perform all calculations. This translates to average processor load of 0.00308 which is significantly worse compared to


Table 4. Regular and divided SINS time and processor load ratio.


Table 5. Regular and divided SINS processor load and time gains over the forward Euler algorithm.

the regular SINS algorithm and even more so compared to the divided SINS algorithm. Processor load and time spent calculating in each step of the regular and the divided SINS algorithms vary from step to step, whereas the time it takes for the forward Euler algorithm to do its calculations can be considered as constant. Relative time gain (T) and processor load gain (PL) of the divided SINS (DSINS) and the regular SINS (RSINS) algorithm over the forward Euler algorithm are presented in Table 5.

#### 5. Conclusion

All times mentioned are given in microseconds. Total amount of time spent in the step of the algorithm is calculated as the sum of user and system time. Average processor load is calcu-

PL <sup>¼</sup> <sup>u</sup> <sup>þ</sup> <sup>s</sup>

where u is the average user time, s is the average system time, t is the time sample duration. The results obtained after time measurements of the regular SINS algorithm are presented in

The results obtained after time measurements of the divided SINS algorithm are presented in

Results presented in the tables are not comparable to the execution times on faster or slower processors. Even though exact times are not comparable when a switch to a different CPU is made, their ratio will still hold. Relative time gain and processor load gain of the divided SINS

For the sake of completeness, forward Euler version of the SINS algorithm that performs all calculations in each timer interrupt was also taken into consideration. Every 2 ms both quaternions are calculated as well as navigation parameters. Basically, this approach has no notable steps, so the previous method of time measurement is not applicable here. Instead, the average time necessary for the calculation of both quaternions and navigation parameters is taken as

On average, it takes 6.16023 μs for the forward Euler algorithm to perform all calculations. This translates to average processor load of 0.00308 which is significantly worse compared to

Algorithm step 0 1 2 34567 Elapsed time (μs) 2.07395 2.17831 3.11257 2.71898 2.04108 2.00986 2.06820 4.82662 Processor load 0.00104 0.00109 0.00156 0.00136 0.00102 0.00100 0.00103 0.00241

Algorithm step 0 1 2 34567 Elapsed time (μs) 2.08052 2.19571 3.09868 3.07395 2.07148 2.10682 2.27033 2.35067 Processor load 0.00104 0.00109 0.00154 0.00153 0.00103 0.00105 0.00113 0.00115

Algorithm step 0 1 2 3 4567

Time (%) +0.32 +0.80 �0.45 +13.05 +1.49 +4.82 +9.77 �51.30 Processor load (%) +0.00 +0.00 �1.3 +12.5 +0.98 +5.00 +9.71 �52.28

<sup>t</sup> (28)

lated as

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Table 2.

Table 3.

over the regular SINS are presented in Table 4.

the performance measure.

Table 2. Measured time of the regular SINS.

Table 3. Measured time of divided SINS.

Table 4. Regular and divided SINS time and processor load ratio.

In this chapter, navigation algorithm based on strapdown inertial navigation system algorithm optimized for coding in eight steps is presented. This algorithm proved to be a good option in situations where time and processor speed are limiting factors. Average time necessary for the regular SINS algorithm to complete all the steps and perform one full calculation is 21.02957 μs, whereas the divided SINS algorithm needs 19.24816 μs to perform the same operation, which scales to 8.47% improvement in time consumption. Even more important than time consumption improvement is the processor load in each timer interval, which is more uniformly distributed across all the steps in the divided SINS algorithm. Uniformly distributed processor load allows for easier design and development of multithreaded applications, as well as more free resources for the control computer to gather information about its surroundings and to issue commands to other devices in the control chain accordingly.

Also this algorithm proved to be mathematically more stable in term of quaternion norm, which mean that there is less error in angle computation and cumulatively in trajectory calculation.

#### Author details

Ivana Todić 1 \* and Vladimir Kuzmanović 2


#### References

[1] Titterton DH, Weston JL. Strapdown Inertial Navigation Technology. IEE Radar, Sonar, Navigation and Avionic Series. Vol. 17. 2nd ed. 2004. 576 p. ISBN: 0-86341-358-7

**Chapter 5**

Provisional chapter

**On Six DOF Relative Orbital Motion of Satellites**

In this chapter, we reveal a dual-tensor-based procedure to obtain exact expressions for the six degree of freedom (6-DOF) relative orbital law of motion in the specific case of two Keplerian confocal orbits. The result is achieved by pure analytical methods in the general case of any leader and deputy motion, without singularities or implying any secular terms. Orthogonal dual tensors play a very important role, with the representation of the solution being, to the authors' knowledge, the shortest approach for describing the complete onboard solution of the 6-DOF orbital motion problem. The solution does not depend on the local-vertical–local-horizontal (LVLH) properties involves that is true in any reference frame of the leader with the origin in its mass center. A representation theorem is provided for the full-body initial value problem. Furthermore, the representation theorems for rotation part and translation part of the relative motion are obtained.

Keywords: relative orbital motion, full body problem, dual algebra, Lie group,

The relative motion between the leader and the deputy in the relative motion is a six-degreesof-freedom (6-DOF) motion engendered by the joining of the relative translational motion with the rotational one. Recently, the modeling of the 6-DOF motion of spacecraft gained a special attention [1–5], similar to the controlling the relative pose of satellite formation that became a very important research subject [6–10]. The approach implies to consider the relative translational and rotational dynamics in the case of chief-deputy spacecraft formation to be modeled

In this chapter we reveal a dual algebra tensor based procedure to obtain exact expressions for the six D.O.F relative orbital law of motion for the case of two Keplerian confocal orbits.

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

On Six DOF Relative Orbital Motion of Satellites

DOI: 10.5772/intechopen.73563

Daniel Condurache

Abstract

1. Introduction

Daniel Condurache

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.73563

Lie algebra, closed form solution

using vector and tensor formalism.


#### **On Six DOF Relative Orbital Motion of Satellites** On Six DOF Relative Orbital Motion of Satellites

DOI: 10.5772/intechopen.73563

Daniel Condurache Daniel Condurache

References

78 Space Flight

[1] Titterton DH, Weston JL. Strapdown Inertial Navigation Technology. IEE Radar, Sonar, Navigation and Avionic Series. Vol. 17. 2nd ed. 2004. 576 p. ISBN: 0-86341-358-7

[2] Salychev O. Inertial Systems in Navigation and Geophysics. 1st ed. Moscow: Bauman

[3] NIMA TR8350.2: Department of Defense World Geodetic System 1984, Its Definition and

[4] AIAA R-004: "Atmospheric and Space Flight Vehicle Coordinate Systems". American

[5] Salychev O. Applied Inertial Navigation: Problems and Solutions. Moscow: Bauman

[6] Savage PG. Strapdown inertial navigation integration algorithm design Part1: Attitude

[7] Savage PG. Strapdown inertial navigation integration algorithm design part 2: Velocity and position algorithms. Journal of Guidance, Control, and dynamics. 1998;21(2):208-221

[8] Siouris GM. Missile Guidance and Control Systems. 1st ed. New York: Springer; 2004. 666 p.

algorithms. Journal of Guidance, Control, and Dynamics. 1998;21(1):19-28

MSTU Press; 1998. 352 p. ISBN: 5-7038-1346-8

MSTU Press; 2004. 306 p. ISBN: 5-7038-2395-1

ISBN: 0-387-00726-1

Relationship with Local Geodetic Systems. 3rd ed

Institute of Aeronautics and Astronautics (AIAA); 1992. 69 p

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.73563

#### Abstract

In this chapter, we reveal a dual-tensor-based procedure to obtain exact expressions for the six degree of freedom (6-DOF) relative orbital law of motion in the specific case of two Keplerian confocal orbits. The result is achieved by pure analytical methods in the general case of any leader and deputy motion, without singularities or implying any secular terms. Orthogonal dual tensors play a very important role, with the representation of the solution being, to the authors' knowledge, the shortest approach for describing the complete onboard solution of the 6-DOF orbital motion problem. The solution does not depend on the local-vertical–local-horizontal (LVLH) properties involves that is true in any reference frame of the leader with the origin in its mass center. A representation theorem is provided for the full-body initial value problem. Furthermore, the representation theorems for rotation part and translation part of the relative motion are obtained.

Keywords: relative orbital motion, full body problem, dual algebra, Lie group, Lie algebra, closed form solution

#### 1. Introduction

The relative motion between the leader and the deputy in the relative motion is a six-degreesof-freedom (6-DOF) motion engendered by the joining of the relative translational motion with the rotational one. Recently, the modeling of the 6-DOF motion of spacecraft gained a special attention [1–5], similar to the controlling the relative pose of satellite formation that became a very important research subject [6–10]. The approach implies to consider the relative translational and rotational dynamics in the case of chief-deputy spacecraft formation to be modeled using vector and tensor formalism.

In this chapter we reveal a dual algebra tensor based procedure to obtain exact expressions for the six D.O.F relative orbital law of motion for the case of two Keplerian confocal orbits.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Orthogonal dual tensors play a very important role, the representation of the solution being, to the authors' knowledge, the shortest approach for describing the complete onboard solution of the six D.O.F relative orbital motion problem. Because the solution does not depend on the LVLH properties involves that is true in any reference frame of the Leader with the origin in its mass center. To obtain this solution, one has to know only the inertial motion of the Leader spacecraft and the initial conditions of the deputy satellite in the local-vertical-local-horizontal (LVLH) frame. For the full body initial value problem, a general representation theorem is given. More, the real and imaginary parts are split and representation theorems for the rotation and translation parts of the relative orbital motion are obtained. Regarding translation, we will prove that this problem is super-integrable by reducing it to the classic Kepler problem.

<sup>R</sup> <sup>¼</sup> <sup>ð</sup><sup>I</sup> <sup>þ</sup> <sup>ε</sup>erÞ<sup>Q</sup> (2)

<sup>2</sup> <sup>¼</sup> exp ð Þ <sup>α</sup>e<sup>u</sup> (3)

On Six DOF Relative Orbital Motion of Satellites http://dx.doi.org/10.5772/intechopen.73563

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81

where <sup>Q</sup> <sup>∈</sup> <sup>S</sup>O<sup>3</sup> and <sup>r</sup><sup>∈</sup> <sup>V</sup><sup>3</sup> are called structural invariants, <sup>ε</sup><sup>2</sup> <sup>¼</sup> 0, <sup>ε</sup> 6¼ 0.

parameterize displacements of rigid bodies.

<sup>s</sup>o<sup>3</sup> <sup>¼</sup> <sup>α</sup><sup>e</sup> <sup>∈</sup> <sup>L</sup> <sup>V</sup>3; <sup>V</sup><sup>3</sup> ð Þ <sup>α</sup><sup>e</sup> ¼ �αe<sup>T</sup> �

defined by the following equation:

and is the inverse of Eq. (4).

the following.

� � n o

Theorem 3. The mapping is well defined and surjective.

i. roto-translation if c 6¼ 0, c<sup>0</sup> 6¼ 0 and c∙c<sup>0</sup> 6¼ 0 ⇔ c

contains the rotation angle α and the translated distance d.

Taking into account the Lie group structure of SO<sup>3</sup> and the result presented in previous theorem, it can be concluded that any orthogonal dual tensor R ∈SO<sup>3</sup> can be used globally

Theorem 2 (Representation Theorem). For any orthogonal dual tensor R defined as in Eq. (2), a dual number α ¼ α þ εd and a dual unit vector u ¼ uþεu<sup>0</sup> can be computed to have the following Eq. [17, 18]:

The parameters α and u are called the natural invariants of R. The unit dual vector u gives the Plücker representation of the Mozzi-Chalses axis [16, 24] while the dual angle α ¼ α þ εd

The Lie algebra of the Lie group SO<sup>3</sup> is the skew-symmetric dual tensor set denoted by

The link between the Lie algebra so3, the Lie group SO3, and the exponential map is given by

exp : so<sup>3</sup> ! SO3,

Any screw axis that embeds a rigid displacement is parameterized by a unit dual vector, whereas the screw parameters (angle of rotation around the screw and the translation along the screw axis) is structured as a dual angle. The computation of the screw axis is bound to the problem of finding the logarithm of an orthogonal dual tensor R, that is a multifunction

log : SO<sup>3</sup> ! so3,

From Theorem 2 and Theorem 3, for any orthogonal dual tensor R, a dual vector c ¼ αu ¼ c þ εc<sup>0</sup> is computed, represents the screw dual vector or Euler dual vector (that includes the screw axis and screw parameters) and the form of c implies that ce ∈ log R. The types of rigid displacements that is parameterized by the Euler dual vector c as below:

� � �

log R ¼ ce ∈ so<sup>3</sup> exp ce

k¼0

αek k!

� � <sup>¼</sup> <sup>R</sup>

� � � � �

n o (5)

� <sup>∈</sup> <sup>R</sup> and j j <sup>c</sup> <sup>∉</sup>εR;

exp ð Þ¼ <sup>α</sup><sup>e</sup> <sup>e</sup>α<sup>e</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup>

, where the internal mapping is <sup>α</sup>e<sup>1</sup>; <sup>α</sup>e<sup>2</sup> h i <sup>¼</sup> <sup>α</sup>eg<sup>1</sup> <sup>α</sup><sup>2</sup> .

<sup>R</sup>ð Þ¼ <sup>α</sup>, <sup>u</sup> <sup>I</sup> <sup>þ</sup> sin <sup>α</sup>e<sup>u</sup> <sup>þ</sup> ð Þ <sup>1</sup> � cos <sup>α</sup> <sup>e</sup><sup>u</sup>

The chapter is structured as following. The second section is dedicated to the rigid body motion parameterization using orthogonal dual tensors, dual quaternions and other different vector parameterization. The Poisson-Darboux problem is extended in dual Lie algebra. In the third section, the state equations for a rigid body motion relative to an arbitrary non-inertial reference frame are determined. Using the obtained result, in the fourth section, the representation theorem and the complete solution for the case of onboard full-body relative orbital motion problem is given. The last section is designated to the conclusions and to the future works.

#### 2. Rigid body motion parameterization using dual Lie algebra

The key notions that will be presented in this section are tensorial, vectorial and non-vectorial parameterizations that can be used to properly describe the rigid-body motion. We discuss the properties of proper orthogonal dual tensorial maps. The proper orthogonal tensorial maps are related with the skew-symmetric tensorial maps via the Darboux–Poisson equation. Orthogonal dual tensorial maps are a powerful instrument in the study of the rigid motion with respect to an inertial and noninertial reference frames. More on dual numbers, dual vectors and dual tensors can be found in [2, 16–23].

#### 2.1. Isomorphism between Lie group of the rigid displacements SE<sup>3</sup> and Lie group of the orthogonal dual tensors SO<sup>3</sup>

Let the orthogonal dual tensor set be denoted by.

$$\underline{\operatorname{SD}}\_3 = \left\{ \underline{\mathbf{R}} \in \mathbf{L}(\underline{\mathbf{V}}\_3, \underline{\mathbf{V}}\_3) \middle| \underline{\operatorname{RR}}^T = \underline{I}, \det \underline{\mathbf{R}} = 1 \right\} \tag{1}$$

where SO<sup>3</sup> is the set of special orthogonal dual tensors and I is the unit orthogonal dual tensor. The internal structure of any orthogonal dual tensor R ∈SO<sup>3</sup> is illustrated in a series of results which were detailed in our previous work [17, 18, 23].

Theorem 1. (Structure Theorem). For any R ∈SO<sup>3</sup> a unique decomposition is viable

$$
\underline{\mathbf{R}} = (\mathbf{I} + \varepsilon \tilde{\boldsymbol{\rho}}) \mathbf{Q} \tag{2}
$$

where <sup>Q</sup> <sup>∈</sup> <sup>S</sup>O<sup>3</sup> and <sup>r</sup><sup>∈</sup> <sup>V</sup><sup>3</sup> are called structural invariants, <sup>ε</sup><sup>2</sup> <sup>¼</sup> 0, <sup>ε</sup> 6¼ 0.

Taking into account the Lie group structure of SO<sup>3</sup> and the result presented in previous theorem, it can be concluded that any orthogonal dual tensor R ∈SO<sup>3</sup> can be used globally parameterize displacements of rigid bodies.

Theorem 2 (Representation Theorem). For any orthogonal dual tensor R defined as in Eq. (2), a dual number α ¼ α þ εd and a dual unit vector u ¼ uþεu<sup>0</sup> can be computed to have the following Eq. [17, 18]:

$$\underline{\mathbf{R}}(\underline{\underline{a}}, \underline{\underline{u}}) = \mathbf{I} + \sin \underline{\underline{a}} \tilde{\underline{u}} + (1 - \cos \underline{\underline{a}}) \tilde{\underline{u}}^2 = \exp \left( \underline{\underline{a}} \tilde{\underline{u}} \right) \tag{3}$$

The parameters α and u are called the natural invariants of R. The unit dual vector u gives the Plücker representation of the Mozzi-Chalses axis [16, 24] while the dual angle α ¼ α þ εd contains the rotation angle α and the translated distance d.

The Lie algebra of the Lie group SO<sup>3</sup> is the skew-symmetric dual tensor set denoted by <sup>s</sup>o<sup>3</sup> <sup>¼</sup> <sup>α</sup><sup>e</sup> <sup>∈</sup> <sup>L</sup> <sup>V</sup>3; <sup>V</sup><sup>3</sup> ð Þ <sup>α</sup><sup>e</sup> ¼ �αe<sup>T</sup> � � � n o , where the internal mapping is <sup>α</sup>e<sup>1</sup>; <sup>α</sup>e<sup>2</sup> h i <sup>¼</sup> <sup>α</sup>eg<sup>1</sup> <sup>α</sup><sup>2</sup> .

The link between the Lie algebra so3, the Lie group SO3, and the exponential map is given by the following.

Theorem 3. The mapping is well defined and surjective.

$$\begin{aligned} \text{exp} &: \underline{\text{so}}\_3 \rightarrow \underline{\text{SO}}\_3 \\ \text{exp}\left(\underline{\tilde{\alpha}}\right) = \underline{\epsilon} \frac{\tilde{\alpha}}{\mathbf{k}} &= \sum\_{\mathbf{k}=0}^{\text{\textquotedblleft}} \frac{\tilde{\alpha}^{\text{\textquotedblleft}}}{\mathbf{k}!} \end{aligned} \tag{4}$$

Any screw axis that embeds a rigid displacement is parameterized by a unit dual vector, whereas the screw parameters (angle of rotation around the screw and the translation along the screw axis) is structured as a dual angle. The computation of the screw axis is bound to the problem of finding the logarithm of an orthogonal dual tensor R, that is a multifunction defined by the following equation:

$$\begin{aligned} \log: \underline{\operatorname{SD}}\_{3} &\to \underline{\operatorname{ao}}\_{3'} \\ \log \underline{\mathbf{R}} = \left\{ \underline{\tilde{\mu}} \in \underline{\operatorname{ao}}\_{3} \middle| \exp \left( \underline{\tilde{\mu}} \right) = \underline{\mathbf{R}} \right\} \end{aligned} \tag{5}$$

and is the inverse of Eq. (4).

Orthogonal dual tensors play a very important role, the representation of the solution being, to the authors' knowledge, the shortest approach for describing the complete onboard solution of the six D.O.F relative orbital motion problem. Because the solution does not depend on the LVLH properties involves that is true in any reference frame of the Leader with the origin in its mass center. To obtain this solution, one has to know only the inertial motion of the Leader spacecraft and the initial conditions of the deputy satellite in the local-vertical-local-horizontal (LVLH) frame. For the full body initial value problem, a general representation theorem is given. More, the real and imaginary parts are split and representation theorems for the rotation and translation parts of the relative orbital motion are obtained. Regarding translation, we will prove that this problem is super-integrable by reducing it to the classic Kepler

The chapter is structured as following. The second section is dedicated to the rigid body motion parameterization using orthogonal dual tensors, dual quaternions and other different vector parameterization. The Poisson-Darboux problem is extended in dual Lie algebra. In the third section, the state equations for a rigid body motion relative to an arbitrary non-inertial reference frame are determined. Using the obtained result, in the fourth section, the representation theorem and the complete solution for the case of onboard full-body relative orbital motion problem is given. The last section is designated to the conclusions and to the future

The key notions that will be presented in this section are tensorial, vectorial and non-vectorial parameterizations that can be used to properly describe the rigid-body motion. We discuss the properties of proper orthogonal dual tensorial maps. The proper orthogonal tensorial maps are related with the skew-symmetric tensorial maps via the Darboux–Poisson equation. Orthogonal dual tensorial maps are a powerful instrument in the study of the rigid motion with respect to an inertial and noninertial reference frames. More on dual numbers, dual vectors and dual

2.1. Isomorphism between Lie group of the rigid displacements SE<sup>3</sup> and Lie group of the

<sup>S</sup>O<sup>3</sup> <sup>¼</sup> <sup>R</sup> <sup>∈</sup> <sup>L</sup> <sup>V</sup>3; <sup>V</sup><sup>3</sup> ð Þ RR<sup>T</sup> <sup>¼</sup> <sup>I</sup>; det<sup>R</sup> <sup>¼</sup> <sup>1</sup>

where SO<sup>3</sup> is the set of special orthogonal dual tensors and I is the unit orthogonal dual tensor.

The internal structure of any orthogonal dual tensor R ∈SO<sup>3</sup> is illustrated in a series of results

Theorem 1. (Structure Theorem). For any R ∈SO<sup>3</sup> a unique decomposition is viable

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2. Rigid body motion parameterization using dual Lie algebra

problem.

80 Space Flight

works.

tensors can be found in [2, 16–23].

Let the orthogonal dual tensor set be denoted by.

which were detailed in our previous work [17, 18, 23].

orthogonal dual tensors SO<sup>3</sup>

From Theorem 2 and Theorem 3, for any orthogonal dual tensor R, a dual vector c ¼ αu ¼ c þ εc<sup>0</sup> is computed, represents the screw dual vector or Euler dual vector (that includes the screw axis and screw parameters) and the form of c implies that ce ∈ log R. The types of rigid displacements that is parameterized by the Euler dual vector c as below:

i. roto-translation if c 6¼ 0, c<sup>0</sup> 6¼ 0 and c∙c<sup>0</sup> 6¼ 0 ⇔ c � � � � � � <sup>∈</sup> <sup>R</sup> and j j <sup>c</sup> <sup>∉</sup>εR;

$$\text{iii.} \quad \text{pure translation if if } \boldsymbol{\upmu} = \mathbf{0} \text{ and } \boldsymbol{\upmu}\_0 \neq \mathbf{0} \Leftrightarrow \left| \underline{\boldsymbol{\upmu}} \right| \in \underline{\boldsymbol{\upmu}} \underline{\boldsymbol{\upmu}}$$

iii. pure rotation if c6¼0 and c∙c0¼0 ⇔ c � � � � � � <sup>∈</sup> <sup>R</sup>.

Also, k k c < 2π, Theorem 2 and Theorem 3 can be used to uniquely recover the screw dual vector c, which is equivalent with computing log R.

Theorem 4. The natural invariants α ¼ α þ εd, u ¼ uþεu<sup>0</sup> can be used to directly recover the structural invariants Q and r from Eq. (2):

$$\begin{aligned} \mathbf{Q} &= \mathbf{I} + \sin \alpha \tilde{\mathbf{u}} + (1 - \cos \alpha) \tilde{\mathbf{u}}^2 \\ \mathbf{p} &= d \mathbf{u} + \sin \alpha \mathbf{u}\_0 + (1 - \cos \alpha) \mathbf{u} \times \mathbf{u}\_0 \end{aligned} \tag{6}$$

2.2.1. The exponential parameterization (the Euler dual vector parameterization)

c ∈ V3 which combined with Theorem 2 and Theorem 3 lead to

R ¼ exp ce

2.2.2. Dual quaternion parameterization

pair of a dual scalar quantity and a free dual vector:

The product of two dual quaternions <sup>q</sup>b<sup>1</sup> <sup>¼</sup> <sup>q</sup>

qb1

<sup>¼</sup> <sup>q</sup>bqb<sup>∗</sup> can be computed. For <sup>q</sup><sup>b</sup>

. The first one composed from pairs q; 0

any <sup>q</sup><sup>b</sup> <sup>∈</sup> <sup>U</sup>, the following equation is valid [17, 20]:

containing the pairs 0; q

Euler parameters [19].

written as <sup>q</sup><sup>b</sup> <sup>¼</sup> <sup>q</sup> <sup>þ</sup> <sup>q</sup>, where <sup>q</sup> <sup>¼</sup> <sup>q</sup>; <sup>0</sup>

<sup>q</sup>b<sup>2</sup> <sup>¼</sup> <sup>q</sup> 1 ∙q <sup>2</sup> � <sup>q</sup><sup>1</sup> ∙q2 ; q 1 <sup>q</sup><sup>2</sup> <sup>þ</sup> <sup>q</sup> 2 <sup>q</sup><sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>1</sup>

quaternion defined by Eq. (12), the conjugate denoted by <sup>q</sup>b<sup>∗</sup> <sup>¼</sup> <sup>q</sup>; �<sup>q</sup>

� � � � �

where

by <sup>q</sup><sup>b</sup> � � � � � � 2

<sup>Q</sup><sup>V</sup><sup>3</sup>

If R ¼ Rð Þ α; u , then we can construct the Euler dual vector (screw dual vector) c ¼ αu,

� � � � � �c<sup>e</sup> <sup>þ</sup> 1 2

sin j j x

8 < :

One of the most important non-vectorial parameterizations for the orthogonal dual tensor SO<sup>3</sup> is given by the dual quaternions [20, 21]. A dual quaternion can be defined as an associated

The set of dual quaternions will be denoted Q and is organized as a R-module of rank 4, if dual

1 ; q1

From the above properties, results that the R-module Q becomes an associative, noncommutative linear dual algebra of rank 4 over the ring of dual numbers. For any dual

nion. Regarded solely as a free R-module, Q contains two remarkable sub-modules: Q<sup>R</sup> and

quaternions. The scalar and the vector parts of a dual unit quaternion are also known as dual

Let denote with U the set of unit quaternions and with U the set of unit dual quaternions. For

� � and <sup>q</sup> <sup>¼</sup> <sup>0</sup>; <sup>q</sup>

� �and <sup>q</sup>b<sup>2</sup> <sup>¼</sup> <sup>q</sup>

j j <sup>x</sup> , j j <sup>x</sup> <sup>∉</sup>ε<sup>R</sup> 1, j j x ∈εR

sinc<sup>2</sup> <sup>c</sup> � � � � � �

� �, <sup>q</sup><sup>∈</sup> <sup>R</sup>, <sup>q</sup><sup>∈</sup> V3 (12)

� � is defined by

� � and the norm denoted

2 ; q2

� <sup>q</sup><sup>2</sup> � � (13)

� <sup>¼</sup> 1, any dual quaternion is called unit dual quater-

� �, <sup>q</sup><sup>∈</sup> <sup>R</sup>, isomorphic with <sup>R</sup>, and the second one,

� �, or <sup>q</sup><sup>b</sup> <sup>¼</sup> <sup>q</sup><sup>b</sup> <sup>þ</sup> <sup>ε</sup>qb0, where <sup>q</sup>b, <sup>q</sup>b<sup>0</sup> are real

� �, <sup>q</sup>∈V3, isomorphic with <sup>V</sup>3. Also, any dual quaternion can be

<sup>2</sup> <sup>c</sup><sup>e</sup> <sup>2</sup> (10)

On Six DOF Relative Orbital Motion of Satellites http://dx.doi.org/10.5772/intechopen.73563

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83

� � <sup>¼</sup> <sup>I</sup> <sup>þ</sup> sinc <sup>c</sup>

sincð Þ¼ j j x

<sup>q</sup><sup>b</sup> <sup>¼</sup> <sup>q</sup>; <sup>q</sup>

quaternion addition and multiplication with dual numbers are considered.

To prove Eq. (6), we need to use Eqs. (2) and (3). If these equations are equal, then the structure of their dual parts leads to the result presented in Eq. (6).

Theorem 5. (Isomorphism Theorem): The special Euclidean group Sð Þ E3; ∙ and SO3; ∙ � � are connected via the isomorphism of the Lie groups

$$\begin{aligned} \Phi: \mathsf{SE}\_{\mathsf{S}} &\rightarrow \underline{\mathsf{SD}}\_{\mathsf{J}'}\\ \Phi(\mathsf{g}) &= (\mathsf{I} + \varepsilon \widetilde{\mathsf{p}}) \mathsf{Q} \end{aligned} \tag{7}$$

where g <sup>¼</sup> <sup>Q</sup> <sup>r</sup> 0 1 � �, <sup>Φ</sup> <sup>∈</sup>SO3, <sup>r</sup><sup>∈</sup> <sup>V</sup><sup>3</sup>:

Proof. For any g1, g<sup>2</sup> ∈SE3, the map defined in Eq. (7) yields

$$\Phi(\mathcal{g}\_1 \cdot \mathcal{g}\_2) = \Phi(\mathcal{g}\_1) \cdot \Phi(\mathcal{g}\_2) \tag{8}$$

Let R ∈SO3. Based on Theorem 1, which ensures a unique decomposition, we can conclude that the only choice for <sup>g</sup>, such that <sup>Φ</sup>ð Þ¼ <sup>g</sup> <sup>R</sup> is <sup>g</sup> <sup>¼</sup> <sup>Q</sup> <sup>r</sup> 0 1 � �. This underlines that <sup>Φ</sup> is a bijection and keeps all the internal operations.

Remark 1: The inverse of Φ is

$$\boldsymbol{\Phi}^{-1} : \underline{\operatorname{SCD}}\_{3} \hookrightarrow \operatorname{S\boxplus}\_{3}; \boldsymbol{\Phi}^{-1}(\underline{\mathbf{R}}) = \begin{bmatrix} \mathbf{Q} & \mathbf{p} \\ \mathbf{0} & 1 \end{bmatrix} \tag{9}$$

where <sup>Q</sup> <sup>¼</sup> Reð Þ <sup>R</sup> , <sup>r</sup> <sup>¼</sup> vect Duð Þ <sup>R</sup> <sup>∙</sup>Q<sup>T</sup> � �.

#### 2.2. Dual tensor-based parameterizations of rigid-body motion

The Lie group SO<sup>3</sup> admits multiple parameterization and few of them will be discussed in this section.

#### 2.2.1. The exponential parameterization (the Euler dual vector parameterization)

If R ¼ Rð Þ α; u , then we can construct the Euler dual vector (screw dual vector) c ¼ αu, c ∈ V3 which combined with Theorem 2 and Theorem 3 lead to

$$\underline{R} = \exp\left(\underline{\tilde{\Psi}}\right) = \underline{I} + \mathrm{sinc}\left|\underline{\Psi}\right|\underline{\tilde{\Psi}} + \frac{1}{2}\mathrm{sinc}^2\frac{\left|\underline{\Psi}\right|}{2}\underline{\tilde{\Psi}}^2\tag{10}$$

where

ii. pure translation if if c ¼ 0 and c<sup>0</sup> 6¼ 0 ⇔ c

vector c, which is equivalent with computing log R.

of their dual parts leads to the result presented in Eq. (6).

, Φ ∈SO3, r∈ V<sup>3</sup>:

Proof. For any g1, g<sup>2</sup> ∈SE3, the map defined in Eq. (7) yields

that the only choice for <sup>g</sup>, such that <sup>Φ</sup>ð Þ¼ <sup>g</sup> <sup>R</sup> is <sup>g</sup> <sup>¼</sup> <sup>Q</sup> <sup>r</sup>

2.2. Dual tensor-based parameterizations of rigid-body motion

bijection and keeps all the internal operations.

where <sup>Q</sup> <sup>¼</sup> Reð Þ <sup>R</sup> , <sup>r</sup> <sup>¼</sup> vect Duð Þ <sup>R</sup> <sup>∙</sup>Q<sup>T</sup> � �.

Φ g1∙g<sup>2</sup>

<sup>Φ</sup>�<sup>1</sup> : <sup>S</sup>O<sup>3</sup> \$ <sup>S</sup>E3; <sup>Φ</sup>�<sup>1</sup>

� � <sup>¼</sup> <sup>Φ</sup> <sup>g</sup><sup>1</sup>

Let R ∈SO3. Based on Theorem 1, which ensures a unique decomposition, we can conclude

The Lie group SO<sup>3</sup> admits multiple parameterization and few of them will be discussed in this

� �∙<sup>Φ</sup> <sup>g</sup><sup>2</sup>

0 1 � �

ð Þ¼ <sup>R</sup> <sup>Q</sup> <sup>r</sup> 0 1 � �

iii. pure rotation if c6¼0 and c∙c0¼0 ⇔ c

structural invariants Q and r from Eq. (2):

connected via the isomorphism of the Lie groups

where g <sup>¼</sup> <sup>Q</sup> <sup>r</sup>

82 Space Flight

0 1 � �

Remark 1: The inverse of Φ is

section.

� � � � � � <sup>∈</sup>εR;

Also, k k c < 2π, Theorem 2 and Theorem 3 can be used to uniquely recover the screw dual

Theorem 4. The natural invariants α ¼ α þ εd, u ¼ uþεu<sup>0</sup> can be used to directly recover the

<sup>Q</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup> sin <sup>α</sup>e<sup>u</sup> <sup>þ</sup> ð Þ <sup>1</sup> � cos <sup>α</sup> <sup>e</sup><sup>u</sup>

r ¼ du þ sin αu<sup>0</sup> þ ð Þ 1 � cos α u�u<sup>0</sup>

To prove Eq. (6), we need to use Eqs. (2) and (3). If these equations are equal, then the structure

Φ : SE<sup>3</sup> ! SO3,

Theorem 5. (Isomorphism Theorem): The special Euclidean group Sð Þ E3; ∙ and SO3; ∙

2

<sup>Φ</sup>ð Þ¼ <sup>g</sup> <sup>ð</sup><sup>I</sup> <sup>þ</sup> <sup>ε</sup>erÞ<sup>Q</sup> (7)

� � (8)

. This underlines that Φ is a

(6)

are

(9)

� �

� � � � � � <sup>∈</sup> <sup>R</sup>.

$$\text{sinc}(|\underline{\mathbf{x}}|) = \begin{cases} \frac{\sin|\underline{\mathbf{x}}|}{|\underline{\mathbf{x}}|}, |\underline{\mathbf{x}}| \notin \varepsilon \mathbb{R} \\\ 1, |\underline{\mathbf{x}}| \in \varepsilon \mathbb{R} \end{cases} \tag{11}$$

#### 2.2.2. Dual quaternion parameterization

One of the most important non-vectorial parameterizations for the orthogonal dual tensor SO<sup>3</sup> is given by the dual quaternions [20, 21]. A dual quaternion can be defined as an associated pair of a dual scalar quantity and a free dual vector:

$$
\widehat{\bf q} = \left(\underline{q}, \underline{\mathbf{q}}\right), \underline{q} \in \mathbb{E}, \underline{\mathbf{q}} \in \underline{\mathbf{v}}\_3 \tag{12}
$$

The set of dual quaternions will be denoted Q and is organized as a R-module of rank 4, if dual quaternion addition and multiplication with dual numbers are considered.

The product of two dual quaternions <sup>q</sup>b<sup>1</sup> <sup>¼</sup> <sup>q</sup> 1 ; q1 � �and <sup>q</sup>b<sup>2</sup> <sup>¼</sup> <sup>q</sup> 2 ; q2 � � is defined by

$$
\hat{\underline{\mathbf{q}}}\_{1}\hat{\underline{\mathbf{q}}}\_{2} = \left(\underline{q}\_{1}\underline{\cdot}\underline{q}\_{2} - \underline{\mathbf{q}}\_{1}\cdot\underline{\mathbf{q}}\_{2}, \underline{q}\_{1}\underline{\mathbf{q}}\_{2} + \underline{q}\_{2}\underline{\mathbf{q}}\_{1} + \underline{\mathbf{q}}\_{1} \times \underline{\mathbf{q}}\_{2}\right) \tag{13}
$$

From the above properties, results that the R-module Q becomes an associative, noncommutative linear dual algebra of rank 4 over the ring of dual numbers. For any dual quaternion defined by Eq. (12), the conjugate denoted by <sup>q</sup>b<sup>∗</sup> <sup>¼</sup> <sup>q</sup>; �<sup>q</sup> � � and the norm denoted by <sup>q</sup><sup>b</sup> � � � � � � 2 <sup>¼</sup> <sup>q</sup>bqb<sup>∗</sup> can be computed. For <sup>q</sup><sup>b</sup> � � � � � � <sup>¼</sup> 1, any dual quaternion is called unit dual quaternion. Regarded solely as a free R-module, Q contains two remarkable sub-modules: Q<sup>R</sup> and <sup>Q</sup><sup>V</sup><sup>3</sup> . The first one composed from pairs q; 0 � �, <sup>q</sup><sup>∈</sup> <sup>R</sup>, isomorphic with <sup>R</sup>, and the second one, containing the pairs 0; q � �, <sup>q</sup>∈V3, isomorphic with <sup>V</sup>3. Also, any dual quaternion can be written as <sup>q</sup><sup>b</sup> <sup>¼</sup> <sup>q</sup> <sup>þ</sup> <sup>q</sup>, where <sup>q</sup> <sup>¼</sup> <sup>q</sup>; <sup>0</sup> � � and <sup>q</sup> <sup>¼</sup> <sup>0</sup>; <sup>q</sup> � �, or <sup>q</sup><sup>b</sup> <sup>¼</sup> <sup>q</sup><sup>b</sup> <sup>þ</sup> <sup>ε</sup>qb0, where <sup>q</sup>b, <sup>q</sup>b<sup>0</sup> are real quaternions. The scalar and the vector parts of a dual unit quaternion are also known as dual Euler parameters [19].

Let denote with U the set of unit quaternions and with U the set of unit dual quaternions. For any <sup>q</sup><sup>b</sup> <sup>∈</sup> <sup>U</sup>, the following equation is valid [17, 20]:

$$
\hat{\underline{\mathbf{q}}} = \left(1 + \varepsilon \frac{1}{2} \hat{\underline{\mathbf{p}}}\right) \hat{\underline{\mathbf{q}}}\tag{14}
$$

Theorem 8. The inverse of the previous fractional order Cayley map, is a multifunction with n

ffiffiffiffiffi qb n 2 q � 1 ffiffiffiffiffi qb n 2 q þ1

not necessarily pass through the origin of reference system. Meanwhile, if j j v ∈ εR the mapping cayn

α

higher order Rodrigues dual vector, while for <sup>k</sup> <sup>¼</sup> f g <sup>1</sup>;…; <sup>n</sup> � <sup>1</sup> the dual vectors <sup>v</sup><sup>k</sup> <sup>¼</sup> tan <sup>α</sup>þ2k<sup>π</sup>

<sup>2</sup><sup>n</sup> , results that <sup>v</sup><sup>k</sup> j j¼ <sup>v</sup><sup>0</sup> j jþ tan <sup>k</sup><sup>π</sup>

Theorem 9. If v∈ V3 is the parameterization of a displacement obtained from Eq. (20), then

1 þ j j v

½ � Xn=2 k¼0

½ � ð Þ n X�<sup>1</sup> <sup>=</sup><sup>2</sup> k¼0

ð Þ �<sup>1</sup> <sup>k</sup> <sup>2</sup><sup>k</sup> n

ð Þ �<sup>1</sup> <sup>k</sup> <sup>2</sup><sup>k</sup> <sup>þ</sup> <sup>1</sup> n

�q<sup>b</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pnð Þ¼ X

qnð Þ¼ X

are the shadow parameterization [25] that can be used to describe the same pose. Based on

<sup>2</sup> <sup>þ</sup> <sup>u</sup> sin <sup>α</sup>

: (18)

On Six DOF Relative Orbital Motion of Satellites http://dx.doi.org/10.5772/intechopen.73563

ð Þ v is the parameterization of roto-

<sup>2</sup> , (19)

u, k ¼ f g 0; 1;…; n � 1 : (20)

<sup>n</sup> , which allows the avoidance of any singularity of type

<sup>2</sup> � �<sup>n</sup> <sup>r</sup> pnð Þþ j j <sup>v</sup> qnð Þ j j <sup>v</sup> <sup>v</sup> � � (21)

� �X<sup>2</sup><sup>k</sup> (22)

� �X<sup>2</sup><sup>k</sup> (23)

2 ð Þ v 85

<sup>2</sup><sup>n</sup> u, which is the

<sup>2</sup><sup>n</sup> u

ð Þ v is the parameterization of a pure rotation about an axis which does

2

n <sup>1</sup>� <sup>v</sup><sup>0</sup> j j tan <sup>k</sup><sup>π</sup> n .

v ¼

Taking into account that a dual number α and a dual vector u exist in order to have

<sup>q</sup><sup>b</sup> <sup>¼</sup> cos

α þ 2kπ 2n

The previous equation contains both the principal parameterization <sup>v</sup><sup>0</sup> <sup>¼</sup> tan <sup>α</sup>

branches f �<sup>1</sup> : <sup>U</sup> ! <sup>V</sup><sup>3</sup> given by

Remark 4: If j j v ∈ R then cayn

from Eq. (18), results that:

translation.

<sup>v</sup><sup>0</sup> j j¼ tan <sup>α</sup>

Re <sup>α</sup> 2n � � <sup>¼</sup> <sup>π</sup>

where

2

is the parameterization of a pure translation. Otherwise, cayn

v ¼ tan

<sup>2</sup><sup>n</sup> and <sup>v</sup><sup>k</sup> j j¼ tan <sup>α</sup>þ2k<sup>π</sup>

If Re <sup>v</sup><sup>0</sup> ð Þ! j j <sup>∞</sup> then Re <sup>v</sup><sup>k</sup> ð Þ!� j j cot <sup>k</sup><sup>π</sup>

<sup>2</sup> þ πℤ.

where <sup>r</sup><sup>∈</sup> <sup>V</sup><sup>3</sup> and <sup>q</sup><sup>b</sup> <sup>∈</sup> <sup>U</sup>. This representation is the quaternionic counterpart to Eq. (2). Also a dual number α and a unit dual vector u exist so that:

$$\hat{\underline{\mathbf{q}}} = \cos\frac{\underline{\alpha}}{2} + \underline{\mathbf{u}}\sin\frac{\underline{\alpha}}{2} = \exp\left(\frac{1}{2}\underline{\alpha}\,\underline{\mathbf{u}}\right). \tag{15}$$

Remark 2: The mapping exp : <sup>V</sup><sup>3</sup> ! <sup>U</sup>, <sup>q</sup><sup>b</sup> <sup>¼</sup> exp <sup>1</sup> <sup>2</sup> Ψ, is well defined and surjective.

Remark 3: The dual unit quaternions set U, by the multiplication of dual quaternions, is a Lie group with V3 being it's associated Lie algebra (with the cross product between dual vectors as the internal operation).

Using the internal structure of any element from SO<sup>3</sup> the following theorem is valid:

Theorem 6. The Lie groups U and SO<sup>3</sup> are linked by a surjective homomorphism

$$\Delta: \underline{\mathbb{U}} \to \underline{\mathbb{S}} \mathbb{D}\_{\mathbb{S}'} \Delta \Big( \underline{q} + \underline{\mathbf{q}} \Big) = \underline{\mathbb{I}} + 2\underline{q}\widetilde{\underline{q}} + 2\widetilde{\underline{q}}^2 \tag{16}$$

Proof. Taking into account that any <sup>q</sup><sup>b</sup> <sup>∈</sup> <sup>U</sup> can be decomposed as in Eq. (15), results that <sup>Δ</sup> <sup>q</sup><sup>b</sup> � � <sup>¼</sup> exp ð Þ <sup>α</sup>e<sup>u</sup> <sup>∈</sup> <sup>S</sup>O3. This shows that relation Eq. (16) is well defined and surjective. Using direct calculus, we can also acknowledge that <sup>Δ</sup> <sup>q</sup>b<sup>2</sup> qb1 � � <sup>¼</sup> <sup>Δ</sup> <sup>q</sup>b<sup>2</sup> � �<sup>Δ</sup> <sup>q</sup>b<sup>1</sup> � �.

An important property of the previous homomorphism is that for <sup>q</sup><sup>b</sup> and �q<sup>b</sup> we can associate the same orthogonal dual tensor, which shows that Eq. (16) is not injective and U is a double cover of SO3.

#### 2.2.3. N-order modified fractional Cayley transform for dual vectors

Next, we present a series of results that are the core of our research. These results are obtained after using a set of Cayley transforms that are different than the ones already reported in literature [17, 25–27].

Theorem 7. The fractional order Cayley map f : V<sup>3</sup> ! U

$$\text{cay}\_{\frac{\pi}{2}}(\underline{\mathbf{y}}) = f(\underline{\mathbf{y}}) = (1 + \underline{\mathbf{y}})^{\frac{\pi}{2}} (1 - \underline{\mathbf{y}})^{-\frac{\pi}{2}}, n \in \mathbb{N}^\* \tag{17}$$

is well defined and surjective.

Proof. Using direct calculus results that fð Þ v f ∗ ð Þ¼ v 1 and j j¼ fð Þ v 1. The surjectivity is proved by the following theorem.

Theorem 8. The inverse of the previous fractional order Cayley map, is a multifunction with n branches f �<sup>1</sup> : <sup>U</sup> ! <sup>V</sup><sup>3</sup> given by

$$\underline{\mathbf{v}} = \frac{\sqrt[n]{\widehat{\underline{\mathbf{q}}}^2} - 1}{\sqrt[n]{\widehat{\underline{\mathbf{q}}}^2} + 1} \,. \tag{18}$$

Remark 4: If j j v ∈ R then cayn 2 ð Þ v is the parameterization of a pure rotation about an axis which does not necessarily pass through the origin of reference system. Meanwhile, if j j v ∈ εR the mapping cayn 2 ð Þ v is the parameterization of a pure translation. Otherwise, cayn 2 ð Þ v is the parameterization of rototranslation.

Taking into account that a dual number α and a dual vector u exist in order to have

$$\underline{\hat{\mathbf{q}}} = \cos\frac{\alpha}{2} + \underline{\mathbf{u}}\sin\frac{\alpha}{2},\tag{19}$$

from Eq. (18), results that:

<sup>q</sup><sup>b</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>ε</sup>

<sup>2</sup> <sup>þ</sup> <sup>u</sup> sin <sup>α</sup>

Using the internal structure of any element from SO<sup>3</sup> the following theorem is valid:

Theorem 6. The Lie groups U and SO<sup>3</sup> are linked by a surjective homomorphism

Δ : U ! SO3, Δ q þ q

dual number α and a unit dual vector u exist so that:

Remark 2: The mapping exp : <sup>V</sup><sup>3</sup> ! <sup>U</sup>, <sup>q</sup><sup>b</sup> <sup>¼</sup> exp <sup>1</sup>

direct calculus, we can also acknowledge that <sup>Δ</sup> <sup>q</sup>b<sup>2</sup>

2.2.3. N-order modified fractional Cayley transform for dual vectors

Theorem 7. The fractional order Cayley map f : V<sup>3</sup> ! U

Proof. Using direct calculus results that fð Þ v f

cayn 2

the internal operation).

<sup>Δ</sup> <sup>q</sup><sup>b</sup> � �

84 Space Flight

cover of SO3.

literature [17, 25–27].

is well defined and surjective.

by the following theorem.

<sup>q</sup><sup>b</sup> <sup>¼</sup> cos

α

where <sup>r</sup><sup>∈</sup> <sup>V</sup><sup>3</sup> and <sup>q</sup><sup>b</sup> <sup>∈</sup> <sup>U</sup>. This representation is the quaternionic counterpart to Eq. (2). Also a

Remark 3: The dual unit quaternions set U, by the multiplication of dual quaternions, is a Lie group with V3 being it's associated Lie algebra (with the cross product between dual vectors as

� �

Proof. Taking into account that any <sup>q</sup><sup>b</sup> <sup>∈</sup> <sup>U</sup> can be decomposed as in Eq. (15), results that

An important property of the previous homomorphism is that for <sup>q</sup><sup>b</sup> and �q<sup>b</sup> we can associate the same orthogonal dual tensor, which shows that Eq. (16) is not injective and U is a double

Next, we present a series of results that are the core of our research. These results are obtained after using a set of Cayley transforms that are different than the ones already reported in

ð Þ¼ <sup>v</sup> <sup>f</sup>ð Þ¼ <sup>v</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>v</sup> <sup>n</sup>

∗

<sup>¼</sup> exp ð Þ <sup>α</sup>e<sup>u</sup> <sup>∈</sup> <sup>S</sup>O3. This shows that relation Eq. (16) is well defined and surjective. Using

qb1 � �

<sup>2</sup> ð Þ <sup>1</sup> � <sup>v</sup> �<sup>n</sup>

1 2 br � �

<sup>2</sup> <sup>¼</sup> exp

1 2 α u � �

<sup>¼</sup> <sup>I</sup> <sup>þ</sup> <sup>2</sup>qe<sup>q</sup> <sup>þ</sup> <sup>2</sup>e<sup>q</sup>

<sup>¼</sup> <sup>Δ</sup> <sup>q</sup>b<sup>2</sup> � �

<sup>Δ</sup> <sup>q</sup>b<sup>1</sup> � � .

<sup>2</sup> Ψ, is well defined and surjective.

<sup>q</sup><sup>b</sup> (14)

: (15)

<sup>2</sup> (16)

<sup>2</sup> , n∈ℕ<sup>∗</sup> (17)

ð Þ¼ v 1 and j j¼ fð Þ v 1. The surjectivity is proved

$$\underline{\mathbf{v}} = \tan \frac{\underline{\alpha} + 2k\pi}{2n} \underline{\mathbf{u}}, k = \{0, 1, \dots, n - 1\}. \tag{20}$$

The previous equation contains both the principal parameterization <sup>v</sup><sup>0</sup> <sup>¼</sup> tan <sup>α</sup> <sup>2</sup><sup>n</sup> u, which is the higher order Rodrigues dual vector, while for <sup>k</sup> <sup>¼</sup> f g <sup>1</sup>;…; <sup>n</sup> � <sup>1</sup> the dual vectors <sup>v</sup><sup>k</sup> <sup>¼</sup> tan <sup>α</sup>þ2k<sup>π</sup> <sup>2</sup><sup>n</sup> u are the shadow parameterization [25] that can be used to describe the same pose. Based on <sup>v</sup><sup>0</sup> j j¼ tan <sup>α</sup> <sup>2</sup><sup>n</sup> and <sup>v</sup><sup>k</sup> j j¼ tan <sup>α</sup>þ2k<sup>π</sup> <sup>2</sup><sup>n</sup> , results that <sup>v</sup><sup>k</sup> j j¼ <sup>v</sup><sup>0</sup> j jþ tan <sup>k</sup><sup>π</sup> n <sup>1</sup>� <sup>v</sup><sup>0</sup> j j tan <sup>k</sup><sup>π</sup> n .

If Re <sup>v</sup><sup>0</sup> ð Þ! j j <sup>∞</sup> then Re <sup>v</sup><sup>k</sup> ð Þ!� j j cot <sup>k</sup><sup>π</sup> <sup>n</sup> , which allows the avoidance of any singularity of type Re <sup>α</sup> 2n � � <sup>¼</sup> <sup>π</sup> <sup>2</sup> þ πℤ.

Theorem 9. If v∈ V3 is the parameterization of a displacement obtained from Eq. (20), then

$$\pm \underline{\hat{\mathbf{q}}} = \frac{1}{\sqrt{\left(1 + \left|\underline{\mathbf{v}}\right|^2\right)^n}} \left[p\_n(\left|\underline{\mathbf{v}}\right|) + q\_n(\left|\underline{\mathbf{v}}\right|) \underline{\mathbf{v}}\right] \tag{21}$$

where

$$p\_n(X) = \sum\_{k=0}^{\lceil n/2 \rceil} (-1)^k \binom{2k}{n} X^{2k} \tag{22}$$

$$q\_n(\mathbf{X}) = \sum\_{k=0}^{\left[ (n-1)/2 \right]} (-1)^k \binom{2k+1}{n} \mathbf{X}^{2k} \tag{23}$$

In Eqs. (22) and (23), ½ �: represents the floor of a number and k n � � are binomial coefficients.

Remark 5. The structure of the polynomials pnð Þ X and qnð Þ X , given by Eqs. (22) and (23), can be used to obtain the following iterative expressions:

$$\begin{aligned} p\_{n+1}(\mathbf{X}) &= p\_n(\mathbf{X}) - \mathbf{X}^2 q\_n(\mathbf{X}) \\ q\_{n+1}(\mathbf{X}) &= q\_n(\mathbf{X}) + q\_n(\mathbf{X}) \\ p\_1(\mathbf{X}) &= 1, q\_1(\mathbf{X}) = \mathbf{1}. \end{aligned} \tag{24}$$

Theorem 10. In a general rigid motion, described by an orthogonal dual tensor function R, the velocity

Let <sup>Φ</sup> <sup>¼</sup> <sup>R</sup>\_ <sup>R</sup>T, then <sup>R</sup>\_ <sup>R</sup><sup>T</sup> <sup>þ</sup> RR\_ <sup>T</sup> <sup>¼</sup> <sup>0</sup>, equivalent with <sup>Φ</sup> ¼ �Φ<sup>T</sup>, which shows that <sup>Φ</sup> <sup>∈</sup> <sup>s</sup>o<sup>R</sup>

The dual vector <sup>ω</sup> <sup>¼</sup> vectR\_ <sup>R</sup><sup>T</sup> is called dual angular velocity of the rigid body and has the

where ω is the instantaneous angular velocity of the rigid body and v ¼ r\_ � ω � r represents the linear velocity of the point of the body that coincides instantaneously with the origin of the

The next Theorem permits the reconstruction of the rigid body motion knowing in any moment the twist of the rigid body that is equivalent with knowing the dual angular velocity

> <sup>R</sup>\_ <sup>¼</sup> <sup>ω</sup><sup>e</sup> <sup>R</sup> Rð Þ¼ t<sup>0</sup> R0, R<sup>0</sup> ∈ SO<sup>3</sup>

Due to the fact that orthogonal dual tensor R completely models the six degree of freedom motion, we can conclude that the Theorem 11 is the dual form of the Poisson-Darboux problem [28] for the case when the rotation tensor is computed from the instantaneous angular velocity. So, in order to recover R, it is necessary to find out how the dual angular

The dual tensor R can be derived from ω, when is positioned in space, or from ω<sup>B</sup>, which

Remark 6. The dual angular velocity vector positioned in the rigid body can be recovered from

<sup>R</sup>\_ <sup>¼</sup> <sup>R</sup>ω<sup>e</sup> <sup>B</sup> Rð Þ¼ t<sup>0</sup> R0, R<sup>0</sup> ∈SO<sup>3</sup>

velocity vector ω behaves in time and also the value of R at time t ¼ t0.

denotes the dual angular velocity vector to be positioned in the rigid body.

(

reference frame. The pair (ω, v) is usually refereed as the twist of the rigid body.

2.3.1. Poisson-Darboux equation in dual Lie algebra

Theorem 11. For any continuous function ω ∈ V<sup>R</sup>

<sup>ω</sup><sup>B</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup>ω, thus transforming Eq. (32) into:

<sup>h</sup>\_ <sup>¼</sup> <sup>Φ</sup>h, <sup>∀</sup>h<sup>∈</sup> <sup>V</sup><sup>3</sup> (29)

On Six DOF Relative Orbital Motion of Satellites http://dx.doi.org/10.5772/intechopen.73563

<sup>Φ</sup>¼R\_ <sup>R</sup><sup>T</sup>: (30)

ω ¼ ω þ εv (31)

<sup>3</sup> a unique dual tensor R ∈ SO<sup>R</sup>

3 .

87

<sup>3</sup> exists so that

(32)

(33)

dual tensor function Φ defined as

is expressed by

form:

[5, 18].

In order to evaluate the usefulness of the iterative expressions, we provide the second to third order polynomials and the resulting dual quaternions and dual orthogonal tensors:

$$\begin{aligned} p\_1(\mathbf{X}) &= 1; q\_1(\mathbf{X}) = 1; \underline{\mathbf{v}} = \tan\frac{\underline{\alpha}}{2}\underline{\mathbf{u}};\\ \pm\widehat{\underline{\mathbf{u}}} &= \frac{1}{\sqrt{\underline{\mathbf{l}} + |\underline{\mathbf{v}}|^2}} [1 + \underline{\mathbf{v}}];\\ \underline{\mathbf{R}} &= \underline{\mathbf{I}} + \frac{2}{1 + |\underline{\mathbf{v}}|^2} \left[\underline{\underline{v}} + \underline{\overline{v}}^2\right];\end{aligned} \tag{25}$$

$$\begin{split} p\_2(\mathbf{X}) &= 1 - \mathbf{X}^2; q\_2 = 2; \underline{\mathbf{v}} = \tan \frac{\underline{\mathbf{u}} + 2k\pi}{4} \underline{\mathbf{u}}; \mathbf{k} = \overline{0, 1};\\ \pm \underline{\hat{\mathbf{u}}} &= \frac{1}{\underline{1} + |\underline{\mathbf{v}}|^2} \left[ 1 - |\underline{\mathbf{v}}|^2 + 2\underline{\mathbf{v}} \right]; \underline{\mathbf{R}} = \underline{\mathbf{I}} + \frac{4}{\left( 1 + |\underline{\mathbf{v}}|^2 \right)^2} \left[ \left( 1 - |\underline{\mathbf{v}}|^2 \right) \tilde{\underline{v}} + 2\underline{\tilde{v}}^2 \right]; \end{split} \tag{26}$$

$$\begin{split} p\_3(X) &= 1 - 3X^2; q\_3 = 3 - X^2; \underline{\mathbf{v}} = \tan\frac{\underline{\alpha} + 2k\pi}{6}\underline{\mathbf{u}};\\ \mathbf{k} &= \overline{0, 2\gamma} \pm \hat{\underline{\mathbf{q}}} = \frac{1}{\sqrt{\left(\underline{1} + |\underline{\mathbf{v}}|^2\right)^3}} \left[1 - 3|\underline{\mathbf{v}}|^2 + \left(3 - |\underline{\mathbf{v}}|^2\right)\underline{\mathbf{v}}\right];\\ \underline{\mathbf{R}} &= \underline{1} + \frac{2\left(3 - |\underline{\mathbf{v}}|^2\right)}{\left(1 + |\underline{\mathbf{v}}|^2\right)^3} \left[\left(1 - 3|\underline{\mathbf{v}}|^2\right)\underline{\overline{\mathbf{z}}} + \left(3 - |\underline{\mathbf{v}}|^2\right)\underline{\overline{\mathbf{z}}}\right]. \end{split} \tag{27}$$

#### 2.3. Poisson-Darboux problems in dual Lie algebra and vector parameterization

Consider the functions <sup>Q</sup> <sup>¼</sup> <sup>Q</sup>ð Þ<sup>t</sup> <sup>∈</sup>SO<sup>R</sup> <sup>3</sup> and <sup>r</sup> <sup>¼</sup> <sup>r</sup>ð Þ<sup>t</sup> <sup>∈</sup>V<sup>R</sup> <sup>3</sup> to be the parametric equations of any rigid motion. Thus, any rigid motion can be parameterized by a curve in SO<sup>3</sup> where <sup>R</sup>ðÞ¼ <sup>t</sup> <sup>ð</sup><sup>I</sup> <sup>þ</sup> <sup>ε</sup>erð ÞÞ <sup>t</sup> <sup>Q</sup>ð Þ<sup>t</sup> , where t is time variable. Let <sup>h</sup><sup>0</sup> embed the Plücker coordinates of a line feature at t ¼ t0. At a time stamp t the line is transformed into:

$$
\underline{\mathbf{h}}(t) = \underline{\mathbf{R}}(t)\underline{\mathbf{h}}\_0 \tag{28}
$$

Theorem 10. In a general rigid motion, described by an orthogonal dual tensor function R, the velocity dual tensor function Φ defined as

$$
\underline{\mathbf{h}} = \underline{\mathbf{Q}} \underline{\mathbf{h}} \,\,\forall \underline{\mathbf{h}} \in \underline{\mathbf{V}}\_3 \tag{29}
$$

is expressed by

In Eqs. (22) and (23), ½ �: represents the floor of a number and

be used to obtain the following iterative expressions:

86 Space Flight

<sup>p</sup>2ð Þ¼ <sup>X</sup> <sup>1</sup> � <sup>X</sup><sup>2</sup>

1 þ j j v

R ¼ I þ

Consider the functions <sup>Q</sup> <sup>¼</sup> <sup>Q</sup>ð Þ<sup>t</sup> <sup>∈</sup>SO<sup>R</sup>

�q<sup>b</sup> <sup>¼</sup> <sup>1</sup>

k n � �

qnð Þ X

α 2 u;

Remark 5. The structure of the polynomials pnð Þ X and qnð Þ X , given by Eqs. (22) and (23), can

qnþ<sup>1</sup>ð Þ¼ <sup>X</sup> qnð Þþ <sup>X</sup> qnð Þ <sup>X</sup> p1ð Þ¼ X 1, q1ð Þ¼ X 1:

In order to evaluate the usefulness of the iterative expressions, we provide the second to third

1 þ j j v 2 <sup>q</sup> ½ � <sup>1</sup> <sup>þ</sup> <sup>v</sup> ;

α þ 2kπ

; R ¼ I þ

<sup>2</sup> � �<sup>3</sup> <sup>r</sup> <sup>1</sup> � <sup>3</sup>j j <sup>v</sup>

<sup>2</sup> � �

<sup>3</sup> and <sup>r</sup> <sup>¼</sup> <sup>r</sup>ð Þ<sup>t</sup> <sup>∈</sup>V<sup>R</sup>

any rigid motion. Thus, any rigid motion can be parameterized by a curve in SO<sup>3</sup> where <sup>R</sup>ðÞ¼ <sup>t</sup> <sup>ð</sup><sup>I</sup> <sup>þ</sup> <sup>ε</sup>erð ÞÞ <sup>t</sup> <sup>Q</sup>ð Þ<sup>t</sup> , where t is time variable. Let <sup>h</sup><sup>0</sup> embed the Plücker coordinates of a line

; q<sup>3</sup> <sup>¼</sup> <sup>3</sup> � <sup>X</sup><sup>2</sup>

1 þ j j v

<sup>2</sup> � �<sup>3</sup> <sup>1</sup> � <sup>3</sup>j j <sup>v</sup>

2.3. Poisson-Darboux problems in dual Lie algebra and vector parameterization

<sup>2</sup> <sup>e</sup><sup>v</sup> <sup>þ</sup> <sup>e</sup><sup>v</sup> <sup>2</sup> h i ;

<sup>4</sup> <sup>u</sup>; <sup>k</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>;

1 þ j j v

; v ¼ tan

4

<sup>2</sup> � �<sup>2</sup> <sup>1</sup> � j j <sup>v</sup>

α þ 2kπ <sup>6</sup> <sup>u</sup>;

h i

<sup>e</sup><sup>v</sup> <sup>þ</sup> <sup>3</sup> � j j <sup>v</sup> <sup>2</sup> � �

<sup>2</sup> h i

<sup>2</sup> <sup>þ</sup> <sup>3</sup> � j j <sup>v</sup> <sup>2</sup> � �

ev

hðÞ¼ t Rð Þt h<sup>0</sup> (28)

:

<sup>2</sup> � �

<sup>2</sup> h i

v

;

<sup>3</sup> to be the parametric equations of

<sup>e</sup><sup>v</sup> <sup>þ</sup> <sup>2</sup>e<sup>v</sup>

;

2 1 þ j j v

pnþ<sup>1</sup>ð Þ¼ <sup>X</sup> pnð Þ� <sup>X</sup> <sup>X</sup><sup>2</sup>

order polynomials and the resulting dual quaternions and dual orthogonal tensors:

p1ð Þ¼ X 1; q1ð Þ¼ X 1; v ¼ tan

�q<sup>b</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R ¼ I þ

; q<sup>2</sup> ¼ 2; v ¼ tan

<sup>k</sup> <sup>¼</sup> <sup>0</sup>, <sup>2</sup>; � <sup>q</sup><sup>b</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 3 � j j v <sup>2</sup> � �

1 þ j j v

feature at t ¼ t0. At a time stamp t the line is transformed into:

<sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>v</sup> h i

<sup>2</sup> 1 � j j v

<sup>p</sup>3ð Þ¼ <sup>X</sup> <sup>1</sup> � <sup>3</sup>X<sup>2</sup>

are binomial coefficients.

(24)

(25)

(26)

(27)

$$
\boldsymbol{\mathfrak{O}} = \underline{\dot{\mathbf{R}}} \underline{\mathbf{R}}^{\mathrm{T}}.\tag{30}
$$

Let <sup>Φ</sup> <sup>¼</sup> <sup>R</sup>\_ <sup>R</sup>T, then <sup>R</sup>\_ <sup>R</sup><sup>T</sup> <sup>þ</sup> RR\_ <sup>T</sup> <sup>¼</sup> <sup>0</sup>, equivalent with <sup>Φ</sup> ¼ �Φ<sup>T</sup>, which shows that <sup>Φ</sup> <sup>∈</sup> <sup>s</sup>o<sup>R</sup> 3 .

The dual vector <sup>ω</sup> <sup>¼</sup> vectR\_ <sup>R</sup><sup>T</sup> is called dual angular velocity of the rigid body and has the form:

$$
\underline{\omega} = \omega + \varepsilon \mathbf{v} \tag{31}
$$

where ω is the instantaneous angular velocity of the rigid body and v ¼ r\_ � ω � r represents the linear velocity of the point of the body that coincides instantaneously with the origin of the reference frame. The pair (ω, v) is usually refereed as the twist of the rigid body.

#### 2.3.1. Poisson-Darboux equation in dual Lie algebra

The next Theorem permits the reconstruction of the rigid body motion knowing in any moment the twist of the rigid body that is equivalent with knowing the dual angular velocity [5, 18].

Theorem 11. For any continuous function ω ∈ V<sup>R</sup> <sup>3</sup> a unique dual tensor R ∈ SO<sup>R</sup> <sup>3</sup> exists so that

$$\begin{aligned} \underline{\dot{\mathbf{R}}} &= \underline{\tilde{\omega}} \, \underline{\mathbf{R}} \\ \underline{\mathbf{R}}(t\_0) &= \underline{\mathbf{R}}\_0, \underline{\mathbf{R}}\_0 \in \underline{\operatorname{SO}}\_3 \end{aligned} \tag{32}$$

Due to the fact that orthogonal dual tensor R completely models the six degree of freedom motion, we can conclude that the Theorem 11 is the dual form of the Poisson-Darboux problem [28] for the case when the rotation tensor is computed from the instantaneous angular velocity. So, in order to recover R, it is necessary to find out how the dual angular velocity vector ω behaves in time and also the value of R at time t ¼ t0.

The dual tensor R can be derived from ω, when is positioned in space, or from ω<sup>B</sup>, which denotes the dual angular velocity vector to be positioned in the rigid body.

Remark 6. The dual angular velocity vector positioned in the rigid body can be recovered from <sup>ω</sup><sup>B</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup>ω, thus transforming Eq. (32) into:

$$\begin{cases} \underline{\dot{\mathbf{R}}} = \underline{\mathbf{R}} \underline{\stackrel{\r}{\omega}}{\underline{\omega}}^{\underline{\mathcal{B}}}\\ \underline{\mathbf{R}}(t\_0) = \underline{\mathbf{R}}\_0, \underline{\mathbf{R}}\_0 \in \underline{\operatorname{SD}}\_3 \end{cases} \tag{33}$$

Eqs. (32) and (33) represent the dual replica of the classical orientation Poisson-Darboux problem [17, 28, 29].

The tensorial Eqs. (32) and (33) are equivalent with 18 scalar differential equations. The previous parameterizations of the orthogonal dual tensors allow us to determine some solutions of smaller dimension in order to solve the dual Poisson- Darboux problem.

#### 2.3.2. Kinematic equation for Euler dual vector parameterization

Consider Ψ ∈V<sup>R</sup> <sup>3</sup> such that R ¼ exp Ψ f. According to the Eq. (10), the Poisson-Darboux problem (32) is equivalent to

$$\begin{cases} \underline{\Psi} = \underline{T\omega} \\ \underline{\Psi}(t\_0) = \underline{\Psi}\_0 \end{cases} \tag{34}$$

2.3.4. Kinematic equation for dual quaternion parameterization

� � <sup>¼</sup> <sup>R</sup>. According to Eq. (16), the Poisson–Darboux problems (32) and

On Six DOF Relative Orbital Motion of Satellites http://dx.doi.org/10.5772/intechopen.73563

(40)

89

(41)

\_ <sup>q</sup><sup>b</sup> <sup>¼</sup> <sup>1</sup> 2 ωqb

\_ <sup>q</sup><sup>b</sup> <sup>¼</sup> <sup>1</sup> 2 <sup>q</sup>bω<sup>B</sup>

8 ><

>:

Eqs. (40) and (41) are equivalent to eight scalar differential equations.

3. Rigid body motion in arbitrary non-inertial frame revised

8 < :

<sup>q</sup>bð Þ¼ <sup>t</sup><sup>0</sup> <sup>q</sup>b<sup>0</sup>

<sup>q</sup>bð Þ¼ <sup>t</sup><sup>0</sup> <sup>q</sup>b<sup>0</sup>

To the author's knowledge, in the field of astrodynamics there aren't many reports on how the motion of rigid body can be studied in arbitrary non-inertial frames. Next, we proposed a dual tensors based model for the motion of the rigid body in arbitrary non-inertial frame. The proposed method eludes the calculus of inertia forces that contributes to the rigid body relative state. So, the free of coordinate state equation of the rigid body motion in arbitrary non-inertial

Let R<sup>D</sup> and R<sup>C</sup> be the dual orthogonal tensors which describe the motion of two rigid bodies

If R is the orthogonal dual tensor which embeds the six degree of freedom relative motion of

<sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup>

rigid body D relative to the rigid body C, then, conforming with Eq. (42):

the dual form of the Euler equation given in [30] results that:

Let ω<sup>C</sup> denote the dual angular velocity of the rigid body C and ω<sup>D</sup> the dual angular velocity of the rigid body D, both being related to inertial reference frame. In the followings, the inertial motion of the rigid body C is considered to be known. If ω is the dual angular velocity of the

<sup>D</sup> being the dual angular velocity vector of the rigid body D in the body frame,

<sup>C</sup>R<sup>D</sup> (42)

ω ¼ ω<sup>D</sup> � ω<sup>C</sup> (43)

Let <sup>q</sup><sup>b</sup> <sup>∈</sup> <sup>U</sup><sup>R</sup> such that <sup>Δ</sup> <sup>q</sup><sup>b</sup>

(33) are equivalent to:

and

where <sup>Δ</sup> <sup>q</sup>b<sup>0</sup>

� � <sup>¼</sup> <sup>R</sup><sup>0</sup>

frame will be obtained.

Considering ω<sup>B</sup>

relative to the inertial frame.

rigid body C relative to rigid body D, then:

where exp Ψ f<sup>0</sup> ¼ R0, and T is the following dual tensor:

$$\underline{T} = \frac{|\underline{\Psi}|}{2} \cot \frac{|\underline{\Psi}|}{2} \underline{I} - \frac{1}{2} \widetilde{\underline{\Psi}} - \frac{1}{2|\underline{\Psi}|} \cot \frac{|\underline{\Psi}|}{2} \widetilde{\underline{\Psi}}^2 \tag{35}$$

The representation of the Poisson-Darboux problem from Eq. (33) is equivalent to

$$\begin{cases} \underline{\Psi} = \underline{T}^{\mathrm{T}} \underline{\underline{\omega}}^{\mathrm{B}} \\ \underline{\underline{\Psi}}(t\_0) = \underline{\underline{\Psi}}\_0 \end{cases} \tag{36}$$

2.3.3. Kinematic equation for high order Rodrigues dual vector parameterization

Let v∈ V<sup>R</sup> <sup>3</sup> such that R ¼ caynv. The problems (32) and (33) are equivalent to:

$$\begin{cases} \begin{array}{l} \dot{\underline{\mathbf{v}}} = \underline{\mathbf{S}} \underline{\omega} \\ \underline{\mathbf{v}}(t\_0) = \underline{\mathbf{v}\_0} \end{array} \tag{37}$$

$$\begin{cases} \dot{\underline{\mathbf{v}}} = \underline{\mathbf{S}}^{\mathrm{T}} \underline{\mathbf{a}}^{\mathrm{B}} \\ \underline{\mathbf{v}}(t\_0) = \underline{\mathbf{v}}\_0 \end{cases} \tag{38}$$

where caynv<sup>0</sup> ¼ R0, and S is the following dual tensor [29]:

$$\underline{\mathbf{S}} = \frac{p\_n |\underline{\mathbf{y}}|}{2q\_n |\underline{\mathbf{y}}|} \underline{\mathbf{I}} - \frac{1}{2} \underline{\tilde{\mathbf{y}}} + \frac{\left(1 + |\mathbf{v}|^2\right) q\_n \left(|\underline{\mathbf{y}}| - np\_n |\underline{\mathbf{y}}|\right)}{2n |\underline{\mathbf{y}}|^2 q\_n |\underline{\mathbf{y}}|} \underline{\mathbf{y}}^2 \tag{39}$$

and the polynomials pn, qn are given by the Eqs. (22)–(24).

Eqs. (34), (36)–(38) are equivalent with six scalar differential equations. This is a minimal parameterization of the Poisson-Darboux problem in dual algebra.

#### 2.3.4. Kinematic equation for dual quaternion parameterization

Let <sup>q</sup><sup>b</sup> <sup>∈</sup> <sup>U</sup><sup>R</sup> such that <sup>Δ</sup> <sup>q</sup><sup>b</sup> � � <sup>¼</sup> <sup>R</sup>. According to Eq. (16), the Poisson–Darboux problems (32) and (33) are equivalent to:

$$\begin{cases} \dot{\underline{\hat{q}}} = \frac{1}{2} \underline{\omega} \hat{\underline{\hat{q}}}\\ \underline{\hat{\underline{q}}}(t\_0) = \hat{\underline{\hat{q}}}\_0 \end{cases} \tag{40}$$

and

Eqs. (32) and (33) represent the dual replica of the classical orientation Poisson-Darboux

The tensorial Eqs. (32) and (33) are equivalent with 18 scalar differential equations. The previous parameterizations of the orthogonal dual tensors allow us to determine some solu-

> <sup>Ψ</sup>\_ <sup>¼</sup> <sup>T</sup><sup>ω</sup> Ψð Þ¼ t<sup>0</sup> Ψ<sup>0</sup>

<sup>Ψ</sup>\_ <sup>¼</sup> <sup>T</sup><sup>T</sup>ω<sup>B</sup> Ψð Þ¼ t<sup>0</sup> Ψ<sup>0</sup>

v\_ ¼ Sω vð Þ¼ t<sup>0</sup> v0 ,

<sup>v</sup>\_ <sup>¼</sup> <sup>S</sup><sup>T</sup>ω<sup>B</sup> vð Þ¼ t<sup>0</sup> v<sup>0</sup>

1 þ j j v <sup>2</sup> � �

Eqs. (34), (36)–(38) are equivalent with six scalar differential equations. This is a minimal

2nj j v 2

qn j j <sup>v</sup> � npnj j <sup>v</sup> � �

qnj j <sup>v</sup> <sup>e</sup><sup>v</sup>

f. According to the Eq. (10), the Poisson-Darboux prob-

<sup>2</sup> <sup>Ψ</sup> f2

<sup>2</sup>j j <sup>Ψ</sup> cot j j <sup>Ψ</sup>

(34)

(35)

(36)

(37)

(38)

<sup>2</sup> (39)

tions of smaller dimension in order to solve the dual Poisson- Darboux problem.

(

<sup>2</sup> <sup>I</sup> � <sup>1</sup> 2 Ψ <sup>f</sup> � <sup>1</sup>

The representation of the Poisson-Darboux problem from Eq. (33) is equivalent to

(

<sup>3</sup> such that R ¼ caynv. The problems (32) and (33) are equivalent to:

(

(

2.3.2. Kinematic equation for Euler dual vector parameterization

f<sup>0</sup> ¼ R0, and T is the following dual tensor:

<sup>2</sup> cot j j <sup>Ψ</sup>

2.3.3. Kinematic equation for high order Rodrigues dual vector parameterization

<sup>T</sup> <sup>¼</sup> j j <sup>Ψ</sup>

where caynv<sup>0</sup> ¼ R0, and S is the following dual tensor [29]:

<sup>I</sup> � <sup>1</sup> 2 <sup>e</sup><sup>v</sup> <sup>þ</sup>

parameterization of the Poisson-Darboux problem in dual algebra.

<sup>S</sup> <sup>¼</sup> pnj j <sup>v</sup> 2qnj j v

and the polynomials pn, qn are given by the Eqs. (22)–(24).

<sup>3</sup> such that R ¼ exp Ψ

problem [17, 28, 29].

88 Space Flight

Consider Ψ ∈V<sup>R</sup>

where exp Ψ

Let v∈ V<sup>R</sup>

lem (32) is equivalent to

$$\begin{cases} \dot{\underline{\hat{q}}} = \frac{1}{2} \hat{\underline{\hat{q}}} \underline{\omega}^{\mathcal{B}} \\ \underline{\hat{q}}(t\_0) = \underline{\hat{q}}\_0 \end{cases} \tag{41}$$

where <sup>Δ</sup> <sup>q</sup>b<sup>0</sup> � � <sup>¼</sup> <sup>R</sup><sup>0</sup>

Eqs. (40) and (41) are equivalent to eight scalar differential equations.

#### 3. Rigid body motion in arbitrary non-inertial frame revised

To the author's knowledge, in the field of astrodynamics there aren't many reports on how the motion of rigid body can be studied in arbitrary non-inertial frames. Next, we proposed a dual tensors based model for the motion of the rigid body in arbitrary non-inertial frame. The proposed method eludes the calculus of inertia forces that contributes to the rigid body relative state. So, the free of coordinate state equation of the rigid body motion in arbitrary non-inertial frame will be obtained.

Let R<sup>D</sup> and R<sup>C</sup> be the dual orthogonal tensors which describe the motion of two rigid bodies relative to the inertial frame.

If R is the orthogonal dual tensor which embeds the six degree of freedom relative motion of rigid body C relative to rigid body D, then:

$$\underline{\mathbf{R}} = \underline{\mathbf{R}}\_{C}^{T} \underline{\mathbf{R}}\_{D} \tag{42}$$

Let ω<sup>C</sup> denote the dual angular velocity of the rigid body C and ω<sup>D</sup> the dual angular velocity of the rigid body D, both being related to inertial reference frame. In the followings, the inertial motion of the rigid body C is considered to be known. If ω is the dual angular velocity of the rigid body D relative to the rigid body C, then, conforming with Eq. (42):

$$
\underline{\omega} = \underline{\omega}\_{\heartsuit} - \underline{\omega}\_{\heartsuit} \tag{43}
$$

Considering ω<sup>B</sup> <sup>D</sup> being the dual angular velocity vector of the rigid body D in the body frame, the dual form of the Euler equation given in [30] results that:

$$
\underline{\mathbf{M}\omega^{B}\_{D}} + \underline{\omega}^{B}\_{D} \times \underline{\mathbf{M}\omega^{B}\_{D}} = \underline{\mathbf{\tau}}^{B} \tag{44}
$$

is a compact form which can be used to model the six D.O.F relative motion problem. In the previous equation the state of the rigid body D in relation with the rigid body C is modeled by the dual tensor R and the dual angular velocities field ω. This initial value problem can be used to study the behavior of the rigid body D in relation with the frame attached to the rigid body C. In Eq. (53), all the vectors are represented in the body frame of C, which shows that the proposed solution is onboard and has the property of being

Next, we present a procedure that allows the decoupling of the proposed solution.

<sup>þ</sup>ω\_ <sup>C</sup>Þ. The result is equivalent with <sup>ω</sup>\_ <sup>∗</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> <sup>ω</sup><sup>C</sup> � <sup>ω</sup> <sup>þ</sup> <sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ <sup>C</sup> ð Þ or

(

<sup>3</sup> , the solution of Eq. (53) emerges from

(

After some steps of algebraic calculus, from Eqs. (54), (55) and (52), results that:

M ω\_ <sup>∗</sup> þ ω<sup>∗</sup> � M ω<sup>∗</sup> ¼ τ<sup>∗</sup>

Where <sup>τ</sup><sup>∗</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup><sup>τ</sup> is the dual torque related to the mass center in the body frame of the rigid

<sup>R</sup>\_ <sup>¼</sup> <sup>ω</sup><sup>e</sup> <sup>R</sup> Rð Þ¼ t<sup>0</sup> R<sup>0</sup>

Making use of Eq. (54), results that <sup>R</sup>ω<sup>∗</sup> <sup>¼</sup> <sup>ω</sup> <sup>þ</sup> <sup>ω</sup>C. If <sup>e</sup> operator used, the previous calculus is transformed into <sup>R</sup>gω<sup>∗</sup> <sup>¼</sup> <sup>ω</sup><sup>e</sup> <sup>þ</sup> <sup>ω</sup><sup>e</sup> <sup>C</sup> <sup>⇔</sup> <sup>R</sup> <sup>ω</sup><sup>e</sup> <sup>∗</sup>R<sup>T</sup> <sup>¼</sup> <sup>R</sup>\_ <sup>R</sup><sup>T</sup> <sup>þ</sup> <sup>ω</sup><sup>e</sup> <sup>C</sup>. After multiplying the last expres-

<sup>R</sup>\_ <sup>¼</sup> <sup>R</sup> <sup>ω</sup><sup>e</sup> <sup>∗</sup> � <sup>ω</sup><sup>e</sup> <sup>C</sup> <sup>R</sup>

Using the variable change Eq. (54), the initial value problem (53) has been decoupled into two

<sup>3</sup> be the unique solution of the following Poisson-Darboux problem:

Rð Þ¼ t<sup>0</sup> R<sup>0</sup>

(

<sup>ω</sup>∗ð Þ¼ <sup>t</sup><sup>0</sup> <sup>ω</sup><sup>0</sup>

<sup>0</sup> ω<sup>0</sup> þ ω<sup>C</sup> ð Þ ð Þ t<sup>0</sup> . Eq. (56) is a dual Euler fixed point classic problem.

∗

In order to describe the solution to Eq. (53), we consider the following change of variable:

<sup>ω</sup><sup>∗</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> <sup>ω</sup> <sup>þ</sup> <sup>ω</sup><sup>C</sup> ð Þ (54)

ω<sup>C</sup> � ω þ ω\_ þ ω\_ <sup>C</sup> ¼ R ω\_ <sup>∗</sup> (55)

<sup>ω</sup> <sup>þ</sup> <sup>ω</sup><sup>C</sup> ð Þþ <sup>R</sup><sup>T</sup> <sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ ð Þ¼� <sup>C</sup> <sup>R</sup><sup>T</sup>ω<sup>e</sup> <sup>ω</sup> <sup>þ</sup> <sup>ω</sup><sup>C</sup> ð Þþ <sup>R</sup><sup>T</sup>ðω\_

On Six DOF Relative Orbital Motion of Satellites http://dx.doi.org/10.5772/intechopen.73563

(56)

91

(57)

(58)

coupled in R and ω.

body D and ω<sup>0</sup>

For any R ∈ SO<sup>R</sup>

Let <sup>R</sup>�ω<sup>C</sup> <sup>∈</sup>SO<sup>R</sup>

<sup>∗</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup>

sion by R, we obtain the initial value problem:

distinct initial value problems (56) and (58).

This change of variable leads to <sup>ω</sup>\_ <sup>∗</sup> <sup>¼</sup> <sup>R</sup>\_ <sup>T</sup>

In Eq. (44) <sup>τ</sup><sup>B</sup> <sup>¼</sup> <sup>F</sup><sup>B</sup> <sup>þ</sup> <sup>ε</sup>τB, where <sup>F</sup><sup>B</sup> the force applied in the mass center and <sup>τ</sup><sup>B</sup> is the torque. Also in Eq. (44), M represents the inertia dual operator, which is given by M ¼ mD d <sup>d</sup><sup>ε</sup> IþεJ, where J is the inertia tensor of the rigid body D related to its mass center and mD is the mass of the rigid body D. Combining <sup>M</sup>�<sup>1</sup> <sup>¼</sup> <sup>J</sup>�<sup>1</sup> <sup>d</sup> <sup>d</sup><sup>ε</sup> <sup>þ</sup> <sup>ε</sup> <sup>1</sup> mD I with Eq. (44) results:

$$
\underline{\dot{\omega}}\_{\rm D}^{\rm B} + \underline{\underline{M}}^{-1} \left( \underline{\omega}\_{\rm D}^{\rm B} \times \underline{\underline{M}} \underline{\omega}\_{\rm D}^{\rm B} \right) = \underline{\underline{M}}^{-1} \underline{\underline{\pi}}^{\rm B} \tag{45}
$$

Taking into account that <sup>ω</sup><sup>D</sup> <sup>¼</sup> <sup>R</sup>ω<sup>B</sup> <sup>D</sup>, the dual angular velocity vector can be computed from

$$
\underline{\omega} = \underline{\mathbf{R}\omega^B\_D} - \underline{\omega}\_{\mathbb{C}} \tag{46}
$$

this through differentiation gives:

$$
\underline{\dot{\omega}} + \dot{\underline{\omega}}\_{\mathcal{C}} = \underline{\dot{\mathbf{R}}} \underline{\omega}\_{\mathcal{D}}^{\mathcal{B}} + \underline{\mathbf{R}} \dot{\underline{\omega}}\_{\mathcal{D}}^{\mathcal{B}} \tag{47}
$$

If the previous equation is multiplied by R<sup>T</sup> , then

$$\underline{\mathbf{R}}^{T}(\underline{\dot{\boldsymbol{\omega}}} + \underline{\dot{\boldsymbol{\omega}}}\_{C}) = \underline{\mathbf{R}}^{T}\underline{\dot{\mathbf{R}}}\underline{\boldsymbol{\omega}}\_{D}^{B} + \underline{\dot{\boldsymbol{\omega}}}\_{D}^{B} \tag{48}$$

which combined with <sup>R</sup>\_ <sup>¼</sup> <sup>ω</sup><sup>e</sup> <sup>R</sup> generates:

$$\underline{\mathbf{R}}^{T}(\underline{\mathbf{\omega}} + \underline{\mathbf{\omega}}\_{\circ}) = \underline{\mathbf{R}}^{T}\underline{\mathbf{\omega}}\underline{\mathbf{R}}\underline{\mathbf{\omega}}\_{\mathrm{D}}^{B} + \underline{\mathbf{\omega}}\_{\mathrm{D}}^{B} \tag{49}$$

After a few steps, Eq. (49) is transformed into

$$
\underline{\dot{\omega}} + \underline{\dot{\omega}}\_{\mathbb{C}} = \underline{\mathbf{R}} \dot{\underline{\omega}}\_{D}^{\mathbb{B}} + \underline{\underline{\omega}} \times \underline{\omega}\_{\mathbb{C}} \tag{50}
$$

which combined with Eq. (45) gives:

$$
\dot{\underline{\omega}} + \dot{\underline{\omega}}\_{\mathbb{C}} = \underline{\underline{\mathbf{R}M}^{-1}} \underline{\underline{\mathbf{r}}}^{B} - \underline{\underline{\mathbf{R}M}^{-1}} \left( \underline{\omega}\_{\mathbb{D}}^{B} \times \underline{\underline{\mathbf{M}} \omega\_{\mathbb{D}}^{B}} \right) + \underline{\omega} \times \underline{\omega}\_{\mathbb{C}} \tag{51}
$$

Because ω<sup>B</sup> <sup>D</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> <sup>ω</sup> � <sup>ω</sup><sup>C</sup> ð Þ, the final equation is:

$$
\underline{\dot{\omega}} + \underline{\dot{\omega}}\_{\circ} = \underline{\mathbf{RM}}^{-1} \left[ \underline{\mathbf{z}}^{\mathrm{B}} - \underline{\mathbf{R}}^{\mathrm{T}} (\underline{\omega} + \underline{\omega}\_{\circ}) \times \underline{\mathbf{MR}}^{\mathrm{T}} (\underline{\omega} + \underline{\omega}\_{\circ}) \right] + \underline{\omega} \times \underline{\omega}\_{\circ} \tag{52}
$$

The system:

$$\begin{cases} \begin{aligned} \dot{\underline{\dot{R}}} &= \underline{\ddot{\alpha}} \underline{\mathbf{R}} \\ \dot{\underline{\omega}} + \dot{\underline{\omega}}\_{\circ} &= \underline{\underline{\mathbf{R}}} \underline{\mathbf{M}}^{-1} [\underline{\underline{\mathbf{R}}}^{T} \underline{\underline{\mathbf{r}}} - \underline{\underline{\mathbf{R}}}^{T} (\underline{\omega} + \underline{\omega}\_{\circ}) \times \\ & \times \underline{\underline{\mathbf{M}}}^{T} (\underline{\underline{\omega}} + \underline{\underline{\omega}}\_{\circ}) \big] + \underline{\underline{\omega}} \times \underline{\underline{\omega}\_{\circ}} \\ \underline{\underline{\omega}}(t\_{0}) &= \underline{\underline{\omega}}\_{0^{\prime}} \underline{\omega}\_{0} \in \underline{V}\_{3} \end{aligned} \tag{53}$$

is a compact form which can be used to model the six D.O.F relative motion problem. In the previous equation the state of the rigid body D in relation with the rigid body C is modeled by the dual tensor R and the dual angular velocities field ω. This initial value problem can be used to study the behavior of the rigid body D in relation with the frame attached to the rigid body C. In Eq. (53), all the vectors are represented in the body frame of C, which shows that the proposed solution is onboard and has the property of being coupled in R and ω.

Next, we present a procedure that allows the decoupling of the proposed solution.

In order to describe the solution to Eq. (53), we consider the following change of variable:

$$
\underline{\omega}\_{\*} = \underline{\mathbf{R}}^{T} (\underline{\omega} + \underline{\omega}\_{\cdot}) \tag{54}
$$

This change of variable leads to <sup>ω</sup>\_ <sup>∗</sup> <sup>¼</sup> <sup>R</sup>\_ <sup>T</sup> <sup>ω</sup> <sup>þ</sup> <sup>ω</sup><sup>C</sup> ð Þþ <sup>R</sup><sup>T</sup> <sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ ð Þ¼� <sup>C</sup> <sup>R</sup><sup>T</sup>ω<sup>e</sup> <sup>ω</sup> <sup>þ</sup> <sup>ω</sup><sup>C</sup> ð Þþ <sup>R</sup><sup>T</sup>ðω\_ <sup>þ</sup>ω\_ <sup>C</sup>Þ. The result is equivalent with <sup>ω</sup>\_ <sup>∗</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> <sup>ω</sup><sup>C</sup> � <sup>ω</sup> <sup>þ</sup> <sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ <sup>C</sup> ð Þ or

$$
\underline{\omega}\_{\uparrow} \times \underline{\omega} + \dot{\underline{\omega}} + \dot{\underline{\omega}}\_{\uparrow} = \underline{\mathbf{R}} \,\dot{\underline{\omega}}\_{\ast} \tag{55}
$$

After some steps of algebraic calculus, from Eqs. (54), (55) and (52), results that:

$$\begin{cases} \underline{\mathbf{M}} \,\underline{\mathbf{\omega}}\_{\*} + \underline{\mathbf{\omega}}\_{\*} \times \underline{\mathbf{M}} \,\underline{\mathbf{\omega}}\_{\*} = \underline{\mathbf{\pi}}\_{\*} \\\\ \underline{\mathbf{\omega}}\_{\*} (t\_{0}) = \underline{\mathbf{\omega}}\_{\*}^{0} \end{cases} \tag{56}$$

Where <sup>τ</sup><sup>∗</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup><sup>τ</sup> is the dual torque related to the mass center in the body frame of the rigid body D and ω<sup>0</sup> <sup>∗</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> <sup>0</sup> ω<sup>0</sup> þ ω<sup>C</sup> ð Þ ð Þ t<sup>0</sup> . Eq. (56) is a dual Euler fixed point classic problem.

For any R ∈ SO<sup>R</sup> <sup>3</sup> , the solution of Eq. (53) emerges from

Mω\_<sup>B</sup>

<sup>D</sup> <sup>þ</sup> <sup>M</sup>�<sup>1</sup> <sup>ω</sup><sup>B</sup>

ω\_ <sup>B</sup>

the rigid body D. Combining <sup>M</sup>�<sup>1</sup> <sup>¼</sup> <sup>J</sup>�<sup>1</sup> <sup>d</sup>

Taking into account that <sup>ω</sup><sup>D</sup> <sup>¼</sup> <sup>R</sup>ω<sup>B</sup>

90 Space Flight

this through differentiation gives:

If the previous equation is multiplied by R<sup>T</sup>

which combined with <sup>R</sup>\_ <sup>¼</sup> <sup>ω</sup><sup>e</sup> <sup>R</sup> generates:

After a few steps, Eq. (49) is transformed into

<sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ <sup>C</sup> <sup>¼</sup> RM�<sup>1</sup>

<sup>D</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> <sup>ω</sup> � <sup>ω</sup><sup>C</sup> ð Þ, the final equation is:

8 >>>>>>><

>>>>>>>:

which combined with Eq. (45) gives:

Because ω<sup>B</sup>

The system:

<sup>D</sup> <sup>þ</sup> <sup>ω</sup><sup>B</sup>

Also in Eq. (44), M represents the inertia dual operator, which is given by M ¼ mD

<sup>d</sup><sup>ε</sup> <sup>þ</sup> <sup>ε</sup> <sup>1</sup>

<sup>ω</sup> <sup>¼</sup> <sup>R</sup>ω<sup>B</sup>

<sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ <sup>C</sup> <sup>¼</sup> <sup>R</sup>\_ <sup>ω</sup><sup>B</sup>

<sup>R</sup><sup>T</sup> <sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ <sup>C</sup> ð Þ¼ <sup>R</sup><sup>T</sup>R\_ <sup>ω</sup><sup>B</sup>

<sup>R</sup><sup>T</sup> <sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ <sup>C</sup> ð Þ¼ <sup>R</sup><sup>T</sup>ω<sup>e</sup> <sup>R</sup>ω<sup>B</sup>

<sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ <sup>C</sup> <sup>¼</sup> <sup>R</sup>ω\_ <sup>B</sup>

<sup>τ</sup><sup>B</sup> � RM�<sup>1</sup> <sup>ω</sup><sup>B</sup>

<sup>R</sup>\_ <sup>¼</sup> <sup>ω</sup><sup>e</sup> <sup>R</sup>

�MR<sup>T</sup> <sup>ω</sup> <sup>þ</sup> <sup>ω</sup><sup>C</sup> ð Þ� þ <sup>ω</sup> � <sup>ω</sup><sup>C</sup> ωð Þ¼ t0 ω0, ω<sup>0</sup> ∈ V<sup>3</sup> Rð Þ¼ t<sup>0</sup> R0, R<sup>0</sup> ∈ SO<sup>3</sup>

<sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ <sup>C</sup> <sup>¼</sup> RM�<sup>1</sup>

, then

<sup>D</sup> � <sup>M</sup>ω<sup>B</sup>

In Eq. (44) <sup>τ</sup><sup>B</sup> <sup>¼</sup> <sup>F</sup><sup>B</sup> <sup>þ</sup> <sup>ε</sup>τB, where <sup>F</sup><sup>B</sup> the force applied in the mass center and <sup>τ</sup><sup>B</sup> is the torque.

where J is the inertia tensor of the rigid body D related to its mass center and mD is the mass of

<sup>D</sup> � <sup>M</sup>ω<sup>B</sup> D � � <sup>¼</sup> <sup>M</sup>�<sup>1</sup>

mD I with Eq. (44) results:

<sup>D</sup> <sup>þ</sup> <sup>R</sup>ω\_ <sup>B</sup>

<sup>D</sup> <sup>þ</sup> <sup>ω</sup>\_ <sup>B</sup>

<sup>D</sup> <sup>þ</sup> <sup>ω</sup>\_ <sup>B</sup>

<sup>D</sup> � <sup>M</sup>ω<sup>B</sup> D

<sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ <sup>C</sup> <sup>¼</sup> RM�<sup>1</sup> <sup>τ</sup><sup>B</sup> � <sup>R</sup><sup>T</sup> <sup>ω</sup> <sup>þ</sup> <sup>ω</sup><sup>C</sup> ð Þ� MR<sup>T</sup> <sup>ω</sup> <sup>þ</sup> <sup>ω</sup><sup>C</sup> ð Þ � � <sup>þ</sup> <sup>ω</sup> � <sup>ω</sup><sup>C</sup> (52)

<sup>½</sup>R<sup>T</sup> <sup>τ</sup> � <sup>R</sup><sup>T</sup> <sup>ω</sup> <sup>þ</sup> <sup>ω</sup><sup>C</sup> ð Þ�

<sup>D</sup>, the dual angular velocity vector can be computed from

<sup>D</sup> � ω<sup>C</sup> (46)

<sup>D</sup> <sup>¼</sup> <sup>τ</sup><sup>B</sup> (44)

τ<sup>B</sup> (45)

<sup>D</sup> (47)

<sup>D</sup> (48)

<sup>D</sup> (49)

(53)

<sup>D</sup> þ ω � ω<sup>C</sup> (50)

� � <sup>þ</sup> <sup>ω</sup> � <sup>ω</sup><sup>C</sup> (51)

d <sup>d</sup><sup>ε</sup> IþεJ,

$$\begin{cases} \dot{\underline{R}} = \tilde{\underline{\omega}} \, \underline{\underline{R}}\\ \underline{\underline{R}}(t\_0) = \underline{\underline{R}}\_0 \end{cases} \tag{57}$$

Making use of Eq. (54), results that <sup>R</sup>ω<sup>∗</sup> <sup>¼</sup> <sup>ω</sup> <sup>þ</sup> <sup>ω</sup>C. If <sup>e</sup> operator used, the previous calculus is transformed into <sup>R</sup>gω<sup>∗</sup> <sup>¼</sup> <sup>ω</sup><sup>e</sup> <sup>þ</sup> <sup>ω</sup><sup>e</sup> <sup>C</sup> <sup>⇔</sup> <sup>R</sup> <sup>ω</sup><sup>e</sup> <sup>∗</sup>R<sup>T</sup> <sup>¼</sup> <sup>R</sup>\_ <sup>R</sup><sup>T</sup> <sup>þ</sup> <sup>ω</sup><sup>e</sup> <sup>C</sup>. After multiplying the last expression by R, we obtain the initial value problem:

$$\begin{cases} \dot{\underline{R}} = \underline{\underline{R}} \,\, \underline{\underline{\omega}}\_{\*} - \underline{\underline{\omega}}\_{C} \,\, \underline{\underline{R}}\\ \qquad \underline{\underline{R}}(t\_{0}) = \underline{\underline{R}}\_{0} \end{cases} \tag{58}$$

Using the variable change Eq. (54), the initial value problem (53) has been decoupled into two distinct initial value problems (56) and (58).

Let <sup>R</sup>�ω<sup>C</sup> <sup>∈</sup>SO<sup>R</sup> <sup>3</sup> be the unique solution of the following Poisson-Darboux problem:

$$\begin{cases} \dot{\underline{\mathbf{R}}} + \widetilde{\underline{\mathbf{a}}}\_{\mathbb{C}} \underline{\mathbf{R}} = 0 \\ \underline{\mathbf{R}}(t\_0) = \mathbf{I} - \varepsilon \widetilde{\mathbf{r}}\_{\mathbb{C}}(t\_0) \end{cases} \tag{59}$$

Different particular cases can be analyzed for Eq. (62):

order Modified Rodrigues Parameter dual vector.

flying, distributed spacecraft missions [3, 4, 6–10].

<sup>ω</sup><sup>C</sup> <sup>¼</sup> \_ f C h<sup>C</sup> hC ¼ 1 r2 C

true anomaly and eC is the eccentricity of the Leader.

<sup>2</sup> u be the Rodrigues dual vector for n = 1:

<sup>S</sup> <sup>¼</sup> <sup>1</sup> � <sup>ξ</sup> � � � � 2 <sup>4</sup> <sup>I</sup> <sup>þ</sup>

<sup>S</sup> <sup>¼</sup> <sup>1</sup> 2 I þ 1 2 eξ þ 1 2 ξ ⊗ ξ

<sup>4</sup> u be the modified Rodrigues dual vector (Wiener-Milenkovic dual vector) for

On Six DOF Relative Orbital Motion of Satellites http://dx.doi.org/10.5772/intechopen.73563 93

1 2 eξ þ 1 2 ξ ⊗ ξ:

The initial value problem (62) is a minimum parameterization of the six degrees of freedom motion problem. The singularity cases can be avoided using the shadow parameters of the n-th

4. A dual tensor formulation of the six degree of freedom relative orbital

The results from the previous paragraphs will be used to study the six degrees of freedom

The relative orbital motion problem may now be considered classical one considering the many scientific papers written on this subject in the last decades. Also, the problem is quite important knowing its numerous applications: rendezvous operations, spacecraft formation

The model of the relative motion consists in two spacecraft flying in Keplerian orbits due to the influence of the same gravitational attraction center. The main problem is to determine the pose of the Deputy satellite relative to a reference frame originated in the Leader satellite center of mass. This non-inertial reference frame, known as "LVLH (Local-Vertical-Local- Horizontal)" is chosen as following: the Cx axis has the same orientation as the position vector of the Leader with respect to an inertial reference frame with the origin in the attraction center; the orientation of the Cz is the same as the Leader orbit angular momentum; the Cy axis completes a right-handed frame. The angular velocity of the LVLH is given by vector ωC, which has the expression:

<sup>r</sup><sup>C</sup> <sup>¼</sup> pC

1 þ eC cos f <sup>C</sup>ð Þt

where pC is the conic parameter, h<sup>C</sup> is the angular momentum of the Leader, f <sup>C</sup>ð Þt being the

<sup>h</sup><sup>C</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> eC cos <sup>f</sup> <sup>C</sup>ð Þ<sup>t</sup> pC � �<sup>2</sup>

> r0 C r0 C

h<sup>C</sup> (64)

(65)

1. Let <sup>ξ</sup> <sup>¼</sup> tan <sup>α</sup>

2. Let <sup>ξ</sup> <sup>¼</sup> tan <sup>α</sup>

motion problem

where vector r<sup>C</sup> is

relative orbital motion problem.

n = 2:

Considering R ¼ R�ω<sup>C</sup> R∗, a representation theorem of the solution of Eq. (53) can be formulated.

Theorem 12. (Representation Theorem). The solution of Eq. (53) results from the application of the tensor R�ω<sup>C</sup> from Eq. (59) to the solution of the classical dual Euler fixed point problem:

$$\begin{cases} \begin{aligned} \dot{\underline{\mathcal{R}}}\_{\*} &= \underline{\mathcal{R}}\_{\*} \underline{\mathcal{U}}\_{\*} \\ \underline{\mathbf{M}} \, \underline{\dot{\omega}}\_{\*} + \underline{\omega}\_{\*} \times \underline{\mathbf{M}} \, \underline{\omega}\_{\*} &= \underline{\mathbf{r}}\_{\*} \\ \underline{\omega}\_{\*}(t\_{0}) &= \underline{\omega}\_{\*0} \end{aligned} \tag{60} \\ \underline{\mathbf{R}}\_{\*}(t\_{0}) &= \underline{\mathbf{R}}\_{\*0} \end{aligned} \tag{60}$$

where <sup>ω</sup>∗<sup>0</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> <sup>0</sup> <sup>ω</sup><sup>0</sup> <sup>þ</sup> <sup>ω</sup><sup>C</sup> ð Þ ð Þ <sup>t</sup><sup>0</sup> , <sup>R</sup>∗<sup>0</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup> <sup>ε</sup>erCð ÞÞ <sup>t</sup><sup>0</sup> <sup>R</sup><sup>0</sup> � , <sup>τ</sup><sup>∗</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup>τ.

Different representations can be considered for the problem (60).

Using dual quaternion representation <sup>R</sup>∗¼<sup>Δ</sup> <sup>q</sup>b<sup>∗</sup> � �, Eq. (60) is equivalent with the following one:

$$\begin{cases} \begin{aligned} \hat{\underline{\mathbf{q}}}^{\*} &= \frac{1}{2} \hat{\underline{\mathbf{q}}}\_{\*} \underline{\mathbf{w}}\_{\*} \\ \underline{\mathbf{M}} \, \underline{\mathbf{w}}\_{\*} + \underline{\mathbf{w}}\_{\*} \times \underline{\mathbf{M}} \, \underline{\mathbf{w}}\_{\*} &= \underline{\mathbf{r}}\_{\*} \\ \underline{\mathbf{w}}\_{\*}(t\_{0}) &= \underline{\mathbf{w}}\_{\*0} \end{aligned} \tag{61} \\ \hat{\underline{\mathbf{q}}}^{\*}(t\_{0}) &= \hat{\underline{\mathbf{q}}}^{\*} \end{aligned} \tag{62}$$

For the n-th order of Cayley transform based representation R<sup>∗</sup> ¼ cayn ξ � �, <sup>ξ</sup> <sup>¼</sup> tan <sup>α</sup> <sup>2</sup><sup>n</sup> u, the Eq. (60) becomes:

$$\begin{cases} \dot{\underline{\xi}} = \underline{\mathsf{S}} \left( \underline{\xi} \right) \underline{\mathsf{\omega}}\_{\*} \\\\ \underline{\mathsf{M}} \,\underline{\mathsf{\omega}}\_{\*} + \underline{\mathsf{\omega}}\_{\*} \times \underline{\mathsf{M}} \,\underline{\mathsf{\omega}}\_{\*} = \underline{\mathsf{\tau}}\_{\*} \\\\ \underline{\mathsf{\omega}}\_{\*} (t\_{0}) = \underline{\mathsf{\omega}}\_{0} \end{cases} \tag{62}$$

where the tensor S is:

$$\underline{S} = \frac{p\_n\left(\left|\underline{\xi}\right|\right)}{2q\_n\left(\left|\underline{\xi}\right|\right)}\underline{I} + \frac{1}{2}\underline{\tilde{\xi}} + \frac{\left(1 + \left|\underline{\xi}\right|^2\right)q\_n\left(\underline{\xi}\right) - np\_n\left(\left|\underline{\xi}\right|\right)}{2n\left|\underline{\xi}\right|^2 q\_n\left(\left|\underline{\xi}\right|\right)}\underline{\xi} \otimes \underline{\xi} \tag{63}$$

when pnð Þ X and qnð Þ X are defined by Eqs. (22) and (23).

Different particular cases can be analyzed for Eq. (62):

<sup>R</sup>\_ <sup>þ</sup> <sup>ω</sup><sup>e</sup> <sup>C</sup> <sup>R</sup> <sup>¼</sup> <sup>0</sup> <sup>R</sup>ð Þ¼ <sup>t</sup><sup>0</sup> <sup>I</sup>�εerCð Þ <sup>t</sup><sup>0</sup>

Theorem 12. (Representation Theorem). The solution of Eq. (53) results from the application of the

R\_

M ω\_ <sup>∗</sup> þ ω<sup>∗</sup> � M ω<sup>∗</sup> ¼ τ<sup>∗</sup>

<sup>∗</sup> <sup>¼</sup> <sup>R</sup>∗ω<sup>e</sup> <sup>∗</sup>

ω∗ð Þ¼ t<sup>0</sup> ω∗<sup>0</sup>

R∗ð Þ¼ t<sup>0</sup> R∗<sup>0</sup>

, Eq. (60) is equivalent with the following one:

� �

� � � <sup>ξ</sup> <sup>⊗</sup> <sup>ξ</sup> (63)

, <sup>ξ</sup> <sup>¼</sup> tan <sup>α</sup>

� , <sup>τ</sup><sup>∗</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup>τ.

\_ <sup>q</sup>b<sup>∗</sup> <sup>¼</sup> <sup>1</sup> 2 qb ∗ ω∗

ω∗ð Þ¼ t<sup>0</sup> ω∗<sup>0</sup>

<sup>q</sup>b<sup>∗</sup>ð Þ¼ <sup>t</sup><sup>0</sup> <sup>q</sup>b<sup>∗</sup><sup>0</sup>

ω∗ð Þ¼ t<sup>0</sup> ω∗<sup>0</sup>

ξð Þ¼ t<sup>0</sup> ξ<sup>0</sup>

qn ξ � �

2n ξ � � � � 2 qn ξ � � �

� npn ξ � � � � � �

M ω\_ <sup>∗</sup> þ ω<sup>∗</sup> � M ω<sup>∗</sup> ¼ τ<sup>∗</sup>

\_ ξ ¼ S ξ � � ω∗

M ω\_ <sup>∗</sup> þ ω<sup>∗</sup> � M ω<sup>∗</sup> ¼ τ<sup>∗</sup>

1 þ ξ � � � � <sup>2</sup> � �

� �

R∗, a representation theorem of the solution of Eq. (53) can be formu-

(59)

(60)

(61)

<sup>2</sup><sup>n</sup> u, the

(62)

(

tensor R�ω<sup>C</sup> from Eq. (59) to the solution of the classical dual Euler fixed point problem:

8 >>>>><

>>>>>:

<sup>0</sup> <sup>ω</sup><sup>0</sup> <sup>þ</sup> <sup>ω</sup><sup>C</sup> ð Þ ð Þ <sup>t</sup><sup>0</sup> , <sup>R</sup>∗<sup>0</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup> <sup>ε</sup>erCð ÞÞ <sup>t</sup><sup>0</sup> <sup>R</sup><sup>0</sup>

8

>>>>>>>><

>>>>>>>>:

8 >>>>>>><

>>>>>>>:

1 2 eξ þ

For the n-th order of Cayley transform based representation R<sup>∗</sup> ¼ cayn ξ

Different representations can be considered for the problem (60).

Using dual quaternion representation <sup>R</sup>∗¼<sup>Δ</sup> <sup>q</sup>b<sup>∗</sup>

Considering R ¼ R�ω<sup>C</sup>

where <sup>ω</sup>∗<sup>0</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup>

Eq. (60) becomes:

where the tensor S is:

S ¼¼

pn ξ � � � � � �

2qn ξ � � � � � � <sup>I</sup> <sup>þ</sup>

when pnð Þ X and qnð Þ X are defined by Eqs. (22) and (23).

lated.

92 Space Flight

1. Let <sup>ξ</sup> <sup>¼</sup> tan <sup>α</sup> <sup>2</sup> u be the Rodrigues dual vector for n = 1:

$$\underline{\mathbf{S}} = \frac{1}{2}\underline{I} + \frac{1}{2}\underline{\tilde{\xi}} + \frac{1}{2}\underline{\xi} \otimes \underline{\mathfrak{S}}$$

2. Let <sup>ξ</sup> <sup>¼</sup> tan <sup>α</sup> <sup>4</sup> u be the modified Rodrigues dual vector (Wiener-Milenkovic dual vector) for n = 2:

$$\underline{\mathbf{S}} = \frac{1 - \left| \underline{\xi} \right|^2}{4} \underline{I} + \frac{1}{2} \underline{\tilde{\xi}} + \frac{1}{2} \underline{\xi} \otimes \underline{\xi} \underline{\omega}$$

The initial value problem (62) is a minimum parameterization of the six degrees of freedom motion problem. The singularity cases can be avoided using the shadow parameters of the n-th order Modified Rodrigues Parameter dual vector.

#### 4. A dual tensor formulation of the six degree of freedom relative orbital motion problem

The results from the previous paragraphs will be used to study the six degrees of freedom relative orbital motion problem.

The relative orbital motion problem may now be considered classical one considering the many scientific papers written on this subject in the last decades. Also, the problem is quite important knowing its numerous applications: rendezvous operations, spacecraft formation flying, distributed spacecraft missions [3, 4, 6–10].

The model of the relative motion consists in two spacecraft flying in Keplerian orbits due to the influence of the same gravitational attraction center. The main problem is to determine the pose of the Deputy satellite relative to a reference frame originated in the Leader satellite center of mass. This non-inertial reference frame, known as "LVLH (Local-Vertical-Local- Horizontal)" is chosen as following: the Cx axis has the same orientation as the position vector of the Leader with respect to an inertial reference frame with the origin in the attraction center; the orientation of the Cz is the same as the Leader orbit angular momentum; the Cy axis completes a right-handed frame. The angular velocity of the LVLH is given by vector ωC, which has the expression:

$$\mathbf{\dot{\omega}\_{\text{C}}} = \dot{f}\_{\text{C}} \frac{\mathbf{h}\_{\text{C}}}{\mathbf{h}\_{\text{C}}} = \frac{1}{\mathbf{r}\_{\text{C}}^{2}} \mathbf{h}\_{\text{C}} = \left[ \frac{1 + \mathbf{e}\_{\text{C}} \cos f\_{\text{C}}(\mathbf{t})}{\mathbf{p}\_{\text{C}}} \right]^{2} \mathbf{h}\_{\text{C}} \tag{64}$$

where vector r<sup>C</sup> is

$$\mathbf{r}\_{\text{C}} = \frac{p\_{\text{C}}}{1 + e\_{\text{C}} \cos f\_{\text{C}}(t)} \frac{\mathbf{r}\_{\text{C}}^{0}}{\mathbf{r}\_{\text{C}}^{0}} \tag{65}$$

where pC is the conic parameter, h<sup>C</sup> is the angular momentum of the Leader, f <sup>C</sup>ð Þt being the true anomaly and eC is the eccentricity of the Leader.

We propose dual tensors based model for the motion and the pose for the mass center of the Deputy in relation with LVLH. Both, the Leader satellite and the Deputy satellite can be considered rigid bodies.

Furthermore, the time variation of r<sup>C</sup> is:

$$\dot{\mathbf{r}}\_{\text{C}} = \frac{\mathbf{e}\_{\text{C}} |\mathbf{h}\_{\text{C}}| \sin f\_{\text{C}}(\mathbf{t})}{\mathbf{p}\_{\text{C}}} \frac{\mathbf{r}\_{\text{C}}^{0}}{\mathbf{r}\_{\text{C}}^{0}} \tag{66}$$

<sup>Q</sup>\_ <sup>¼</sup> <sup>ω</sup><sup>e</sup> <sup>Q</sup>

�JQ<sup>T</sup>ð Þ� þ <sup>ω</sup>þω<sup>c</sup> <sup>ω</sup>�ω<sup>c</sup> ωð Þ¼ t0 ω0, ω<sup>0</sup> ∈ V3 Qð Þ¼ t0 Q0, Q<sup>0</sup> ∈SO<sup>3</sup>

which has the solution Q ¼ Qð Þt , the real tensor Q being the attitude of Deputy in relation with LVLH. In Eq. (72), ω is the angular velocity of the Deputy in relation with LVLH, ω<sup>c</sup> is the angular velocity of LVLH, τ is the resulting torque of the forces applied on the Deputy in relation with is mass center, J is the inertia tensor of the Deputy in relation with its mass center. The angular velocity of Deputy in respect to LVLH at time t0 is denoted with ω<sup>0</sup> and Q<sup>0</sup> is the

Consider now the dual part of Eq. (53). Taking into account the internal structure of R, which is given by Eq. (2), after some basic algebraic calculus we obtain a second initial value problem that models the translation of the Deputy satellite mass center with respect to the LVLH

> €þ 2ω<sup>c</sup> � r\_ þ ω<sup>c</sup> � ð Þþ ω<sup>c</sup> � r ω\_ <sup>c</sup> � rþ <sup>þ</sup> <sup>μ</sup>

where μ > 0 is the gravitational parameter of the attraction center and r0, v0 represent the relative position and relative velocity vectors of the mass center of the Deputy spacecraft with

> Q ¼ R�ωCQ<sup>∗</sup> r ¼ R�ω<sup>C</sup> r<sup>∗</sup> � r<sup>c</sup>

where Q<sup>∗</sup> and r<sup>∗</sup> are the solutions of the the classical Euler fixed point problem and, respectively,

<sup>Q</sup>\_ <sup>∗</sup> <sup>¼</sup> <sup>Q</sup>∗ωe<sup>∗</sup> Jω\_ <sup>∗</sup> þ ω<sup>∗</sup> � Jω∗¼τ<sup>∗</sup>

Q∗ð Þ¼ t<sup>0</sup> Q<sup>0</sup>

<sup>0</sup> ð Þ ω<sup>0</sup> þ ωcð Þ t<sup>0</sup>

<sup>ω</sup>∗ð Þ¼ <sup>t</sup><sup>0</sup> <sup>Q</sup><sup>T</sup>

j j <sup>r</sup>cþ<sup>r</sup> <sup>3</sup> ð Þ� <sup>r</sup>cþ<sup>r</sup> <sup>μ</sup>

r3 c rc¼0

rð Þ¼ t<sup>0</sup> r0,r\_ð Þ¼ t<sup>0</sup> v<sup>0</sup>

<sup>½</sup>Q<sup>T</sup>τ�Q<sup>T</sup>ð Þ� <sup>ω</sup>þω<sup>c</sup>

(72)

95

On Six DOF Relative Orbital Motion of Satellites http://dx.doi.org/10.5772/intechopen.73563

(73)

(74)

(75)

<sup>ω</sup>\_ <sup>þ</sup> <sup>ω</sup>\_ <sup>c</sup> <sup>¼</sup> QJ�<sup>1</sup>

8 >>>>>><

>>>>>>:

orientation of Deputy in respect to LVLH at time t0.

r

8 >>>><

>>>>:

respect to LVLH at the initial moment of time t0 ≥ 0.

Based on the representation theorem 12, the following theorem results.

Theorem 14. The solutions of problems Eqs. (72) and (73) are given by

8 >>><

>>>:

reference frame:

Kepler's problem:

and

In order to a more easy to read list of notations, for t ¼ t<sup>0</sup> there will be used the followings:

$$\mathbf{u}\_{\text{C}}^{0} = \left[\frac{1 + \mathbf{e}\mathbf{c}\cos f\_{\text{C}}(\mathbf{t}\_{\text{0}})}{\mathbf{p}\_{\text{C}}}\right]^{2} \mathbf{h}\_{\text{C}} \tag{67}$$

$$\dot{\mathbf{r}}\_{\text{C}}^{0} = \frac{\mathbf{e}\_{\text{C}} |\mathbf{h}\_{\text{C}}| \sin f\_{\text{C}}(\mathbf{t}\_{0})}{\mathbf{p}\_{\text{C}}} \frac{\mathbf{r}\_{\text{C}}^{0}}{\mathbf{r}\_{\text{C}}^{0}} \tag{68}$$

where <sup>r</sup><sup>0</sup> C r0 C is the unity vector of the X-axis from LVLH.

The full-body relative orbital motion is described by Eq. (53) where the dual angular velocity of the Chief satellite is:

$$
\underline{\mathbf{u}}\underline{\mathbf{v}} = \mathbf{u}\mathbf{c} + \varepsilon(\dot{\mathbf{r}}\mathbf{c} + \mathbf{u}\mathbf{c} \times \mathbf{r}\mathbf{c})\tag{69}
$$

and the dual torque related to the mass center of Deputy satellite is:

$$\underline{\mathbf{r}} = -\frac{\mu}{\left|\mathbf{r}\_c + \mathbf{r}\right|^3} (\mathbf{r}\_c + \mathbf{r}) + \varepsilon \mathbf{r}.\tag{70}$$

The representation theorem (Theorem 12) is applied in this case using the conditions (66)–(69), the solution of the Poisson-Darboux problem (59) is:

$$\underline{\mathbf{R}}\_{-\underline{\boldsymbol{\omega}}\_{\complement}} = \left(\mathbf{I} - \varepsilon \tilde{r}\_{\complement}(t)\right) \left(\mathbf{I} - \sin f\_c^0 \frac{\tilde{\mathbf{h}}\_{\complement}}{\mathbf{h}\_{\complement}} + \left(1 - \cos f\_c^0\right) \frac{\tilde{\mathbf{h}}\_{\complement}^2}{\mathbf{h}\_{\complement}^2}\right). \tag{71}$$

In (71), we've noted hc ¼ k k hc and f 0 <sup>c</sup> ¼ f <sup>c</sup>ð Þ� t f <sup>c</sup>ð Þ t<sup>0</sup> :

Theorem 13. (Representation Theorem of the full body relative orbital motion). The solution of Eq. (53) results from the application of the tensor R�ω<sup>C</sup> from Eq. (71) to the solution of the classical dual Euler fixed point problem (60).

#### 4.1. The rotational and translational parts of the relative orbital motion

The complete solution of Eq. (53) can be recovered in two steps.

Consider first the real part of Eq. (53). This leads to an initial value problem:

$$\begin{cases} \dot{\mathbf{Q}} = \tilde{\mathbf{\omega}} \mathbf{Q} \\ \dot{\boldsymbol{\omega}} + \dot{\boldsymbol{\omega}}\_{c} = \mathbf{Q} \mathbf{J}^{-1} [\mathbf{Q}^{\mathrm{T}} \mathbf{\tau} - \mathbf{Q}^{\mathrm{T}} (\boldsymbol{\omega} + \boldsymbol{\omega}\_{c}) \times \\ \quad \times \mathbf{J} \mathbf{Q}^{\mathrm{T}} (\boldsymbol{\omega} + \boldsymbol{\omega}\_{c})] + \boldsymbol{\omega} \times \boldsymbol{\omega}\_{c} \\ \quad \quad \boldsymbol{\omega} (\mathbf{t}\_{0}) = \boldsymbol{\omega}\_{0}, \boldsymbol{\omega}\_{0} \in \mathbf{V}\_{3} \\ \mathbf{Q} (\mathbf{t}\_{0}) = \mathbf{Q}\_{0'} \mathbf{Q}\_{0} \in \mathbf{S} \mathbf{0}\_{3} \end{cases} \tag{72}$$

which has the solution Q ¼ Qð Þt , the real tensor Q being the attitude of Deputy in relation with LVLH. In Eq. (72), ω is the angular velocity of the Deputy in relation with LVLH, ω<sup>c</sup> is the angular velocity of LVLH, τ is the resulting torque of the forces applied on the Deputy in relation with is mass center, J is the inertia tensor of the Deputy in relation with its mass center. The angular velocity of Deputy in respect to LVLH at time t0 is denoted with ω<sup>0</sup> and Q<sup>0</sup> is the orientation of Deputy in respect to LVLH at time t0.

Consider now the dual part of Eq. (53). Taking into account the internal structure of R, which is given by Eq. (2), after some basic algebraic calculus we obtain a second initial value problem that models the translation of the Deputy satellite mass center with respect to the LVLH reference frame:

$$\begin{cases} \ddot{\mathbf{r}} + 2\boldsymbol{\omega}\_c \times \dot{\mathbf{r}} + \boldsymbol{\omega}\_c \times (\boldsymbol{\omega}\_c \times \mathbf{r}) + \dot{\boldsymbol{\omega}}\_c \times \mathbf{r} + \\ \\ \quad + \frac{\mu}{\left| \mathbf{r}\_c + \mathbf{r} \right|^3} (\mathbf{r}\_c + \mathbf{r}) - \frac{\mu}{r\_c^3} \mathbf{r}\_c = 0 \\ \quad \quad \mathbf{r}(t\_0) = \mathbf{r}\_0, \dot{\mathbf{r}}(t\_0) = \mathbf{v}\_0 \end{cases} \tag{73}$$

where μ > 0 is the gravitational parameter of the attraction center and r0, v0 represent the relative position and relative velocity vectors of the mass center of the Deputy spacecraft with respect to LVLH at the initial moment of time t0 ≥ 0.

Based on the representation theorem 12, the following theorem results.

Theorem 14. The solutions of problems Eqs. (72) and (73) are given by

$$\begin{aligned} \mathbf{Q} &= \mathbf{R}\_{-\omega \mathbf{c}} \mathbf{Q}\_{\*} \\ \mathbf{r} &= \mathbf{R}\_{-\omega \mathbf{c}} \mathbf{r}\_{\*} - \mathbf{r}\_{\mathbf{c}} \end{aligned} \tag{74}$$

where Q<sup>∗</sup> and r<sup>∗</sup> are the solutions of the the classical Euler fixed point problem and, respectively, Kepler's problem:

$$\begin{cases} \begin{aligned} \dot{\mathbf{Q}}\_{\*} &= \mathbf{Q}\_{\*} \widetilde{\boldsymbol{\omega}}\_{\*} \\ \int \dot{\boldsymbol{\omega}}\_{\*} + \boldsymbol{\omega}\_{\*} \times \mathbf{J} \boldsymbol{\omega}\_{\*} &= \mathbf{\tau}\_{\*} \\ \boldsymbol{\omega}\_{\*}(t\_{0}) &= \mathbf{Q}\_{0}^{\mathrm{T}} (\boldsymbol{\omega}\_{0} + \boldsymbol{\omega}\_{c}(t\_{0})) \\ \mathbf{Q}\_{\*}(t\_{0}) &= \mathbf{Q}\_{0} \end{aligned} \end{cases} \tag{75}$$

and

We propose dual tensors based model for the motion and the pose for the mass center of the Deputy in relation with LVLH. Both, the Leader satellite and the Deputy satellite can be

> <sup>r</sup>\_<sup>C</sup> <sup>¼</sup> eCj j <sup>h</sup><sup>C</sup> sin <sup>f</sup> <sup>C</sup>ð Þ<sup>t</sup> pC

In order to a more easy to read list of notations, for t ¼ t<sup>0</sup> there will be used the followings:

<sup>C</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> eC cos <sup>f</sup> <sup>C</sup>ð Þ t0 pC � �<sup>2</sup>

<sup>C</sup> <sup>¼</sup> eCj j <sup>h</sup><sup>C</sup> sin <sup>f</sup> <sup>C</sup>ð Þ t0 pC

The full-body relative orbital motion is described by Eq. (53) where the dual angular velocity of

The representation theorem (Theorem 12) is applied in this case using the conditions (66)–(69),

<sup>c</sup> ¼ f <sup>c</sup>ð Þ� t f <sup>c</sup>ð Þ t<sup>0</sup> :

Theorem 13. (Representation Theorem of the full body relative orbital motion). The solution of Eq. (53) results from the application of the tensor R�ω<sup>C</sup> from Eq. (71) to the solution of the classical dual

0 c ehC hc

þ 1 � cos f

!

ω0

r\_ 0

is the unity vector of the X-axis from LVLH.

and the dual torque related to the mass center of Deputy satellite is:

the solution of the Poisson-Darboux problem (59) is:

<sup>R</sup>�ω<sup>C</sup> <sup>¼</sup> �

In (71), we've noted hc ¼ k k hc and f

Euler fixed point problem (60).

<sup>τ</sup> ¼ � <sup>μ</sup>

<sup>I</sup> � <sup>ε</sup>erCð ÞÞ <sup>t</sup> <sup>I</sup> � sin <sup>f</sup>

0

4.1. The rotational and translational parts of the relative orbital motion

Consider first the real part of Eq. (53). This leads to an initial value problem:

The complete solution of Eq. (53) can be recovered in two steps.

r0 C r0 C

> r0 C r0 C

ω<sup>C</sup> ¼ ω<sup>C</sup> þ εð Þ r\_<sup>C</sup> þ ω<sup>C</sup> � r<sup>C</sup> (69)

j j <sup>r</sup><sup>c</sup> <sup>þ</sup> <sup>r</sup> <sup>3</sup> ð Þþ <sup>r</sup><sup>c</sup> <sup>þ</sup> <sup>r</sup> <sup>ε</sup>τ: (70)

0 c � �eh<sup>C</sup>

2

: (71)

hc 2 (66)

(68)

h<sup>C</sup> (67)

considered rigid bodies.

94 Space Flight

where <sup>r</sup><sup>0</sup> C r0 C

the Chief satellite is:

Furthermore, the time variation of r<sup>C</sup> is:

$$\begin{cases} \ddot{\mathbf{r}}\_{\*} + \frac{\mu}{r\_{\*}^{3}} \mathbf{r}\_{\*} = \mathbf{0};\\ \mathbf{r}\_{\*}(t\_{0}) = \mathbf{r}\_{c}^{0} + \mathbf{r}\_{0};\\ \dot{\mathbf{r}}\_{\*}(t\_{0}) = \dot{\mathbf{r}}\_{\mathbb{C}}^{0} + \mathbf{v}\_{0} + \mathbf{u}\_{\mathbb{C}}^{0} \times \left(\mathbf{r}\_{\mathbb{C}}^{0} + \mathbf{r}\_{0}\right) \end{cases} \tag{76}$$

Using the above results, the free of coordinate state equation of the rigid body motion in

On Six DOF Relative Orbital Motion of Satellites http://dx.doi.org/10.5772/intechopen.73563 97

The results are applied in order to offer a coupled (rotational and translational motion) state equation and a representation theorem for the onboard complete solution of full body relative orbital motion problem. The obtained results interest the domains of the spacecraft formation

flying, rendezvous operation, autonomous mission and control theory.

arbitrary non-inertial frame is obtained.

Nomenclature

a real number a dual number

a real vector a dual vector A real tensor A dual tensor

V<sup>3</sup> real vectors set V<sup>3</sup> dual vectors set

f <sup>c</sup> true anomaly

pc conic parameter

L V3;V<sup>3</sup> ð Þ dual tensor set

<sup>q</sup><sup>b</sup> real quaternion <sup>q</sup><sup>b</sup> dual quaternion R real numbers set R dual numbers set

SO<sup>3</sup> orthogonal real tensors set

SO<sup>3</sup> orthogonal dual tensor set

<sup>3</sup> time depending real vectorial functions

<sup>3</sup> time depending dual vectorial functions

h<sup>c</sup> specific angular momentum of the leader satellite

<sup>e</sup><sup>a</sup> skew-symmetric dual tensor corresponding to the dual vector <sup>a</sup>

V<sup>R</sup>

V<sup>R</sup>

where

$$\mathbf{R}\_{-\omega\mathbf{c}} = \mathbf{I} - \sin f\_c^0 \frac{\tilde{\mathbf{h}}\_{\mathbb{C}}}{\left| \mathbf{h}\_{\mathbb{C}} \right|} + \left( 1 - \cos f\_c^0 \right) \frac{\tilde{\mathbf{h}}\_{\mathbb{C}}}{\left| \mathbf{h}\_{\mathbb{C}} \right|^2} \tag{77}$$

and r<sup>c</sup> is given by Eq. (65).

Remark 7: The problems (72) and (73) are coupled because, in general case, the torque τ depends of the position vector r.

The relative velocity of the translation motion may be computed as:

$$\mathbf{v} = \mathbf{R}\_{-\omega \mathbf{c}} \dot{\mathbf{r}}\_{\*} - \widehat{\boldsymbol{\omega}\_{c}} \mathbf{R}\_{-\omega \mathbf{c}} \mathbf{r}\_{\*} - \frac{\mathbf{c}\_{c} |\mathbf{h}\_{c}| \sin f\_{c}(t)}{p\_{c}} \frac{\mathbf{r}\_{c}^{0}}{r\_{c}^{0}} \tag{78}$$

This result shows a very interesting property of the translational part of the relative orbital motion problem (73). We have proven that this problem is super-integrable by reducing it to the classic Kepler problem [11, 12, 31, 32]. The solution of the translational part of the relative orbital motion problem is expressed thus:

$$\mathbf{r} = \mathbf{r}(t, t\_0, \mathbf{r}\_0, \mathbf{v}\_0); \mathbf{v} = \mathbf{v}(t, t\_0, \mathbf{r}\_0, \mathbf{v}\_0) \tag{79}$$

The exact closed form, free of coordinate, solution of the translational motion can be found in [11, 12, 31, 32, 34].

#### 5. Conclusions

The chapter proposes a new method for the determination of the onboard complete solution to the full-body relative orbital motion problem.

Therefore, the isomorphism between the Lie group of the rigid displacements SE<sup>3</sup> and the Lie group of the orthogonal dual tensors SO<sup>3</sup> is used. It is obtained a Poisson-Darboux like problem written in the Lie algebra of the group SO3, an algebra that is isomorphic with the Lie algebra of the dual vectors. Different vectorial and non-vectorial parameterizations (obtained with n-th order Cayley-like transforms) permit the reduction of the Poisson-Darboux problem in dual Lie algebra to the simpler problems in the space of the dual vectors or dual quaternions.

Using the above results, the free of coordinate state equation of the rigid body motion in arbitrary non-inertial frame is obtained.

The results are applied in order to offer a coupled (rotational and translational motion) state equation and a representation theorem for the onboard complete solution of full body relative orbital motion problem. The obtained results interest the domains of the spacecraft formation flying, rendezvous operation, autonomous mission and control theory.

#### Nomenclature

r €<sup>∗</sup> <sup>þ</sup> <sup>μ</sup> r3 ∗ r<sup>∗</sup> ¼ 0;

0

r\_∗ð Þ¼ t<sup>0</sup> r\_

R�ω<sup>C</sup> ¼ I � sin f

The relative velocity of the translation motion may be computed as:

8 >>><

>>>:

where

96 Space Flight

and r<sup>c</sup> is given by Eq. (65).

[11, 12, 31, 32, 34].

5. Conclusions

quaternions.

depends of the position vector r.

orbital motion problem is expressed thus:

the full-body relative orbital motion problem.

<sup>r</sup>∗ð Þ¼ <sup>t</sup><sup>0</sup> <sup>r</sup><sup>0</sup>

0 c ehC j j h<sup>c</sup>

<sup>C</sup> <sup>þ</sup> <sup>v</sup><sup>0</sup> <sup>þ</sup> <sup>ω</sup><sup>0</sup>

Remark 7: The problems (72) and (73) are coupled because, in general case, the torque τ

<sup>v</sup> <sup>¼</sup> <sup>R</sup>�ω<sup>C</sup> <sup>r</sup>\_∗�ωf<sup>c</sup>R�ω<sup>C</sup> <sup>r</sup>∗� ec <sup>h</sup><sup>c</sup> j j sin <sup>f</sup> <sup>c</sup>ð Þ<sup>t</sup>

This result shows a very interesting property of the translational part of the relative orbital motion problem (73). We have proven that this problem is super-integrable by reducing it to the classic Kepler problem [11, 12, 31, 32]. The solution of the translational part of the relative

The exact closed form, free of coordinate, solution of the translational motion can be found in

The chapter proposes a new method for the determination of the onboard complete solution to

Therefore, the isomorphism between the Lie group of the rigid displacements SE<sup>3</sup> and the Lie group of the orthogonal dual tensors SO<sup>3</sup> is used. It is obtained a Poisson-Darboux like problem written in the Lie algebra of the group SO3, an algebra that is isomorphic with the Lie algebra of the dual vectors. Different vectorial and non-vectorial parameterizations (obtained with n-th order Cayley-like transforms) permit the reduction of the Poisson-Darboux problem in dual Lie algebra to the simpler problems in the space of the dual vectors or dual

<sup>c</sup> þ r0;

<sup>C</sup> � <sup>r</sup><sup>0</sup>

þ 1 � cos f

<sup>C</sup> þ r<sup>0</sup> � �

0 c � � eh<sup>C</sup>

pc

2

r0 c r0 c

r ¼ rð Þ t; t0;r0; v<sup>0</sup> ; v¼vð Þ t; t0;r0; v<sup>0</sup> (79)

j j <sup>h</sup><sup>c</sup> <sup>2</sup><sup>Þ</sup> (77)

(76)

(78)



#### Author details

Daniel Condurache

Address all correspondence to: daniel.condurache@tuiasi.ro

Technical University of Iasi, Iaşi, Romania

#### References

[1] Condurache D, Burlacu A. Onboard exact solution to the full-body relative orbital motion problem. AIAA Journal of Guidance, Control, and Dynamics. 2016;39(12):2638-2648. DOI: 10.2514/1.G000316

[9] Sinclair AJ, Hurtado JE, Junkins JL. Application of the Cayley form to general spacecraft motion. Journal of Guidance, Control, and Dynamics. 2006;29(2):368-373. DOI: 10.2514/

On Six DOF Relative Orbital Motion of Satellites http://dx.doi.org/10.5772/intechopen.73563 99

[10] Yamanaka K, Ankersen F. New state transition matrix for relative motion on an arbitrary elliptical orbit. Journal of Guidance, Control, and Dynamics. 2002;25(1):60-66. DOI:

[11] Condurache D, Martinusi V. Kepler's problem in rotating reference frames. Part 1: Prime integrals, vectorial regularization. Journal of Guidance, Control, and Dynamics. 2007;

[12] Condurache D, Martinusi V. Kepler's problem in rotating reference frames. Part 2: Relative orbital motion. Journal of Guidance, Control, and Dynamics. 2007;30(1):201-213.

[13] Condurache D, Martinusi V. Relative spacecraft motion in a central force field. Journal of

[14] Condurache D, Martinusi V. A Novel Hypercomplex Solution to Kepler's Problem. Vol. 19. Publications of the Astronomy Department of the Eötvös University (PADEU); 2007.

[15] Condurache D, Martinusi V. Hypercomplex eccentric anomaly in the unified solution of the relative orbital motion. Advances in the Astronautical Sciences. 2010;135:281-300.

[16] Angeles J. The application of dual algebra to kinematic analysis. Computational Methods

[17] Condurache D, Burlacu A. Dual Lie algebra representations of the rigid body motion. AIAA/AAS Astrodynamics Specialist Conference, AIAA Paper, San Diego; 2014. pp.

[18] Condurache D, Burlacu A. Dual tensors based solutions for rigid body motion parameterization. Mechanism and Machine Theory. 2014;74:390-412. DOI: 10.1016/j.mechmachtheory.

[19] Condurache D, Burlacu A. Recovering dual Euler parameters from feature-based representation of motion. Advances in Robot Kinematics. 2014:295-305. DOI: 10.1007/978-3-

[20] Pennestri E, Valentini PP. Dual quaternions as a tool for rigid body motion analysis: A tutorial with an application to biomechanics. The Archive of Mechanical Engineering. 2010;

[21] Pennestri E, Valentini PP. Linear Dual Algebra Algorithms and their Application to Kinematics, Multibody Dynamics: Computational Methods and Applications. Vol. 122009.

in Mechanical Systems. 1998;161:3-32. DOI: 10.1007/978-3-662-03729-4\_1

Guidance, Control, and Dynamics. 2007;30(3):873-876. DOI: 10.2514/1.26361

1.9910

10.2514/2.4875

DOI: 10.2514/1.20470

ISSN: 0065-3438

2013.12.016

319-06698-1\_31

30(1):192-200. DOI: 10.2514/1.20466

pp. 201-213. ISBN: 963 463 557

2014–4347. DOI: 10.2514/6.2014-4347

LVII:184-205. DOI: 10.2478/v10180-010-0010-2

pp. 207-229. DOI: 10.1007/978-1-4020-8829-2\_11


[9] Sinclair AJ, Hurtado JE, Junkins JL. Application of the Cayley form to general spacecraft motion. Journal of Guidance, Control, and Dynamics. 2006;29(2):368-373. DOI: 10.2514/ 1.9910

SO<sup>R</sup>

98 Space Flight

SO<sup>R</sup>

<sup>3</sup> time depending real tensorial functions

<sup>3</sup> time depending dual tensorial functions

Address all correspondence to: daniel.condurache@tuiasi.ro

[1] Condurache D, Burlacu A. Onboard exact solution to the full-body relative orbital motion problem. AIAA Journal of Guidance, Control, and Dynamics. 2016;39(12):2638-2648.

[2] Condurache D, Burlacu A. On six D.O.F relative orbital motion parameterization using rigid bases of dual vectors. Advances in the Astronautical Sciences. 2013;150:2293-2312 [3] Filipe N, Tsiotras P. Adaptive model-independent tracking of rigid body position and attitude motion with mass and inertia matrix identification using dual quaternions. AIAA Guidance, Navigation, and Control (GNC) Conference, Boston, Massachusetts; 2013.

[4] Segal S, Gurfil P. Effect of kinematic rotation-translation coupling on relative spacecraft translational dynamics. Journal of Guidance, Control, and Dynamics. 2009;32(3):1045-

[5] Condurache D. Poisson-Darboux problems's extended in dual Lie algebra. AAS/AIAA

[6] Alfriend K, Vadali S, Gurfil P, How J, Breger L. Spacecraft Formation Flying. New York:

[7] Carter TE. New form for the optimal rendezvous equations near a Keplerian orbit. Journal of Guidance, Control, and Dynamics. 1990;13(1):183-186. DOI: 10.2514/3.20533 [8] Gim D-W, Alfriend KT. State transition matrix of relative motion for the perturbed noncircular reference orbit. Journal of Guidance, Control, and Dynamics. 2003;26(6):956-

Astrodynamics Specialist Conference, Stevenson, WA, USA; 2017

Elsevier; 1999. pp. 227-232. DOI: 10.1016/B978-0-7506-8533-7.00214-1

U unit quaternions set

Author details

Daniel Condurache

References

U unit dual quaternions set

Technical University of Iasi, Iaşi, Romania

DOI: 10.2514/1.G000316

DOI: 10.2514/6.2013-5173

1050. DOI: 10.2514/1.39320

971. DOI: 10.2514/2.6924


[22] Fischer I. Dual-Number Methods in Kinematics. Statics and Dynamics: CRC Press; 1998. pp. 1-9. ISBN: 9780849391156

**Chapter 6**

Provisional chapter

**Consensus-Based Attitude Maneuver of Multi-**

Consensus-Based Attitude Maneuver of Multi-spacecraft

DOI: 10.5772/intechopen.71506

Some space missions involve cooperative multi-vehicle teams, for such purposes as interferometry and optimal sensor coverage, for example, NASA Terrestrial Planet Finder Mission. Cooperative navigation introduces extra constraints of exclusion zones between the spacecraft to protect them from damaging each other. This is in addition to external exclusion constraints introduced by damaging or blinding celestial objects. This work presents a quaternion-based attitude consensus protocol, using the communication topology of the team of spacecraft. The resulting distributed Laplacians of their communication graph are applied by semidefinite programming (SDP), to synthesize a series of time-varying optimal stochastic matrices. The matrices are used to generate various cooperative attitude maneuvers from the initial attitudes of the spacecraft. Exclusion constraints are satisfied by quaternion-based quadratically constrained attitude control (Q-CAC), where both static and dynamic exclusion zones are identified every time step, expressed as time-varying linear matrix inequalities (LMI) and solved

Keywords: attitude maneuvre, consensus, exclusion, optimization, LMI

Some current space missions already demanded the deployment of teams of spacecraft which cooperate synergistically for such purposes as interferometry and sensor coverage [1, 2]; and many future missions will. Activities such as interferometry and sensor coverage require cooperative attitude control (AC)—the process of making a team of spacecraft, for example, satellites to point toward a specific direction of interest. This makes attitude control an essential part of space missions [3]. Apart from spacecraft, AC is also important in the navigation of aircraft and robots; therefore, it has been studied extensively in the literature,

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

**spacecraft with Exclusion Constraints**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.71506

by semidefinite programming.

with Exclusion Constraints

Innocent Okoloko

Innocent Okoloko

Abstract

1. Introduction

for example [4–11].


#### **Consensus-Based Attitude Maneuver of Multispacecraft with Exclusion Constraints** Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints

DOI: 10.5772/intechopen.71506

Innocent Okoloko Innocent Okoloko

[22] Fischer I. Dual-Number Methods in Kinematics. Statics and Dynamics: CRC Press; 1998.

[23] Condurache D, Burlacu A. Orthogonal dual tensor method for solving the AX= XB sensor calibration problem. Mechanism and Machine Theory. 2016;104:382-404. DOI: 10.1016/j.

[25] Tsiotras P, Junkins JL, Schaub H. Higher-order Cayley transforms with applications to attitude representations. Journal of Guidance, Control, and Dynamics. 1997;20(3):528-534

[26] Vasilescu FH. Quaternionic Cayley transform. Journal of Functional Analysis. 1999;164:

[28] Darboux G. Lecons sur la Theorie Generale des Surfaces et les Applications Geometriques du Calcul Infinitesimal. Paris: Gauthier-Villars; 1887. pp. 175-179. ark:/13960/t2h70912j

[29] Condurache D, Burlacu A. Fractional order Cayley transforms for dual quaternions based pose representation. Advances in the Astronautical Sciences. 2016;156:1317-1339

[30] Brodsky V, Shoham M. Dual numbers representation of rigid body dynamics. Mechanism and Machine Theory. 1999;34(5):693-718. DOI: 10.1016/S0094-114X(98)00049-4

[31] Condurache D, Martinusi V. Foucault pendulum-like problems: A tensorial approach. International Journal of Non-Linear Mechanics. 2008;43(8):743-760. DOI: 10.1016/j.ijnonlin

[32] Condurache D, Martinusi V. Exact solution to the relative orbital motion in eccentric orbits. Solar System Research. 2009;43(1):41-52. DOI: 10.1134/S0038094609010043

[33] Tanygin S. Attitude parameterizations as higher-dimensional map projections. Journal of

[34] Condurache D, Martinusi V. Quaternionic exact solution to the relative orbital motion problem. Journal of Guidance, Control, and Dynamics. 2010;33(4):1035-1047. DOI: 10.

[35] Condurache D, Martinusi V. Exact solution to the relative orbital motion in a central force field, 2nd international symposium on systems and control in aeronautics and astronau-

[36] Gurfil P, Kasdin JN. Nonlinear modeling of spacecraft relative motion in the configuration space. Journal of Guidance, Control, and Dynamics. 2004;27(1):154-157. DOI: 10.2514/

Guidance, Control, and Dynamics. 2012;35(1):13-24. DOI: 10.2514/1.54085

tics, Shenzhen, PRC; 2008. DOI: 10.1109/ISSCAA.2008.4776296

[27] Selig JM. Cayley Maps for SE(3). 12th IFToMM World Congress, Besancon, 2007

[24] Angeles J. Fundamentals of Robotic Mechanical Systems. Springer; 2014

pp. 1-9. ISBN: 9780849391156

100 Space Flight

mechmachtheory.2016.06.002

134-162. MR 2000d:47015

mec.2008.03.009

2514/1.47782

1.9343

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.71506

#### Abstract

Some space missions involve cooperative multi-vehicle teams, for such purposes as interferometry and optimal sensor coverage, for example, NASA Terrestrial Planet Finder Mission. Cooperative navigation introduces extra constraints of exclusion zones between the spacecraft to protect them from damaging each other. This is in addition to external exclusion constraints introduced by damaging or blinding celestial objects. This work presents a quaternion-based attitude consensus protocol, using the communication topology of the team of spacecraft. The resulting distributed Laplacians of their communication graph are applied by semidefinite programming (SDP), to synthesize a series of time-varying optimal stochastic matrices. The matrices are used to generate various cooperative attitude maneuvers from the initial attitudes of the spacecraft. Exclusion constraints are satisfied by quaternion-based quadratically constrained attitude control (Q-CAC), where both static and dynamic exclusion zones are identified every time step, expressed as time-varying linear matrix inequalities (LMI) and solved by semidefinite programming.

Keywords: attitude maneuvre, consensus, exclusion, optimization, LMI

#### 1. Introduction

Some current space missions already demanded the deployment of teams of spacecraft which cooperate synergistically for such purposes as interferometry and sensor coverage [1, 2]; and many future missions will. Activities such as interferometry and sensor coverage require cooperative attitude control (AC)—the process of making a team of spacecraft, for example, satellites to point toward a specific direction of interest. This makes attitude control an essential part of space missions [3]. Apart from spacecraft, AC is also important in the navigation of aircraft and robots; therefore, it has been studied extensively in the literature, for example [4–11].

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

Basically, AC is a challenging problem, which becomes more challenging when multiple spacecraft are involved, in highly dynamic environments, and subject to external constraints such as blinding celestial objects such as the sun or some bright stars, which can damage onboard sensitive instruments. In addition, because of the close packing of spacecraft in a team, each of which has protruding appendages (e.g. thrusters and antennae), they must be careful with each other when changing attitude, in order to avoid collision with each other. When there is such a team of networked spacecraft which can communicate, then consensus theory based on graph Laplacians can be applied to achieve cooperation among them [12, 13].

qi

norm constraints.

is achieved.

(t0), of a set of communicating spacecraft SCi, generate a sequence of attitude consensus trajectories that drive the team to a consensus attitude q(tf) while satisfying avoidance and

Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints

http://dx.doi.org/10.5772/intechopen.71506

103

The problem stated above consists of two parts: consensus and avoidance. For the consensus problem, it is desired to drive the attitudes of all SCi to a collective consensus attitude or to various formation attitudes. Consensus attitude means that each SCi should eventually point to the same direction, which is the average of the initial quaternions. Formation attitudes means SCi should finally point to various patterns, for example, each spacecraft can point at 5<sup>o</sup> away from each other about the z-axis. This we developed by introducing relative offset quaternions in the consensus framework. The second problem, avoidance constraints, is also important, because SCi usually have appendages, for example, some SCi have thrusters that emit hot plumes (plume impingement), and some have instruments that can be damaged by blinding

However, the ordinary consensus protocol violates the non-linearity of quaternion kinematics and the quaternion norm preserving requirement and therefore cannot be applied directly with quaternion dynamics. Also, the protocol ordinarily does not solve the problem of collision avoidance in adversarial situations. Thus, this chapter consists of aspects of our previous works [7–10], where we developed a consensus theory of quaternions, augmented with Q-CAC-based collision avoidance mechanisms. We employed an optimization approach and cast the problems as a semidefinite program (SDP), augmented with some convex quadratic constraints (avoidance), written as linear matrix inequalities (LMI). The quaternion consensus protocol computes consensus attitude trajectories each time step, and the Q-CAC avoidance procedure decides which of the computed trajectories are safe to follow or not. Unsafe trajectories are discarded, and a new set of quaternion vectors that avoid collision is generated. The cycle repeats until consensus

To understand the avoidance (exclusion) problem, let us illustrate with a simpler single-SCi single-obstacle scenario as shown in Figure 1. In the figure, the SCi must avoid (exclude) the

Figure 1. Constrained attitude control problem for a single-sc single-exclusion scenario. SCi must avoid (exclude) the Sun

celestial objects or by the appendage of another team member.

Sun while rotating a photosensitive instrument from q<sup>0</sup> to qf.

while rotating a photosensitive instrument from q<sup>0</sup> to qf.

The most common method of representing spacecraft attitude dynamics is by unit quaternions, mainly because quaternions do not encounter the singularities associated with other representations such as Euler angles and the Modified Rodriques Parameters (MRP). However, the non-linearity of quaternion dynamics makes it difficult to apply Laplacian-like dynamics directly to quaternions.

We shall now consider some previous work on constrained attitude control (CAC). A brief survey of the main method attitude representation is in [4]. Ref. [5] considers quadratically constrained attitude control (Q-CAC), where the exclusion problems are formulated as a quadratic optimization problem and solved using linear matrix inequalities (LMIs) and semidefinite programming (SDP). It was solved for a single-spacecraft single obstacle in [5] and for two spacecraft in [6]. In [7] an attempt was made to extend [5, 6] to more than two spacecraft and obstacles. In [7–10], was extended to multiple spacecraft multiple obstacles in different coordinate frames (as the case of real spacecraft will be). An attempt was made in [11] to reduce the control torques required for effective attitude stabilization from three to two. This is applicable to underactuated spacecraft. [12] applies a consensus-based approach to distributed attitude alignment of a team of communicating spacecraft flying in formation, while [14] applies a Laplacian-based protocol to leader-follower attitude control of a team of spacecraft using the modified Rodriquez parameters.

Among the plethora of AC algorithms, only our works [7–10] apply consensus theory directly to quaternions, and only [5–10] tackle the problem of avoidance constraints. In addition, among the works [5–10] only [8–10] were developed for spacecraft in different coordinate frames, which has direct practical implementation. The contributions of this chapter are therefore aspects of our previous works [7–10], which include the following: (i) the development of a quaternion consensus protocol, (ii) incorporating dynamic avoidance constraints into the consensus framework using Q-CAC, (iii) mathematical convergence analysis for the quaternion-based consensus framework and (iv) solving the problem for the realistic scenario of multiple spacecraft in different coordinate frames, thus making it more suitable for practical implementation.

Note: the words obstacle, avoidance, exclusion and exclusion vector may be used interchangeably in this chapter. Table 1 lists frequently used notation in this chapter.

#### 2. Problem statement

The problem of multi-spacecraft attitude control with avoidance constraints can be stated as follows. Given the initial positions xi (t0)∈ R<sup>3</sup> i = 1⋯n, initial attitudes represented by quaternions qi (t0), of a set of communicating spacecraft SCi, generate a sequence of attitude consensus trajectories that drive the team to a consensus attitude q(tf) while satisfying avoidance and norm constraints.

Basically, AC is a challenging problem, which becomes more challenging when multiple spacecraft are involved, in highly dynamic environments, and subject to external constraints such as blinding celestial objects such as the sun or some bright stars, which can damage onboard sensitive instruments. In addition, because of the close packing of spacecraft in a team, each of which has protruding appendages (e.g. thrusters and antennae), they must be careful with each other when changing attitude, in order to avoid collision with each other. When there is such a team of networked spacecraft which can communicate, then consensus theory based on graph Laplacians can be applied to achieve cooperation among them [12, 13]. The most common method of representing spacecraft attitude dynamics is by unit quaternions, mainly because quaternions do not encounter the singularities associated with other representations such as Euler angles and the Modified Rodriques Parameters (MRP). However, the non-linearity of quaternion dynamics makes it difficult to apply Laplacian-like dynamics directly to quaternions. We shall now consider some previous work on constrained attitude control (CAC). A brief survey of the main method attitude representation is in [4]. Ref. [5] considers quadratically constrained attitude control (Q-CAC), where the exclusion problems are formulated as a quadratic optimization problem and solved using linear matrix inequalities (LMIs) and semidefinite programming (SDP). It was solved for a single-spacecraft single obstacle in [5] and for two spacecraft in [6]. In [7] an attempt was made to extend [5, 6] to more than two spacecraft and obstacles. In [7–10], was extended to multiple spacecraft multiple obstacles in different coordinate frames (as the case of real spacecraft will be). An attempt was made in [11] to reduce the control torques required for effective attitude stabilization from three to two. This is applicable to underactuated spacecraft. [12] applies a consensus-based approach to distributed attitude alignment of a team of communicating spacecraft flying in formation, while [14] applies a Laplacian-based protocol to leader-follower attitude control of a team of spacecraft

Among the plethora of AC algorithms, only our works [7–10] apply consensus theory directly to quaternions, and only [5–10] tackle the problem of avoidance constraints. In addition, among the works [5–10] only [8–10] were developed for spacecraft in different coordinate frames, which has direct practical implementation. The contributions of this chapter are therefore aspects of our previous works [7–10], which include the following: (i) the development of a quaternion consensus protocol, (ii) incorporating dynamic avoidance constraints into the consensus framework using Q-CAC, (iii) mathematical convergence analysis for the quaternion-based consensus framework and (iv) solving the problem for the realistic scenario of multiple spacecraft in

different coordinate frames, thus making it more suitable for practical implementation.

in this chapter. Table 1 lists frequently used notation in this chapter.

Note: the words obstacle, avoidance, exclusion and exclusion vector may be used interchangeably

The problem of multi-spacecraft attitude control with avoidance constraints can be stated as

(t0)∈ R<sup>3</sup> i = 1⋯n, initial attitudes represented by quaternions

using the modified Rodriquez parameters.

102 Space Flight

2. Problem statement

follows. Given the initial positions xi

The problem stated above consists of two parts: consensus and avoidance. For the consensus problem, it is desired to drive the attitudes of all SCi to a collective consensus attitude or to various formation attitudes. Consensus attitude means that each SCi should eventually point to the same direction, which is the average of the initial quaternions. Formation attitudes means SCi should finally point to various patterns, for example, each spacecraft can point at 5<sup>o</sup> away from each other about the z-axis. This we developed by introducing relative offset quaternions in the consensus framework. The second problem, avoidance constraints, is also important, because SCi usually have appendages, for example, some SCi have thrusters that emit hot plumes (plume impingement), and some have instruments that can be damaged by blinding celestial objects or by the appendage of another team member.

However, the ordinary consensus protocol violates the non-linearity of quaternion kinematics and the quaternion norm preserving requirement and therefore cannot be applied directly with quaternion dynamics. Also, the protocol ordinarily does not solve the problem of collision avoidance in adversarial situations. Thus, this chapter consists of aspects of our previous works [7–10], where we developed a consensus theory of quaternions, augmented with Q-CAC-based collision avoidance mechanisms. We employed an optimization approach and cast the problems as a semidefinite program (SDP), augmented with some convex quadratic constraints (avoidance), written as linear matrix inequalities (LMI). The quaternion consensus protocol computes consensus attitude trajectories each time step, and the Q-CAC avoidance procedure decides which of the computed trajectories are safe to follow or not. Unsafe trajectories are discarded, and a new set of quaternion vectors that avoid collision is generated. The cycle repeats until consensus is achieved.

To understand the avoidance (exclusion) problem, let us illustrate with a simpler single-SCi single-obstacle scenario as shown in Figure 1. In the figure, the SCi must avoid (exclude) the Sun while rotating a photosensitive instrument from q<sup>0</sup> to qf.

Figure 1. Constrained attitude control problem for a single-sc single-exclusion scenario. SCi must avoid (exclude) the Sun while rotating a photosensitive instrument from q<sup>0</sup> to qf.

Let v<sup>I</sup> cami ð Þ<sup>t</sup> denote the unit camera vector in <sup>F</sup><sup>I</sup> SCi corresponding to the SCi's attitude qi (as defined in Table 1), and let vI obsi ð Þt be the attitude quaternion of the obstacle to be avoided (in this case the Sun). Exclusion requires the time evolution of camera vector v<sup>I</sup> cami from vI cami ð Þ t<sup>0</sup> to vI cami tf to avoid vI obsi ð Þt all times with a minimum angular separation of ∅. The requirement is.

$$\Theta(t) \succeq \bigotimes \tag{1}$$

3. Mathematical background

Table 1. Frequently used notations in this chapter.

3.1. Quaternion-based rotational dynamics

[15]. The quaternion is a four-element vector:

chapter. More comprehensive study and analysis are in [10].

Notation Meaning

FI SCi

F<sup>B</sup> SCi

vB

vI

vI

vB

vI

(xij)

A~ Cone avoidance constraint matrix R<sup>i</sup> Rotation matrix corresponding to qi

obsi Vector of obstacle in <sup>F</sup><sup>B</sup>

obsi Vector of obstacle in <sup>F</sup><sup>I</sup>

cami Vector of the SCi's camera in F<sup>B</sup>

cami Vector of the SCi's camera in F<sup>I</sup>

⊗ Kronecker multiplication operator ⊙ Quaternion multiplication operator ⊖ Quaternion difference operator

obsi :<sup>j</sup> Vector of the j

t<sup>0</sup> Initial time tf Final time

x<sup>i</sup> Position vector of SCi, SCi

x Stacked vector of n position vectors

off Offset vector between i and j xoff Stacked vector of n offset vectors

Fixed coordinate (Inertial) frame with origin at SCi's center

SCi

SCi

th obstacle in F<sup>I</sup>

Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints

Rotational coordinate (Body) frame with origin at SCi's center

http://dx.doi.org/10.5772/intechopen.71506

105

SCi

SCi

SCi

In this section, we shall briefly consider the two basic mathematical theories relevant to this

<sup>C</sup> The consensus space for <sup>q</sup>, <sup>C</sup> <sup>¼</sup> <sup>q</sup>jq<sup>1</sup> <sup>¼</sup> <sup>q</sup><sup>2</sup> <sup>¼</sup>; <sup>⋯</sup>; <sup>¼</sup> <sup>q</sup><sup>n</sup>

Because quaternions are free from the problems of singularities inherent in Euler angles and most other ways of representing rotations, it is convenient to use unit quaternions to represent the attitude of a rigid body rotating in three-dimensional space (such as spacecraft or satellite)

or

$$\begin{aligned} \boldsymbol{v}\_{cam\_i}^l(t)^T \boldsymbol{v}\_{obs\_i}^l(t) &\leq \cos \mathcal{Q}, \\ \forall t \in \left[t\_0, t\_f\right] \end{aligned} \tag{2}$$

The constraint is a non-convex quadratic constraint; it was convexified in [4], which made it possible to be represented as a LMI using the quaternion attitude constraint formulation developed in [3] for a single-spacecraft single-obstacle scenario. In [4], v<sup>I</sup> obs was static, while vI cami ð Þt was evolving; both vectors were in the same coordinate frame. Although solving it in the same coordinate frame somewhat simplified the solution, it was not suitable for practical implementation because, in reality, the obstacle and spacecraft operate in different coordinate frames. Next, we present the basic mathematical preliminaries.



Table 1. Frequently used notations in this chapter.

#### 3. Mathematical background

Let v<sup>I</sup> cami

104 Space Flight

vI cami tf

or

vI cami

q�i or q<sup>i</sup> ∗

defined in Table 1), and let vI

obsi

to avoid vI

ð Þ<sup>t</sup> denote the unit camera vector in <sup>F</sup><sup>I</sup>

obsi

this case the Sun). Exclusion requires the time evolution of camera vector v<sup>I</sup>

vI cami ð Þt T vI obsi

developed in [3] for a single-spacecraft single-obstacle scenario. In [4], v<sup>I</sup>

qi Attitude quaternion vector of SCi, SCi, qi

Ω, Π Quaternion dynamics plant matrix

P Laplacian-like stochastic matrix I<sup>n</sup> Then n � n identity matrix

<sup>S</sup><sup>m</sup> The set of <sup>m</sup> � <sup>m</sup> positive definite matrices

q Stacked vector of more than one quaternion vectors qoff Stacked vector of more than one offset quaternion vectors

P Quaternion dynamics Laplacian-like plant matrix

frames. Next, we present the basic mathematical preliminaries.

Notation Meaning SCi, SCi Spacecraft i

qi Vector part of qi

qi� Antisymmetric of qi

ω Angular velocity τ Control torque J Inertia matrix L Laplacian matrix

∀t∈ t0; tf

SCi corresponding to the SCi's attitude qi (as

θð Þt ≥ ∅ (1)

(2)

cami from vI

cami ð Þ t<sup>0</sup> to

obs was static, while

= [q<sup>1</sup> q<sup>2</sup> q3| q4]

T

ð Þt be the attitude quaternion of the obstacle to be avoided (in

ð Þt all times with a minimum angular separation of ∅. The requirement is.

ð Þt ≤ cos ∅,

The constraint is a non-convex quadratic constraint; it was convexified in [4], which made it possible to be represented as a LMI using the quaternion attitude constraint formulation

ð Þt was evolving; both vectors were in the same coordinate frame. Although solving it in the same coordinate frame somewhat simplified the solution, it was not suitable for practical implementation because, in reality, the obstacle and spacecraft operate in different coordinate

Conjugate of qi

, <sup>q</sup><sup>i</sup> <sup>¼</sup> <sup>q</sup><sup>1</sup> <sup>q</sup><sup>2</sup> <sup>q</sup><sup>3</sup> <sup>T</sup>

> In this section, we shall briefly consider the two basic mathematical theories relevant to this chapter. More comprehensive study and analysis are in [10].

#### 3.1. Quaternion-based rotational dynamics

Because quaternions are free from the problems of singularities inherent in Euler angles and most other ways of representing rotations, it is convenient to use unit quaternions to represent the attitude of a rigid body rotating in three-dimensional space (such as spacecraft or satellite) [15]. The quaternion is a four-element vector:

$$q = \begin{bmatrix} q\_1 \ q\_2 \ q\_3 |q\_4| \end{bmatrix}^T. \tag{3}$$

The dynamics of the rotational (angular) velocity ω<sup>i</sup>

J i <sup>2</sup> � J i 3 � �ω

J i <sup>3</sup> � J i 1 � �ω

J i <sup>1</sup> � J i 2 � �ω

<sup>0</sup> <sup>J</sup> i 2 J i 1 ωi i 2ωi <sup>3</sup> <sup>þ</sup> <sup>τ</sup><sup>i</sup> 1

� �=<sup>J</sup>

i 3ωi <sup>1</sup> <sup>þ</sup> <sup>τ</sup><sup>i</sup> 2

� �=<sup>J</sup>

i 1ωi <sup>2</sup> <sup>þ</sup> <sup>τ</sup><sup>i</sup> 3

� �=<sup>J</sup>

<sup>3</sup> � <sup>J</sup>

<sup>3</sup> <sup>0</sup> � <sup>J</sup>


ð Þ¼ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>I</sup><sup>3</sup> <sup>þ</sup> <sup>Δ</sup>tΥ<sup>i</sup>

i

ð Þ k þ 1 I<sup>4</sup>

The typical task of controller synthesis is to determine the torque τ<sup>i</sup>

ð Þ<sup>k</sup> � �ω<sup>i</sup>

i 3 J i 1 ωi 2

i 1 J i 2 ωi 1

ω\_ i 1

ω\_ i 2

ω\_ i 3

¼

where ω<sup>i</sup>

vector form yields.

2 6 4

3.2. Basic consensus theory

(vehicle) i be x<sup>i</sup>

i

J i 3 J i 2 ωi

J i 1 J i 3 ωi <sup>2</sup> � <sup>J</sup> i 2 J i 3 ωi <sup>1</sup> 0

Euler's first-order discretization of Eq. (10) is

<sup>j</sup> is the rotational velocity, J

<sup>0</sup><sup>4</sup>�<sup>3</sup> � <sup>Δ</sup><sup>t</sup>

ωi

�Δt J<sup>i</sup> � ��<sup>1</sup> <sup>I</sup><sup>3</sup> <sup>0</sup><sup>3</sup>�<sup>4</sup>

2 Πi


of qi is

Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints

i 1

i 2

i 3

> ωi 1 ωi 2 ωi 3

<sup>j</sup> is the moment of inertia, and τ<sup>i</sup>

th rigid body along the three principal axes j = 1, 2, 3. Combining Eqs. (9) and (11) in stacked

τi ð Þk ωi ð Þ k þ 1

qi ð Þ k þ 2


, and x is the stacked vector of all the states of the vehicles. For systems modeled

3 7 5 2 6 4

The problem of consensus theory is to create distributed protocols based on communication graphs which can drive the states of a team of communicating agents to a common state or an agreed state. Where the agents i (i = 1, ⋯, n) are represented by vertices of the communication graph; the edges of the graph are the communication links between them. Let the state of agent

1=J i

ð Þþ <sup>k</sup> <sup>Δ</sup>t J<sup>i</sup> � ��<sup>1</sup>

3 7 5 <sup>1</sup> 0 0

0 1=J i <sup>2</sup> 0

0 01=J


τi

<sup>¼</sup> <sup>I</sup><sup>3</sup> <sup>þ</sup> <sup>Δ</sup>tΥ<sup>i</sup>

ð Þ<sup>k</sup> � �ω<sup>i</sup>

" #


qi ð Þ k þ 1

i 3

τi 1

ð Þk , (11)

<sup>j</sup> is the control torque, of the

ð Þk

that stabilizes the system.

(12)

(10)

107

http://dx.doi.org/10.5772/intechopen.71506

τi 2

τi 3

Here, [q<sup>1</sup> q<sup>2</sup> q3] <sup>T</sup> is the vector part, representing the axis of rotation in the Cartesian (x, y, z) coordinates, and q<sup>4</sup> is a scalar part, representing the angle of rotation of the quaternion in degrees. The difference between two quaternions q<sup>1</sup> and q<sup>2</sup> can be represented in multiplication terms as.

$$\begin{split} q^d &= q^1 \odot q^{-2} = q^1 \odot \left[ -q\_1^2 - q\_2^2 - q\_3^2 - q\_4^2 \right]^T \\ &= Q^2 q^1, \end{split} \tag{4}$$

where q�<sup>2</sup> is the conjugate of q<sup>2</sup> . We used ⊙ here as a quaternion multiplication operator. And Q<sup>2</sup> is defined as

$$\mathbf{Q}^{i} = \begin{bmatrix} q\_4^i & q\_3^i & -q\_2^i & -q\_1^i \\ -q\_3^i & q\_4^i & q\_1^i & -q\_2^i \\ q\_2^i & -q\_1^i & q\_4^i & -q\_3^i \\ q\_1^i & q\_2^i & q\_3^i & q\_4^i \end{bmatrix} \tag{5}$$

Eq. (4) means that q<sup>d</sup> is the rotation quaternion that originally transformed q<sup>1</sup> to q<sup>2</sup> or, alternatively, qd is a rotation quaternion that can transform q<sup>1</sup> to q<sup>2</sup> .

The rotational dynamics for the i th quaternion is.

$$
\sigma^i = \frac{1}{2} \Omega^i \sigma^i = \frac{1}{2} \Pi^i \omega^i \tag{6}
$$

where

$$\begin{aligned} \Omega^i = \begin{bmatrix} 0 & \omega\_3^i & -\omega\_2^i & \omega\_1^i \\ -\omega\_3^i & 0 & \omega\_1^i & \omega\_2^i \\ \omega\_2^i & -\omega\_1^i & 0 & \omega\_3^i \\ -\omega\_1^i & -\omega\_2^i & -\omega\_3^i & 0 \end{bmatrix} \tag{7} \\ \Omega^i = \begin{bmatrix} -q\_4^i & q\_3^i & -q\_2^i \\ -q\_3^i & -q\_4^i & q\_1^i \\ 0 & 0 & \end{bmatrix} \tag{8} \tag{8} \tag{8}$$

$$II^i = \begin{bmatrix} -q\_3^i & -q\_4^i & q\_1^i \\ q\_2^i & -q\_1^i & -q\_4^i \\ q\_1^i & q\_2^i & q\_3^i \end{bmatrix} \tag{8}$$

are the plant matrices of quaternion dynamics.

Euler's first-order discretization of Eq. (6) yields

$$q^i(k+1) = I\_4 q^i(k) + \frac{\Delta t}{2} \Omega^i(k) \\ q^i(k) = q^i(k) + \frac{\Delta t}{2} \Pi^i(k) \omega^i(k). \tag{9}$$

The dynamics of the rotational (angular) velocity ω<sup>i</sup> of qi is

q ¼ q<sup>1</sup> q<sup>2</sup> q3jq<sup>4</sup> � �<sup>T</sup>

coordinates, and q<sup>4</sup> is a scalar part, representing the angle of rotation of the quaternion in degrees. The difference between two quaternions q<sup>1</sup> and q<sup>2</sup> can be represented in multiplication terms as.

<sup>⊙</sup> �q<sup>2</sup>

<sup>q</sup><sup>d</sup> <sup>¼</sup> <sup>q</sup><sup>1</sup>

<sup>¼</sup> <sup>Q</sup><sup>2</sup> q1 ,

<sup>Q</sup><sup>i</sup> <sup>¼</sup>

<sup>Ω</sup><sup>i</sup> <sup>¼</sup>

are the plant matrices of quaternion dynamics. Euler's first-order discretization of Eq. (6) yields

ð Þ¼ <sup>k</sup> <sup>þ</sup> <sup>1</sup> I4q<sup>i</sup>

qi

tively, qd is a rotation quaternion that can transform q<sup>1</sup> to q<sup>2</sup>

<sup>⊙</sup> <sup>q</sup>�<sup>2</sup> <sup>¼</sup> <sup>q</sup><sup>1</sup>

qi <sup>4</sup> qi

qi <sup>2</sup> �qi

qi <sup>1</sup> qi

th quaternion is.

<sup>q</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> 2 Ωi <sup>q</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> 2 Πi

�ω<sup>i</sup>

<sup>Π</sup><sup>i</sup> <sup>¼</sup>

ð Þþ <sup>k</sup> <sup>Δ</sup><sup>t</sup> <sup>2</sup> <sup>Ω</sup><sup>i</sup> ð Þ<sup>k</sup> qi

�ω<sup>i</sup>

ωi <sup>2</sup> �ω<sup>i</sup>

�qi <sup>3</sup> qi

<sup>T</sup> is the vector part, representing the axis of rotation in the Cartesian (x, y, z)

� �<sup>T</sup>

<sup>2</sup> �qi 1

<sup>1</sup> �qi 2

<sup>4</sup> �qi 3

<sup>3</sup> qi 4

.

. We used ⊙ here as a quaternion multiplication operator. And

<sup>1</sup> � <sup>q</sup><sup>2</sup> <sup>2</sup> � <sup>q</sup><sup>2</sup> <sup>3</sup> � <sup>q</sup><sup>2</sup> 4

<sup>3</sup> �q<sup>i</sup>

<sup>4</sup> q<sup>i</sup>

<sup>1</sup> q<sup>i</sup>

<sup>2</sup> q<sup>i</sup>

Eq. (4) means that q<sup>d</sup> is the rotation quaternion that originally transformed q<sup>1</sup> to q<sup>2</sup> or, alterna-

0 ω<sup>i</sup>

<sup>1</sup> �ω<sup>i</sup>

�qi <sup>4</sup> q<sup>i</sup>

�qi <sup>3</sup> �q<sup>i</sup>

> qi <sup>2</sup> �q<sup>i</sup>

> qi <sup>1</sup> q<sup>i</sup>

<sup>3</sup> �ω<sup>i</sup>

<sup>2</sup> �ω<sup>i</sup>

<sup>1</sup> 0 ω<sup>i</sup>

<sup>3</sup> �q<sup>i</sup> 2

<sup>4</sup> q<sup>i</sup> 1

<sup>1</sup> �q<sup>i</sup> 4

<sup>2</sup> q<sup>i</sup> 3

ð Þ¼ <sup>k</sup> <sup>q</sup><sup>i</sup>

<sup>3</sup> 0 ω<sup>i</sup>

<sup>2</sup> ω<sup>i</sup> 1

<sup>1</sup> ω<sup>i</sup> 2

<sup>3</sup> 0

ð Þþ <sup>k</sup> <sup>Δ</sup><sup>t</sup> <sup>2</sup> <sup>Π</sup><sup>i</sup> ð Þ<sup>k</sup> <sup>ω</sup><sup>i</sup>

3

Here, [q<sup>1</sup> q<sup>2</sup> q3]

106 Space Flight

Q<sup>2</sup> is defined as

where

where q�<sup>2</sup> is the conjugate of q<sup>2</sup>

The rotational dynamics for the i

: (3)

ω<sup>i</sup> (6)

(4)

(5)

(7)

(8)

ð Þk : (9)

$$
\begin{aligned}
\begin{bmatrix}
\dot{\omega}\_{1}^{i} \\
\dot{\omega}\_{2}^{i} \\
\dot{\omega}\_{3}^{i}
\end{bmatrix} &= \begin{bmatrix}
\left( (\dot{f}\_{2}^{i} - \dot{f}\_{3}^{i})\omega\_{2}^{i}\omega\_{3}^{i} + \dot{\pi}\_{1}^{i} \right) / f\_{1}^{i} \\
\left( (\dot{f}\_{3}^{i} - \dot{f}\_{1}^{i})\omega\_{3}^{i}\omega\_{1}^{i} + \dot{\pi}\_{2}^{i} \right) / f\_{2}^{i} \\
\left( (\dot{f}\_{1}^{i} - \dot{f}\_{2}^{i})\omega\_{1}^{i}\omega\_{2}^{i} + \dot{\pi}\_{3}^{i} \right) / f\_{3}^{i} \\
\end{bmatrix} \\
&= \begin{bmatrix}
0 & \frac{\dot{f}\_{2}^{i}}{\dot{f}\_{1}^{i}}\omega\_{3}^{i} & -\frac{\dot{f}\_{3}^{i}}{\dot{f}\_{1}^{i}}\omega\_{2}^{i} \\
\frac{\dot{f}\_{2}^{i}}{\dot{f}\_{2}^{i}}\omega\_{3}^{i} & 0 & -\frac{\dot{f}\_{1}^{i}}{\dot{f}\_{2}^{i}}\omega\_{1}^{i} \\
\end{bmatrix} \begin{bmatrix}
\omega\_{1}^{i} \\
\omega\_{2}^{i} \\
\omega\_{3}^{i}
\end{bmatrix} + \underbrace{\begin{bmatrix}
1/f\_{1}^{i} & 0 & 0 \\
0 & 1/f\_{1}^{i} & 0 \\
0 & 0 & 1/f\_{1}^{i} & 0
\end{bmatrix}}\_{\left(f\_{3}^{i}\right)^{-1}}.
\end{aligned}
\tag{10}
$$

Euler's first-order discretization of Eq. (10) is

$$
\omega^i(k+1) = \left(\mathbf{I}\_3 + \Delta t \Upsilon^i(k)\right) \omega^i(k) + \Delta t \left(\mathbf{j}^i\right)^{-1} \tau^i(k), \tag{11}
$$

where ω<sup>i</sup> <sup>j</sup> is the rotational velocity, J i <sup>j</sup> is the moment of inertia, and τ<sup>i</sup> <sup>j</sup> is the control torque, of the i th rigid body along the three principal axes j = 1, 2, 3. Combining Eqs. (9) and (11) in stacked vector form yields.

$$\underbrace{\begin{bmatrix} -\Delta t \left( \dot{l}^i \right)^{-1} & \mathbf{I}\_3 & \mathbf{0}\_{3 \times 4} \\\\ \mathbf{0}\_{4 \times 3} & -\frac{\Delta t}{2} \Pi \dot{r}(k+1) & \mathbf{I}\_4 \end{bmatrix}}\_{\dot{F}(k)} \underbrace{\begin{bmatrix} \tau^i(k) \\\\ a^i(k+1) \\\\ \tilde{q}^i(k+2) \end{bmatrix}}\_{\mathcal{L}^i(k+1)} = \underbrace{\begin{bmatrix} (\mathbf{I}\_3 + \Delta t \Upsilon^i(k))a^i(k) \\\\ \dot{q}^i(k+1) \end{bmatrix}}\_{\mathbf{y}^i(k)} \tag{12}$$

The typical task of controller synthesis is to determine the torque τ<sup>i</sup> that stabilizes the system.

#### 3.2. Basic consensus theory

The problem of consensus theory is to create distributed protocols based on communication graphs which can drive the states of a team of communicating agents to a common state or an agreed state. Where the agents i (i = 1, ⋯, n) are represented by vertices of the communication graph; the edges of the graph are the communication links between them. Let the state of agent (vehicle) i be x<sup>i</sup> , and x is the stacked vector of all the states of the vehicles. For systems modeled by first-order dynamics, the following first-order consensus protocol (or similar protocols) has been proposed, for example [16, 17]:

$$
\dot{\mathbf{x}}(t) = -\mathbf{L}\left(\mathbf{x}(t) - \mathbf{x}^{\text{off}}\right). \tag{13}
$$

qi <sup>k</sup>þ<sup>1</sup> <sup>¼</sup> qi

Laplacian-like behavior, and Λ<sup>i</sup>

where Λ<sup>i</sup>

pencil [18].

the claim.

that

<sup>k</sup> � <sup>Δ</sup>t yΛ<sup>i</sup>

PðÞ¼ t

Moreover, Azi = λiBzi for i = 1, ⋯, n, where λ<sup>i</sup> = ai/bi.

values; the rest of its eigenvalues are strictly positive.

Theorem 2: The time-varying system Eq. (14) achieves consensus.

<sup>1</sup>ð Þ� <sup>t</sup> <sup>Λ</sup><sup>i</sup>

the collective quaternion consensus dynamics Eq. (14). The components of P(t) are

<sup>0</sup> <sup>⋯</sup> <sup>Λ</sup><sup>n</sup>ð Þ<sup>t</sup>


<sup>Λ</sup><sup>1</sup>ð Þ<sup>t</sup> <sup>⋯</sup> <sup>0</sup> ⋮ ⋱⋮

<sup>2</sup>ð Þ<sup>t</sup> <sup>⋯</sup> � <sup>Λ</sup><sup>i</sup>

nents are chosen by the optimization process. For analysis purposes, we shall now reconsider

where Γ is composed of components of the Laplacian L = [lij] (i, j = 1, ⋯, n), which gives P(t) its

Theorem 1: For a symmetric-definite pencil A � λB, there exists a nonsingular Z= [z1, ⋯, zn] such

Lemma 1: For any time t, the eigenvalues of P(t) are γiηi(t). Here, γ<sup>i</sup> are the eigenvalues of Γ and ηi(t) the eigenvalues of Λ(t). It can therefore be observed that P(t) has only four zero eigen-

Proof: To find the eigenvalues of P(t), consider a scalar λ such that for some nonzero vector z:

<sup>Γ</sup><sup>z</sup> <sup>¼</sup> λΛ�<sup>1</sup>

Eq. (20) defines a symmetric-definite generalized eigenvalue problem (SDGEP), where <sup>Γ</sup> � <sup>λ</sup>Λ�<sup>1</sup>

defines a matrix pencil. Theorem 1 therefore immediately implies that the eigenvalues of P(t) are γiηi(t). It is also easy to observe (or show numerically) that due to the property of the Laplacian matrix L, P(t) has positive eigenvalues except for four zero eigenvalues. This proves

(t) > 0 is as previously defined. We now present the proof of stability of P(t), that is, that Eq. (14) does indeed achieve consensus. Different versions of all the theorems, lemmas and proofs in this section had been presented in [7–10]. Let us begin by recalling the following standard result on a matrix

h i


<sup>y</sup>ð Þt

(t) > 0 is an unknown positive definite optimization matrix variable, whose compo-

qT <sup>1</sup> ð Þ<sup>t</sup> qT

Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints

l11I<sup>4</sup> ⋯ l1nI<sup>4</sup> ⋮ ⋱⋮ ln1I<sup>4</sup> ⋯ lnnI<sup>4</sup>


ZTA<sup>Z</sup> <sup>¼</sup> diagð Þ¼ <sup>a</sup>1; <sup>⋯</sup>; an DA, (18)

ZTB<sup>Z</sup> <sup>¼</sup> diagð Þ¼ <sup>b</sup>1; <sup>⋯</sup>; bn DB: (19)

ð Þt z: (20)

<sup>2</sup> ð Þ<sup>t</sup> <sup>⋯</sup>q<sup>T</sup> <sup>y</sup> ð Þt h i, (16)

http://dx.doi.org/10.5772/intechopen.71506

(17)

109

(t)

We know that consensus has been achieved when <sup>k</sup>xi � xj k! (xij) off as <sup>t</sup>! <sup>∞</sup>, <sup>∀</sup><sup>i</sup> 6¼ <sup>j</sup>. A more comprehensive analysis of the mathematical basis of graph theoretic consensus theory can be found in [10].

Now we state the limitations of consensus theory that motivates our work. First, the basic consensus protocol Eq. (13) does not admit quaternions directly because quaternion dynamics are highly nonlinear. It violates quaternion unit norm requirements, and therefore we cannot practically apply Eq. (6) with consensus directly. To extend Eq. (13) to attitude quaternions, we proposed the following consensus protocol for quaternions [7–10]:

$$\dot{\mathbf{q}}(t) = -\mathbf{P}(t) \left( \mathbf{q}(t) \Theta \mathbf{q}^{-\text{eff}} \right). \tag{14}$$

Here, P(t) is a Laplacian-like stochastic matrix whose values are partially unknown, but a Laplacian-like structure is imposed on it by optimization, and q(t)=[q<sup>1</sup> (t), q<sup>2</sup> (t)⋯q<sup>n</sup> (t)]<sup>T</sup> . We present more analysis of P(t) in the "Solutions" section.

#### 4. Solutions

We present a four-step solution to the problem statement in Section 2 [7–10], listed as follows: (1) development of a consensus protocol for quaternions, (2) development of collision avoidance behavior for quaternion consensus, (3) determining obstacle vectors in different coordinate frames and (4) integration of quaternion consensus with Q-CAC avoidance.

#### 4.1. Development of a consensus protocol for quaternions

To handle the difficulty of non-linearity in quaternion kinematics, we develop a consensus protocol especially for quaternions. We adopt an optimization approach and cast the problem as a semidefinite program, which is subject to convex quadratic constraints, stated as linear matrix inequalities (LMI). Based on the current communication graph of any SCi, a series of Laplacian-like matrices P<sup>i</sup> (t) are synthesized each time step to drive qi (t) to consensus while satisfying quaternion kinematics:

$$\dot{q}^i(t) - \mathbf{P}^i(t) \left[ q\_1^T(t) q\_2^T(t) \cdots q\_y^T(t) \right]\_{\prime} \tag{15}$$

where q<sup>T</sup> <sup>1</sup> ð Þ<sup>t</sup> <sup>q</sup><sup>T</sup> <sup>2</sup> ð Þ<sup>t</sup> <sup>⋯</sup>qT <sup>y</sup> ð Þt are the quaternions of the y other neighboring SC which SCi can communicate with at time t. Euler's first-order discretization of Eq. (15) is

Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints http://dx.doi.org/10.5772/intechopen.71506 109

$$\boldsymbol{q}\_{k+1}^{i} = \boldsymbol{q}\_{k}^{i} - \Delta t \underbrace{\left[\boldsymbol{y}\boldsymbol{\Lambda}\_{1}^{i}(\mathbf{t}) - \boldsymbol{\Lambda}\_{2}^{i}(\mathbf{t})\cdots - \boldsymbol{\Lambda}\_{y}^{i}(\mathbf{t})\right]}\_{\mathbf{P}^{\prime}(\mathbf{t})} \left[\boldsymbol{q}\_{1}^{T}(t)\boldsymbol{q}\_{2}^{T}(t)\cdots \boldsymbol{q}\_{y}^{T}(t)\right],\tag{16}$$

where Λ<sup>i</sup> (t) > 0 is an unknown positive definite optimization matrix variable, whose components are chosen by the optimization process. For analysis purposes, we shall now reconsider the collective quaternion consensus dynamics Eq. (14). The components of P(t) are

$$\mathbf{P}(t) = \underbrace{\begin{bmatrix} \Lambda^1(t) & \cdots & 0\\ \vdots & & \ddots & \vdots\\ 0 & & \cdots & \Lambda^n(t) \end{bmatrix}}\_{\Lambda(t)} \begin{bmatrix} l\_{11}\mathbf{I}\_4 & \cdots & l\_{1n}\mathbf{I}\_4\\ \vdots & & \ddots & \vdots\\ l\_{n1}\mathbf{I}\_4 & \cdots & l\_{nn}\mathbf{I}\_4 \end{bmatrix} \tag{17}$$

where Γ is composed of components of the Laplacian L = [lij] (i, j = 1, ⋯, n), which gives P(t) its Laplacian-like behavior, and Λ<sup>i</sup> (t) > 0 is as previously defined.

We now present the proof of stability of P(t), that is, that Eq. (14) does indeed achieve consensus. Different versions of all the theorems, lemmas and proofs in this section had been presented in [7–10]. Let us begin by recalling the following standard result on a matrix pencil [18].

Theorem 1: For a symmetric-definite pencil A � λB, there exists a nonsingular Z= [z1, ⋯, zn] such that

$$\mathbf{Z}^{\mathsf{T}} \mathbf{A} \mathbf{Z} = \text{diag}(a\_1, \dots, a\_n) = D\_{A\prime} \tag{18}$$

$$\mathbf{Z}^{\mathsf{T}} \mathbf{B} \mathbf{Z} = \text{diag}(b\_1, \dots, b\_n) = D\_\mathsf{B}. \tag{19}$$

Moreover, Azi = λiBzi for i = 1, ⋯, n, where λ<sup>i</sup> = ai/bi.

by first-order dynamics, the following first-order consensus protocol (or similar protocols) has

comprehensive analysis of the mathematical basis of graph theoretic consensus theory can be

Now we state the limitations of consensus theory that motivates our work. First, the basic consensus protocol Eq. (13) does not admit quaternions directly because quaternion dynamics are highly nonlinear. It violates quaternion unit norm requirements, and therefore we cannot practically apply Eq. (6) with consensus directly. To extend Eq. (13) to attitude quaternions, we

Here, P(t) is a Laplacian-like stochastic matrix whose values are partially unknown, but a

We present a four-step solution to the problem statement in Section 2 [7–10], listed as follows: (1) development of a consensus protocol for quaternions, (2) development of collision avoidance behavior for quaternion consensus, (3) determining obstacle vectors in different coordinate frames and (4) integration of quaternion consensus with Q-CAC

To handle the difficulty of non-linearity in quaternion kinematics, we develop a consensus protocol especially for quaternions. We adopt an optimization approach and cast the problem as a semidefinite program, which is subject to convex quadratic constraints, stated as linear matrix inequalities (LMI). Based on the current communication graph of any SCi, a series of

(t) are synthesized each time step to drive qi

<sup>2</sup> ð Þ<sup>t</sup> <sup>⋯</sup>q<sup>T</sup> <sup>y</sup> ð Þt

<sup>y</sup> ð Þt are the quaternions of the y other neighboring SC which SCi can

h i

ð Þ<sup>t</sup> qT <sup>1</sup> ð Þ<sup>t</sup> <sup>q</sup><sup>T</sup>

<sup>x</sup>\_ðÞ¼� <sup>t</sup> L xð Þ� <sup>t</sup> <sup>x</sup>off � �: (13)

off as <sup>t</sup>! <sup>∞</sup>, <sup>∀</sup><sup>i</sup> 6¼ <sup>j</sup>. A more

(t), q<sup>2</sup>

(t)⋯q<sup>n</sup>

(t) to consensus while

, (15)

(t)]<sup>T</sup> . We

k! (xij )

<sup>q</sup>\_ðÞ¼� <sup>t</sup> <sup>P</sup>ð Þ<sup>t</sup> <sup>q</sup>ð Þ<sup>t</sup> <sup>Θ</sup>q�off � �: (14)

� xj

been proposed, for example [16, 17]:

found in [10].

108 Space Flight

4. Solutions

avoidance.

where q<sup>T</sup>

Laplacian-like matrices P<sup>i</sup>

<sup>1</sup> ð Þ<sup>t</sup> <sup>q</sup><sup>T</sup>

satisfying quaternion kinematics:

<sup>2</sup> ð Þ<sup>t</sup> <sup>⋯</sup>qT

We know that consensus has been achieved when <sup>k</sup>xi

present more analysis of P(t) in the "Solutions" section.

4.1. Development of a consensus protocol for quaternions

q\_ i ð Þ� <sup>t</sup> <sup>P</sup><sup>i</sup>

communicate with at time t. Euler's first-order discretization of Eq. (15) is

proposed the following consensus protocol for quaternions [7–10]:

Laplacian-like structure is imposed on it by optimization, and q(t)=[q<sup>1</sup>

Lemma 1: For any time t, the eigenvalues of P(t) are γiηi(t). Here, γ<sup>i</sup> are the eigenvalues of Γ and ηi(t) the eigenvalues of Λ(t). It can therefore be observed that P(t) has only four zero eigenvalues; the rest of its eigenvalues are strictly positive.

Proof: To find the eigenvalues of P(t), consider a scalar λ such that for some nonzero vector z:

$$I\mathbf{z} = \lambda \Lambda^{-1}(t)\mathbf{z}.\tag{20}$$

Eq. (20) defines a symmetric-definite generalized eigenvalue problem (SDGEP), where <sup>Γ</sup> � <sup>λ</sup>Λ�<sup>1</sup> (t) defines a matrix pencil. Theorem 1 therefore immediately implies that the eigenvalues of P(t) are γiηi(t). It is also easy to observe (or show numerically) that due to the property of the Laplacian matrix L, P(t) has positive eigenvalues except for four zero eigenvalues. This proves the claim.

Theorem 2: The time-varying system Eq. (14) achieves consensus.

Proof: For simplicity, we shall assume no offsets are defined, that is, qoff = 0 (or (qoff) i = [0 0 0 1]<sup>T</sup> ∀ i). By consensus theory, when q has entered the consensus space C = {q|q<sup>1</sup> = q2 =, ⋯, =qn }, then q\_ = 0 (i.e. no vehicles are moving anymore). C is the nullspace of P(t), that is, the set of all q such that P(t)q = 0. Therefore, q stays in C once it enters there.

Suppose that q has not entered C (i.e. q\_ 6¼ 0), then consider a Lyapunov candidate function V = q<sup>T</sup> Γq; V > 0 unless q∈ C. Then:

$$\begin{split} \dot{V} &= \mathbf{q}^T \Gamma \dot{\mathbf{q}} + \dot{\mathbf{q}}^T \Gamma \mathbf{q} \\ &= -\mathbf{q}^T \Gamma \mathbf{P}(t) \mathbf{q} - \mathbf{q}^T \mathbf{P}(t) \Gamma \mathbf{q} \\ &= -\mathbf{q}^T \Gamma \Lambda(t) \Gamma \mathbf{q} - \mathbf{q}^T \Gamma \Lambda \Gamma \mathbf{q} \\ &= -2\mathbf{q}^T \Gamma \Lambda(t) \Gamma \mathbf{q} \\ &= -2s^T \Lambda(t) s\_\prime \end{split} \tag{21}$$

μ q<sup>i</sup>

qi

2 6 4

A~ i j ðÞ¼ t

ð Þ<sup>t</sup> <sup>T</sup> <sup>þ</sup> vI

obsi:j ð Þ<sup>t</sup> <sup>v</sup><sup>B</sup> cami

bjðÞ¼� <sup>t</sup> vB

djðÞ¼ <sup>t</sup> <sup>v</sup><sup>B</sup>

� �, so it is used to find a collision-free vI

cami

cami ð Þt T vI obsi:j

where

and

for j = 1, ⋯, m.

ponding to vI

point of vI

it defines vI

avoided by SCi.

ð Þt ≥ ∅∀t ∈ t0; tf

coordinate frame and that v<sup>I</sup>

frame of SCj but defined in F<sup>I</sup>

obsj

ensure that <sup>μ</sup>I<sup>4</sup> <sup>þ</sup> <sup>A</sup><sup>~</sup> <sup>i</sup>

vI obsi:j AjðÞ¼ <sup>t</sup> <sup>v</sup><sup>B</sup>

cami ð Þ<sup>t</sup> vI obsi:j

j

obsj (defined in <sup>F</sup><sup>I</sup>

of SC2, whereby the intersection defines vI

changes its attitude from q<sup>0</sup> to qf

Eq. (24) defines the set of attitude quaternions qi

ð Þt is positive definite.

However, the solution presented above assumes that vB

obsi:j

SCj ) (vI

SCi

evolves. If both spacecraft are close enough, then vector vI

, v<sup>I</sup>

such a practical issue, we present a mechanism to calculate v<sup>I</sup>

frames relative to Earth. A thruster attached to SC1 body frame is at vI

ð Þt T

ð Þ<sup>t</sup> <sup>μ</sup>I<sup>4</sup> <sup>þ</sup> <sup>A</sup><sup>~</sup> <sup>i</sup>

j ð Þt � ��<sup>1</sup>

Ajð Þt bjð Þt bjð Þ<sup>t</sup> <sup>T</sup> djð Þ<sup>t</sup>

" #

3 7 5

Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints

∈ R<sup>4</sup>�<sup>4</sup>

cami ð Þt T vI obsi:j

cami

ð Þt is static, so t is constant. In reality, this is not so. To address

obsi:<sup>j</sup> means the obstacle vector originated from the rotating

). This is essentially a mechanism to determine the intersection

ð Þ<sup>t</sup> <sup>T</sup> and vI

cami

ð Þt with the sphere of radius r, centerd on SCi. If indeed such an intersection exists,

obsi:<sup>j</sup> which can be used to define an attitude constraint represented as Eq. (24) to be

cam2 must avoid the cone created around vI

The scenario is illustrated in Figure 2, whereSC1 and SC2 are shown in their different coordinate

around SC1 and SC2 are spheres representing the coordinate frames from which their attitude

ð Þ<sup>t</sup> <sup>T</sup> � vB

ðÞ� <sup>t</sup> <sup>v</sup><sup>I</sup> obsi:j ≥ 0: (25)

http://dx.doi.org/10.5772/intechopen.71506

111

, (26)

ð Þþ t cos θ � �I3, (27)

ð Þt , (28)

ð Þt , (29)

ð Þt . In Eq. (25), μ is chosen to

cami ð Þt T

) corres-

ð Þt are in the same

SCi

, while the circles

� �.

obs2:<sup>1</sup>∀t ∈ t0; tf

(t) to satisfy the constraint v<sup>I</sup>

obsi:j

obsi:<sup>j</sup> (defined in <sup>F</sup><sup>I</sup>

obs1

obs1 may intersect a point on the sphere

obs2:<sup>1</sup> in the frame of SC2. The requirement is that as SC2

where s = Γq 6¼ 0 for q∉C, which implies that q approaches a point in C as t !∞. This proves the claim. Eq. (21) is true as long as L is nonempty, that is, some vehicles can sense, see or communicate with each other all the time.

#### 4.2. Development of collision avoidance behavior for quaternion consensus

Eq. (15) or (16) will indeed generate a consensus q<sup>i</sup> (t) for any SCi, but the system still needs to determine whether the trajectory is safe or not. This brings us to the issue of avoidance. Any rigid appendage attached to the body of SCi, for example, a camera, whose direction vector is vI cami in inertial frame, can be transformed to the spacecraft fixed body frame by the rotation:

$$
\boldsymbol{\upsilon}\_{\text{cam}\_{i}}^{B}(\mathbf{t}) = \mathcal{R}\_{i}^{-1}(\mathbf{t}) \boldsymbol{\upsilon}\_{\text{cam}\_{i}}^{l}(\mathbf{t}). \tag{22}
$$

where

$$\mathcal{R}\_i(t) = \left( \left( 2\eta\_4^i(t) \right)^2 - 1 \right) \mathbf{I}\_3 + 2\overline{q}^i(t)\overline{q}^i(t)^T - 2\eta\_4^i(t)\overline{q}^i(t)^\times \tag{23}$$

is the rotation matrix corresponding to the q i (t) at time t; qi (t) � is the antisymmetric matrix [19]. For a simpler analysis, let us consider a single SCi with a single camera, vI cami , and m (possibly, timevarying) obstacles, v<sup>I</sup> obsi:j ð Þ <sup>j</sup> <sup>¼</sup> <sup>1</sup>; <sup>⋯</sup>; <sup>m</sup> , defined in <sup>F</sup><sup>I</sup> SCi . We want vI cami to avoid all vI obsi:<sup>j</sup> when SCi is re-orientating. Then following Eq. (3), the resulting attitude constraint of Eq. (2) can be written as

$$q^i(t)^T \tilde{A}\_j^i(t) q^i(t) \le 0. \tag{24}$$

Its LMI equivalent [5] is

Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints http://dx.doi.org/10.5772/intechopen.71506 111

$$\begin{bmatrix} \mu & q^i(t)^T \\\\ q^i(t) & \left(\mu \mathbf{I}\_4 + \tilde{A}\_j^i(t)\right)^{-1} \end{bmatrix} \ge 0. \tag{25}$$

where

i =

(21)

(t) for any SCi, but the system still needs to

ð Þt : (22)

� is the antisymmetric matrix [19]. For

ð Þt � (23)

, and m (possibly, time-

obsi:<sup>j</sup> when SCi is

=, ⋯, =qn },

Proof: For simplicity, we shall assume no offsets are defined, that is, qoff = 0 (or (qoff)

then q\_ = 0 (i.e. no vehicles are moving anymore). C is the nullspace of P(t), that is, the set of all

Suppose that q has not entered C (i.e. q\_ 6¼ 0), then consider a Lyapunov candidate function

¼ �q<sup>T</sup>ΓPð Þ<sup>t</sup> <sup>q</sup> � <sup>q</sup><sup>T</sup>Pð Þ<sup>t</sup> <sup>Γ</sup>q, ¼ �q<sup>T</sup>ΓΛð Þ<sup>t</sup> <sup>Γ</sup><sup>q</sup> � <sup>q</sup><sup>T</sup>ΓΛΓq,

where s = Γq 6¼ 0 for q∉C, which implies that q approaches a point in C as t !∞. This proves the claim. Eq. (21) is true as long as L is nonempty, that is, some vehicles can sense, see or

determine whether the trajectory is safe or not. This brings us to the issue of avoidance. Any rigid appendage attached to the body of SCi, for example, a camera, whose direction vector is

> <sup>i</sup> ð Þ<sup>t</sup> vI cami

<sup>I</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>qi

(t) at time t; qi

SCi

re-orientating. Then following Eq. (3), the resulting attitude constraint of Eq. (2) can be written as

ð Þ<sup>t</sup> <sup>q</sup><sup>i</sup>

ð Þ<sup>t</sup> <sup>T</sup> � <sup>2</sup>q<sup>i</sup>

(t)

. We want vI

<sup>4</sup>ð Þ<sup>t</sup> qi

cami

cami to avoid all vI

ð Þt ≤ 0: (24)

cami in inertial frame, can be transformed to the spacecraft fixed body frame by the rotation:

ðÞ¼ <sup>t</sup> <sup>R</sup>�<sup>1</sup>

� 1

i

[0 0 0 1]<sup>T</sup> ∀ i). By consensus theory, when q has entered the consensus space C = {q|q<sup>1</sup> = q2

<sup>V</sup>\_ <sup>¼</sup> <sup>q</sup><sup>T</sup>Γq\_ <sup>þ</sup> <sup>q</sup>\_ <sup>T</sup>Γq,

¼ �2q<sup>T</sup>ΓΛð Þ<sup>t</sup> <sup>Γ</sup>q, ¼ �2sTΛð Þ<sup>t</sup> s,

4.2. Development of collision avoidance behavior for quaternion consensus

vB cami

<sup>4</sup>ð Þ<sup>t</sup> <sup>2</sup>

a simpler analysis, let us consider a single SCi with a single camera, vI

ð Þ <sup>j</sup> <sup>¼</sup> <sup>1</sup>; <sup>⋯</sup>; <sup>m</sup> , defined in <sup>F</sup><sup>I</sup>

qi ð Þ<sup>t</sup> TA<sup>~</sup> <sup>i</sup> j ð Þ<sup>t</sup> qi

q such that P(t)q = 0. Therefore, q stays in C once it enters there.

Γq; V > 0 unless q∈ C. Then:

communicate with each other all the time.

Eq. (15) or (16) will indeed generate a consensus q<sup>i</sup>

<sup>R</sup>iðÞ¼ <sup>t</sup> <sup>2</sup>qi

is the rotation matrix corresponding to the q

obsi:j

V = q<sup>T</sup>

110 Space Flight

vI

where

varying) obstacles, v<sup>I</sup>

Its LMI equivalent [5] is

$$\tilde{A}\_{\dot{\jmath}}^{i}(t) = \begin{bmatrix} A\_{\dot{\jmath}}(t) & b\_{\dot{\jmath}}(t) \\\\ b\_{\dot{\jmath}}(t)^{T} & d\_{\dot{\jmath}}(t) \end{bmatrix} \in \mathbb{R}^{4 \times 4},\tag{26}$$

and

$$A\_{\boldsymbol{j}}(t) = \boldsymbol{\upsilon}\_{\text{cam}\_{i}}^{\mathcal{B}}(t)\boldsymbol{\upsilon}\_{\text{obs}\_{i};j}^{l}(t)^{T} + \boldsymbol{\upsilon}\_{\text{obs}\_{i};j}^{l}(t)\boldsymbol{\upsilon}\_{\text{cam}\_{i}}^{\mathcal{B}}(t)^{T} - \left(\boldsymbol{\upsilon}\_{\text{cam}\_{i}}^{\mathcal{B}}(t)^{T}\boldsymbol{\upsilon}\_{\text{obs}\_{i};j}^{l}(t) + \cos\Theta\right)\mathbf{I}\_{3\prime} \tag{27}$$

$$b\_j(t) = -\upsilon\_{cam\_i}^B(t) \times \upsilon\_{obs\_i,j}^I(t),\tag{28}$$

$$d\_j(t) = \boldsymbol{v}\_{cam\_l}^{\mathcal{B}}(t)^T \boldsymbol{v}\_{abs\_j}^{l}(t),\tag{29}$$

for j = 1, ⋯, m.

Eq. (24) defines the set of attitude quaternions qi (t) to satisfy the constraint v<sup>I</sup> cami ð Þt T vI obsi:j ð Þt ≥ ∅∀t ∈ t0; tf � �, so it is used to find a collision-free vI cami ð Þt . In Eq. (25), μ is chosen to ensure that <sup>μ</sup>I<sup>4</sup> <sup>þ</sup> <sup>A</sup><sup>~</sup> <sup>i</sup> j ð Þt is positive definite.

However, the solution presented above assumes that vB cami ð Þ<sup>t</sup> <sup>T</sup> and vI obsi:j ð Þt are in the same coordinate frame and that v<sup>I</sup> obsi:j ð Þt is static, so t is constant. In reality, this is not so. To address such a practical issue, we present a mechanism to calculate v<sup>I</sup> obsi:<sup>j</sup> (defined in <sup>F</sup><sup>I</sup> SCi ) corresponding to vI obsj (defined in <sup>F</sup><sup>I</sup> SCj ) (vI obsi:<sup>j</sup> means the obstacle vector originated from the rotating frame of SCj but defined in F<sup>I</sup> SCi ). This is essentially a mechanism to determine the intersection point of vI obsj ð Þt with the sphere of radius r, centerd on SCi. If indeed such an intersection exists, it defines vI obsi:<sup>j</sup> which can be used to define an attitude constraint represented as Eq. (24) to be avoided by SCi.

The scenario is illustrated in Figure 2, whereSC1 and SC2 are shown in their different coordinate frames relative to Earth. A thruster attached to SC1 body frame is at vI obs1 , while the circles around SC1 and SC2 are spheres representing the coordinate frames from which their attitude evolves. If both spacecraft are close enough, then vector vI obs1 may intersect a point on the sphere of SC2, whereby the intersection defines vI obs2:<sup>1</sup> in the frame of SC2. The requirement is that as SC2 changes its attitude from q<sup>0</sup> to qf , v<sup>I</sup> cam2 must avoid the cone created around vI obs2:<sup>1</sup>∀t ∈ t0; tf � �.

4.4. Integration of quaternion consensus with Q-CAC avoidance

generated sequence is safe or not. If the next safe quaternion trajectory qi

using the normal quaternion dynamics Eq. (12). Otherwise, Eq. (25) adjusts the qi

and angular velocity ω<sup>i</sup>

<sup>k</sup> � <sup>Δ</sup>tP<sup>i</sup>

ð Þt T

ð Þ<sup>t</sup> <sup>μ</sup>I<sup>4</sup> <sup>þ</sup> <sup>A</sup><sup>~</sup> <sup>i</sup>

<sup>0</sup><sup>4</sup>�<sup>3</sup> � <sup>Δ</sup><sup>t</sup>

<sup>k</sup>þ<sup>1</sup> � qi k � � <sup>¼</sup> <sup>0</sup>,

μ qi

ð Þ<sup>t</sup> <sup>q</sup><sup>i</sup> k,

> 3 7 5 ≥ 0:

j ð Þt � ��<sup>1</sup>

�Δt J<sup>i</sup> � ��<sup>1</sup> <sup>I</sup><sup>3</sup> <sup>0</sup><sup>3</sup>�<sup>4</sup>

2 Πi <sup>k</sup>þ<sup>1</sup> <sup>I</sup><sup>4</sup>

5.1. Q-CAC avoidance in different coordinate frames without consensus

q1

q2


safe, which will be close to but not be exactly qi

the control torque τ<sup>i</sup>

following constraints:

qi <sup>k</sup>þ<sup>1</sup> <sup>¼</sup> qi

qi <sup>k</sup>T q<sup>i</sup>

2 6 4

2 6 4

5. Simulation results

qi

and YALMIP [22] running inside Matlab®.

each other. Their initial quaternions are q<sup>1</sup>

qi

qi

The integration of the quaternion consensus protocol with the Q-CAC collision avoidance in different coordinate frames is a two-stage process. First, the quaternion consensus protocol generates a set of consensus quaternion trajectories using Eq. (15) or (16). Then Eq. (25) tests whether the

Using semidefinite programming, the solutions presented previously are cast as an optimization problem, augmented with a set of LMI constraints and solved for collision-free consensus quaternion trajectories. We consider the algorithm in discrete time. Given the initial attitude

(0) of SCi, (i = 1, ⋯, n), find a sequence of consensus quaternion trajectories that satisfies the

3 7 5

We shall present only three results for attitude multi-path planning in different coordinate frames due to limitation of space. These results will partly be found in [7–10]. For the SDP programming and simulation, we used the available optimization software tools SeDuMi [21]

In this experiment SC1and SC2 are changing their orientation to point an instrument to Earth. They are close to each other, and their thrusters can cause plume impingements to damage

<sup>f</sup> <sup>¼</sup> ½ � <sup>0</sup>:2269 0:0421 0:9567 0:<sup>1776</sup> <sup>T</sup>

<sup>0</sup> ¼ ½0001�

<sup>0</sup> <sup>¼</sup> <sup>q</sup><sup>2</sup>

<sup>f</sup> <sup>¼</sup> ½ � <sup>000</sup>:9903 0:<sup>1387</sup> <sup>T</sup>:

τi k ωi kþ1 qi kþ2


<sup>¼</sup> <sup>I</sup><sup>3</sup> <sup>þ</sup> <sup>Δ</sup>tY<sup>i</sup>

qi kþ1

" #


k � �ω<sup>i</sup>

k

<sup>T</sup>. The desired final quaternions are.

to rotate the SCi optimally to q<sup>i</sup>

Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints

safe. The cycle repeats until consensus is achieved.

safe has been determined,

http://dx.doi.org/10.5772/intechopen.71506

safe can be determined by

unsafe to generate a

113

(31)

(32)

Figure 2. Q-CAC problem in different frames. SC2 must maneuver from q<sup>0</sup> to qf, while vI cam2 must avoid <sup>v</sup><sup>I</sup> obs2 :<sup>1</sup> by at least ∅ ∀ t ∈[t0, tf].

#### 4.3. Determination of obstacle vectors in different coordinate frames

Pursuing the issue of practicality further, given SCi in F<sup>I</sup> SCi and SCj in <sup>F</sup><sup>I</sup> SCj with emanating vectors, an intersection between vectors emanating from F<sup>I</sup> SCj with the sphere centered on <sup>F</sup><sup>I</sup> SCi can be determined, either by using onboard sensors or by application of computational geometry. Given a line segment [p1, p2], originating at p<sup>1</sup> and terminating at p2, a point p = [px py pz] T on [p1, p2] can be tested for intersection with a sphere centered at an external point p<sup>3</sup> with radius r [20]. Therefore, for any v<sup>I</sup> obsj ð Þ<sup>t</sup> in <sup>F</sup><sup>I</sup> SCj , if an intersection point p(t) exists at time t with the sphere centered on F<sup>I</sup> SCi with radius <sup>r</sup>, then <sup>v</sup><sup>I</sup> obsi:j ðÞ¼ t p tð Þ; otherwise, one can set vI obsi:j ðÞ¼� <sup>t</sup> vI cami ð Þt to show that no constraint violation has occurred. The value of r will thus depend on the current application but must be proportional to the urgency of avoiding obstacle vectors originating from other spacecraft. The above formulation effectively completes the decentralization of the avoidance problem which has already been partly decentralized by Eq. (16). Eq. (16) will be written in a semidefinite optimization program, which gives us the privilege to apply further constraints. Therefore, the norm constraints required by quaternion kinematics can be enforced as follows:

$$\boldsymbol{q}\_{k}^{\boldsymbol{i}^{\mathrm{T}}} \left( \boldsymbol{q}\_{k+1}^{i} - \boldsymbol{q}\_{k}^{i} \right) = \mathbf{0} \tag{30}$$

Essentially, Eq. (30) is the discrete time version of qi (t) T q\_ i (t) = 0 or q(t) T q\_(t) = 0. This guarantees that qi (t) T qi (t) = 1 or q(t) T <sup>q</sup>(t) = <sup>n</sup> for nSC, iff <sup>k</sup>qi (0)k = 1 ∀ i.

#### 4.4. Integration of quaternion consensus with Q-CAC avoidance

The integration of the quaternion consensus protocol with the Q-CAC collision avoidance in different coordinate frames is a two-stage process. First, the quaternion consensus protocol generates a set of consensus quaternion trajectories using Eq. (15) or (16). Then Eq. (25) tests whether the generated sequence is safe or not. If the next safe quaternion trajectory qi safe has been determined, the control torque τ<sup>i</sup> and angular velocity ω<sup>i</sup> to rotate the SCi optimally to q<sup>i</sup> safe can be determined by using the normal quaternion dynamics Eq. (12). Otherwise, Eq. (25) adjusts the qi unsafe to generate a qi safe, which will be close to but not be exactly qi safe. The cycle repeats until consensus is achieved.

Using semidefinite programming, the solutions presented previously are cast as an optimization problem, augmented with a set of LMI constraints and solved for collision-free consensus quaternion trajectories. We consider the algorithm in discrete time. Given the initial attitude qi (0) of SCi, (i = 1, ⋯, n), find a sequence of consensus quaternion trajectories that satisfies the following constraints:

$$\begin{aligned} \mathbf{q}\_{k+1}^{i} &= \dot{\mathbf{q}}\_{k}^{i} - \Delta t \mathbf{P}^{i}(t) \mathbf{q}\_{k}^{i} \\ \dot{\mathbf{q}}\_{k}^{i} \mathbf{T} \left(\mathbf{q}\_{k+1}^{i} - \dot{q}\_{k}^{i}\right) &= 0, \\ \begin{bmatrix} \mu & q\_{i}(t)^{\mathrm{T}} \\ q\_{i}(t) & \left(\mu \mathbf{I}\_{4} + \tilde{A}\_{j}^{i}(t)\right)^{-1} \end{bmatrix} &\geq 0. \\ \begin{bmatrix} -\Delta t \left(\boldsymbol{f}^{i}\right)^{-1} & \mathbf{I}\_{3} & \mathbf{0}\_{3\times 4} \\ \mathbf{0}\_{4\times 3} & -\frac{\Delta t}{2} \Pi\_{k+1}^{i} & \mathbf{I}\_{4} \end{bmatrix} &\underbrace{\begin{bmatrix} \boldsymbol{\pi}\_{k}^{i} \\ \boldsymbol{\omega}\_{k+1}^{i} \\ \boldsymbol{\omega}\_{k+2}^{i} \end{bmatrix}}\_{\mathbf{\tilde{x}}\_{k+1}^{i}} &= \underbrace{\begin{bmatrix} \left(\mathbf{I}\_{3} + \Delta t \boldsymbol{Y}\_{k}^{i}\right) \boldsymbol{\omega}\_{k}^{i} \\ \boldsymbol{q}\_{k+1}^{i} \\ \boldsymbol{q}\_{k}^{i} \end{bmatrix}}\_{\mathbf{\tilde{y}}\_{k}^{i}} \end{aligned} \tag{31}$$

#### 5. Simulation results

4.3. Determination of obstacle vectors in different coordinate frames

Figure 2. Q-CAC problem in different frames. SC2 must maneuver from q<sup>0</sup> to qf, while vI

can be determined, either by using onboard sensors or by application of computational geometry. Given a line segment [p1, p2], originating at p<sup>1</sup> and terminating at p2, a point p = [px py pz]

on [p1, p2] can be tested for intersection with a sphere centered at an external point p<sup>3</sup>

depend on the current application but must be proportional to the urgency of avoiding obstacle vectors originating from other spacecraft. The above formulation effectively completes the decentralization of the avoidance problem which has already been partly decentralized by Eq. (16). Eq. (16) will be written in a semidefinite optimization program, which gives us the privilege to apply further constraints. Therefore, the norm constraints required by quaternion

> <sup>k</sup>þ<sup>1</sup> � qi k

> > (t) T q\_ i

(0)k = 1 ∀ i.

ð Þ<sup>t</sup> in <sup>F</sup><sup>I</sup> SCj

SCi with radius <sup>r</sup>, then <sup>v</sup><sup>I</sup>

obsj

qi T <sup>k</sup> <sup>q</sup><sup>i</sup>

<sup>q</sup>(t) = <sup>n</sup> for nSC, iff <sup>k</sup>qi

SCi and SCj in <sup>F</sup><sup>I</sup>

<sup>¼</sup> <sup>0</sup> (30)

T

(t) = 0 or q(t)

obsi:j

ð Þt to show that no constraint violation has occurred. The value of r will thus

SCj with the sphere centered on <sup>F</sup><sup>I</sup>

cam2 must avoid <sup>v</sup><sup>I</sup>

, if an intersection point p(t) exists at time t

ðÞ¼ t p tð Þ; otherwise, one can set

q\_(t) = 0. This guarantees

SCj with emanating

obs2 :<sup>1</sup> by at least

SCi

T

Pursuing the issue of practicality further, given SCi in F<sup>I</sup>

vectors, an intersection between vectors emanating from F<sup>I</sup>

with radius r [20]. Therefore, for any v<sup>I</sup>

kinematics can be enforced as follows:

(t) = 1 or q(t)

Essentially, Eq. (30) is the discrete time version of qi

T

with the sphere centered on F<sup>I</sup>

cami

ðÞ¼� <sup>t</sup> vI

vI obsi:j

∅ ∀ t ∈[t0, tf].

112 Space Flight

that qi (t) T qi We shall present only three results for attitude multi-path planning in different coordinate frames due to limitation of space. These results will partly be found in [7–10]. For the SDP programming and simulation, we used the available optimization software tools SeDuMi [21] and YALMIP [22] running inside Matlab®.

#### 5.1. Q-CAC avoidance in different coordinate frames without consensus

In this experiment SC1and SC2 are changing their orientation to point an instrument to Earth. They are close to each other, and their thrusters can cause plume impingements to damage each other. Their initial quaternions are q<sup>1</sup> <sup>0</sup> <sup>¼</sup> <sup>q</sup><sup>2</sup> <sup>0</sup> ¼ ½0001� <sup>T</sup>. The desired final quaternions are.

$$\begin{aligned} q\_f^1 &= \begin{bmatrix} 0.2269 \ 0.0421 \ 0.9567 \ 0.1776 \end{bmatrix}^\top\\ q\_f^2 &= \begin{bmatrix} 0 \ 0 \ 0.9903 \ 0.1387 \end{bmatrix}^\top. \end{aligned} \tag{32}$$

Three thrusters of SC1 in F<sup>B</sup> SC1 are

$$\begin{aligned} \boldsymbol{\upsilon}\_{\text{obs}\_{1}1}^{B} &= \begin{bmatrix} -0.2132 - 0.0181 \ 0.9768 \end{bmatrix}^{T} \\ \boldsymbol{\upsilon}\_{\text{obs}\_{1}2}^{B} &= \begin{bmatrix} 0.314 \ 0.283 - 0.906 \end{bmatrix}^{T} \\ \boldsymbol{\upsilon}\_{\text{obs}\_{1}3}^{B} &= \begin{bmatrix} -0.112 - 0.133 - 0.985 \end{bmatrix}^{T} . \end{aligned} \tag{33}$$

The initial positions are

0.4807 � 0.2396 0.2112]<sup>T</sup>

The relative offsets are defined as

graph and (c) attitude consensus graph.

FI

FI

FI

A set of initial quaternions were randomly generated, with the following data:

qI

q2

q3

This proves that consensus is indeed achieved by Eq. (16).

5.3. Consensus-based attitude formation acquisition with avoidance

from each other about the z-axis. The previous set of initial data for q<sup>i</sup>

plumes emanating from each of the two other SC by an angle of 30<sup>o</sup>

q off

q off

q off

<sup>1</sup> <sup>¼</sup> ½ � <sup>0001</sup> <sup>T</sup>

SC1 ¼ �½ � <sup>202</sup> <sup>T</sup>

SC2 <sup>¼</sup> ½ � <sup>0</sup>:<sup>502</sup> <sup>T</sup>

(35)

115

(36)

offsets

(37)

SCi were used.

<sup>0</sup> and F<sup>I</sup>

.

SC3 <sup>¼</sup> ½ � <sup>302</sup> <sup>T</sup>:

<sup>0</sup> ¼ �½ � <sup>0</sup>:5101 0:<sup>6112</sup> � <sup>0</sup>:<sup>3187</sup> � <sup>0</sup>:<sup>5145</sup> <sup>T</sup>

<sup>0</sup> ¼ �½ � <sup>0</sup>:9369 0:<sup>2704</sup> � <sup>0</sup>:<sup>1836</sup> � <sup>0</sup>:<sup>124</sup> <sup>T</sup>

Figure 4 (a) shows the solution trajectories while (b) shows the avoidance graph; no constraints are not violated; (c) shows the consensus graph. The final consensus quaternion is qf = [�0.8167

This experiment is to test the capability of the quaternion consensus algorithm in attitude formation acquisition. SCi (i = 1, 2, 3) will maneuver to a consensus formation attitude, with relative offset quaternions defined to enable the sensitive instruments to point at 30<sup>o</sup>

Like the previous experiment, it is desired that the sensitive instruments avoid the thruster

<sup>2</sup> <sup>¼</sup> ½ � <sup>000</sup>:2588 0:<sup>9659</sup> <sup>T</sup>

Figure 4. (a) Reorientation to consensus attitude with intervehicle thruster plume avoidance, (b) avoidance constraints

<sup>3</sup> <sup>¼</sup> ½ � <sup>000</sup>:5 0:<sup>866</sup> <sup>T</sup>:

, which is the normalized average of the initial attitude quaternions.

Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints

http://dx.doi.org/10.5772/intechopen.71506

<sup>0</sup> <sup>¼</sup> ½ � <sup>0</sup>:<sup>1448</sup> � <sup>0</sup>:1151 0:1203 0:<sup>9753</sup> <sup>T</sup>:

A single thruster of SC2 in F<sup>B</sup> SC2 is

$$\boldsymbol{\sigma}\_{\text{abs}}^{\mathcal{B}} = \begin{bmatrix} 0.02981 \ 0.0819 \ 0.9962 \end{bmatrix}^{\mathcal{T}}.\tag{34}$$

It is desired that vI obs2 avoid vI obs1:<sup>1</sup> by 50<sup>o</sup> and avoid vI obs1:<sup>2</sup> and vI obs1:<sup>3</sup> by 30<sup>o</sup> , while both are maneuvering to their desired final attitudes. The trajectories obtained are shown in Figure 3 (a) and (b). This experiment demonstrates that when both constraints are in conflict, the avoidance constraint is superior to the desired final quaternion constraint. As seen from (a), SC2 cannot reconfigure exactly to the desired q<sup>2</sup> <sup>f</sup> due to the satisfaction of the avoidance constraints. This can be resolved by changing either the position of SC2 or SC1.

#### 5.2. Consensus with Q-CAC avoidance in different coordinate frames

In this experiment SCi (i = 1, 2, 3) will maneuver to a consensus attitude. Each carries a sensitive instrument vI cami , pointing in the direction SCi's initial attitude quaternion. In addition, each SCi has only one thruster pointing to the opposite (rear) of SCi's initial attitude. It is desired that the time evolution of the attitude trajectory of the sensitive instrument avoids the thruster plumes emanating from each of the two other SC by 30<sup>o</sup> . From the generated initial quaternions, there is possibility of intersection of the thrusters of SC1 and SC3, with SC2, and the thruster of SC2 may impinge on SC1 or SC3 at any time k.

Figure 3. (a) shows the avoidance between thrusters of SC1and SC2 during reorientation to Earth: SC2 cannot reconfigure to the desired q<sup>2</sup> <sup>f</sup> due to the avoidance constraints. Note that vI obs2 :<sup>1</sup>, vI obs2 :<sup>2</sup> and <sup>v</sup><sup>I</sup> obs2 :<sup>3</sup> are the points of intersections of vI obs1 :<sup>1</sup>, vI obs1 :2 and v<sup>I</sup> obs1 :<sup>3</sup> with SC2. (b) Satisfaction of avoidance constraints: the sudden jumps to and from �1 indicate times when any of vI obs1 :<sup>1</sup>, vI obs1 :<sup>2</sup> and <sup>v</sup><sup>I</sup> obs1 :<sup>3</sup> lost intersection with the sphere of SC2 and therefore was replaced with �vI obs1 :i , i ¼ 1, ⋯, 3.

The initial positions are

(33)

, while both are

Three thrusters of SC1 in F<sup>B</sup>

114 Space Flight

A single thruster of SC2 in F<sup>B</sup>

It is desired that vI

instrument vI

the desired q<sup>2</sup>

obs1 :<sup>2</sup> and <sup>v</sup><sup>I</sup>

and v<sup>I</sup>

vI obs1 :<sup>1</sup>, vI cami

SC1 are

vB

vB

vB

SC2 is

obs2 avoid vI

cannot reconfigure exactly to the desired q<sup>2</sup>

vB

This can be resolved by changing either the position of SC2 or SC1.

plumes emanating from each of the two other SC by 30<sup>o</sup>

thruster of SC2 may impinge on SC1 or SC3 at any time k.

<sup>f</sup> due to the avoidance constraints. Note that vI

5.2. Consensus with Q-CAC avoidance in different coordinate frames

obs1:<sup>1</sup> ¼ �½ � <sup>0</sup>:<sup>2132</sup> � <sup>0</sup>:0181 0:<sup>9768</sup> <sup>T</sup>

obs1:<sup>3</sup> ¼ �½ � <sup>0</sup>:<sup>112</sup> � <sup>0</sup>:<sup>133</sup> � <sup>0</sup>:<sup>985</sup> <sup>T</sup>:

obs2 <sup>¼</sup> ½ � <sup>0</sup>:02981 0:0819 0:<sup>9962</sup> <sup>T</sup>: (34)

obs1:<sup>3</sup> by 30<sup>o</sup>

. From the generated initial quater-

obs2 :<sup>3</sup> are the points of intersections of vI

obs1 :i

, i ¼ 1, ⋯, 3.

obs1 :<sup>1</sup>, vI obs1 :2

<sup>f</sup> due to the satisfaction of the avoidance constraints.

obs1:<sup>2</sup> and vI

obs1:<sup>2</sup> <sup>¼</sup> ½ � <sup>0</sup>:314 0:<sup>283</sup> � <sup>0</sup>:<sup>906</sup> <sup>T</sup>

obs1:<sup>1</sup> by 50<sup>o</sup> and avoid vI

maneuvering to their desired final attitudes. The trajectories obtained are shown in Figure 3 (a) and (b). This experiment demonstrates that when both constraints are in conflict, the avoidance constraint is superior to the desired final quaternion constraint. As seen from (a), SC2

In this experiment SCi (i = 1, 2, 3) will maneuver to a consensus attitude. Each carries a sensitive

has only one thruster pointing to the opposite (rear) of SCi's initial attitude. It is desired that the time evolution of the attitude trajectory of the sensitive instrument avoids the thruster

nions, there is possibility of intersection of the thrusters of SC1 and SC3, with SC2, and the

Figure 3. (a) shows the avoidance between thrusters of SC1and SC2 during reorientation to Earth: SC2 cannot reconfigure to

obs2 :<sup>1</sup>, vI

obs1 :<sup>3</sup> lost intersection with the sphere of SC2 and therefore was replaced with �vI

obs1 :<sup>3</sup> with SC2. (b) Satisfaction of avoidance constraints: the sudden jumps to and from �1 indicate times when any of

obs2 :<sup>2</sup> and <sup>v</sup><sup>I</sup>

, pointing in the direction SCi's initial attitude quaternion. In addition, each SCi

$$\begin{aligned} \mathcal{F}\_{\text{SC}\_1}^{l} &= \begin{bmatrix} -\text{2 0 2} \end{bmatrix}^{T} \\ \mathcal{F}\_{\text{SC}\_2}^{l} &= \begin{bmatrix} 0.5 \ 0 \ 2 \end{bmatrix}^{T} \\ \mathcal{F}\_{\text{SC}\_3}^{l} &= \begin{bmatrix} 3 \ 0 \ 2 \end{bmatrix}^{T} . \end{aligned} \tag{35}$$

A set of initial quaternions were randomly generated, with the following data:

$$q\_0^l = \begin{bmatrix} -0.5101 \ 0.6112 - 0.3187 - 0.5145 \end{bmatrix}^T$$

$$q\_0^2 = \begin{bmatrix} -0.9369 \ 0.2704 - 0.1836 - 0.124 \end{bmatrix}^T \tag{36}$$

$$q\_0^3 = \begin{bmatrix} 0.1448 - 0.1151 \ 0.1203 \ 0.9753 \end{bmatrix}^T.$$

Figure 4 (a) shows the solution trajectories while (b) shows the avoidance graph; no constraints are not violated; (c) shows the consensus graph. The final consensus quaternion is qf = [�0.8167 0.4807 � 0.2396 0.2112]<sup>T</sup> , which is the normalized average of the initial attitude quaternions. This proves that consensus is indeed achieved by Eq. (16).

#### 5.3. Consensus-based attitude formation acquisition with avoidance

This experiment is to test the capability of the quaternion consensus algorithm in attitude formation acquisition. SCi (i = 1, 2, 3) will maneuver to a consensus formation attitude, with relative offset quaternions defined to enable the sensitive instruments to point at 30<sup>o</sup> offsets from each other about the z-axis. The previous set of initial data for q<sup>i</sup> <sup>0</sup> and F<sup>I</sup> SCi were used. Like the previous experiment, it is desired that the sensitive instruments avoid the thruster plumes emanating from each of the two other SC by an angle of 30<sup>o</sup> .

The relative offsets are defined as

$$\begin{aligned} \boldsymbol{q}\_1^{\text{off}} &= \begin{bmatrix} \boldsymbol{0} \ \boldsymbol{0} \ \boldsymbol{0} \ \boldsymbol{1} \end{bmatrix}^T\\ \boldsymbol{q}\_2^{\text{off}} &= \begin{bmatrix} \boldsymbol{0} \ \boldsymbol{0} \ \boldsymbol{0} \ \boldsymbol{0}.2588 \ \boldsymbol{0}.9659 \end{bmatrix}^T\\ \boldsymbol{q}\_3^{\text{off}} &= \begin{bmatrix} \boldsymbol{0} \ \boldsymbol{0} \ \boldsymbol{0}.5 \ \boldsymbol{0}.866 \end{bmatrix}^T. \end{aligned} \tag{37}$$

Figure 4. (a) Reorientation to consensus attitude with intervehicle thruster plume avoidance, (b) avoidance constraints graph and (c) attitude consensus graph.

Figure 5. (a) Reorientation to consensus formation attitude with intervehicle thruster plume avoidance, (b) avoidance constraints graph and (c) attitude consensus graph.

Figure 5 (a) shows the trajectories, while (b) shows the avoidance graph; no constraints are violated. Finally, (c) shows the consensus graph. The final consensus quaternions are.

$$\begin{aligned} q\_f^1 &= \begin{bmatrix} -0.6926 \ 0.6468 - 0.2798 \ 0.1541 \end{bmatrix}^T\\ q\_f^2 &= \begin{bmatrix} -0.8364 \ 0.4455 - 0.2303 \ 0.2212 \end{bmatrix}^T\\ q\_f^3 &= \begin{bmatrix} -0.9232 \ 0.2138 - 0.1652 \ 0.2733 \end{bmatrix}^T. \end{aligned} \tag{38}$$

References

12.460942

825959

conle.2014.04.008

2000. p. 29-30. DOI: ISSN/ISBN:03796566

tory, CALTECH technical report; 1998 NA p. DOI: NA

46(3):1097-1109. DOI: 10.1109/TAES.2010.5545176

2012. p. 1-10. DOI: 10.1109/AERO.2012.6187119

able from: http://hdl.handle.net/10019.1/71905

matic Control. 1991;36(10):1148-1162. DOI: 10.1109/9.90228

[1] Blackwood G, Lay O, Deininger B, Gudim M, Ahmed A, Duren R, Noeckerb C, Barden B. The StarLight mission: A formation-flying stellar interferometer. In: SPIE 4852, Interferometry in Space; 22 August; Waikoloa, Hawaii. SPIE Digital Library; 2002. DOI: 10.1117/

Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints

http://dx.doi.org/10.5772/intechopen.71506

117

[2] Beichman CA. NASA's terrestrial planet finder. In: Darwin and Astronomy: The Infrared Space Interferometer; 17–19 November; Stockholm, Sweden. The Netherlands: Noordwijk;

[3] Ahmed A, Alexander J, Boussalis D, Breckenridge W, Macala G, Mesbahi M, Martin MS, Singh G, Wong E. Cassini Control Analysis Book. Pasadena, CA: Jet Propulsion Labora-

[4] Wen JT, Kreutz-Delgado K. The attitude control problem. IEEE Transactions on Auto-

[5] Kim Y, Mesbahi M. Quadratically constrained attitude control via semidefinite programming. IEEE Transactions on Automatic Control. 2004;49:731-735. DOI: 10.1109/TAC.2004.

[6] Kim Y, Mesbahi M, Singh G, Hadaegh FY. On the convex parameterization of constrained spacecraft reorientation. IEEE Transactions on Aerospace and Electronic Systems. 2010;

[7] Okoloko I, Kim Y. Distributed constrained attitude and position control using graph laplacians. In: ASME Dynamic Systems and Control Conference; 13–15 September; Cam-

[8] Okoloko I, Kim Y. Attitude synchronization of multiple spacecraft with cone avoidance constraints. In: IEEE Aerospace Conference; 3–10 March; Big Sky, Montana. IEEEXplore;

[9] Okoloko I, Kim Y. Attitude synchronization of multiple spacecraft with cone avoidance constraints. Systems & Control Letters. 2014;69:73-79. DOI: https://doi.org/10.1016/j.sys

[10] Okoloko I. Multi-Path Planning and Multi-Body Constrained Attitude Control [dissertation]. Stellenbosch, South Africa: PhD Thesis: Stellenbosch University; 2012. p. 185. Avail-

[11] Bullo F, Murray R M, Sarti A. Control on the sphere and reduced attitude stabilization. In: IFAC symposium on nonlinear control systems; 25-28 June; Tahoe City, CA. Elsevier;

[12] Ren W. Distributed attitude alignment in spacecraft formation flying. International Journal of Adaptive Control and Signal Processing. 2006;21(2–3):95-113. DOI: 10.1002/acs.916

1995. p. 495-501. DOI: https://doi.org/10.1016/S1474-6670(17)46878-9

bridge, Massachusetts. ASME; 2010. p. 377-383. DOI: 10.1115/DSCC2010-4036

The differences of these quaternions are 30<sup>o</sup> apart about the same axis. Clearly, the algorithm is capable of attitude formation acquisition with avoidance.

#### 6. Conclusion

In this chapter, a method of consensus with quaternion-based attitude maneuver with avoidance, of multiple networked communicating spacecraft, was presented. The presentation is composed of aspects of solutions we previously developed, by combining consensus theory and Q-CAC optimization theory. The solutions enable a team of spacecraft to point to the same direction or to various formation patterns, while they avoid an arbitrary number of attitude obstacles or exclusion zones in any coordinate frames. The proof of stability of the Laplacian-like dynamics was also presented. Simulation results also demonstrated the effectiveness of the algorithm. We hope to implement the algorithms using rotorcraft and specialized hardware.

#### Author details

Innocent Okoloko

Address all correspondence to: okoloko@ieee.org

Department of Electrical Engineering, Universidad de Ingenieria y Tecnologia, Lima, Peru

#### References

Figure 5 (a) shows the trajectories, while (b) shows the avoidance graph; no constraints are

Figure 5. (a) Reorientation to consensus formation attitude with intervehicle thruster plume avoidance, (b) avoidance

<sup>f</sup> ¼ �½ � <sup>0</sup>:6926 0:<sup>6468</sup> � <sup>0</sup>:2798 0:<sup>1541</sup> <sup>T</sup>

<sup>f</sup> ¼ �½ � <sup>0</sup>:8364 0:<sup>4455</sup> � <sup>0</sup>:2303 0:<sup>2212</sup> <sup>T</sup>

(38)

<sup>f</sup> ¼ �½ � <sup>0</sup>:9232 0:<sup>2138</sup> � <sup>0</sup>:1652 0:<sup>2733</sup> <sup>T</sup>:

The differences of these quaternions are 30<sup>o</sup> apart about the same axis. Clearly, the algorithm is

In this chapter, a method of consensus with quaternion-based attitude maneuver with avoidance, of multiple networked communicating spacecraft, was presented. The presentation is composed of aspects of solutions we previously developed, by combining consensus theory and Q-CAC optimization theory. The solutions enable a team of spacecraft to point to the same direction or to various formation patterns, while they avoid an arbitrary number of attitude obstacles or exclusion zones in any coordinate frames. The proof of stability of the Laplacian-like dynamics was also presented. Simulation results also demonstrated the effectiveness of the algorithm. We hope to implement the algorithms using rotorcraft and special-

Department of Electrical Engineering, Universidad de Ingenieria y Tecnologia, Lima, Peru

violated. Finally, (c) shows the consensus graph. The final consensus quaternions are.

q1

constraints graph and (c) attitude consensus graph.

q2

q3

capable of attitude formation acquisition with avoidance.

6. Conclusion

116 Space Flight

ized hardware.

Author details

Innocent Okoloko

Address all correspondence to: okoloko@ieee.org


[13] Fax AJ. Optimal and Cooperative Control of Vehicle Formations [thesis]. Pasadena, CA: PhD Thesis: CALTECH; 2002. p. 135. Available from: thesis.library.caltech.edu/4230/1/ Fax\_ja\_2002.pdf

**Chapter 7**

Provisional chapter

**Mars Networks-Based Navigation: Observability and**

DOI: 10.5772/intechopen.73605

In order to achieve more scientific returns for Mars, future Mars landers will be required to land at certain landing point with special scientific interest. Therefore, autonomous navigation is indispensable during the Mars approach, entry, and landing phase. However, the number of beacons or the Mars orbiters which can provide the navigation service is so limited and the line-of-sight visibility cannot be guaranteed during the landing period. So the navigation scheme especially the beacon configuration has to be optimized in order to efficiently use the limited navigation information. This chapter aims to analyze the feasibility and optimize the performance of the Mars Networks-based navigation scheme for the Mars pinpoint landing. The observability of navigation system is used as an index describing the navigation capability. Focusing on the relationship between the configuration of radio beacons and observability, the Fisher information matrix is introduced to analytically derive the degree of observability, which gives valuable conclusions for navigation system design. In order to improve the navigation performance, the navigation scheme is optimized by beacon configuration optimization, which gives the best locations of beacons (or the best orbit of navigation orbiters). This is the main approach to

As the most similar planet to the Earth in the Solar system, Mars is considered as an ideal target for planetary exploration [1, 2]. Since the 1960s, humans have investigated the Mars exploration missions in the near distance. With the development of aerospace science and

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Mars Networks-Based Navigation: Observability and

**Optimization**

Optimization

Shengying Zhu

Abstract

1. Introduction

Shengying Zhu

Zhengshi Yu, Pingyuan Cui, Rui Xu and

Zhengshi Yu, Pingyuan Cui, Rui Xu and

http://dx.doi.org/10.5772/intechopen.73605

improve the navigation capability.

Keywords: Mars networks, navigation, observability, optimization

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter


#### **Mars Networks-Based Navigation: Observability and Optimization** Mars Networks-Based Navigation: Observability and Optimization

DOI: 10.5772/intechopen.73605

Zhengshi Yu, Pingyuan Cui, Rui Xu and Shengying Zhu Zhengshi Yu, Pingyuan Cui, Rui Xu and Shengying Zhu

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.73605

#### Abstract

[13] Fax AJ. Optimal and Cooperative Control of Vehicle Formations [thesis]. Pasadena, CA: PhD Thesis: CALTECH; 2002. p. 135. Available from: thesis.library.caltech.edu/4230/1/

[14] Dimarogonas DV, Tsiotras P, Kyriakopoulos KJ. Leader-follower cooperative attitude control of multiple rigid bodies. Systems and Control Letters, DOI. 2009;58(6):429-435

[15] Kuipers JB. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality. 1st ed. Princeton, NJ: Princeton University Press; 2002.

[16] Peng L, Zhao Y, Tian B, Zhang J, Bing-Hong W, Hai-Tao Z, Zhou T. Consensus of selfdriven agents with avoidance of collisions. Physical Review. 2009;79(E). DOI: 10.1103/

[17] Olfati-Saber R. Flocking for multi-agent dynamic systems: Algorithms and theory. IEEE Transactions on Automatic Control. 2006;51(3):401-420. DOI: 10.1109/TAC.2005.864190

[18] Golub GH, Van Loan CF. Matrix Computations. 3rd ed. Baltimore, MD: Johns Hopkins

[19] Hughes PC. Spacecraft Attitude Dynamics. 2nd ed. Mineola, NY: Dover Publications Inc;

[20] Eberly DH. 3D Game Engine Design: A Practical Approach to Real-Time Computer Graphics. 2nd ed. London, UK: Taylor & Francis; 2012. p. 1015. DOI: ISBN: 978-0-12-229063-3

[21] Sturm JF. Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optimization methods and software. 1998;11(12):625-653. DOI: http://dx.doi.org/10.1080/

[22] Lofberg J. Yalmip: A toolbox for modelling and optimization in Matlab. In: IEEE CACSD Conference; 2–4 Sept.; Taipei, Taiwan. IEEEXplore; 2004. p. 284-289. DOI: 10.1109/CACSD.

Fax\_ja\_2002.pdf

118 Space Flight

https://doi.org/10.1016/j.sysconle.2009.02.002

University Press; 1996. p. 699. ISBN-13: 978–0801854149

p. 371. ISBN-13:978–0691102986

2004. p. 592. ISBN13:9780486439259

PhysRevE.79.026113

10556789908805766

2004.1393890

In order to achieve more scientific returns for Mars, future Mars landers will be required to land at certain landing point with special scientific interest. Therefore, autonomous navigation is indispensable during the Mars approach, entry, and landing phase. However, the number of beacons or the Mars orbiters which can provide the navigation service is so limited and the line-of-sight visibility cannot be guaranteed during the landing period. So the navigation scheme especially the beacon configuration has to be optimized in order to efficiently use the limited navigation information. This chapter aims to analyze the feasibility and optimize the performance of the Mars Networks-based navigation scheme for the Mars pinpoint landing. The observability of navigation system is used as an index describing the navigation capability. Focusing on the relationship between the configuration of radio beacons and observability, the Fisher information matrix is introduced to analytically derive the degree of observability, which gives valuable conclusions for navigation system design. In order to improve the navigation performance, the navigation scheme is optimized by beacon configuration optimization, which gives the best locations of beacons (or the best orbit of navigation orbiters). This is the main approach to improve the navigation capability.

Keywords: Mars networks, navigation, observability, optimization

#### 1. Introduction

As the most similar planet to the Earth in the Solar system, Mars is considered as an ideal target for planetary exploration [1, 2]. Since the 1960s, humans have investigated the Mars exploration missions in the near distance. With the development of aerospace science and

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

technology, the manner of Mars exploration has shifted from flyby/orbiting to landing and roving explorations. Considering scientific returns and exploration capabilities, Mars landing exploration is also essential and is one of the most popular tasks of human deep space exploration in the near future. The representative Mars landing missions including NASA's Viking 1 and 2, Mars Pathfinder (MPF), Mars Exploration Rovers (MER, including the Spirit and Opportunity rovers), Phoenix, Mars Science Laboratory (MSL, including the Curiosity rover), and ESA's Mars Express/Beagle 2 mission. All of these greatly inspire the development of advanced guidance, navigation, and control (GNC) technologies.

capability of Mars network can be further increased. Focusing on how to fulfill the function of a Mars network, Ely firstly established the basic principle to design a constellation for navigation [10]. Then, taking the Mean of the Position Accuracy Response Time (MPART) as the performance index, the constellation configuration was optimized [11]. In [12], the number of orbiters and the coverage was considered to design the Martian navigation constellations envisaged in the ESA's Martian Constellation for Precise Object Location program. The optimization method of the above researches is inherited from the Global Positioning System (GPS). The global navigation performance was emphasized. For the limited amount of Mars orbiters, global coverage is difficult to realized, and local navigation performance should be investigated thoroughly for specific missions. Moreover, the effect of geometric configuration of the Mars network on the navigation performance should be revealed clearly. Inspired by these requirements, Yu et al. optimized the orbits of Mars orbiters in the observability point of view, and tried to explain the relationship between the configuration of beacons and orbiters

Mars Networks-Based Navigation: Observability and Optimization

http://dx.doi.org/10.5772/intechopen.73605

121

To optimize the configuration of the radio beacons, a performance index should be firstly setup. The observability of the navigation system is selected as the performance index since it reflects the navigation capability directly. A lot of work has investigated the observability of linear and nonlinear dynamic systems [6, 15–17]. However, the analytic relationship between geometric configuration and observability has never been revealed. According to Cramér-Rao inequality [18], the inverse of the Fisher Information Matrix (FIM) estimates the lower bound of the estimation error. Therefore, FIM can be used to quantify the observability of the navigation system [19–21]. In this circumstance, some valuable analytic conclusions about the navi-

Based on the requirement of the navigation optimization for Mars pinpoint landing, this chapter discusses the design and optimization of the Mars Networks-based navigation during Mars entry phase. Firstly, the Mars Networks-based navigation scheme is introduced, and the dynamic model and the observation model are given. Based on the navigation system, the observability of the Mars entry navigation analysis, and the analysis methods based on the quadratic approximation and Fisher information matrix are proposed. The relationship between the observability and the beacon configuration is derived, and the theoretically optimal configuration is given. Considering the constraints of Mars entry scenario, the ground beacons and the orbit of Mars orbiters are optimized based on observability based on an entry trajectory. The simulations also indicate the improved navigation

In the dynamical model with respect to a stationary atmosphere of a rotating planet, the 6 dimensional states x of the entry vehicle include r (radius from the center of Mars to the

and the navigation capability [13, 14].

gation design can thus be obtained.

2. Mars networks-based navigation scheme

2.1. Dynamic model of Mars entry phase

performance.

During the past 50 years of Mars exploration, 46 Mars exploration spacecraft have been launched. The overall success rate is only 41.3% though. Furthermore, among the 20 Mars landing attempts, only 7 robotic rovers were successful. The success rate for Mars landing missions is only 35%. Among the failed landing missions, most failures occur during the landing phase. The pinpoint landing has to be based on the precise autonomous navigation technology.

In the entry phase of a Mars landing, the lander is covered by a heat shield which blocks the optical sensor measurement, causing that all landers relied on the Inertial Measurement Unit (IMU) recursion. The initial errors of the lander cannot be corrected by IMU data. Even worse, the recursion errors using IMU are accumulated due to the sensor bias and noise. To overcome the incapability of IMU, the Mars Network-based Mars entry navigation is developed based on high frequency radio communication between the lander with ground or orbiting radio beacons [3–5]. Involving the radio measurement date into a navigation filter, the position and velocity of the lander can be optimally estimated.

The Mars Network-based Mars entry navigation is faced with two challenging. One is that the geometric configuration of the radio beacons affects the navigation performance. The other is that the available beacons at present are very limited. Considering these two factors, effort should be devoted to optimizing the configuration of radio beacons to maximize the function of the limited beacons. In [7], the navigation accuracy from the Extend Kalman Filter (EKF) by processing the radio measurements is analyzed, and the optimal configuration of ground beacons is selected among potential beacon position. Yu focused on the navigation observability and take it as a performance index to optimize the configuration of radio beacons [8]. The research on ground beacons, to some extent, inspired the future Mars landing navigation. However, the practice application of ground beacon-based navigation is hardly applied in practice. The first concern is that no ground beacon is available. Even if several beacons are distributed on Mars surface, it's still a tough job to place them exactly at the optimal locations. Moreover, the accurate positions of the beacons are hardly obtained accurately. Considering the immovability of ground beacons, the potential location areas are constrained by the line-of-sight visibility, resulting in an unsatisfactory beacon configuration during the entry phase.

As a substitution of ground radio beacons, the Mars orbiters which can also serve as beacons for Mars Network-Based Navigation are of more practice value. Currently, the operational orbiter around Mars includes 2001 Mars Odyssey and 2005 Mars Reconnaissance Orbiter. With another forthcoming spacecraft Mars Atmosphere and Volatile Evolution (MAVEN) [9], the capability of Mars network can be further increased. Focusing on how to fulfill the function of a Mars network, Ely firstly established the basic principle to design a constellation for navigation [10]. Then, taking the Mean of the Position Accuracy Response Time (MPART) as the performance index, the constellation configuration was optimized [11]. In [12], the number of orbiters and the coverage was considered to design the Martian navigation constellations envisaged in the ESA's Martian Constellation for Precise Object Location program. The optimization method of the above researches is inherited from the Global Positioning System (GPS). The global navigation performance was emphasized. For the limited amount of Mars orbiters, global coverage is difficult to realized, and local navigation performance should be investigated thoroughly for specific missions. Moreover, the effect of geometric configuration of the Mars network on the navigation performance should be revealed clearly. Inspired by these requirements, Yu et al. optimized the orbits of Mars orbiters in the observability point of view, and tried to explain the relationship between the configuration of beacons and orbiters and the navigation capability [13, 14].

To optimize the configuration of the radio beacons, a performance index should be firstly setup. The observability of the navigation system is selected as the performance index since it reflects the navigation capability directly. A lot of work has investigated the observability of linear and nonlinear dynamic systems [6, 15–17]. However, the analytic relationship between geometric configuration and observability has never been revealed. According to Cramér-Rao inequality [18], the inverse of the Fisher Information Matrix (FIM) estimates the lower bound of the estimation error. Therefore, FIM can be used to quantify the observability of the navigation system [19–21]. In this circumstance, some valuable analytic conclusions about the navigation design can thus be obtained.

Based on the requirement of the navigation optimization for Mars pinpoint landing, this chapter discusses the design and optimization of the Mars Networks-based navigation during Mars entry phase. Firstly, the Mars Networks-based navigation scheme is introduced, and the dynamic model and the observation model are given. Based on the navigation system, the observability of the Mars entry navigation analysis, and the analysis methods based on the quadratic approximation and Fisher information matrix are proposed. The relationship between the observability and the beacon configuration is derived, and the theoretically optimal configuration is given. Considering the constraints of Mars entry scenario, the ground beacons and the orbit of Mars orbiters are optimized based on observability based on an entry trajectory. The simulations also indicate the improved navigation performance.

#### 2. Mars networks-based navigation scheme

#### 2.1. Dynamic model of Mars entry phase

technology, the manner of Mars exploration has shifted from flyby/orbiting to landing and roving explorations. Considering scientific returns and exploration capabilities, Mars landing exploration is also essential and is one of the most popular tasks of human deep space exploration in the near future. The representative Mars landing missions including NASA's Viking 1 and 2, Mars Pathfinder (MPF), Mars Exploration Rovers (MER, including the Spirit and Opportunity rovers), Phoenix, Mars Science Laboratory (MSL, including the Curiosity rover), and ESA's Mars Express/Beagle 2 mission. All of these greatly inspire the development

During the past 50 years of Mars exploration, 46 Mars exploration spacecraft have been launched. The overall success rate is only 41.3% though. Furthermore, among the 20 Mars landing attempts, only 7 robotic rovers were successful. The success rate for Mars landing missions is only 35%. Among the failed landing missions, most failures occur during the landing phase. The pinpoint landing has to be based on the precise autonomous navigation

In the entry phase of a Mars landing, the lander is covered by a heat shield which blocks the optical sensor measurement, causing that all landers relied on the Inertial Measurement Unit (IMU) recursion. The initial errors of the lander cannot be corrected by IMU data. Even worse, the recursion errors using IMU are accumulated due to the sensor bias and noise. To overcome the incapability of IMU, the Mars Network-based Mars entry navigation is developed based on high frequency radio communication between the lander with ground or orbiting radio beacons [3–5]. Involving the radio measurement date into a navigation filter, the position and

The Mars Network-based Mars entry navigation is faced with two challenging. One is that the geometric configuration of the radio beacons affects the navigation performance. The other is that the available beacons at present are very limited. Considering these two factors, effort should be devoted to optimizing the configuration of radio beacons to maximize the function of the limited beacons. In [7], the navigation accuracy from the Extend Kalman Filter (EKF) by processing the radio measurements is analyzed, and the optimal configuration of ground beacons is selected among potential beacon position. Yu focused on the navigation observability and take it as a performance index to optimize the configuration of radio beacons [8]. The research on ground beacons, to some extent, inspired the future Mars landing navigation. However, the practice application of ground beacon-based navigation is hardly applied in practice. The first concern is that no ground beacon is available. Even if several beacons are distributed on Mars surface, it's still a tough job to place them exactly at the optimal locations. Moreover, the accurate positions of the beacons are hardly obtained accurately. Considering the immovability of ground beacons, the potential location areas are constrained by the line-of-sight visibility, resulting in an unsatisfactory beacon configuration

As a substitution of ground radio beacons, the Mars orbiters which can also serve as beacons for Mars Network-Based Navigation are of more practice value. Currently, the operational orbiter around Mars includes 2001 Mars Odyssey and 2005 Mars Reconnaissance Orbiter. With another forthcoming spacecraft Mars Atmosphere and Volatile Evolution (MAVEN) [9], the

of advanced guidance, navigation, and control (GNC) technologies.

velocity of the lander can be optimally estimated.

technology.

120 Space Flight

during the entry phase.

In the dynamical model with respect to a stationary atmosphere of a rotating planet, the 6 dimensional states x of the entry vehicle include r (radius from the center of Mars to the vehicle's center of mass), θ (longitude), ϕ (latitude), V (relative velocity), γ (flight path angle), and Ψ (heading angle, with Ψ = 0 as due east). The motion of the entry vehicle is governed by the following state equations:

$$\begin{aligned} \dot{r} &= V \sin \gamma \\ \dot{\theta} &= V \cos \gamma \cos \Psi / (r \cos \phi) \\ \dot{\phi} &= V \cos \gamma \sin \Psi / r \\ \dot{V} &= -d - g \sin \gamma \\ \dot{\gamma} &= \left[ l \cos \sigma - \left( g - V^2 / r \right) \cos \gamma \right] / V + 2\omega \left( \tan \gamma \sin \Psi \cos \phi - \sin \phi \right) \\ \dot{\Psi} &= -\left( l \sin \sigma + V^2 \cos^2 \gamma \cos \Psi \tan \phi / r \right) / (V \cos \gamma) + 2\omega \cos \Psi \cos \phi \end{aligned} \tag{1}$$

In the equation, σ is the banking angle, which is fixed at 0 in the following analysis. ω refers to the rotation rate of Mars. For simplicity, the second order terms of ω are neglected, which is feasible because the value of ω is quite small. Then the gravity acceleration g, lift and drag accelerations l and d are given by

$$\mathbf{g} = \mu/r^2\tag{2}$$

The relative velocity model is given by

<sup>V</sup> is the velocity measurement noise.

mized based on the observability analysis.

Consider the following nonlinear system:

3. Observability of the navigation system

3.1. Observability analysis based on the quadratic approximation

vector. Define <sup>h</sup> : <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>m</sup> as the nonlinear measurement operator. The Lie algebra is an efficient tool for observability analysis. For the k

> L<sup>k</sup>þ<sup>1</sup> <sup>f</sup> hj <sup>¼</sup> <sup>X</sup><sup>n</sup>

∇Lk

<sup>f</sup> h ¼ ∇Lk

i¼1

<sup>f</sup> hj <sup>¼</sup> <sup>∂</sup>Lk

<sup>f</sup> h<sup>1</sup> � �<sup>T</sup>

∂L<sup>k</sup> f hj ∂xi

k ¼ 0, 1, ⋯ j ¼ 1, 2⋯, m

f hj ∂x<sup>1</sup>

Regarding the zero-order Lie derivative of the jth measurement function hj as hj itself, the

<sup>f</sup> <sup>i</sup> <sup>¼</sup> <sup>∇</sup>Lk

, <sup>⋯</sup>, <sup>∂</sup>Lk

, ⋯, ∇Lk

� �<sup>T</sup> � �<sup>T</sup>

" #

<sup>f</sup> hjf,

f hj ∂xn

<sup>f</sup> hm

th measurement function, which can be expressed as Lk

with respect to the state equation f can be computed as:

<sup>f</sup> hj is defined as

∇Lk

and ε<sup>i</sup>

j

The differential of Lk

<sup>f</sup> h is given as

matrix∇Lk

yVi <sup>¼</sup> Vi <sup>þ</sup> <sup>ε</sup><sup>i</sup>

where Vi is the real line-of-sight relative velocity between the lander and the i

<sup>V</sup> <sup>¼</sup> dRi=dt <sup>þ</sup> <sup>ε</sup><sup>i</sup>

With different radio beacons come different navigation scenarios. Without losing the generality, the observation model can be summarized as y ¼ h xð Þ. Obviously, both radio measurements in Eqs. (6) and (7) are nonlinear. Moreover, the navigation performance is closely related to the geometric configuration of radio beacons. Therefore, the beacon configuration needs to be opti-

> <sup>Σ</sup> : <sup>x</sup>\_ <sup>¼</sup> f xð Þ <sup>y</sup> <sup>¼</sup> h xð Þ �

where x∈ R<sup>n</sup> is the n-dimensional state vector and y∈ R<sup>m</sup> is the m-dimensional observation

<sup>V</sup> (7)

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Mars Networks-Based Navigation: Observability and Optimization

th radio beacon,

123

(8)

<sup>f</sup> hj

(9)

(10)

(11)

th order Lie derivative of the

<sup>f</sup> hj, the k + 1th order Lie derivative Lkþ<sup>1</sup>

$$l = 0.5 \rho V^2 \mathcal{C}\_l \mathcal{S}/m\tag{3}$$

$$d = 0.5\rho V^2 \mathbb{C}\_d \text{S/m} \tag{4}$$

where μ is the Martian gravitational constant. S and m denotes the reference area and mass of the entry vehicle, and Cl and Cd are the lift and drag coefficients respectively. Furthermore, the Mars atmospheric density r is approximated by the conventional exponential model

$$\rho = \rho\_0 \exp[(r\_0 - r)/h\_s] \tag{5}$$

where <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>4</sup> kg/m3 is the reference density, <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>3437</sup>:2 km is the reference radial position, and hs ¼ 7500 m refers to the atmospheric scale height. The dynamical model of the entry vehicle is abbreviated as x\_ ¼ f xð Þ.

#### 2.2. Observation model

The radio ranging and velocity data between the lander and the radio beacon can be measured through radio communication, given by

$$\begin{aligned} y\_{R\_i} &= R\_i + \varepsilon\_R^i \\ &= \sqrt{\left(\mathbf{x}\_{\mathcal{B}}^i - \mathbf{x}\right)^2 + \left(y\_{\mathcal{B}}^i - \mathbf{y}\right)^2 + \left(z\_{\mathcal{B}}^i - z\right)^2} + \varepsilon\_R^i \\ &\ge = r\cos\phi\cos\theta, \quad y = r\cos\phi\sin\theta, \quad z = r\sin\phi \end{aligned} \tag{6}$$

where Ri is the real range between the lander and the i th beacon, xi <sup>B</sup>, y<sup>i</sup> <sup>B</sup>, and z<sup>i</sup> <sup>B</sup> represent respectively the triaxial position components of the beacon, and ε<sup>i</sup> <sup>R</sup> is the radio ranging measurement noise.

The relative velocity model is given by

vehicle's center of mass), θ (longitude), ϕ (latitude), V (relative velocity), γ (flight path angle), and Ψ (heading angle, with Ψ = 0 as due east). The motion of the entry vehicle is governed by

<sup>=</sup><sup>r</sup> � �cos<sup>γ</sup> � �=<sup>V</sup> <sup>þ</sup> <sup>2</sup><sup>ω</sup> tanγsinΨcos<sup>ϕ</sup> � sin<sup>ϕ</sup> � �

In the equation, σ is the banking angle, which is fixed at 0 in the following analysis. ω refers to the rotation rate of Mars. For simplicity, the second order terms of ω are neglected, which is feasible because the value of ω is quite small. Then the gravity acceleration g, lift and drag

g ¼ μ=r

where μ is the Martian gravitational constant. S and m denotes the reference area and mass of the entry vehicle, and Cl and Cd are the lift and drag coefficients respectively. Furthermore, the

where <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>4</sup> kg/m3 is the reference density, <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>3437</sup>:2 km is the reference radial position, and hs ¼ 7500 m refers to the atmospheric scale height. The dynamical model of the

The radio ranging and velocity data between the lander and the radio beacon can be measured

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x ¼ r cosϕcosθ, y ¼ r cosϕsinθ, z ¼ r sinϕ

<sup>B</sup> � <sup>z</sup> � �<sup>2</sup> <sup>q</sup>

<sup>þ</sup> zi

<sup>þ</sup> yi <sup>B</sup> � <sup>y</sup> � �<sup>2</sup>

<sup>l</sup> <sup>¼</sup> <sup>0</sup>:5rV<sup>2</sup>

<sup>d</sup> <sup>¼</sup> <sup>0</sup>:5rV<sup>2</sup>

Mars atmospheric density r is approximated by the conventional exponential model

<sup>2</sup> (2)

ClS=m (3)

CdS=m (4)

r ¼ r0exp ð Þ r<sup>0</sup> � r =hs ½ � (5)

<sup>þ</sup> <sup>ε</sup><sup>i</sup> R

th beacon, xi

<sup>B</sup>, y<sup>i</sup>

<sup>B</sup>, and z<sup>i</sup>

<sup>R</sup> is the radio ranging measure-

cos<sup>2</sup>γcosΨtanϕ=<sup>r</sup> � �=ð Þþ <sup>V</sup>cos<sup>γ</sup> <sup>2</sup>ωcosΨcos<sup>ϕ</sup>

(1)

(6)

<sup>B</sup> represent

the following state equations:

122 Space Flight

accelerations l and d are given by

entry vehicle is abbreviated as x\_ ¼ f xð Þ.

through radio communication, given by

yRi <sup>¼</sup> Ri <sup>þ</sup> <sup>ε</sup><sup>i</sup>

where Ri is the real range between the lander and the i

xi <sup>B</sup> � <sup>x</sup> � �<sup>2</sup>

respectively the triaxial position components of the beacon, and ε<sup>i</sup>

¼

R

2.2. Observation model

ment noise.

r\_ ¼ Vsinγ

<sup>θ</sup>\_ <sup>¼</sup> <sup>V</sup>cosγcosΨ<sup>=</sup> <sup>r</sup>cos<sup>ϕ</sup> � �

<sup>ϕ</sup>\_ <sup>¼</sup> <sup>V</sup>cosγsinΨ=<sup>r</sup> <sup>V</sup>\_ ¼ �<sup>d</sup> � <sup>g</sup>sin<sup>γ</sup> <sup>γ</sup>\_ <sup>¼</sup> <sup>l</sup>cos<sup>σ</sup> � <sup>g</sup> � <sup>V</sup><sup>2</sup>

<sup>Ψ</sup>\_ ¼ � <sup>l</sup>sin<sup>σ</sup> <sup>þ</sup> <sup>V</sup><sup>2</sup>

$$y\_{V\_i} = V\_i + \varepsilon\_V^t = d\mathbb{R}\_i/dt + \varepsilon\_V^t \tag{7}$$

where Vi is the real line-of-sight relative velocity between the lander and the i th radio beacon, and ε<sup>i</sup> <sup>V</sup> is the velocity measurement noise.

With different radio beacons come different navigation scenarios. Without losing the generality, the observation model can be summarized as y ¼ h xð Þ. Obviously, both radio measurements in Eqs. (6) and (7) are nonlinear. Moreover, the navigation performance is closely related to the geometric configuration of radio beacons. Therefore, the beacon configuration needs to be optimized based on the observability analysis.

#### 3. Observability of the navigation system

#### 3.1. Observability analysis based on the quadratic approximation

Consider the following nonlinear system:

$$\Sigma: \begin{cases} \dot{\mathfrak{x}} = f(\mathfrak{x}) \\ \mathfrak{y} = h(\mathfrak{x}) \end{cases} \tag{8}$$

where x∈ R<sup>n</sup> is the n-dimensional state vector and y∈ R<sup>m</sup> is the m-dimensional observation vector. Define <sup>h</sup> : <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>m</sup> as the nonlinear measurement operator.

The Lie algebra is an efficient tool for observability analysis. For the k th order Lie derivative of the j th measurement function, which can be expressed as Lk <sup>f</sup> hj, the k + 1th order Lie derivative Lkþ<sup>1</sup> <sup>f</sup> hj with respect to the state equation f can be computed as:

$$\begin{aligned} L\_f^{k+1} h\_{\dot{l}} &= \sum\_{i=1}^n \frac{\partial L\_f^k h\_{\dot{l}}}{\partial \mathbf{x}\_i} f\_i = \nabla L\_f^k h\_{\dot{l}} f\_{\dot{r}}\\ k &= 0, 1, \cdots \quad \dot{j} = 1, 2 \cdots, m \end{aligned} \tag{9}$$

The differential of Lk <sup>f</sup> hj is defined as

$$\nabla L\_f^k h\_{\dot{f}} = \begin{bmatrix} \frac{\partial L\_f^k h\_{\dot{f}}}{\partial \mathbf{x}\_1}, & \cdots, & \frac{\partial L\_f^k h\_{\dot{f}}}{\partial \mathbf{x}\_n} \end{bmatrix} \tag{10}$$

Regarding the zero-order Lie derivative of the jth measurement function hj as hj itself, the matrix∇Lk <sup>f</sup> h is given as

$$\nabla L\_f^k \mathbf{h} = \left[ \left( \nabla L\_f^k \mathbf{h}\_1 \right)^T, \quad \cdots, \quad \left( \nabla L\_f^k \mathbf{h}\_m \right)^T \right]^T \tag{11}$$

It is proven that the dynamical system Σ at state x<sup>0</sup> is locally observable if the observability matrix O<sup>Σ</sup> given below has the rank of n.

$$\mathbf{O}\_{\Sigma} = \left[ \left( \nabla L\_f^0 \mathbf{h} \right)^T, \quad \left( \nabla L\_f^1 \mathbf{h} \right)^T, \quad \cdots, \quad \left( \nabla L\_f^{n-1} \mathbf{h} \right)^T \right]\_{\mathbf{x} = \mathbf{x}\_0}^T \tag{12}$$

O<sup>Σ</sup> ¼ ∇L<sup>0</sup>

¼ J<sup>0</sup> L � �<sup>T</sup>

, ⋯, J<sup>k</sup> Lm � �<sup>T</sup> h i<sup>T</sup>

Ol

where J<sup>k</sup>

Obtaining J<sup>0</sup>

<sup>L</sup> ¼ J<sup>k</sup> L1 � �<sup>T</sup>

the computation cost.

Lj and H<sup>0</sup>

the performance index, given by

observability.

f h � �<sup>T</sup>

> , J<sup>1</sup> L � �<sup>T</sup>

> > .

Linearize the dynamical and observation model by first-order approximation

Construct the observability matrix according to the linear system theory

<sup>Σ</sup> <sup>¼</sup> Jh ð ÞT, JhJf

approximation of Lie derivatives and the linearization of state equation.

<sup>δ</sup> <sup>¼</sup> <sup>1</sup> condð Þ O<sup>Σ</sup>

higher order terms of <sup>x</sup> � <sup>x</sup><sup>0</sup> may appear when computing Lkþ<sup>1</sup>

, ∇L<sup>1</sup> f h � �<sup>T</sup>

� �<sup>T</sup> h i<sup>T</sup>

� �<sup>T</sup> � �<sup>T</sup>

, ⋯, J<sup>n</sup>�<sup>1</sup> L

differential of h is needed here to compute the Jacobian and Hessian matrices, reducing largely

f ≈ f <sup>0</sup> þ Jfð Þ x � x<sup>0</sup> h ≈ h<sup>0</sup> þ Jhð Þ x � x<sup>0</sup>

� �<sup>T</sup>

The Hessian matrix is involved in the quadratic approximation, improving the accuracy of observability analysis compared with the linearized observability analysis. However, the

is approximated to a higher order. In this case, the predetermined presentation form in Eq. (13) is no longer valid. One way to defeat this case is to increase the approximation order of Lie derivatives. Note that tensor calculus can be involved and the computation complexity is increased. Thus, the trade between accuracy and computation cost is balanced by the quadratic

In the optimization of observability, the condition number of observability matrix is selected as

condð Þ¼ <sup>M</sup> <sup>σ</sup>maxð Þ <sup>M</sup>

where σmax and σmin are, respectively, the maximum and minimum singular value of the matrix. The condition number measures the singularity of the matrix. A larger condition number means a more singular matrix. Here we take the inverse of the condition number to quantify the system

> <sup>¼</sup> <sup>σ</sup>minð Þ <sup>O</sup><sup>Σ</sup> σmaxð Þ O<sup>Σ</sup>

, ⋯, ∇Ln�<sup>1</sup>

Lj, the observability matrix can be iteratively calculated. Only 2nd order

, ⋯, J<sup>n</sup>�<sup>1</sup>

� �<sup>T</sup> � �<sup>T</sup>

<sup>h</sup> Jf

<sup>f</sup> hj in Eq. (9) if the state equation

<sup>σ</sup>minð Þ <sup>M</sup> (20)

<sup>f</sup> h

� � � � � x¼x<sup>0</sup>

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Mars Networks-Based Navigation: Observability and Optimization

(17)

125

(18)

(19)

(21)

It's a heavy burden to calculate the observability matrix in Eq. (12) due to the existence of high order differential, especially for the 6-dimensional dynamics of Mars entry phase which requires the calculation of 5th order Lie derivatives. Next, a quadratic approximation method is developed to simplify the computation of the observability matrix.

First of all, the quadratic approximation of the kth order Lie derivative Lk <sup>f</sup> hj is given as

$$L\_f^k h\_\rangle \approx L\_f^k h\_{\slash} + f\_{L\not\slash}^k(\mathbf{x} - \mathbf{x}\_0) + \frac{1}{2}(\mathbf{x} - \mathbf{x}\_0)^T \mathbf{H}\_{L\not\slash}^k(\mathbf{x} - \mathbf{x}\_0) \tag{13}$$

where Lk <sup>f</sup> hj<sup>0</sup> is the value of Lk <sup>f</sup> hj at x0, and J<sup>k</sup> <sup>L</sup>and H<sup>k</sup> <sup>L</sup> refer to, respectively, the Jacobian and Hessian matrix of Lk <sup>f</sup> hj at x0. The linearized state equation is given by

$$f \approx f\_0 + f\_f(\mathbf{x} - \mathbf{x}\_0) \tag{14}$$

in which f <sup>0</sup> refers to the value of f at x0, and Jf is the Jacobi matrix of f at x0.

According to Eq. (9) and Eq. (13), the relationship between the kth and k + 1th order Lie derivative can be rewritten as

$$\begin{aligned} \mathbf{L}\_{f}^{k+1}h\_{j} &= \nabla \mathbf{L}\_{f}^{k}h\_{j} \cdot \mathbf{f} \\ &= \left[\mathbf{I}\_{lj}^{k} + \frac{1}{2}(\mathbf{x} - \mathbf{x}\_{0})^{T} \left(\mathbf{H}\_{lj}^{k} + \left(\mathbf{H}\_{lj}^{k}\right)^{T}\right)\right] \left[\mathbf{f}\_{0} + \mathbf{J}\_{f}(\mathbf{x} - \mathbf{x}\_{0})\right] \\ &+ \left[\mathbf{J}\_{lj}^{k}f\_{f} + \frac{1}{2}\left(\left[\mathbf{H}\_{lj}^{k} + \left(\mathbf{H}\_{lj}^{k}\right)^{T}\right] \mathbf{f}\_{0}\right)^{T}\right] (\mathbf{x} - \mathbf{x}\_{0}) + \frac{1}{2}(\mathbf{x} - \mathbf{x}\_{0})^{T} \left[\mathbf{H}\_{lj}^{k} + \left(\mathbf{H}\_{lj}^{k}\right)^{T}\right] \mathbf{J}\_{f}(\mathbf{x} - \mathbf{x}\_{0}) \\ &= \mathbf{L}\_{f}^{k+1}h\_{\varnothing} + \mathbf{J}\_{lj}^{k+1}(\mathbf{x} - \mathbf{x}\_{0}) + (\mathbf{x} - \mathbf{x}\_{0})^{T}\mathbf{H}\_{lj}^{k+1}(\mathbf{x} - \mathbf{x}\_{0}) \end{aligned} \tag{15}$$

This leads to

$$\begin{aligned} \mathbf{L}\_f^{k+1} \mathbf{h}\_{\not p} &= \mathbf{J}\_{Lj}^k \mathbf{f}\_0 \\ \mathbf{J}\_{Lj}^{k+1} &= \mathbf{J}\_{Lj}^k \mathbf{J}\_f + \frac{1}{2} \left( \left[ \mathbf{H}\_{Lj}^k + \left( \mathbf{H}\_{Lj}^k \right)^T \right] \mathbf{f}\_0 \right)^T \\ \mathbf{H}\_{Lj}^{k+1} &= \frac{1}{2} \left[ \mathbf{H}\_{Lj}^k + \left( \mathbf{H}\_{Lj}^k \right)^T \right] \mathbf{I}\_f \end{aligned} \tag{16}$$

The observability matrix can be computed as

$$\begin{aligned} \mathbf{O}\_{\Sigma} &= \left[ \left( \nabla L\_f^0 \mathbf{h} \right)^T, \quad \left( \nabla L\_f^1 \mathbf{h} \right)^T, \quad \cdots, \quad \left( \nabla L\_f^{n-1} \mathbf{h} \right)^T \right]^T \bigg|\_{\mathbf{x} = \mathbf{x}\_0} \\ &= \left[ \left( f\_L^0 \right)^T, \quad \left( f\_L^1 \right)^T, \quad \cdots, \quad \left( f\_L^{n-1} \right)^T \right]^T \end{aligned} \tag{17}$$

where J<sup>k</sup> <sup>L</sup> ¼ J<sup>k</sup> L1 � �<sup>T</sup> , ⋯, J<sup>k</sup> Lm � �<sup>T</sup> h i<sup>T</sup> .

It is proven that the dynamical system Σ at state x<sup>0</sup> is locally observable if the observability

� �<sup>T</sup> � �<sup>T</sup>

It's a heavy burden to calculate the observability matrix in Eq. (12) due to the existence of high order differential, especially for the 6-dimensional dynamics of Mars entry phase which requires the calculation of 5th order Lie derivatives. Next, a quadratic approximation method

> 1 2

<sup>L</sup>and H<sup>k</sup>

ð Þ x � x<sup>0</sup>

<sup>T</sup>H<sup>k</sup>

, ⋯, ∇Ln�<sup>1</sup>

<sup>f</sup> h

� � � � � x¼x<sup>0</sup>

<sup>f</sup> hj is given as

Ljð Þ x � x<sup>0</sup> (13)

<sup>L</sup> refer to, respectively, the Jacobian and

f ≈ f <sup>0</sup> þ Jfð Þ x � x<sup>0</sup> (14)

<sup>¼</sup> <sup>J</sup><sup>k</sup> Ljf <sup>0</sup>

Lj <sup>þ</sup> <sup>H</sup><sup>k</sup> Lj � �<sup>T</sup> � �

Jfð Þ x � x<sup>0</sup>

(15)

(16)

<sup>T</sup> H<sup>k</sup>

f 0

(12)

matrix O<sup>Σ</sup> given below has the rank of n.

O<sup>Σ</sup> ¼ ∇L<sup>0</sup>

Lk <sup>f</sup> hj ≈ Lk

<sup>f</sup> hj<sup>0</sup> is the value of Lk

where Lk

124 Space Flight

L<sup>k</sup>þ<sup>1</sup>

<sup>f</sup> hj <sup>¼</sup> <sup>∇</sup>L<sup>k</sup>

<sup>¼</sup> <sup>J</sup><sup>k</sup> Lj þ 1 2

<sup>þ</sup> <sup>J</sup><sup>k</sup> LjJf þ 1 <sup>2</sup> <sup>H</sup><sup>k</sup>

<sup>¼</sup> Lkþ<sup>1</sup>

This leads to

Hessian matrix of Lk

derivative can be rewritten as

<sup>f</sup> hj � f

<sup>f</sup> hj<sup>0</sup> <sup>þ</sup> <sup>J</sup><sup>k</sup>þ<sup>1</sup>

ð Þ x � x<sup>0</sup>

<sup>T</sup> H<sup>k</sup>

� �<sup>T</sup> � � � �

Lj <sup>þ</sup> <sup>H</sup><sup>k</sup> Lj � �<sup>T</sup> � �

Lj ð Þþ x � x<sup>0</sup> ð Þ x � x<sup>0</sup>

Lkþ<sup>1</sup> <sup>f</sup> hj<sup>0</sup> <sup>¼</sup> <sup>J</sup><sup>k</sup>

J<sup>k</sup>þ<sup>1</sup> Lj <sup>¼</sup> <sup>J</sup><sup>k</sup>

H<sup>k</sup>þ<sup>1</sup> Lj <sup>¼</sup> <sup>1</sup>

The observability matrix can be computed as

� �<sup>T</sup> " #

Lj <sup>þ</sup> <sup>H</sup><sup>k</sup> Lj

f 0

f h � �<sup>T</sup>

is developed to simplify the computation of the observability matrix.

<sup>f</sup> hj<sup>0</sup> <sup>þ</sup> <sup>J</sup><sup>k</sup>

First of all, the quadratic approximation of the kth order Lie derivative Lk

<sup>f</sup> hj at x0, and J<sup>k</sup>

in which f <sup>0</sup> refers to the value of f at x0, and Jf is the Jacobi matrix of f at x0.

Ljð Þþ x � x<sup>0</sup>

<sup>f</sup> hj at x0. The linearized state equation is given by

According to Eq. (9) and Eq. (13), the relationship between the kth and k + 1th order Lie

ð Þþ x � x<sup>0</sup>

Lj ð Þ x � x<sup>0</sup>

Hk Lj <sup>þ</sup> <sup>H</sup><sup>k</sup> Lj � �<sup>T</sup> � �

Lj <sup>þ</sup> <sup>H</sup><sup>k</sup> Lj � �<sup>T</sup> � �

<sup>T</sup>H<sup>k</sup>þ<sup>1</sup>

Ljf <sup>0</sup>

LjJf þ 1 2

<sup>2</sup> <sup>H</sup><sup>k</sup>

f <sup>0</sup> þ Jfð Þ x � x<sup>0</sup> h i

> 1 2

ð Þ x � x<sup>0</sup>

� �<sup>T</sup>

Jf

, ∇L<sup>1</sup> f h � �<sup>T</sup>

> Obtaining J<sup>0</sup> Lj and H<sup>0</sup> Lj, the observability matrix can be iteratively calculated. Only 2nd order differential of h is needed here to compute the Jacobian and Hessian matrices, reducing largely the computation cost.

Linearize the dynamical and observation model by first-order approximation

$$\begin{aligned} f &\approx f\_0 + \mathbf{J}\_f(\mathbf{x} - \mathbf{x}\_0) \\ \mathbf{h} &\approx \mathbf{h}\_0 + \mathbf{J}\_h(\mathbf{x} - \mathbf{x}\_0) \end{aligned} \tag{18}$$

Construct the observability matrix according to the linear system theory

$$\mathbf{O}\_{\Sigma}^{l} = \left[ \begin{pmatrix} \mathbf{J}\_{h} \end{pmatrix}^{T}, \quad \begin{pmatrix} \mathbf{J}\_{h} \mathbf{J}\_{f} \end{pmatrix}^{T}, \quad \cdots, \quad \begin{pmatrix} \mathbf{J}\_{h}^{n-1} \mathbf{J}\_{f} \end{pmatrix}^{T} \right]^{T} \tag{19}$$

The Hessian matrix is involved in the quadratic approximation, improving the accuracy of observability analysis compared with the linearized observability analysis. However, the higher order terms of <sup>x</sup> � <sup>x</sup><sup>0</sup> may appear when computing Lkþ<sup>1</sup> <sup>f</sup> hj in Eq. (9) if the state equation is approximated to a higher order. In this case, the predetermined presentation form in Eq. (13) is no longer valid. One way to defeat this case is to increase the approximation order of Lie derivatives. Note that tensor calculus can be involved and the computation complexity is increased. Thus, the trade between accuracy and computation cost is balanced by the quadratic approximation of Lie derivatives and the linearization of state equation.

In the optimization of observability, the condition number of observability matrix is selected as the performance index, given by

$$mod(\mathcal{M}) = \frac{\sigma\_{\text{max}}(\mathcal{M})}{\sigma\_{\text{min}}(\mathcal{M})} \tag{20}$$

where σmax and σmin are, respectively, the maximum and minimum singular value of the matrix. The condition number measures the singularity of the matrix. A larger condition number means a more singular matrix. Here we take the inverse of the condition number to quantify the system observability.

$$\delta = \frac{1}{cond(\mathbf{O}\_{\Sigma})} = \frac{\sigma\_{\min}(\mathbf{O}\_{\Sigma})}{\sigma\_{\max}(\mathbf{O}\_{\Sigma})} \tag{21}$$

Obviously, the observability degree δ is in the interval [0, 1]. When δ ¼ 0, the observability matrix is rank defect, and the navigation system is locally unobservable. When δ > 0, the observability is full rank, indicating an observable navigation system.

#### 3.2. Observability analysis based on the fisher information matrix

Without loss of generality, we will consider the nonlinear observation models

$$y\_i = h\_i(\mathbf{x}) + \varepsilon\_i \quad i = 1, \cdots, N \tag{22}$$

observability. Quantify the observability by the determinant of FIM detð Þ¼ <sup>F</sup> <sup>Q</sup>

3

1 λi > 3 P 3 i¼1 λi <sup>¼</sup> <sup>3</sup>

Mars Networks-Based Navigation: Observability and Optimization

i¼1

In this subsection, the system observability using only range measurements between the lander and ground beacons is analyzed. Since no velocity information is included in Eq. (6), only the observability of the position vector is studied. The cases with different amount of beacons are

trð Þ <sup>P</sup> <sup>≥</sup> tr <sup>F</sup>�<sup>1</sup> � � <sup>¼</sup> <sup>X</sup>

4. Observability analysis of Mars networks-based navigation

4.1. Observability analysis using only range measurements

<sup>F</sup><sup>1</sup> <sup>¼</sup> <sup>σ</sup>�<sup>2</sup> R1

∂R1ð Þr ∂r

The rank of the matrix N1is only one. Solving the following equation

Clearly, the eigenvalues of N<sup>1</sup> are given by twice repeated 0 and n<sup>2</sup>

<sup>w</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup> n1<sup>z</sup>

the eigenvalues of <sup>F</sup><sup>1</sup> are given by <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>λ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>, <sup>λ</sup><sup>3</sup> <sup>¼</sup> <sup>σ</sup>�<sup>2</sup>

Next, we have the eigenvector corresponding to λ<sup>3</sup>

component along the vector n<sup>1</sup> can be observable.

∂R1ð Þr ∂r � �<sup>T</sup>

n1x, n1y, n1<sup>z</sup> � �<sup>T</sup> <sup>¼</sup> <sup>1</sup>

The vector w<sup>3</sup> corresponds to the observable state combination, and means that only the state

<sup>¼</sup> <sup>3</sup>σ<sup>2</sup>

According to Eq. (28), the lower bound of estimation errors can be obtained as

3 trð Þ F<sup>1</sup> <sup>¼</sup> <sup>σ</sup>�<sup>2</sup> R1n1n<sup>T</sup>

<sup>R</sup><sup>1</sup> .

n1<sup>z</sup>

<sup>1</sup> � <sup>σ</sup>�<sup>2</sup>

detð Þ¼ λI<sup>3</sup>�<sup>3</sup> � N<sup>1</sup> 0 (30)

<sup>1</sup><sup>x</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup>

<sup>1</sup><sup>y</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup>

<sup>R</sup><sup>1</sup> (32)

n<sup>1</sup> (31)

Eq. (28) means that the trace of FIM measures the lower bound of estimation errors.

ing relationship can be obtained.

studied.

4.2. One-beacon case

In this case, the FIM is given by

3 i¼1

http://dx.doi.org/10.5772/intechopen.73605

trð Þ <sup>F</sup> (28)

<sup>R</sup>1N<sup>1</sup> (29)

<sup>1</sup><sup>z</sup> ¼ 1. Therefore,

λi. The follow-

127

This equation may describe the measurement of relative range and range-rate according to Eq. (6) and (7). Meanwhile, in order to investigate the impact of different measurement methods on the observability of position and velocity of the entry vehicle separately, the 3-dimensional state x may be r or v of the entry vehicle. The likelihood function of x is defined as the joint probability density function of multiple measurements given by

$$L(y\_1, \dots, y\_3 | \mathbf{x}) = \prod\_{i=1}^{N} \frac{1}{\sqrt{2\pi}\sigma\_i} \exp\left(-\frac{1}{2}\sigma\_i^{-2} \left\| y\_i - h\_i(\mathbf{x}) \right\|^2\right) \tag{23}$$

Then, take the negative of the natural log of Eq. (23) and omitting the terms not related to x, and the loss function can be derived as

$$J(\mathbf{x}) = \frac{1}{2} \sum\_{i=1}^{N} \sigma\_i^{-2} \left\| y\_i - h\_i(\mathbf{x}) \right\|^2 \tag{24}$$

Find a state vector to minimize Jð Þx and the state vector is the optimal estimation of the lander's states. The FIM of the state is given by

$$F = E\left\{\frac{\partial^2}{\partial \mathbf{x} \partial \mathbf{x}^T} J(\mathbf{x})\right\} = \sum\_{i=1}^N \sigma\_i^{-2} \frac{\partial h\_i(\mathbf{x})}{\partial \mathbf{x}} \left(\frac{\partial h\_i(\mathbf{x})}{\partial \mathbf{x}}\right)^T \tag{25}$$

The estimate error covariance and FIM satisfy the following equation

$$P \ge F^{-1} \tag{26}$$

where <sup>P</sup> is the estimate error covariance, and " <sup>≥</sup> " means that (<sup>P</sup> � <sup>F</sup>�<sup>1</sup> ) is positive semidefinite. According to Eq. (26), the FIM can be used to evaluate the lower bound of the estimation error covariance, and further the system observability. Give the trace of F�<sup>1</sup> in Eq. (27).

$$\text{tr}\left(\mathbf{F}^{-1}\right) = \sum\_{i=1}^{3} \frac{1}{\lambda\_i} \tag{27}$$

where λ<sup>i</sup> ð Þ i ¼ 1; 2; 3 are the eigenvalues of F. It's illustrated from Eq. (27) that larger eigenvalues of the FIM leads to smaller trace of estimation error covariance and stronger system observability. Quantify the observability by the determinant of FIM detð Þ¼ <sup>F</sup> <sup>Q</sup> 3 i¼1 λi. The following relationship can be obtained.

$$\text{tr}(\mathbf{P}) \ge \text{tr}\{\mathbf{F}^{-1}\} = \sum\_{i=1}^{3} \frac{1}{\lambda\_i} > \frac{3}{\frac{3}{\lambda\_i}} = \frac{3}{\text{tr}(F)}\tag{28}$$

Eq. (28) means that the trace of FIM measures the lower bound of estimation errors.

#### 4. Observability analysis of Mars networks-based navigation

#### 4.1. Observability analysis using only range measurements

In this subsection, the system observability using only range measurements between the lander and ground beacons is analyzed. Since no velocity information is included in Eq. (6), only the observability of the position vector is studied. The cases with different amount of beacons are studied.

#### 4.2. One-beacon case

Obviously, the observability degree δ is in the interval [0, 1]. When δ ¼ 0, the observability matrix is rank defect, and the navigation system is locally unobservable. When δ > 0, the

This equation may describe the measurement of relative range and range-rate according to Eq. (6) and (7). Meanwhile, in order to investigate the impact of different measurement methods on the observability of position and velocity of the entry vehicle separately, the 3-dimensional state x may be r or v of the entry vehicle. The likelihood function of x is defined

yi ¼ hið Þþ x εi, i ¼ 1, ⋯, N (22)

<sup>i</sup> yi � hið Þ<sup>x</sup> � � �

� �

� 2

<sup>2</sup> (24)

) is positive semidefinite.

(23)

(25)

(27)

observability is full rank, indicating an observable navigation system.

3.2. Observability analysis based on the fisher information matrix

Without loss of generality, we will consider the nonlinear observation models

as the joint probability density function of multiple measurements given by

N

1 ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> <sup>σ</sup><sup>i</sup>

i¼1

Then, take the negative of the natural log of Eq. (23) and omitting the terms not related to x,

σ�<sup>2</sup>

Find a state vector to minimize Jð Þx and the state vector is the optimal estimation of the

<sup>¼</sup> <sup>X</sup> N

i¼1

According to Eq. (26), the FIM can be used to evaluate the lower bound of the estimation error

3

1 λi

i¼1

tr <sup>F</sup>�<sup>1</sup> � � <sup>¼</sup> <sup>X</sup>

where λ<sup>i</sup> ð Þ i ¼ 1; 2; 3 are the eigenvalues of F. It's illustrated from Eq. (27) that larger eigenvalues of the FIM leads to smaller trace of estimation error covariance and stronger system

σ�<sup>2</sup> i

∂hið Þx ∂x

∂hið Þx ∂x � �<sup>T</sup>

P ≥ F�<sup>1</sup> (26)

exp � <sup>1</sup> 2 σ�<sup>2</sup>

<sup>i</sup> yi � hið Þ<sup>x</sup> � � �

�

i¼1

Jð Þ¼ x 1 2 X N

<sup>∂</sup>x∂x<sup>T</sup> <sup>J</sup>ð Þ<sup>x</sup> � �

The estimate error covariance and FIM satisfy the following equation

where <sup>P</sup> is the estimate error covariance, and " <sup>≥</sup> " means that (<sup>P</sup> � <sup>F</sup>�<sup>1</sup>

covariance, and further the system observability. Give the trace of F�<sup>1</sup> in Eq. (27).

L y1; <sup>⋯</sup>; <sup>y</sup>3j<sup>x</sup> � � <sup>¼</sup> <sup>Y</sup>

and the loss function can be derived as

126 Space Flight

lander's states. The FIM of the state is given by

<sup>F</sup> <sup>¼</sup> <sup>E</sup> <sup>∂</sup><sup>2</sup>

In this case, the FIM is given by

$$\sigma\_1 \mathbf{F}\_1 = \sigma\_{R1}^{-2} \frac{\partial \mathcal{R}\_1(\mathbf{r})}{\partial \mathbf{r}} \left(\frac{\partial \mathcal{R}\_1(\mathbf{r})}{\partial \mathbf{r}}\right)^T = \sigma\_{R1}^{-2} \mathfrak{n}\_1 \mathfrak{n}\_1^T \equiv \sigma\_{R1}^{-2} \mathbf{N}\_1 \tag{29}$$

The rank of the matrix N1is only one. Solving the following equation

$$\det(\lambda I\_{3 \times 3} - \mathbf{N}\_1) = 0 \tag{30}$$

Clearly, the eigenvalues of N<sup>1</sup> are given by twice repeated 0 and n<sup>2</sup> <sup>1</sup><sup>x</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup> <sup>1</sup><sup>y</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup> <sup>1</sup><sup>z</sup> ¼ 1. Therefore, the eigenvalues of <sup>F</sup><sup>1</sup> are given by <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>λ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>, <sup>λ</sup><sup>3</sup> <sup>¼</sup> <sup>σ</sup>�<sup>2</sup> <sup>R</sup><sup>1</sup> .

Next, we have the eigenvector corresponding to λ<sup>3</sup>

$$\mathfrak{w}\_3 = \frac{1}{n\_{1z}} \begin{bmatrix} n\_{1x} & n\_{1y} & n\_{1z} \end{bmatrix}^T = \frac{1}{n\_{1z}} \mathfrak{n}\_1 \tag{31}$$

The vector w<sup>3</sup> corresponds to the observable state combination, and means that only the state component along the vector n<sup>1</sup> can be observable.

According to Eq. (28), the lower bound of estimation errors can be obtained as

$$\frac{3}{\text{tr}(F\_1)} = 3\sigma\_{R1}^2\tag{32}$$

Eq. (32) means the lower bound of estimation errors is higher than the estimation accuracy. In another word, the estimation accuracy cannot be higher than the measurement accuracy. Note that, even if multiple beacons are involved in the navigation system, the observability is still deteriorated if the beacons are located in similar direction.

#### 4.3. Two-beacon case

Assume two non-collinear beacons, the FIM in Eq. (25) is derived by

$$F\_2 = \sum\_{i=1}^2 \sigma\_{Ri}^{-2} \frac{\partial R\_i(\mathbf{r})}{\partial \mathbf{r}} \left(\frac{\partial R\_i(\mathbf{r})}{\partial \mathbf{r}}\right)^T = \sum\_{i=1}^2 \sigma\_{Ri}^{-2} \mathfrak{n}\_i \mathfrak{n}\_i^T \tag{33}$$

detð Þ¼ F<sup>N</sup>

nants are listed in Table 1.

The lower bound of estimation errors is derived as

endlessly by only increasing the number of beacons.

The FIM of vehicle's velocity using range-rate data is given by

4.5.1. Observability analysis of vehicle's velocity

3 trð Þ F<sup>N</sup>

detð Þ <sup>F</sup><sup>N</sup> <sup>≤</sup> <sup>σ</sup>�<sup>6</sup>

X 1 ≤ k3<k2<k<sup>1</sup> ≤ N

σ�<sup>2</sup> Rk<sup>1</sup> σ�<sup>2</sup> Rk<sup>2</sup> σ�<sup>2</sup>

The detailed derivation can be found in Ref. [14]. From Eq. (7), we can know that more radio beacons, no matter where they are, increase the determinant of the FIM, thus increase the system observability. To analyze the maximum value of detð Þ F<sup>N</sup> , Eq. (37) is reorganized as

1 ≤ k3<k2<k<sup>1</sup> ≤ N

where σRmin is the minimum value among σRi. The selection of the direction of radio beacons to

<sup>n</sup><sup>k</sup><sup>1</sup> � <sup>n</sup><sup>k</sup><sup>2</sup> � <sup>n</sup><sup>k</sup><sup>3</sup> ½ � ð Þ <sup>2</sup>

<sup>R</sup>min <sup>X</sup>

maximize the observability can be described by the following optimization problem

1 ≤ k3<k2<k<sup>1</sup> ≤ N

subject to k k n<sup>i</sup> ¼ 1, i ¼ 1, ⋯, N

Note that the locations of radio beacons are not constrained. In cases with three beacons, the determinant of F<sup>3</sup> is maximized if and only if n1, n2, and n<sup>3</sup> are orthogonal to each other. However, no analytic results can be obtained when there are more than three beacons. Thus, a Genetic Algorithm is exploited to solve the optimization problem. The maximum determi-

According to the results in Table 1, the relationship between the maximum determinant and

ix <sup>þ</sup> <sup>n</sup><sup>2</sup>

The change of lower bound of estimation errors with number of beacons is shown in Figure 1. It's shown that with more beacons comes more accurate estimation. However, the increasing rate of accuracy is slowed down, indicating that the navigation accuracy cannot be improved

3σ�<sup>2</sup> Rmin � �<sup>3</sup>

iy <sup>þ</sup> <sup>n</sup><sup>2</sup> iz � � ≥ 3σ<sup>2</sup> Rmin

detð Þ <sup>F</sup><sup>N</sup> max <sup>¼</sup> <sup>N</sup>

the number of beacons can be induced by an exponential formulation, given by

<sup>¼</sup> <sup>3</sup> P N i¼1 σ�<sup>2</sup> Ri n<sup>2</sup>

4.5. Observability analysis of the navigation using range-rate measurements

max X

Rk<sup>3</sup> <sup>n</sup><sup>k</sup><sup>1</sup> � <sup>n</sup><sup>k</sup><sup>2</sup> � <sup>n</sup><sup>k</sup><sup>3</sup> ½ � ð Þ <sup>2</sup> (37)

http://dx.doi.org/10.5772/intechopen.73605

Mars Networks-Based Navigation: Observability and Optimization

<sup>n</sup><sup>k</sup><sup>1</sup> � <sup>n</sup><sup>k</sup><sup>2</sup> � <sup>n</sup><sup>k</sup><sup>3</sup> ½ � ð Þ <sup>2</sup> (38)

(39)

129

(40)

<sup>N</sup> (41)

Involving one more measurement, the rank of F<sup>2</sup> is increased to two. The observable state combinations can be obtained by solving the eigenvalue and eigenvector. In this case, the eigenvalues of F<sup>2</sup> are given by λ<sup>1</sup> ¼ λ<sup>2</sup> 6¼ 0, λ<sup>3</sup> ¼ 0. The eigenvector corresponding to the zero eigenvalue is obtained as

$$\mathfrak{w}\_{3} = \begin{bmatrix} \frac{n\_{1y}n\_{2} - n\_{1z}n\_{2y}}{n\_{1z}n\_{2y} - n\_{1y}n\_{2z}} & \frac{n\_{1z}n\_{2z} - n\_{1z}n\_{2z}}{n\_{1z}n\_{2y} - n\_{1y}n\_{2z}} & 1 \end{bmatrix}^{T} = \frac{1}{n\_{1x}n\_{2y} - n\_{1y}n\_{2x}} \mathfrak{n}\_{1} \times \mathfrak{n}\_{2} \tag{34}$$

The vector w<sup>3</sup> gives the unobservable state component which is in the direction perpendicular to the plane constructed by n<sup>1</sup> and n2. From an opposite view, all state components in plane are observable.

Since, in this case, the observability matrix is still zero, the navigation system is unobservable. According to Eq. (28), the lower bound of the estimation errors can be obtained as

$$\frac{3}{\text{tr}(F\_2)} = \frac{3}{\sum\_{i=1}^{2} \sigma\_{Ri}^{-2} \left( n\_{ix}^2 + n\_{iy}^2 + n\_{iz}^2 \right)} = \frac{3}{\sum\_{i=1}^{2} \sigma\_{Ri}^{-2}} \ge \frac{3 \sigma\_{R\text{min}}^2}{2} \tag{35}$$

where σRmin is the smaller standard deviation among σR<sup>1</sup> and σR2. It's known by comparing Eqs. (32) and (35) that the estimation accuracy can be improved by using one more radio beacon.

#### 4.4. More-than-two-beacon case

In this case, the FIM is given by

$$F\_N = \sum\_{i=1}^N \sigma\_{Ri}^{-2} \frac{\partial \mathcal{R}\_i(\mathbf{r})}{\partial \mathbf{r}} \left(\frac{\partial \mathcal{R}\_i(\mathbf{r})}{\partial \mathbf{r}}\right)^T = \sum\_{i=1}^N \sigma\_{Ri}^{-2} \mathfrak{n}\_i \mathfrak{n}\_i^T, \quad N \ge 3 \tag{36}$$

The matrix F<sup>N</sup> has a full rank, indicating an observable system. The determinant of F<sup>N</sup> is given in Eq. (37).

$$\det(\mathbf{F}\_N) = \sum\_{1 \le k\_3 < k\_2 < k\_1 \le N} \sigma\_{Rk\_1}^{-2} \sigma\_{Rk\_2}^{-2} \sigma\_{Rk\_3}^{-2} \left[ \mathfrak{n}\_{k\_1} \cdot (\mathfrak{n}\_{k\_2} \times \mathfrak{n}\_{k\_3}) \right]^2 \tag{37}$$

The detailed derivation can be found in Ref. [14]. From Eq. (7), we can know that more radio beacons, no matter where they are, increase the determinant of the FIM, thus increase the system observability. To analyze the maximum value of detð Þ F<sup>N</sup> , Eq. (37) is reorganized as

$$\det(F\_N) \le \sigma\_{R\text{min}}^{-6} \sum\_{1 \le k\_3 < k\_2 < k\_1 \le N} \left[ \mathfrak{n}\_{k\_1} \cdot (\mathfrak{n}\_{k\_2} \times \mathfrak{n}\_{k\_3}) \right]^2 \tag{38}$$

where σRmin is the minimum value among σRi. The selection of the direction of radio beacons to maximize the observability can be described by the following optimization problem

$$\begin{aligned} \text{max} & \sum\_{1 \le k\_3 < k\_2 < k\_1 \le N} \left[ \mathfrak{n}\_{k\_1} \cdot (\mathfrak{n}\_{k\_2} \times \mathfrak{n}\_{k\_3}) \right]^2 \\ \text{subject to} & ||\mathfrak{n}\_i|| = 1, \quad i = 1, \dots, N \end{aligned} \tag{39}$$

Note that the locations of radio beacons are not constrained. In cases with three beacons, the determinant of F<sup>3</sup> is maximized if and only if n1, n2, and n<sup>3</sup> are orthogonal to each other. However, no analytic results can be obtained when there are more than three beacons. Thus, a Genetic Algorithm is exploited to solve the optimization problem. The maximum determinants are listed in Table 1.

According to the results in Table 1, the relationship between the maximum determinant and the number of beacons can be induced by an exponential formulation, given by

$$\left(\det(F\_N)\_{\max} = \left(\frac{N}{3\sigma\_{R\text{min}}^{-2}}\right)^3\right)^3 \tag{40}$$

The lower bound of estimation errors is derived as

Eq. (32) means the lower bound of estimation errors is higher than the estimation accuracy. In another word, the estimation accuracy cannot be higher than the measurement accuracy. Note that, even if multiple beacons are involved in the navigation system, the observability is still

> ∂Rið Þr ∂r � �<sup>T</sup>

Involving one more measurement, the rank of F<sup>2</sup> is increased to two. The observable state combinations can be obtained by solving the eigenvalue and eigenvector. In this case, the eigenvalues of F<sup>2</sup> are given by λ<sup>1</sup> ¼ λ<sup>2</sup> 6¼ 0, λ<sup>3</sup> ¼ 0. The eigenvector corresponding to the zero

The vector w<sup>3</sup> gives the unobservable state component which is in the direction perpendicular to the plane constructed by n<sup>1</sup> and n2. From an opposite view, all state components in plane are

Since, in this case, the observability matrix is still zero, the navigation system is unobservable.

iy <sup>þ</sup> <sup>n</sup><sup>2</sup> iz � � <sup>¼</sup> <sup>3</sup>

where σRmin is the smaller standard deviation among σR<sup>1</sup> and σR2. It's known by comparing Eqs. (32) and (35) that the estimation accuracy can be improved by using one more radio

> ∂Rið Þr ∂r � �<sup>T</sup>

The matrix F<sup>N</sup> has a full rank, indicating an observable system. The determinant of F<sup>N</sup> is given

<sup>¼</sup> <sup>X</sup> N

i¼1 σ�<sup>2</sup> Ri nin<sup>T</sup>

According to Eq. (28), the lower bound of the estimation errors can be obtained as

ix <sup>þ</sup> <sup>n</sup><sup>2</sup>

<sup>¼</sup> <sup>3</sup> P 2 i¼1 σ�<sup>2</sup> Ri n<sup>2</sup>

> ∂Rið Þr ∂r

<sup>n</sup>1xn2y�n1yn2<sup>x</sup> , <sup>1</sup>

<sup>¼</sup> <sup>X</sup> 2

i¼1

<sup>¼</sup> <sup>1</sup>

P 2 i¼1 σ�<sup>2</sup> Ri

n1xn2<sup>y</sup> � n1yn2<sup>x</sup>

≥ 3σ<sup>2</sup> Rmin

σ�<sup>2</sup> Ri nin<sup>T</sup>

<sup>i</sup> (33)

n<sup>1</sup> � n<sup>2</sup> (34)

<sup>2</sup> (35)

<sup>i</sup> , N ≥ 3 (36)

deteriorated if the beacons are located in similar direction.

<sup>F</sup><sup>2</sup> <sup>¼</sup> <sup>X</sup> 2

<sup>w</sup><sup>3</sup> <sup>¼</sup> <sup>n</sup>1yn2z�n1zn2<sup>y</sup>

3 trð Þ F<sup>2</sup>

<sup>F</sup><sup>N</sup> <sup>¼</sup> <sup>X</sup> N

i¼1 σ�<sup>2</sup> Ri

i¼1

Assume two non-collinear beacons, the FIM in Eq. (25) is derived by

σ�<sup>2</sup> Ri

<sup>n</sup>1xn2y�n1yn2<sup>x</sup> , <sup>n</sup>1zn2x�n1xn2<sup>z</sup>

h i<sup>T</sup>

∂Rið Þr ∂r

4.3. Two-beacon case

128 Space Flight

eigenvalue is obtained as

observable.

beacon.

in Eq. (37).

4.4. More-than-two-beacon case

In this case, the FIM is given by

$$\frac{3}{\text{tr}(F\_N)} = \frac{3}{\sum\_{i=1}^N \sigma\_{Ri}^{-2} \left(n\_{ix}^2 + n\_{iy}^2 + n\_{ix}^2\right)} \ge \frac{3\sigma\_{R\text{min}}^2}{N} \tag{41}$$

The change of lower bound of estimation errors with number of beacons is shown in Figure 1. It's shown that with more beacons comes more accurate estimation. However, the increasing rate of accuracy is slowed down, indicating that the navigation accuracy cannot be improved endlessly by only increasing the number of beacons.

#### 4.5. Observability analysis of the navigation using range-rate measurements

#### 4.5.1. Observability analysis of vehicle's velocity

The FIM of vehicle's velocity using range-rate data is given by

$$F\_N = \sum\_{i=1}^N \sigma\_{Vi}^{-2} \frac{\partial V\_i(\mathbf{r}, \mathbf{v})}{\partial \mathbf{v}} \left(\frac{\partial V\_i(\mathbf{r}, \mathbf{v})}{\partial \mathbf{v}}\right)^T = \sum\_{i=1}^N \sigma\_{Vi}^{-2} \mathfrak{n}\_i \mathfrak{n}\_i^T, \quad N \ge 1 \tag{42}$$

<sup>L</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> Ri

observable cases are focused on.

cases has a similar format as Eq. (37)

X 1 ≤ k3<k2<k<sup>1</sup> ≤ N

cases using only range measurements.

vx n<sup>2</sup>

vx 1 þ

vx þ vy þ vz � �<sup>2</sup>

3

<sup>¼</sup> <sup>3</sup> X N

i¼1 σ�<sup>2</sup> Vi V<sup>i</sup> <sup>T</sup>V<sup>i</sup>

8 ><

>:

σ�<sup>2</sup> Vk<sup>1</sup> σ�<sup>2</sup> Vk<sup>2</sup> σ�<sup>2</sup>

<sup>¼</sup> <sup>3</sup>

iy <sup>þ</sup> <sup>n</sup><sup>2</sup>

1 <sup>2</sup> <sup>n</sup><sup>2</sup> iy <sup>þ</sup> <sup>n</sup><sup>2</sup> iz � � � � <sup>þ</sup> vy <sup>1</sup> <sup>þ</sup>

The lower bound of estimation errors in this case is evaluated by

iz � � � vynixniy � vznixniz h i<sup>2</sup>

iy h i � � <sup>2</sup>

�vxnixniz � vyniyniz <sup>þ</sup> vz <sup>n</sup><sup>2</sup>

detð Þ¼ F<sup>N</sup>

3 trð Þ F<sup>N</sup>

> X N

σ�<sup>2</sup> Vi R2 i

σ�<sup>2</sup> Vi R2 i

> σ�<sup>2</sup> Vi R2 i

i¼1

X N

i¼1

3 2 X N

i¼1

2

>

>

≥

3 X N

i¼1 σ�<sup>2</sup> Vi v2 R2 i

n2 iy <sup>þ</sup> <sup>n</sup><sup>2</sup>

<sup>F</sup><sup>N</sup> <sup>¼</sup> <sup>X</sup> N

i¼1 σ�<sup>2</sup> Vi ViV<sup>i</sup>

2 6 4

�niynix <sup>n</sup><sup>2</sup>

iz �nixniy �nixniz

ix �niyniz

<sup>V</sup>min <sup>X</sup> 1 ≤ k3<k2<k<sup>1</sup> ≤ N

þ �vxnixniy <sup>þ</sup> vy <sup>n</sup><sup>2</sup>

ix <sup>þ</sup> <sup>n</sup><sup>2</sup> iy 3 7

Mars Networks-Based Navigation: Observability and Optimization

<sup>5</sup> (44)

131

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T, N ≥ 1 (45)

<sup>V</sup><sup>k</sup><sup>1</sup> � <sup>V</sup><sup>k</sup><sup>2</sup> � <sup>V</sup><sup>k</sup><sup>3</sup> ½ � ð Þ <sup>2</sup>

ix <sup>þ</sup> <sup>n</sup><sup>2</sup> iz � � � vzniyniz � �<sup>2</sup>

> 1 <sup>2</sup> <sup>n</sup><sup>2</sup> ix <sup>þ</sup> <sup>n</sup><sup>2</sup>

, N <sup>≥</sup> <sup>3</sup> (46)

þ

9 >=

>;

(47)

iz <sup>þ</sup> <sup>n</sup><sup>2</sup>

�niznix �nizniy <sup>n</sup><sup>2</sup>

The FIM here is much more complicated than that in Section 4.1 due to the involvement of both range and velocity information in FIM. Define V<sup>i</sup> ¼ Liv, the following equation can be obtained.

When there are one or two beacons, the FIM is rank defect, and the navigation system is also unobservable. With three or more beacons comes the full-rank FIM. In this section, only the

It is also concluded that the determinant of FIM will be zero if only one or two beacons is used, which indicates that the position of entry vehicle will be observable if more than two beacons are used. Furthermore, we focus on three-beacon and more-than-three beacon cases. Compare Eq. (46) with Eq. (36), we can find that the determinant of FIM for range-rate measurement

It's shown that Eq. (46) has the similar format with Eq. (37). Thus, the change of the observability with the number of radio beacons is similar with the results in Table 1. However, due to involving relative range and velocity information, the optimal geometric configuration is different with the

Vk<sup>3</sup> <sup>V</sup><sup>k</sup><sup>1</sup> � <sup>V</sup><sup>k</sup><sup>2</sup> � <sup>V</sup><sup>k</sup><sup>3</sup> ½ � ð Þ <sup>2</sup> <sup>≤</sup> <sup>σ</sup>�<sup>6</sup>

ix <sup>þ</sup> <sup>n</sup><sup>2</sup>

3

1 <sup>2</sup> <sup>n</sup><sup>2</sup> ix <sup>þ</sup> <sup>n</sup><sup>2</sup> iz � � � � <sup>þ</sup> vz <sup>1</sup> <sup>þ</sup>

iy � � � � � �<sup>2</sup> ( )

Eq. (42) has a similar form with Eq. (36) which describes the FIM of position. The only difference lies in the measurement deviation. Hence the same conclusion of the observability of velocity can be obtained as that in Section 4.1. The detailed analysis is omitted here.


Table 1. Maximum determinants of FIM related to different number of beacons.

Figure 1. Lower bound of estimation errors with beacon number.

#### 4.6. Observability analysis of vehicle's position

Using the range-rate measurements, the FIM of the lander's position is derived as

$$F\_N = \sum\_{i=1}^N \sigma\_{Vi}^{-2} \frac{\partial V\_i(\mathbf{r}, \mathbf{v})}{\partial \mathbf{r}} \left(\frac{\partial V\_i(\mathbf{r}, \mathbf{v})}{\partial \mathbf{r}}\right)^T = \sum\_{i=1}^N \sigma\_{Vi}^{-2} \mathbf{L}\_i \mathbf{v} \mathbf{v}^T \mathbf{L}\_i^T, \quad N \ge 1 \tag{43}$$

where L<sup>i</sup> is given by

Mars Networks-Based Navigation: Observability and Optimization http://dx.doi.org/10.5772/intechopen.73605 131

$$\mathbf{L}\_{i} = \frac{1}{R\_{i}} \begin{bmatrix} n\_{iy}^{2} + n\_{iz}^{2} & -n\_{ix}n\_{iy} & -n\_{ix}n\_{iz} \\ -n\_{iy}n\_{ix} & n\_{iz}^{2} + n\_{ix}^{2} & -n\_{iy}n\_{iz} \\ -n\_{iz}n\_{ix} & -n\_{ix}n\_{iy} & n\_{ix}^{2} + n\_{iy}^{2} \end{bmatrix} \tag{44}$$

The FIM here is much more complicated than that in Section 4.1 due to the involvement of both range and velocity information in FIM. Define V<sup>i</sup> ¼ Liv, the following equation can be obtained.

$$F\_N = \sum\_{i=1}^N \sigma\_{Vi}^{-2} \mathbf{V}\_i \mathbf{V}\_i^T \quad \text{ } N \ge 1 \tag{45}$$

When there are one or two beacons, the FIM is rank defect, and the navigation system is also unobservable. With three or more beacons comes the full-rank FIM. In this section, only the observable cases are focused on.

It is also concluded that the determinant of FIM will be zero if only one or two beacons is used, which indicates that the position of entry vehicle will be observable if more than two beacons are used. Furthermore, we focus on three-beacon and more-than-three beacon cases. Compare Eq. (46) with Eq. (36), we can find that the determinant of FIM for range-rate measurement cases has a similar format as Eq. (37)

$$\det(\mathbf{F}\_N) = \sum\_{1 \le k\_l < k\_l < k\_l \le N} \sigma\_{\mathcal{V}k\_l}^{-2} \sigma\_{\mathcal{V}k\_l}^{-2} [\mathbf{V}\_{k\_l} \cdot (\mathbf{V}\_{k\_l} \times \mathbf{V}\_{k\_l})]^2 \le \sigma\_{V\text{min}}^{-6} \sum\_{1 \le k\_l < k\_l < k\_l \le N} [\mathbf{V}\_{k\_l} \cdot (\mathbf{V}\_{k\_l} \times \mathbf{V}\_{k\_l})]^2, N \ge 3 \tag{46}$$

It's shown that Eq. (46) has the similar format with Eq. (37). Thus, the change of the observability with the number of radio beacons is similar with the results in Table 1. However, due to involving relative range and velocity information, the optimal geometric configuration is different with the cases using only range measurements.

The lower bound of estimation errors in this case is evaluated by

<sup>F</sup><sup>N</sup> <sup>¼</sup> <sup>X</sup> N

130 Space Flight

4.6. Observability analysis of vehicle's position

Figure 1. Lower bound of estimation errors with beacon number.

i¼1 σ�<sup>2</sup> Vi

<sup>F</sup><sup>N</sup> <sup>¼</sup> <sup>X</sup> N

where L<sup>i</sup> is given by

Using the range-rate measurements, the FIM of the lander's position is derived as

∂Við Þ r; v ∂r � �<sup>T</sup> <sup>¼</sup> <sup>X</sup> N

i¼1 σ�<sup>2</sup> Vi Livv<sup>T</sup>L<sup>T</sup>

<sup>i</sup> , N ≥ 1 (43)

∂Við Þ r; v ∂r

i¼1 σ�<sup>2</sup> Vi

∂Við Þ r; v ∂v

3 1.000 σ�<sup>6</sup>

4 2.3704 σ�<sup>6</sup>

5 4.6296 σ�<sup>6</sup>

6 8.0000 σ�<sup>6</sup>

7 12.7037 σ�<sup>6</sup>

8 18.9630 σ�<sup>6</sup>

Table 1. Maximum determinants of FIM related to different number of beacons.

∂Við Þ r; v ∂v � �<sup>T</sup>

Eq. (42) has a similar form with Eq. (36) which describes the FIM of position. The only difference lies in the measurement deviation. Hence the same conclusion of the observability

of velocity can be obtained as that in Section 4.1. The detailed analysis is omitted here.

Number of beacons Maximum determinant of FIM

<sup>¼</sup> <sup>X</sup> N

i¼1 σ�<sup>2</sup> Vi nin<sup>T</sup>

Rmin

Rmin

Rmin

Rmin

Rmin

Rmin

<sup>i</sup> , N ≥ 1 (42)

$$\begin{split} \frac{\mathcal{G}}{\text{tr}(\mathbf{F}\_{N})} &= \sum\_{i=1}^{N} \sigma\_{ii}^{2} \mathbf{V}\_{i}^{T} \mathbf{V}\_{i} \\ &= \frac{1}{N} \sum\_{i=1}^{N} \frac{\sigma\_{ij}^{2}}{R\_{i}^{2}} \left[ \begin{bmatrix} \upsilon\_{z} \left( n\_{y}^{2} + n\_{z}^{2} \right) - \upsilon\_{y} n\_{ix} n\_{y} - \upsilon\_{z} n\_{ix} n\_{iz} \end{bmatrix}^{2} + \begin{bmatrix} -\upsilon\_{x} n\_{ix} n\_{y} + \upsilon\_{y} \left( n\_{ix}^{2} + n\_{iz}^{2} \right) - \upsilon\_{z} n\_{iy} n\_{iz} \end{bmatrix}^{2} + 1 \right] \\ &> \frac{\mathcal{N}}{\sum\_{i=1}^{N} \frac{\sigma\_{ij}^{2}}{R\_{i}^{2}} \left\{ \left[ \upsilon\_{z} \left( 1 + \frac{1}{2} \left( n\_{y}^{2} + n\_{iz}^{2} \right) \right) + \upsilon\_{y} \left( 1 + \frac{1}{2} \left( n\_{ix}^{2} + n\_{iz}^{2} \right) \right) + \upsilon\_{z} \left( 1 + \frac{1}{2} \left( n\_{ix}^{2} + n\_{iy}^{2} \right) \right) \right]^{2} \right\} \\ &> \frac{3}{3 \sum\_{i=1}^{N} \frac{\sigma\_{ij}^{2}}{R\_{i}^{2}} \left( \upsilon\_{x} + \upsilon\_{y} + \upsilon\_{z} \right)^{2} \\ &\ge \frac{2}{3 \sum\_{i=1}^{N} \sigma\_{ii}^{2}} \frac{\sigma\_{i}^{2}}{R\_{i}^{2}} \end{split} \tag{47}$$

where v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 <sup>x</sup> þ v<sup>2</sup> <sup>y</sup> þ v<sup>2</sup> z q is the lander's velocity value. Obviously, more radio beacons lead to more accurate estimation. Since the value of relative range is much bigger than relative velocity, the lower bound of estimation errors using range-rate date is larger than that using range data. Besides, it's concluded that more accurate range-rate measurement, closer relative range, and slower velocity can realizer more accurate position estimation.

#### 5. Orbit optimization based on observability analysis

#### 5.1. Optimization of navigation using ground beacons

The configuration radio beacons is expressed by the following set

$$\mathcal{C} = \{ \mathfrak{p}\_{\mathcal{B}}^i | i = 1, \dots, l \} \tag{48}$$

where Ω is the set of the areas of radio beacons that satisfy the visibility during the whole entry phase. In this optimization problem, the global optimization algorithm is selected to obtain the

Mars Networks-Based Navigation: Observability and Optimization

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133

It is assumed that the Mars entry phase lasts approximately 240 seconds. The entry trajectory

Three radio ranging measurements at a certain time can geometrically determine the position of the lander. Thus the navigation scenario with three beacons is first analyzed with respect to

The optimal three beacons are located close to the edge of both sides of the visible area. The beacon on the east side is almost along the entry trajectory, while the west two beacons are separated on the north and south side of the entry trajectory. The observability degree in this situation calculated by different methods is illustrated in Figure 5, and the computation time

Figure 5 shows a huge undulation in observability degree during the Marts entry phase. The maximum and minimum value are 1:<sup>413</sup> � 10-8 and 2:<sup>945</sup> � 10-7 respectively. Considering the machine precision, the navigation system is observable only if the observability exceeds

Initial state rð Þ0 km θð Þ0 deg φð Þ0 deg Vð Þ0 m/s γð Þ0 deg ψð Þ0 deg Value 3518.2 �89.872 �28.02 5515 �11.8 5.156

the observability. The optimal locations of beacons are displayed in Figure 4.

optimal beacon configuration.

for each method is listed in Table 3.

Figure 2. Principle of the line-of-sight visibility.

Table 2. Initial states of the lander.

The initial states of the lander are listed in Table 2.

and the corresponding visible area are shown in Figure 3.

where p<sup>i</sup> <sup>B</sup> <sup>¼</sup> xi <sup>B</sup>, yi <sup>B</sup>, zi B � �<sup>T</sup> is the position of the i th beacon. Considering the time-varying observability, the minimum value of the observability in the entry phase is taken as the optimization performance index.

$$D(\mathbf{C}) = \min\_{\mathbf{x} \in \mathcal{T}\_{\mathbf{x}}} \delta \tag{49}$$

To realize the Mars network-based navigation, the visibility of the beacons to the lander should be guaranteed. Define two unit vectors as follows

$$\mathfrak{n}\_{Bi} = \frac{\begin{bmatrix} \mathfrak{x}\_{\mathcal{B}^\*}^i & \mathfrak{y}\_{\mathcal{B}^\*}^i & \mathfrak{z}\_{\mathcal{B}}^i \end{bmatrix}^T}{\sqrt{\left(\mathfrak{x}\_{\mathcal{B}}^i\right)^2 + \left(\mathfrak{y}\_{\mathcal{B}}^i\right)^2 + \left(\mathfrak{z}\_{\mathcal{B}}^i\right)^2}}, \quad \mathfrak{n}\_{\mathbb{C}} = \frac{\begin{bmatrix} \tilde{\boldsymbol{x}}, \tilde{\boldsymbol{y}}.\tilde{\boldsymbol{z}} \end{bmatrix}^T}{\sqrt{\tilde{\boldsymbol{x}}^2 + \tilde{\boldsymbol{y}}^2 + \tilde{\boldsymbol{z}}^2}} \tag{50}$$

where ½ � ~x, ~y,~z <sup>T</sup> is the relative position vector from the lander to the radio beacon, obtained as

$$\left[\tilde{\mathbf{x}}, \tilde{\mathbf{y}}, \tilde{\mathbf{z}}\right]^{T} = \left[\mathbf{x}, \quad \mathbf{y}, \quad \mathbf{z}\right]^{T} - \left[\mathbf{x}\_{\text{B}}^{i} \quad \mathbf{y}\_{\text{B}}^{i} \quad \mathbf{z}\_{\text{B}}^{i}\right]^{T} \tag{51}$$

To guarantee the visibility, the two vectors in Eq. (50) should satisfy

$$\arccos\left(\mathfrak{n}\_{\text{Bi}} \cdot \mathfrak{n}\_{\text{C}}\right) < \frac{\pi}{2}, \quad \mathfrak{x} \in T\_{\mathfrak{x}} \tag{52}$$

The schematic of visibility is shown in Figure 2.

The optimization problem of beacon configuration is given as

$$\begin{aligned} \max \quad & D\left(\mathsf{C}\right) \\ \text{s.t.} \quad & p\_{\mathsf{B}}^{i} \in \mathsf{Q}, \quad i = 1, \ldots, l \end{aligned} \tag{53}$$

where Ω is the set of the areas of radio beacons that satisfy the visibility during the whole entry phase. In this optimization problem, the global optimization algorithm is selected to obtain the optimal beacon configuration.

The initial states of the lander are listed in Table 2.

where v ¼

132 Space Flight

where p<sup>i</sup>

where ½ � ~x, ~y,~z

<sup>B</sup> <sup>¼</sup> xi

zation performance index.

<sup>B</sup>, yi

<sup>B</sup>, zi B

be guaranteed. Define two unit vectors as follows

<sup>n</sup>Bi <sup>¼</sup> xi

xi B � �<sup>2</sup>

½ � ~x,~y,~z

The optimization problem of beacon configuration is given as

The schematic of visibility is shown in Figure 2.

To guarantee the visibility, the two vectors in Eq. (50) should satisfy

<sup>B</sup>, yi

<sup>B</sup>, zi B

<sup>T</sup> <sup>¼</sup> ½ � x, y, <sup>z</sup>

arc cos nBi ð Þ � n<sup>C</sup> <

max D ð Þ C s:t: p<sup>i</sup>

� �<sup>T</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>þ</sup> yi B � �<sup>2</sup>

� �<sup>T</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>y</sup> þ v<sup>2</sup> z

slower velocity can realizer more accurate position estimation.

5.1. Optimization of navigation using ground beacons

5. Orbit optimization based on observability analysis

The configuration radio beacons is expressed by the following set

<sup>C</sup> <sup>¼</sup> <sup>p</sup><sup>i</sup> B �

is the position of the i

observability, the minimum value of the observability in the entry phase is taken as the optimi-

Dð Þ¼ C min x∈Tx

To realize the Mars network-based navigation, the visibility of the beacons to the lander should

<sup>þ</sup> <sup>z</sup><sup>i</sup> B � �<sup>2</sup> <sup>q</sup> , <sup>n</sup><sup>C</sup> <sup>¼</sup> ½ � <sup>~</sup>x,~y,~<sup>z</sup>

<sup>T</sup> is the relative position vector from the lander to the radio beacon, obtained as

<sup>B</sup>, yi

<sup>B</sup>, zi B

<sup>T</sup> � <sup>x</sup><sup>i</sup>

π

is the lander's velocity value. Obviously, more radio beacons lead to

�<sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>⋯</sup>; <sup>l</sup> � � (48)

th beacon. Considering the time-varying

δ (49)

<sup>x</sup>~<sup>2</sup> <sup>þ</sup> <sup>y</sup>~<sup>2</sup> <sup>þ</sup> <sup>~</sup>z<sup>2</sup> <sup>p</sup> (50)

� �<sup>T</sup> (51)

<sup>2</sup> , <sup>x</sup> <sup>∈</sup>Tx (52)

<sup>B</sup> <sup>∈</sup> <sup>Ω</sup>, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, l (53)

T ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

more accurate estimation. Since the value of relative range is much bigger than relative velocity, the lower bound of estimation errors using range-rate date is larger than that using range data. Besides, it's concluded that more accurate range-rate measurement, closer relative range, and

v2 <sup>x</sup> þ v<sup>2</sup>

q

It is assumed that the Mars entry phase lasts approximately 240 seconds. The entry trajectory and the corresponding visible area are shown in Figure 3.

Three radio ranging measurements at a certain time can geometrically determine the position of the lander. Thus the navigation scenario with three beacons is first analyzed with respect to the observability. The optimal locations of beacons are displayed in Figure 4.

The optimal three beacons are located close to the edge of both sides of the visible area. The beacon on the east side is almost along the entry trajectory, while the west two beacons are separated on the north and south side of the entry trajectory. The observability degree in this situation calculated by different methods is illustrated in Figure 5, and the computation time for each method is listed in Table 3.

Figure 5 shows a huge undulation in observability degree during the Marts entry phase. The maximum and minimum value are 1:<sup>413</sup> � 10-8 and 2:<sup>945</sup> � 10-7 respectively. Considering the machine precision, the navigation system is observable only if the observability exceeds

Figure 2. Principle of the line-of-sight visibility.


Table 2. Initial states of the lander.

<sup>1</sup> 10-16. The observability degree during the entire entry phase passes through the threshold, and thus, the navigation system is observable. The minimum degree of observability occurs at the beginning of the entry phase when the entry vehicle is at its greatest distance from radio beacons, while the maximum degree of observability occurs when the entry vehicle approaches two beacons on the west side. In order to explain the evolution of the degree of observability. An

Figure 3. Entry trajectory and the visible area.

index related to the geometric configuration of the lander and radio beacons is given in Eq. (54)

Analysis approach Computation time, s

Method with Lie algebra >10,000 Linearization method 1.3987 Method based on quadratic approximation 2.1558

where ni, nj, and n<sup>k</sup> are the unit vectors from the beacon to the lander, N is the number of beacons. The evolution of index I is displayed in Figure 6, showing an identical variation trend

The observability degree obtained from the three methods is quite close to each other. However, the method based on Lie algebra consumes the most time. The linearization method provides the largest deviations, especially at the peak time, indicating a relatively low accuracy. The proposed quadratic approximation method achieves a performance balance in accuracy and complexity. To analyze the navigation accuracy, the Extended Kalman Filter (EKF) is

n<sup>i</sup> � n<sup>j</sup> � n<sup>k</sup>

� � � � <sup>2</sup> (54)

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<sup>I</sup> <sup>¼</sup> <sup>X</sup> 1 ≤ i<j<k ≤ N

with observability degree and backing up the observability analysis conclusion.

to explain the evolution of the degree of observability.

Table 3. Computation time for each approach.

Figure 5. Degree of observability for the optimal scenario with three beacons.

Figure 4. Optimal configuration for the scenario with three beacons.

Figure 5. Degree of observability for the optimal scenario with three beacons.


Table 3. Computation time for each approach.

<sup>1</sup> 10-16. The observability degree during the entire entry phase passes through the threshold, and thus, the navigation system is observable. The minimum degree of observability occurs at the beginning of the entry phase when the entry vehicle is at its greatest distance from radio beacons, while the maximum degree of observability occurs when the entry vehicle approaches two beacons on the west side. In order to explain the evolution of the degree of observability. An

Figure 3. Entry trajectory and the visible area.

134 Space Flight

Figure 4. Optimal configuration for the scenario with three beacons.

index related to the geometric configuration of the lander and radio beacons is given in Eq. (54) to explain the evolution of the degree of observability.

$$I = \sum\_{1 \le i < j < k \le N} \left[ \mathfrak{n}\_i \cdot \left( \mathfrak{n}\_j \times \mathfrak{n}\_k \right) \right]^2 \tag{54}$$

where ni, nj, and n<sup>k</sup> are the unit vectors from the beacon to the lander, N is the number of beacons. The evolution of index I is displayed in Figure 6, showing an identical variation trend with observability degree and backing up the observability analysis conclusion.

The observability degree obtained from the three methods is quite close to each other. However, the method based on Lie algebra consumes the most time. The linearization method provides the largest deviations, especially at the peak time, indicating a relatively low accuracy. The proposed quadratic approximation method achieves a performance balance in accuracy and complexity. To analyze the navigation accuracy, the Extended Kalman Filter (EKF) is

rB

(

vB

(

Figure 7. Navigation results for the optimal scenario with three beacons.

<sup>i</sup><sup>0</sup> ¼ ð Þ RM þ ai cosf <sup>i</sup>

<sup>i</sup><sup>0</sup> ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ=ð Þ RM þ ai <sup>p</sup> sin <sup>f</sup> <sup>i</sup>

where RM is the radius of Mars, ai is the orbit altitude, and P<sup>i</sup> and Q<sup>i</sup> are given by

<sup>P</sup><sup>i</sup> <sup>¼</sup> cosΩi, sinΩ<sup>i</sup> ½ � , <sup>0</sup> <sup>T</sup>

Ið Þ¼ e

P<sup>i</sup> þ ð Þ RM þ ai sinf <sup>i</sup>

<sup>Q</sup><sup>i</sup> ¼ �sinΩicosii, cosΩicosii, sinii ½ �<sup>T</sup>

Given the initial states of the Mars obiter, the subsequent states can be obtained by propagating the two-body dynamics. Likewise, the trajectory of the lander can be also obtained by propagating the entry dynamics. To evaluate the overall performance of the observability of the entry phase, the integration of the observability is taken as the performance index, given by

> ð tf

t¼0

Qi

Qi

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O tð Þdt (57)

(55)

137

(56)

<sup>P</sup><sup>i</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ=ð Þ RM þ ai <sup>p</sup> cos <sup>f</sup> <sup>i</sup>

Figure 6. The value of I for the optimal scenario with three beacons.


Table 4. Errors of initial states.

used to estimate the lander's states. The range measurement error is assumed to be Gaussian white noise with a standard deviation of 100 m. The initial errors are listed in Table 4. The estimation errors and the 1-sigma uncertainty bounds are depicted in Figure 7.

It's illustrated that θ and ϕ have the most accurate estimation and the fastest convergence. The convergence of the states V, γ, and Ψ is relatively slow at the beginning of the Mars entry phase due to the weak observability. With the increase of the observability degree comes the rapid convergence of the uncertainty bounds and the state estimation errors from about 90 to 115 seconds. The max deceleration of the lander also contributes to the rapid convergence.

#### 5.2. Optimization of navigation using Mars orbiters

Compared with ground beacons, the Mars obiters are constrained by the orbital dynamics, which is considered to be two-body dynamics here. In this subsection, the initial states of the Mars orbiters are considered as the optimized variables. Furthermore, assuming that the Mars orbiters moves in a circular orbit, the variables to be optimized are simplified as inclination i, longitude of ascending node Ω, and the true anomaly f. The initial states of the orbiter can be expressed by the optimized variables, given by

$$\begin{cases} \mathbf{r}\_{i0}^{B} = (\mathbf{R}\_{M} + a\_{i}) \text{cos} f\_{i} \mathbf{P}\_{i} + (\mathbf{R}\_{M} + a\_{i}) \text{sin} f\_{i} \mathbf{Q}\_{i} \\\\ \mathbf{v}\_{i0}^{B} = -\sqrt{\mu/(\mathbf{R}\_{M} + a\_{i})} \text{sin} f\_{i} \mathbf{P}\_{i} + \sqrt{\mu/(\mathbf{R}\_{M} + a\_{i})} \text{cos} f\_{i} \mathbf{Q}\_{i} \end{cases} \tag{55}$$

where RM is the radius of Mars, ai is the orbit altitude, and P<sup>i</sup> and Q<sup>i</sup> are given by

$$\begin{cases} \mathbf{P}\_i = \begin{bmatrix} \cos \Omega\_i & \sin \Omega\_i & \mathbf{0} \end{bmatrix}^T \\\\ \mathbf{Q}\_i = \begin{bmatrix} -\sin \Omega\_i \cos i\_\flat & \cos \Omega\_i \cos i\_\flat & \sin i\_\flat \end{bmatrix}^T \end{cases} \tag{56}$$

Given the initial states of the Mars obiter, the subsequent states can be obtained by propagating the two-body dynamics. Likewise, the trajectory of the lander can be also obtained by propagating the entry dynamics. To evaluate the overall performance of the observability of the entry phase, the integration of the observability is taken as the performance index, given by

$$I(\mathbf{e}) = \int\_{t=0}^{t\_f} O(t)dt\tag{57}$$

Figure 7. Navigation results for the optimal scenario with three beacons.

used to estimate the lander's states. The range measurement error is assumed to be Gaussian white noise with a standard deviation of 100 m. The initial errors are listed in Table 4. The

Initial state rð Þ0 m θð Þ0 deg φð Þ0 deg Vð Þ0 m/s γð Þ0 deg ψð Þ0 deg Error 1000 0.2 0.2 10 0.2 0.2

It's illustrated that θ and ϕ have the most accurate estimation and the fastest convergence. The convergence of the states V, γ, and Ψ is relatively slow at the beginning of the Mars entry phase due to the weak observability. With the increase of the observability degree comes the rapid convergence of the uncertainty bounds and the state estimation errors from about 90 to 115 seconds. The max deceleration of the lander also contributes to the rapid convergence.

Compared with ground beacons, the Mars obiters are constrained by the orbital dynamics, which is considered to be two-body dynamics here. In this subsection, the initial states of the Mars orbiters are considered as the optimized variables. Furthermore, assuming that the Mars orbiters moves in a circular orbit, the variables to be optimized are simplified as inclination i, longitude of ascending node Ω, and the true anomaly f. The initial states of the orbiter can be

estimation errors and the 1-sigma uncertainty bounds are depicted in Figure 7.

5.2. Optimization of navigation using Mars orbiters

Figure 6. The value of I for the optimal scenario with three beacons.

Table 4. Errors of initial states.

136 Space Flight

expressed by the optimized variables, given by

where tf is the final time of entry phase, e ¼ f g e1, ⋯, e<sup>n</sup> , e<sup>i</sup> ¼ Ωi, ii, f <sup>i</sup> � �<sup>T</sup> denotes the optimization variables. Similar to ground beacon-based navigation, the visibility between the lander and the Mars obiters should be also guaranteed. Define two angles as follows:

$$\begin{cases} \theta\_0 = \arccos\left(\frac{R\_M}{||r||}\right) \\\\ \theta\_{1i} = \arccos\left(\frac{R\_M}{||r\_i^B||}\right) \end{cases} \tag{58}$$

The navigation scenario with three Mars orbiters is analyzed. The nominal orbit altitude of the

O ¼ detð Þ¼ N<sup>3</sup> ½ � n<sup>1</sup> � ð Þ n<sup>2</sup> � n<sup>3</sup>

At a certain epoch, the maximum value of O is 1 when and only when three unit vectors n1, n2, and n<sup>3</sup> are orthogonal to each other. Considering the overall observability of the entry phase, the orbits of the Mars orbiters are optimized and shown in Figure 9, and the optimal initial

It's shown that the three orbiters keep a relatively stable configuration, and stays orthogonal approximately to each other. The value of maximized performance index is 237.963. The observability almost reaches the maximum value all the time during the Mars entry phase. The comparison of Mars obiters-based navigation and ground beacon-based navigation is

The fixed ground beacons have limited locations due to the visibility constrain and the geometric configuration cannot remain optimal during the entry phase. Thus, the observability is undulated to a large extent. The Mars orbiters overcome this defect with its moving property. To show straightforward the geometric configuration., the observability degree is close to maximum value at each epoch during the Mars entry phase. The angles between the vectors

It's shown that, using the ground beacons, the angles between the three vectors change dramatically in the entry phase. The optimal configuration can be met only at the epoch of 75 s. However, for the orbiter-based navigation scheme, n1, n2, and n<sup>3</sup> are almost orthogonal throughout the entry phase. The advantages of orbiter-based navigation scheme in the config-

Next, 500-time Monte Carlo simulations of navigation systems based on EKF are carried out. The initial position and velocity have standard deviations of 1 km and 0.5 m/s respectively. The measurement error is set to be 50 m, and considered as Gaussian white noise. The simulation

Since no information of entry vehicle's velocity is provided from range measurements, the convergence of velocity estimation is not as quick as position estimation. A much better navigation performance can be achieved by the Mars orbiter-based navigation. It can be

State Value Unit x �3.92 km y �3103.37 km z �1665.41 km vx 5775.31 m/s vy 1124.27 m/s vz 1175.48 m/s

performed. The observability degree of these two scenarios is shown in Figure 10.

uration and observability performance improve the navigation capability.

<sup>2</sup> (62)

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Mars Networks-Based Navigation: Observability and Optimization

three orbiters is 725 km. The observability is quantified by

elements are listed in Table 6.

n1, n2, and n<sup>3</sup> are depicted in Figure 11.

results are shown in Figure 12.

Table 5. Initial states of the lander.

The angle between the position vectors r and r<sup>B</sup> <sup>i</sup> is given by

$$\theta\_i = \arccos\left(\frac{r\_i^B \cdot r}{||r\_i^B|| ||r||}\right) \tag{59}$$

The visibility requires that

$$\theta\_i < \theta\_0 + \theta\_{1i} \quad t \in \begin{bmatrix} 0, & t\_f \end{bmatrix} \tag{60}$$

The schematic of the visibility is illustrated in Figure 8. The gray part represents the area in which the Mars orbiter is invisible to the lander.

Then the orbit optimization problem is given by

$$\begin{aligned} \max \qquad & I(\mathbf{e}) = \int\_{t=0}^{t\_f} O(t)dt\\ \text{subject to} \qquad & \theta\_i < \theta\_0 + \theta\_{1i}, \quad t \in \left[0, \quad t\_f\right], \quad i = 1, \dots, n \end{aligned} \tag{61}$$

In the optimization problem, the performance index cannot be expressed explicitly by the optimization variables, and the gradient cannot be obtained. Thus, the heuristic global optimization algorithm is chosen to solve the optimization problem. The lander's initial states are listed in Table 5 with the assumption of a ballistic entry having a banking angle of zero. The duration of entry phase is setup as 240 seconds.

Figure 8. The schematic of the visibility.

The navigation scenario with three Mars orbiters is analyzed. The nominal orbit altitude of the three orbiters is 725 km. The observability is quantified by

$$O = \det(\mathbf{N}\_3) = \left[\mathfrak{n}\_1 \cdot (\mathfrak{n}\_2 \times \mathfrak{n}\_3)\right]^2 \tag{62}$$

At a certain epoch, the maximum value of O is 1 when and only when three unit vectors n1, n2, and n<sup>3</sup> are orthogonal to each other. Considering the overall observability of the entry phase, the orbits of the Mars orbiters are optimized and shown in Figure 9, and the optimal initial elements are listed in Table 6.

It's shown that the three orbiters keep a relatively stable configuration, and stays orthogonal approximately to each other. The value of maximized performance index is 237.963. The observability almost reaches the maximum value all the time during the Mars entry phase. The comparison of Mars obiters-based navigation and ground beacon-based navigation is performed. The observability degree of these two scenarios is shown in Figure 10.

The fixed ground beacons have limited locations due to the visibility constrain and the geometric configuration cannot remain optimal during the entry phase. Thus, the observability is undulated to a large extent. The Mars orbiters overcome this defect with its moving property. To show straightforward the geometric configuration., the observability degree is close to maximum value at each epoch during the Mars entry phase. The angles between the vectors n1, n2, and n<sup>3</sup> are depicted in Figure 11.

It's shown that, using the ground beacons, the angles between the three vectors change dramatically in the entry phase. The optimal configuration can be met only at the epoch of 75 s. However, for the orbiter-based navigation scheme, n1, n2, and n<sup>3</sup> are almost orthogonal throughout the entry phase. The advantages of orbiter-based navigation scheme in the configuration and observability performance improve the navigation capability.

Next, 500-time Monte Carlo simulations of navigation systems based on EKF are carried out. The initial position and velocity have standard deviations of 1 km and 0.5 m/s respectively. The measurement error is set to be 50 m, and considered as Gaussian white noise. The simulation results are shown in Figure 12.

Since no information of entry vehicle's velocity is provided from range measurements, the convergence of velocity estimation is not as quick as position estimation. A much better navigation performance can be achieved by the Mars orbiter-based navigation. It can be


Table 5. Initial states of the lander.

where tf is the final time of entry phase, e ¼ f g e1, ⋯, e<sup>n</sup> , e<sup>i</sup> ¼ Ωi, ii, f <sup>i</sup>

8 >>>><

>>>>:

The angle between the position vectors r and r<sup>B</sup>

which the Mars orbiter is invisible to the lander. Then the orbit optimization problem is given by

duration of entry phase is setup as 240 seconds.

Figure 8. The schematic of the visibility.

max Ið Þ¼ e

ð tf

O tð Þdt

t¼0

subject to θ<sup>i</sup> < θ<sup>0</sup> þ θ1i, t∈ 0, tf

The visibility requires that

138 Space Flight

lander and the Mars obiters should be also guaranteed. Define two angles as follows:

θ<sup>0</sup> ¼ arccos

θ1<sup>i</sup> ¼ arccos

θ<sup>i</sup> ¼ arccos

θ<sup>i</sup> < θ<sup>0</sup> þ θ1i, t∈ 0, tf

The schematic of the visibility is illustrated in Figure 8. The gray part represents the area in

In the optimization problem, the performance index cannot be expressed explicitly by the optimization variables, and the gradient cannot be obtained. Thus, the heuristic global optimization algorithm is chosen to solve the optimization problem. The lander's initial states are listed in Table 5 with the assumption of a ballistic entry having a banking angle of zero. The

optimization variables. Similar to ground beacon-based navigation, the visibility between the

RM k kr � �

> RM rB i � � � �

<sup>i</sup> is given by

!

rB <sup>i</sup> � r rB i � � � �k kr

!

� �<sup>T</sup>

� � (60)

� �, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, n

denotes the

(58)

(59)

(61)

Figure 9. The optimal orbits of three orbiters.


Figure 11. Angles between three unit vectors in two navigation schemes.

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Figure 12. 1σ error bounds of states in two navigation schemes.

Table 6. Initial orbit elements of three orbiters.

Figure 10. Degree of observability in two navigation schemes.

Figure 11. Angles between three unit vectors in two navigation schemes.

Figure 9. The optimal orbits of three orbiters.

140 Space Flight

Table 6. Initial orbit elements of three orbiters.

Figure 10. Degree of observability in two navigation schemes.

Orbit element Obiter 1 Orbiter 2 Orbiter 3 Ω (deg) 49.329 16.136 36.562 i (deg) 24.209 35.889 18.901 f (deg) 240.219 256.141 229.294

Figure 12. 1σ error bounds of states in two navigation schemes.

concluded that the configuration of orbiters is a main contributor to the navigation performance. The Mars orbiter-based navigation, which can achieve a better configuration, is more practical for Mars entry navigation.

[5] Lightsey EG, Mogensen A, Burkhart PD, Ely TA, Duncan C. Real-time navigation for Mars missions using the Mars network. Journal of Spacecraft and Rockets. 2008;45:519-533 [6] Lévesque JF, de Lafontaine J. Innovative navigation schemes for state and parameter estimation during Mars entry. Journal of Guidance, Control, and Dynamics. 2007;30(1):

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143

[7] Pastor R, Bishop RH, Gay RS, Striepe SA. Mars entry navigation from EKF processing of beacon data. In: AIAA/AAS Astrodynamics Specialist Conference; 14–17 August; Denver,

[8] Yu Z, Cui P, Zhu S. Observability-based beacon configuration optimization for Mars entry navigation. Journal of Guidance, Control, and Dynamics. 2015;38(4):643-650

[9] Chamberlain N, Gladden R, Bruvold K. MAVEN relay operations concept. In: 2012 IEEE

[10] Ely TA, Anderson R, Bar-Sever YE, et al. Mars network constellation design drivers and strategies. In: AAS/AIAA Astrodynamics Specialist Conference; 16–19 August; Girwood,

[11] Ely TA. Optimal orbits for sparse constellations of mars navigation satellites. In: AAS/ AIAA Spaceflight Mechanics Meeting; 11–15 February; Santa Barbara, CA; 2001

[12] Pirondini F, Fernández AJ. A new approach to the design of navigation constellations around Mars: The MARCO POLO evolutionary system. In: the 57th International Astro-

[13] Yu Z, Zhu S, Cui P. Orbit optimization of mars orbiters for entry navigation: From an

[14] Yu Z, Cui P, Zhu S. On the observability of Mars entry navigation using radiometric

[15] Maessen DC, Gill E. Relative state estimation and observability for formation flying

[16] Hermann R, Krener AJ. Nonlinear controllability and observability. IEEE Transactions on

[17] Lall S, Marsden JE, Glavaški S. A subspace approach to balanced truncation for model reduction of nonlinear control systems. International Journal of Robust and Nonlinear

[18] Crassidis J, Junkins J. Optimal Estimation of Dynamic Systems. 2nd ed. Chapman & Hall/

[19] Sun D, Crassidis J. Observability analysis of six-degree-of-freedom configuration determination using vector observations. Journal of Guidance, Control, and Dynamics. 2002;

satellites in the presence of sensor noise. Acta Astronautica. 2013;82:129-136

169-184

CO; 2000. p. AIAA 2000-4426

AK; 1999. p. AAS 99-301

Aerospace Conference; 3–10 March; Bigsky, MT; 2012

nautical Congress; 02–06 October; Valencia, Spain; 2006

Automatic Control. 1977;22(5):728-740

Control. 2002;12(6):519-535

CRC: Boca Raton, FL; 2011

25(6):1149-1157

observability point of view. Acta Astronautica. 2015;111:136-145

measurements. Advances in Space Research. 2014;54(8):1513-1524

#### 6. Conclusions

This chapter introduced the Mars Networks-based navigation for the Mars entry phase. Based on the navigation scheme, the observability of the navigation system was analyzed using the proposed two novel observability analysis methods. Furthermore, the beacon configuration was optimized based on observability considering the line-of-sight constraints were concluded that the beacon configuration is a main contributor to the Mars Networks-based navigation. The observability analysis showed that an improved behavior of observability and more flexibility of beacon configuration determination can be achieved using more beacons. Navigation also demonstrated this conclusion. Meanwhile, compared with the ground beacons, Mars orbiters may be a better choice as Mars Network which gives a more accurate navigation result.

#### Author details

Zhengshi Yu1,2\*, Pingyuan Cui1,2, Rui Xu1,2 and Shengying Zhu1,2

\*Address all correspondence to: yuzhengshibit@qq.com

1 School of Aerospace Engineering, Beijing Institute of Technology, Beijing, China

2 Key Laboratory of Autonomous Navigation and Control for Deep Space Exploration, Ministry of Industry and Information Technology, Beijing, China

#### References


[5] Lightsey EG, Mogensen A, Burkhart PD, Ely TA, Duncan C. Real-time navigation for Mars missions using the Mars network. Journal of Spacecraft and Rockets. 2008;45:519-533

concluded that the configuration of orbiters is a main contributor to the navigation performance. The Mars orbiter-based navigation, which can achieve a better configuration, is more

This chapter introduced the Mars Networks-based navigation for the Mars entry phase. Based on the navigation scheme, the observability of the navigation system was analyzed using the proposed two novel observability analysis methods. Furthermore, the beacon configuration was optimized based on observability considering the line-of-sight constraints were concluded that the beacon configuration is a main contributor to the Mars Networks-based navigation. The observability analysis showed that an improved behavior of observability and more flexibility of beacon configuration determination can be achieved using more beacons. Navigation also demonstrated this conclusion. Meanwhile, compared with the ground beacons, Mars orbiters

may be a better choice as Mars Network which gives a more accurate navigation result.

1 School of Aerospace Engineering, Beijing Institute of Technology, Beijing, China

2 Key Laboratory of Autonomous Navigation and Control for Deep Space Exploration,

[1] Yu Z, Cui P, Crassidis J. Design and optimization of navigation and guidance techniques for Mars pinpoint landing: Review and prospect. Progress in Aerospace Sciences. 2017;

[2] Cui P, Yu Z, Zhu S. Research progress and prospect of autonomous navigation tech-

[3] Edwards CD, Adams JT, Bell DJ, et al. Strategies for telecommunications and navigation

[4] Hastrup RC, Bell DJ, Cesarone RJ, et al. Mars network for enabling low-cost missions.

niques for Mars entry phase. Journal of Astronautics. 2013;34(4):447-456

in support of Mars exploration. Acta Astronautica. 2001;48:661-668

Zhengshi Yu1,2\*, Pingyuan Cui1,2, Rui Xu1,2 and Shengying Zhu1,2

Ministry of Industry and Information Technology, Beijing, China

\*Address all correspondence to: yuzhengshibit@qq.com

practical for Mars entry navigation.

6. Conclusions

142 Space Flight

Author details

References

94C:82-94

Acta Astronautica. 2003;52:227-235


[20] Lee W, Bang H, Leeghim H. Cooperative localization between small UAVs using a combination of heterogeneous sensors. Aerospace Science and Technology. 2013;27(1): 105-111

**Section 4**

**Spacecraft Propulsion**

[21] Cui P, Yu Z, Zhu S, Ai G. Real-time navigation for Mars final approach using X-ray pulsars. In: AIAA Guidance, Navigation, and Control Conference and Exhibit; 19–22 August; Boston, MA; 2013. p. AIAA 2013-5204
