# Angular Trajectory Design for MR-SPS Using Bezier Shaping Approach
## 2 Model Description
### Sun and Earth Orientation Analysis  
Regardless of the dynamic constraints, the real-time and precise orientation to the sun for the arrays and to the rectenna for the antenna can be achieved by adjusting the attitude of the MR-SPS and in this case, the energy receiving efficiency reaches the peak. The energy receiving efficiency is defined as the area weighted average of the cosine of the angle between the normal direction of the arrays and the solar direction. As shown in Figure 2, five kinds of reference frame are introduced to describe the attitude of the MR-SPS. 1. The origin \(o_{i}\) of the geocentric inertial reference frame \(o_{i}\)-\(x_{i}y_{i}z_{i}\) is located at the center of mass of the earth, the \(x_{i}\)-axis is defined along the direction of vernal equinox, the \(z_{i}\)-axis is normal to the equatorial plane and points to the north pole, and the \(y_{i}\)-axis completes a right-handed coordinate system;
2. The origin \(o_{o}\) of the orbital reference frame \(o_{o}\)-\(x_{o}y_{o}z_{o}\) is at the center of mass of the main structure and antenna, the \(z_{o}\)-axis points to the center of the earth, \(y_{o}\)-axis is consistent with the negative normal direction of the orbital plane, and the \(x_{o}\)-axis completes a right-handed system;
3. The origin \(o_{a}\) of the main structure body-fixed reference frame \(o_{a}\)-\(x_{a}y_{a}z_{a}\) coincides with \(o_{o}\), the three axes are aligned with the principal axes of inertia, the \(y_{a}\)-axis is along the longest side of the central truss, and the \(z_{a}\)-axis is perpendicular to the antenna surface. 4. The origin \(o_{\bar{c}j}\) of the \(j\)th intermediate reference frame \(o_{\bar{c}j}\)-\(x_{\bar{c}j}y_{\bar{c}j}z_{\bar{c}j}\) is the hinged point of the \(j\)th solar array and the three axes are parallel to those of the main structure frame;
5. The origin \(o_{\bar{s}j}\) of the \(j\)th array body-fixed reference frame \(o_{\bar{s}j}\)-\(x_{\bar{s}j}y_{\bar{s}j}z_{\bar{s}j}\) is at the center of mass of the \(j\)th solar array, the three axes are aligned with the principal axes of inertia, and \(y_{\bar{s}j}\)-axis is parallel to \(y_{a}\)-axis. Figure 1: Structure of the MW-level MR-SPS. \(\varphi\) is the right ascension of the SPS, \(\alpha\) is the angle that the sun runs on the ecliptic from the vernal equinox, and \(\beta\) is the obliquity of the ecliptic. Limited by the area of the rectenna and the distance between the SPS and the rectenna, the antenna must point to the rectenna precisely, and in other words, the \(z_{a}\)-axis must be parallel to the \(z_{o}\)-axis at any time. Because the direction of the \(z_{a}\)-axis is determined, only one-axis motion is considered for the main structure. After rotating \(\gamma\) around \(z_{o}\)-axis, \(o_{o}\)-\(x_{o}y_{o}z_{o}\) coincides with \(o_{a}\)-\(x_{o}y_{a}z_{a}\). \(\theta_{j}\) is the rotation angle of the \(j\)th array around the main truss from the initial position where \(o_{ij}\)-\(x_{j}y_{j}z_{j}\) coincides with \(o_{ij}\)-\(x_{j}y_{j}z_{j}\). \(\hat{\pi}\) denotes the unit vector of the vector \(\mathbf{x}\) or the \(x\)-axis. The position unit vector of the sun relative to the earth center is \(\hat{r}_{s}=\left[ \begin{array}{cc}\cos\alpha&\sin\alpha\cos\beta&\sin \alpha\sin\beta\end{array} \right]^{T}\), and the unit vector from the earth center to the centroid of the main structure and antenna is \(\hat{\mathbf{r}}_{a}=\left[\cos\varphi&\sin\varphi&0\right]^{T}\). The two unit vectors are both expressed in the inertial frame. The \(z_{o}\)-axis is parallel to the \(z_{a}\)-axis, so \(\hat{\mathbf{z}}_{a}=-\hat{\mathbf{r}}_{a}\). For a spacecraft in GEO, it is obvious that \(\hat{\mathbf{y}}_{o}=\left[ \begin{array}{cc}0&0&-1\end{array} \right]^{T}\) and \(\hat{\mathbf{x}}_{o}=\left[ \begin{array}{cc}-\sin\varphi&\cos\varphi&0\end{array} \right] ^{T}\) expressed in the inertial frame.