3. Geometry of a combination of polynomial and trigonometric functions

These type functions can be used in defining the orbits of artificial satellites (and celestial bodies). Also, the numerical example part of this chapter, to estimate those type functions, will be inspected and applied on a real example. To foresee a model for any problem we should interpret the model parameter and comprehend the geometry of the model (Figure 1).

With respect to independent variable time t, a combination function of p ¼ 1 degree polynomial and order q ¼ 1 trigonometric function(s) [a combination of polynomial degree and trigonometric order (CPT)] to be estimated in the chapter is:

$$\phi\_{\rangle} = a\_{\phi} + b\_{\phi} \cdot t\_{\circ} + c\_{\phi} \cdot \sin \left( d\_{\phi} + e\_{\phi} \cdot t\_{\circ} \right), \tag{12}$$

$$\phi\_{\rangle} \in \{X\_{\circ}, Y\_{\circ}, Z\_{\circ}, S\_{\circ}\}, \quad j \in \{1, 2, \ldots, n\ \ \}.$$

In this chapter, the functions ϕ<sup>j</sup> are the coordinate components Xj;Yj;Zj

precise orbit file and the geometric distances Sj ¼

Figure 2. Earth (GR) and space-fixed (Υ) coordinates for an artificial satellite.

part) (Figure 2)?

t<sup>0</sup> of the data) is carried out by:

R<sup>3</sup> θ<sup>j</sup> � � <sup>¼</sup>

Xγ,j ¼ R<sup>3</sup> θ<sup>j</sup>

xGR,j ¼ R<sup>3</sup> �θ<sup>j</sup>

2 6 4

rotation matrix around the third axis (Figure 2).

approximate values of unknowns for a CPT is:

� � incoming from a

as a function of the compo-

� �, (13b)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� � <sup>x</sup>GR,j, <sup>θ</sup><sup>j</sup> ¼ �wE tj, (13a)

Xj Yj Zj

<sup>3</sup> θ<sup>j</sup> � � <sup>¼</sup> <sup>R</sup>�<sup>1</sup>

, xGR,j ¼

<sup>3</sup> θ<sup>j</sup>

xj yj zj 3 7 5 GR :

2 6 4

� � <sup>¼</sup> <sup>R</sup><sup>T</sup>

2 6 4 <sup>j</sup> <sup>þ</sup> <sup>Z</sup><sup>2</sup> j

On Non-Linearity and Convergence in Non-Linear Least Squares

http://dx.doi.org/10.5772/intechopen.76313

63

X2 <sup>j</sup> <sup>þ</sup> <sup>Z</sup><sup>2</sup>

q

nents. However, nonperiodic earth-fixed coordinates (GR) in the SP3 file should be transformed to the periodic space-fixed coordinates (Υ); why is Eq. (12) is suitable for the space-fixed coordinates, not earth-fixed ones (as seen from Figure 3 in the numerical example

For this propose, an easy transformation into any epoch (e.g., it can be taken as the first epoch

3 7 <sup>5</sup> ,Xγ,j <sup>¼</sup>

where wE and R<sup>3</sup> are in order of the angular velocity of earth and well-known orthogonal

A solution of nonlinear Eq. (12) is realized in the following order. Linearizing Eq. (12) by Taylor expansion and omitting the terms greater than or equal to quadratic ones, the linear equation system as given by Eq. (8) is obtained. The explicit form of the Eq. (8) with respect to the

� � <sup>X</sup>γ,j, <sup>R</sup><sup>3</sup> �θ<sup>j</sup>

cos θ<sup>j</sup> sin θ<sup>j</sup> 0 � sin θ<sup>j</sup> cos θ<sup>j</sup> 0 0 01

where tj; ϕ<sup>j</sup> are data given. In Eq. (12), translation <sup>a</sup><sup>ϕ</sup> and slope <sup>b</sup><sup>ϕ</sup> are elements of a line equation which is a first-order polynomial of CPT function. The other model parameters in the trigonometric part of Eq. (12) are defined as an amplitude cϕ, and an initial phase d<sup>ϕ</sup> and a frequency (or angular velocity) e<sup>ϕ</sup> ¼ 2π=T<sup>ϕ</sup> (a period Tϕ) of a wave (Figure 1).

Figure 1. The geometry of a first-degree and first-order combination of polynomial and trigonometric (CPT) function.

Figure 2. Earth (GR) and space-fixed (Υ) coordinates for an artificial satellite.

Relationships between nonlinearity and LS in a multidimensional surface have been shown by Teunissen et al. [1, 2]. The authors argued the relation on some simple examples and gave some analytical solutions for them. But, they highlighted that those types of analytical solutions have not been given for every problem and emphasized that suitable Taylor expansions

3. Geometry of a combination of polynomial and trigonometric functions

These type functions can be used in defining the orbits of artificial satellites (and celestial bodies). Also, the numerical example part of this chapter, to estimate those type functions, will be inspected and applied on a real example. To foresee a model for any problem we should

With respect to independent variable time t, a combination function of p ¼ 1 degree polynomial and order q ¼ 1 trigonometric function(s) [a combination of polynomial degree and

ϕ<sup>j</sup> ¼ a<sup>ϕ</sup> þ b<sup>ϕ</sup> tj þ c<sup>ϕ</sup> sin d<sup>ϕ</sup> þ e<sup>ϕ</sup> tj

equation which is a first-order polynomial of CPT function. The other model parameters in the trigonometric part of Eq. (12) are defined as an amplitude cϕ, and an initial phase d<sup>ϕ</sup> and a

Figure 1. The geometry of a first-degree and first-order combination of polynomial and trigonometric (CPT) function.

, j<sup>∈</sup> f g <sup>1</sup>; <sup>2</sup>;…; <sup>n</sup> :

are data given. In Eq. (12), translation a<sup>ϕ</sup> and slope b<sup>ϕ</sup> are elements of a line

, (12)

interpret the model parameter and comprehend the geometry of the model (Figure 1).

trigonometric order (CPT)] to be estimated in the chapter is:

where tj; ϕ<sup>j</sup> 

62 Optimization Algorithms - Examples

ϕ<sup>j</sup> ∈ Xj;Yj;Zj; Sj

frequency (or angular velocity) e<sup>ϕ</sup> ¼ 2π=T<sup>ϕ</sup> (a period Tϕ) of a wave (Figure 1).

have been useful to the solution not being transformed into the analytical ones.

In this chapter, the functions ϕ<sup>j</sup> are the coordinate components Xj;Yj;Zj � � incoming from a precise orbit file and the geometric distances Sj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X2 <sup>j</sup> <sup>þ</sup> <sup>Z</sup><sup>2</sup> <sup>j</sup> <sup>þ</sup> <sup>Z</sup><sup>2</sup> j q as a function of the components. However, nonperiodic earth-fixed coordinates (GR) in the SP3 file should be transformed to the periodic space-fixed coordinates (Υ); why is Eq. (12) is suitable for the space-fixed coordinates, not earth-fixed ones (as seen from Figure 3 in the numerical example part) (Figure 2)?

For this propose, an easy transformation into any epoch (e.g., it can be taken as the first epoch t<sup>0</sup> of the data) is carried out by:

$$\mathbf{X}\_{\mathbf{\gamma},\dot{\mathbf{j}}} = \mathbf{R}\_3(\theta\_{\dot{\mathbf{j}}}) \ \mathbf{x}\_{\text{GR},\dot{\mathbf{j}}} \qquad \qquad \qquad \quad \theta\_{\dot{\mathbf{j}}} = -w\_{\mathbf{E}} \ t\_{\mathbf{j}\dot{\mathbf{j}}} \tag{13a}$$

$$\mathbf{x}\_{\text{GR},\dot{\boldsymbol{\gamma}}} = \mathbf{R}\_3 \left( -\boldsymbol{\theta}\_{\dot{\boldsymbol{\gamma}}} \right) \ \mathbf{X}\_{\boldsymbol{\gamma},\dot{\boldsymbol{\gamma}}} \qquad \qquad \mathbf{R}\_3 \left( -\boldsymbol{\theta}\_{\dot{\boldsymbol{\gamma}}} \right) = \mathbf{R}\_3^T \left( \boldsymbol{\theta}\_{\dot{\boldsymbol{\gamma}}} \right) = \mathbf{R}\_3^{-1} \left( \boldsymbol{\theta}\_{\dot{\boldsymbol{\gamma}}} \right) , \tag{13b}$$

$$\mathbf{R}\_3(\boldsymbol{\theta}\_j) = \begin{bmatrix} \cos \theta\_j & \sin \theta\_j & 0 \\ -\sin \theta\_j & \cos \theta\_j & 0 \\ 0 & 0 & 1 \end{bmatrix}, \mathbf{X}\_{\boldsymbol{\gamma},j} = \begin{bmatrix} \mathbf{X}\_j \\ \mathbf{Y}\_j \\ \mathbf{Z}\_j \end{bmatrix}\_{\boldsymbol{\gamma}}, \mathbf{x}\_{\text{GR},j} = \begin{bmatrix} \mathbf{x}\_j \\ \mathbf{y}\_j \\ \mathbf{z}\_j \end{bmatrix}\_{\boldsymbol{\text{GR}}}.$$

where wE and R<sup>3</sup> are in order of the angular velocity of earth and well-known orthogonal rotation matrix around the third axis (Figure 2).

A solution of nonlinear Eq. (12) is realized in the following order. Linearizing Eq. (12) by Taylor expansion and omitting the terms greater than or equal to quadratic ones, the linear equation system as given by Eq. (8) is obtained. The explicit form of the Eq. (8) with respect to the approximate values of unknowns for a CPT is:

$$\mathbf{A}\_{\rangle} = \begin{bmatrix} 1 & t\_{\rangle} & \sin \begin{pmatrix} d\_0 + f\_0 & t\_{\rangle} \end{pmatrix} & c\_0 \cos \begin{pmatrix} d\_0 + e\_0 & t\_{\rangle} \end{pmatrix} & t\_{\rangle} c\_0 \cos \begin{pmatrix} d\_0 + e\_0 & t\_{\rangle} \end{pmatrix} \end{bmatrix},\tag{14a}$$

$$\mathbf{1}\_{j} = \left[\phi\_{j} - \begin{pmatrix} a\_{0} + b\_{0} \ t\_{j} + c\_{0} & \sin\left(d\_{0} + e\_{0} \ t\_{j}\right) \end{pmatrix}\right], \ j \in \{\ 1, 2, ..., n\ \ \}. \tag{14b}$$

We can use a recursive solution for Eq. (14) instead of the batch solution as Eq. (11) because of its solution velocity.

$$\widehat{\mathbf{Q}} = \mathbf{Q} \left( \sum\_{j=1}^{n} \mathbf{A}\_{j}^{T} \mathbf{l}\_{j} \right), \qquad \mathbf{Q} = \left( \sum\_{j=1}^{n} \mathbf{A}\_{j}^{T} \mathbf{A}\_{j} \right)^{-1} = \left( \sum\_{j=1}^{n} \mathbf{A}\_{j}^{T} \mathbf{A}\_{j} \right)^{+}. \tag{15}$$

Continuation of the solution of Eq. (15) can be performed according to Eq. (11). The model given by Eq. (12) is a simple model to determine the satellite orbit motions. For more complicated models, the readers can utilize [19–25] resources.
