1. Introduction

Attitude control is the process of making a spacecraft, e.g. a satellite to point toward a specific direction of interest, and attitude path planning is an essential part of space missions. Some current and future space missions require the deployment of teams of spacecraft for such purposes as interferometry and sensor coverage, e.g. [1, 2]. The general problem of attitude control (AC) is important, not only in the navigation of satellites but also of other spacecraft [3], aircraft, and robots. For this reason, the topic has been studied extensively in the literature, e.g. [4–10].

Attitude path planning is a challenging problem and becomes more challenging when it involves multiple spacecraft. First, they are moving at very high speed in highly dynamic

© 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.


environments, subject to external constraints such as blinding celestial objects, which can damage onboard sensors. Secondly, because they are in a team, they must be careful with each other when changing attitude, so as not to collide with each other and damage appendages. We consider a team of networked spacecraft, which share some common objectives, where

Multi-Spacecraft Attitude Path Planning Using Consensus with LMI-Based Exclusion Constraints

http://dx.doi.org/10.5772/intechopen.71580

47

Spacecraft attitude dynamics is usually represented by unit quaternions because quaternion dynamics do not encounter the singularities associated with other representations. However, quaternion dynamics are non-linear, which makes it difficult to apply Laplacian-like dynamics

Next, we consider some previous work on constrained attitude path planning. In [4], attitude control was formulated as a quadratically constrained optimization problem. Linear matrix inequalities (LMIs) and semidefinite programming (SDP) were employed to solve it for a multiple spacecraft scenario in [6]. In [10], spacecraft attitude stabilization on a sphere was studied. The control torques required for effective attitude stabilization were reduced from three to two. In [12], a consensus-based approach was applied in distributed attitude alignment of a team of communicating spacecraft flying in formation. In [13], a Laplacian-based protocol implemented using the modified Rodriquez parameters (MRP) was employed in leader

However, none of these aforementioned works apply consensus theory directly to quaternions, except our previous works [7–9]. In addition, only [4, 6–9] tackle the important problem of attitude cone avoidance constraints. Moreover, the works [4, 6, 7] were developed for spacecraft in the same coordinate frame, which does not have a direct practical implementation

To handle the difficulty of nonlinearity in quaternion kinematics, we cast the Q-CAC problem as a semidefinite program, which is subject to convex quadratic constraints, stated as LMI. Then a series of Laplacian-like matrices are synthesized, which satisfy the constraints and enables the spacecraft achieve consensus with exclusion. We employed available optimization software tools such as Sedumi [14] and Yalmip [15] running inside

Moreover, the solution presented here was developed for the realistic scenario of spacecraft in different coordinate frames, making it practical to implement directly. Therefore, the contributions of this chapter are aspects of our previous works [7–9], which are: (1) development of a quaternion consensus protocol; (2) incorporating dynamic cone avoidance constraints into the consensus framework; (3) providing a mathematical convergence analysis for the quaternionbased consensus framework; (4) extending the approach to multiple spacecraft in any coordi-

The rest of the chapter is organized as follows: The problem statement is in Section 2, followed by brief mathematical preliminaries in Section 3. The solution technique and convergence analysis are in Section 4, numerical simulations in Section 5, and conclusion in Section 6. Notations frequently used in this chapter are listed in Table 1. The words obstacle, avoidance,

nate frames, thereby making it more suitable for practical implementation.

exclusion, exclusion vector may be used interchangeably in this chapter.

consensus theory based on graph Laplacians can be applied [11, 12].

directly to quaternions.

following attitude control of spacecraft.

unless developed further.

MATLAB®, for simulation.

Table 1. Frequently used notations in this chapter.

environments, subject to external constraints such as blinding celestial objects, which can damage onboard sensors. Secondly, because they are in a team, they must be careful with each other when changing attitude, so as not to collide with each other and damage appendages. We consider a team of networked spacecraft, which share some common objectives, where consensus theory based on graph Laplacians can be applied [11, 12].

Notation Meaning SCi, SCi Spacecraft i

46 Advanced Path Planning for Mobile Entities

q<sup>i</sup> Vector part of q<sup>i</sup>

<sup>q</sup><sup>i</sup>� Antisymmetric of <sup>q</sup><sup>i</sup>

ω Angular velocity τ Control torque J Inertia matrix L Laplacian matrix

q�i or q<sup>i</sup> ∗

FI SCi

F<sup>B</sup> SCi

vB

vI

vI

vB

vI

(xij )

q<sup>i</sup> 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

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

xi Position vector of SCi, SCi

x Stacked vector of n position vectors

<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> qn

off Offset vector between i and j xoff Stacked vector of n offset vectors

obsi :<sup>j</sup> Vector of the j

t<sup>0</sup> Initial time tf Final time

Table 1. Frequently used notations in this chapter.

<sup>S</sup><sup>m</sup> The set of <sup>m</sup> � <sup>m</sup> positive definite matrices A~ Cone avoidance constraint matrix R<sup>i</sup> Rotation matrix corresponding to qi

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

Conjugate of q<sup>i</sup>

, qi <sup>¼</sup> <sup>q</sup><sup>1</sup> <sup>q</sup><sup>2</sup> <sup>q</sup><sup>3</sup> <sup>T</sup>

Fixed coordinate (Inertial) frame with origin at SCi's center

SCi

SCi

th obstacle in F<sup>I</sup>

Rotational coordinate (Body) frame with origin at SCi's center

SCi

SCi

SCi

= [q<sup>1</sup> q<sup>2</sup> q3| q4]

T

Spacecraft attitude dynamics is usually represented by unit quaternions because quaternion dynamics do not encounter the singularities associated with other representations. However, quaternion dynamics are non-linear, which makes it difficult to apply Laplacian-like dynamics directly to quaternions.

Next, we consider some previous work on constrained attitude path planning. In [4], attitude control was formulated as a quadratically constrained optimization problem. Linear matrix inequalities (LMIs) and semidefinite programming (SDP) were employed to solve it for a multiple spacecraft scenario in [6]. In [10], spacecraft attitude stabilization on a sphere was studied. The control torques required for effective attitude stabilization were reduced from three to two. In [12], a consensus-based approach was applied in distributed attitude alignment of a team of communicating spacecraft flying in formation. In [13], a Laplacian-based protocol implemented using the modified Rodriquez parameters (MRP) was employed in leader following attitude control of spacecraft.

However, none of these aforementioned works apply consensus theory directly to quaternions, except our previous works [7–9]. In addition, only [4, 6–9] tackle the important problem of attitude cone avoidance constraints. Moreover, the works [4, 6, 7] were developed for spacecraft in the same coordinate frame, which does not have a direct practical implementation unless developed further.

To handle the difficulty of nonlinearity in quaternion kinematics, we cast the Q-CAC problem as a semidefinite program, which is subject to convex quadratic constraints, stated as LMI. Then a series of Laplacian-like matrices are synthesized, which satisfy the constraints and enables the spacecraft achieve consensus with exclusion. We employed available optimization software tools such as Sedumi [14] and Yalmip [15] running inside MATLAB®, for simulation.

Moreover, the solution presented here was developed for the realistic scenario of spacecraft in different coordinate frames, making it practical to implement directly. Therefore, the contributions of this chapter are aspects of our previous works [7–9], which are: (1) development of a quaternion consensus protocol; (2) incorporating dynamic cone avoidance constraints into the consensus framework; (3) providing a mathematical convergence analysis for the quaternionbased consensus framework; (4) extending the approach to multiple spacecraft in any coordinate frames, thereby making it more suitable for practical implementation.

The rest of the chapter is organized as follows: The problem statement is in Section 2, followed by brief mathematical preliminaries in Section 3. The solution technique and convergence analysis are in Section 4, numerical simulations in Section 5, and conclusion in Section 6. Notations frequently used in this chapter are listed in Table 1. The words obstacle, avoidance, exclusion, exclusion vector may be used interchangeably in this chapter.
