# Angular Trajectory Design for MR-SPS Using Bezier Shaping Approach
## 2 Model Description
### Attitude Dynamic Model  
The Newton-Euler approach based on the balance of all the forces and torques acting on each part of the structure is widely used to solve inverse dynamics of robots. The method avoids complicated derivatives with respect to generalized coordinates, which is  
Figure 2: The definitions of the five reference frames. common with Lagrangian-based approaches [17]. It can be adopted to model the complex dynamics of the MR-SPS. The force balance equations of the central body and the \(j\)th (\(j\) = 1, 2) solar array are listed as follows:  
\[\textbf{F}_{soj}+\textbf{F}_{sij}=m_{sj}\textbf{C}_{sj}\textbf{C}_{a}\textbf{a}_{ j}=m_{sj}\textbf{C}_{sj}\textbf{C}_{a}(\tilde{\textbf{r}}_{a}+\textbf{C}_{a}^{T} \dot{\omega}_{a}^{\times}\textbf{r}_{cj}+\textbf{C}_{a}^{T}\omega_{a}^{\times} \omega_{a}^{\times}\textbf{r}_{cj}) \tag{5}\]  
\[\textbf{F}_{so}+\sum_{k=1}^{2}\textbf{F}_{ak}=m_{a}\textbf{C}_{a}\textbf{a}_{ a}=m_{a}\textbf{C}_{a}\tilde{\textbf{r}}_{a} \tag{6}\]  
\[\textbf{F}_{sij}=-\textbf{C}_{sj}\textbf{F}_{aij} \tag{7}\]  
where \(m_{a}\) and \(m_{sj}\) are the mass of the central body and \(j\)th solar array respectively. \(\textbf{C}_{sj}\) is the rotation matrix from the central body frame to the \(j\)th solar array frame, and \(\textbf{C}_{a}\) denotes the orientation of the central body frame respect to the inertial frame. \(\textbf{F}_{ao}\) and \(\textbf{F}_{sij}\) are external forces acting on the central body and the \(j\)th solar array frame. \(\textbf{F}_{aij}\) and \(\textbf{F}_{sij}\) are the forces exerted on the central body by the \(j\)th solar array and on the \(j\)th solar array by the central body. \(\textbf{r}_{a}\) is the position vector of the central body and \(\textbf{r}_{sj}\) is the vector from the centroid of the central body to the centroid of the \(j\)th solar array expressed in the inertial frame. \(\textbf{\omega}_{a}\) denotes the angular velocity of central body respect to inertial frame expressed in the central body frame. The notation \(\textbf{x}^{\times}\) for \(\textbf{x}=\left[x_{1}\quad x_{2}\quad x_{3}\right]^{T}\) represents the skew symmetric matrix:  
\[\textbf{x}^{\times}=\left[ \begin{array}{ccc}0&-x_{3}&x_{2}\\ x_{3}&0&-x_{1}\\ -x_{2}&x_{1}&0\end{array} \right]\]  
According to Equations (5)-(7), forces exerted on the central body by the \(j\)th solar array are given as follows:  
\[\textbf{F}_{aij}=\textbf{A}_{o}^{j}+\textbf{A}_{\dot{\omega}_{a}}^{j}\dot{ \textbf{\omega}}_{a}+\textbf{A}_{other}^{j} \tag{8}\]  
where  
\[\textbf{A}_{o}^{j}=\frac{1}{m_{a}+\sum\limits_{k=1}^{2}m_{sk}}((m_{a}+\sum \limits_{k=1}^{n}m_{sk})\textbf{C}_{sj}^{T}\textbf{F}_{soj}-m_{sj}\sum\limits _{k=1}^{2}\textbf{C}_{sk}^{T}\textbf{F}_{sok}-m_{sj}\textbf{F}_{ao})\]  
\[\textbf{A}_{\dot{\omega}_{a}}^{j}=\frac{m_{sj}}{m_{a}+\sum\limits_{k=1}^{2}m_ {sk}}(-\sum\limits_{k=1}^{2}(m_{sk}\textbf{r}_{ck})+(m_{a}+\sum\limits_{k=1}^ {n}m_{sk})\textbf{r}_{cj})^{\times}\]  
\[\textbf{A}_{other}^{j}=\frac{m_{sj}\omega_{a}^{\times}\omega_{a}^{\times}}{m_{ a}+\sum\limits_{k=1}^{2}m_{sk}}(\sum\limits_{k=1}^{2}(m_{sk}\textbf{r}_{ck})-(m_{a}+ \sum\limits_{k=1}^{n}m_{sk})\textbf{r}_{cj})\]  
The external forces acting on the MR-SPS include gravity, solar pressure, and microwave radiation force. The magnitude of solar pressure and microwave radiation pressure is far less than that of gravity, so they can be ignored. The external can be expressed as follows:  
\[\textbf{F}_{ao}=-\frac{\mu m_{a}}{\left|\textbf{r}_{a}\right|^{3}}\textbf{C} _{a}\textbf{r}_{a} \tag{9}\]  
\[\textbf{F}_{soj}=-\frac{\mu m_{sj}}{\left|\textbf{r}_{a}+\textbf{C}_{a}^{T} \textbf{r}_{cj}\right|^{3}}(\textbf{C}_{sj}\textbf{C}_{a}\textbf{r}_{a}+ \textbf{C}_{sj}\textbf{r}_{cj}) \tag{10}\]  
where \(\mu\) denotes the gravitational coefficient of the earth.