# Angular Trajectory Design for MR-SPS Using Bezier Shaping Approach
## 3 Trajectory Planning for the MR-SPS Considering Dynamic Constraints  
### State Approximation Using Bezier Curve  
In Bezier shaping approach, it is assumed in advance that the angular trajectory of the MR-SPS follows Bezier curve functions. The approximation of \(\gamma\) and \(\theta_{j}\) can be expanded by applying Bezier curve functions. To avoid repetition, only the processing approach for \(\gamma\)is given, and \(\theta_{j}\) can be processed in the same way. The approximation of \(\gamma\) is expanded as follows:  
\[\gamma(\tau)=\sum_{k=0}^{n_{\gamma}}B_{\gamma,k}(\tau)P_{\gamma,k} \tag{20}\]  
where \(\tau=t/T\) (\(0\leq\tau\leq 1\)) is the scaled time, and \(T\) is the flight time; \(n_{\gamma}\) is the order of the Bezier curve function. \(P_{\gamma,k}\) (\(k\in[0,n_{\gamma}]\)) are the Bezier coefficients; \(B_{\gamma,k}(\tau)\) are the Bezier basis functions given by:  
\[B_{\gamma,k}(\tau)=\frac{n_{\gamma}!}{k!(n_{\gamma}-k)!}\tau^{k}(1-\tau)^{n_{ \gamma}-k} \tag{21}\]  
The first and second \(\tau\)-derivative of \(\gamma\) can be derived as follows:  
\[\gamma^{\prime}(\tau)=\sum_{k=0}^{n_{\gamma}}B^{\prime}_{\gamma,k}(\tau)P_{ \gamma,k} \tag{22}\]  
\[\gamma^{\prime\prime}(\tau)=\sum_{k=0}^{n_{\gamma}}B^{\prime\prime}_{\gamma,k }(\tau)P_{\gamma,k} \tag{23}\]  
where \(B^{\prime}_{\gamma,k}(\tau)\) and \(B^{\prime\prime}_{\gamma,k}(\tau)\) are first and second \(\tau\)-derivative of the Bezier basis functions that can be derived according to Equation (20) as:  
\[B^{\prime}_{\gamma,k}(\tau)=\left\{ \begin{array}{ll}-n_{\gamma}(1-\tau)^{n _{\gamma}-1}&k=0\\ \frac{n_{\gamma}!}{(k-1)!(n_{\gamma}-k)!}\tau^{k-1}(1-\tau)^{n_{\gamma}-k}&k =0\\ -\frac{n_{\gamma}!}{k!(n_{\gamma}-k-1)!}\tau^{k}(1-\tau)^{n_{\gamma}-k-1}&k \in[1,n_{\gamma}-1]\\ n_{\gamma}\tau^{n_{\gamma}-1}&k=n_{r}\end{array} \right. \tag{24}\]  
\[B^{\prime\prime}_{\gamma,k}(\tau)=\left\{ \begin{array}{ll}n_{\gamma}(n_{ \gamma}-1)(1-\tau)^{n_{\gamma}-2}&k=0\\ n_{\gamma}(n_{\gamma}-1)(n_{\gamma}-2)\tau(1-\tau)^{n_{\gamma}-3}-2n_{\gamma} (n_{\gamma}-1)(1-\tau)^{n_{\gamma}-2}&k=1\\ \frac{n_{\gamma}!}{(k-2)!(n_{\gamma}-k)!}\tau^{k-2}(1-\tau)^{n_{\gamma}-k}- \frac{2n_{\gamma}!}{(k-1)!(n_{\gamma}-k-1)!}\tau^{k-1}(1-\tau)^{n_{\gamma}-k- 1}&\\ +\frac{n_{\gamma}!}{k!(n_{\gamma}-k-2)!}\tau^{k}(1-\tau)^{n_{\gamma}-k-2}&k \in[2,n_{\gamma}-2]\\ n_{\gamma}(n_{\gamma}-1)(n_{\gamma}-2)\tau^{n_{\gamma}-3}(1-\tau)-2n_{\gamma} (n_{\gamma}-1)\tau^{n_{\gamma}-2}&k=n_{\gamma}-1\\ n_{\gamma}(n_{\gamma}-1)\tau^{n_{\gamma}-2}&k=n_{\gamma}\end{array} \right. \tag{25}\]  
The start time is selected as when the right ascension between the sun and the SPS is about \(90^{\circ}\) because the value of \(\gamma\) is equal to the inclination of the sun at equatorial plane and \(\dot{\gamma}\) is equal to 0. After a solar day (86400s) instead of the orbital period considering the movement of the sun, \(\gamma\) and \(\dot{\gamma}\) are almost equal to the ones at start time. So, \(T\) is chosen as a solar day. To ensure the convergence, the following boundary conditions are required to be satisfied:  
\[ \begin{array}{ll}\gamma(\tau=0)=\gamma_{0}&\gamma(\tau=1)=\gamma_{f}\\ \gamma^{\prime}(\tau=0)=T\dot{\gamma}_{0}&\gamma^{\prime}(\tau=1)=T\dot{ \gamma}_{f}\end{array}  \tag{26}\]  
where \(\dot{x}\) denotes the derivative of \(x\) with respect to time \(t\); \(x^{\prime}\) denotes the derivative of \(x\) with respect to scaled time \(\tau\). \(\gamma_{0}\), \(\gamma_{f}\), \(\dot{\gamma}_{0}\), and \(\dot{\gamma}_{f}\) can be calculated according to Equations (1)-(4) so that the approximation of \(\gamma\)\(\dot{\gamma}\) is continuous after a solar day. Substituting \(\tau=0\) and \(\tau=1\) into Equations (20)-(22) and (26), four Bezier coefficients \(P_{\gamma,0}\), \(P_{\gamma,1}\), \(P_{\gamma,n_{\gamma}-1}\), and \(P_{\gamma,n_{\gamma}}\) can be determined as follows:  
\[