# Novel Pointing and Stabilizing Manipulator for Optical Space Payloads
## Appendix A  
This section introduced the method to express the kinetic energy of the manipulator's legs with \(\vv{q}_{p}\) and \(\vv{q}_{b}\). To improve model accuracy while retaining model simplicity, the tiny movable parts such as joint balls would be neglected. As shown in Figure 3, each fine leg is divided into upper and lower parts and linked by a prism joint, and the prism joint in each coarse leg keeps locked. Therefore, the kinematic energy of branch chains includes  
kinematic energy of the upper parts fine leg,  
\[\begin{array}{rl}T_{Ifu}&=\frac{1}{2}\sum_{i=1}^{3}\left(m_{fui}\vv{r}_{fui} \vv{r}_{fui}\vv{r}_{fui}\vv{r}_{fui}\vv{r}_  
kinematic energy of coarse leg,  
\[ \begin{array}{ll}T_{lc}&=\frac{1}{2}\sum_{i=1}^{3}\biggl{(}m_{ci}\overrightarrow{v }_{ci}^{\mathsf{T}}\overrightarrow{v}_{ci}^{\mathsf{T}}\overrightarrow{v}_{ci} +\overrightarrow{\omega}_{ci}^{\mathsf{T}}\mathbf{I}_{ci}\overrightarrow{ \omega}_{ci}\biggr{)}\\ &=\frac{1}{2}\sum_{i=1}^{3}\biggl{(}m_{ci}\biggl{(}\overrightarrow{v}_{bci} +\overrightarrow{\omega}_{ci}\times\overrightarrow{\rho}_{ci}\biggr{)}^{ \mathsf{T}}\biggl{(}\overrightarrow{v}_{bci}+\overrightarrow{\omega}_{ci} \times\overrightarrow{\rho}_{ci}\biggr{)}+\overrightarrow{\omega}_{ci}^{ \mathsf{T}}\mathbf{I}_{ci}\overrightarrow{\omega}_{ci}\biggr{)}\end{array}  \tag{10}\]  
where \(\overrightarrow{v}_{fui}(i=1,2,3)\), \(\overrightarrow{v}_{fdi}(i=1,2,3)\), \(\overrightarrow{v}_{ci}(i=1,2,3)\), respectively, represent the centroid velocity of each part. \(\overrightarrow{\omega}_{fi}(i=1,2,3)\) and \(\overrightarrow{\omega}_{ci}(i=1,2,3)\), respectively, represent the angular velocity of fine leg and that of the coarse leg. The definition of each inertial parameter and joint-point velocity can be found in Figure 3.  
Linear velocities \(\overrightarrow{v}_{bci}(i=1,2,3)\), \(\overrightarrow{v}_{pfi}(i=1,2,3)\) and \(\overrightarrow{v}_{bfi}(i=1,2,3)\) could be represented by \(\overrightarrow{q}_{p}\) and \(\overrightarrow{q}_{b}\):  
\[ \begin{array}{ll}\left\{\begin{array}{ll}\overrightarrow{v}_{bci}&= \overrightarrow{t}_{b}+\overrightarrow{\omega}_{b}\times\overrightarrow{r}_{ bci}=\mathbf{J}_{bci}\overrightarrow{q}_{b},&i=1,2,3\\ \overrightarrow{v}_{pfi}&=\overrightarrow{t}_{p}+\overrightarrow{\omega}_{p} \times\overrightarrow{r}_{pfi}=\mathbf{J}_{pfi}\overrightarrow{q}_{p},&i=1,2,3\\ \overrightarrow{v}_{bfi}&=\overrightarrow{t}_{b}+\overrightarrow{\omega}_{b} \times\overrightarrow{r}_{bfi}=\mathbf{J}_{bfi}\overrightarrow{q}_{b},&i=1,2,3\end{array}\right.\end{array}  \tag{11}\]  
The transformation matrices \(\mathbf{J}_{bci}(i=1,2,3)\), \(\mathbf{J}_{pfi}(i=1,2,3)\), and \(\mathbf{J}_{bfi}(i=1,2,3)\) are matrices with similar forms. Taking \(\mathbf{J}_{bci}(i=1,2,3)\) as an example, it has the form of  
\[\mathbf{J}_{bci}=\left[\mathbf{E}_{3}\quad-\widetilde{\mathbf{r}}_{bci}\right] \tag{12}\]  
where \(\widetilde{\mathbf{r}}_{bci}\) is the antisymmetric matrix corresponding to the vector \(\overrightarrow{r}_{bci}\). According to the structure of the universal joint shown in Figure 11, considering that each leg is linked by the universal joint, the angular velocity of each leg must fulfill  
\[ \begin{array}{ll}\left\{\begin{array}{ll}\overrightarrow{\omega}_{ci}^{ \mathsf{T}}\overrightarrow{u}_{lci}=0;&i=1,2,3\\ \overrightarrow{\omega}_{fi}^{\mathsf{T}}\overrightarrow{u}_{lfi}=0;&i=1,2,3 \end{array}\right.\end{array}  \tag{13}\]  
The definition of \(\overrightarrow{n}_{lci}(i=1,2,3)\) and \(\overrightarrow{n}_{lfi}(i=1,2,3)\) can be found in Figure 11. The formula could be equivalently rewritten as  
\[