# Novel Pointing and Stabilizing Manipulator for Optical Space Payloads
## Appendix A
\begin{array}{ll}\left\{\begin{array}{ll}\overrightarrow{\omega}_{ci}= \biggl{(}\mathbf{E}_{3}-\overrightarrow{n}_{lci}\overrightarrow{n}_{lci}^{ \mathsf{T}}\overrightarrow{u}_{lci}\biggr{)}\overrightarrow{\omega}_{ci}= \mathbf{P}_{lci}^{\perp}\overrightarrow{\omega}_{ci};&i=1,2,3\\ \overrightarrow{\omega}_{fi}=\biggl{(}\mathbf{E}_{3}-\overrightarrow{n}_{lfi} \overrightarrow{u}_{lfi}\overrightarrow{u}_{lfi}\biggr{)}\overrightarrow{ \omega}_{fi}=\mathbf{P}_{lfi}^{\perp}\overrightarrow{\omega}_{fi};&i=1,2,3 \end{array}\right.\end{array}  \tag{14}\]  
As shown in Figure 11, it could be approximately deemed that each rod is perpendicular to two pivots of the universal joint. So there is the approximation  
\[\left\{ \begin{array}{ll}\overrightarrow{n}_{lci}\approx\overrightarrow{n}_ {ci},&\mathbf{P}_{lci}^{\perp}\approx\mathbf{P}_{ci}^{\perp}=\widetilde{n}_{ ci}\overrightarrow{n}_{ci}^{\mathsf{T}};&i=1,2,3\\ \overrightarrow{n}_{lfi}\approx\overrightarrow{n}_{fi},&\mathbf{P}_{lfi}^{ \perp}\approx\mathbf{P}_{fi}^{\perp}=\widetilde{n}_{fi}\overrightarrow{n}_{ fi};&i=1,2,3\end{array} \right. \tag{15}\]  
In addition, the velocity of each joint point fulfills that  
\[\left\{ \begin{array}{ll}\overrightarrow{v}_{pci}=\overrightarrow{v}_{bci} +\overrightarrow{\omega}_{ci}\times l_{ci}^{\prime}\overrightarrow{n}_{ci},&i=1,2,3\\ \overrightarrow{v}_{pfi}=\overrightarrow{v}_{bfi}+\overrightarrow{\omega}_{fi} \times l_{fi}\overrightarrow{n}_{fi}+\dot{l}_{fi}\overrightarrow{n}_{fi};&i=1,2,3 \end{array} \right. \tag{16}\]  
Combining Formulas (A1)-(A5), (A11), (A12), and (A15), the kinetic energy of all legs could be rephrased as  
\[T_{lfu}+T_{lfd}+T_{lc}=\frac{1}{2}\left( \begin{matrix}\dot{\vec{\dot{q}}}_{p}& \dot{\vec{\dot{q}}}_{b}\end{matrix} \right) \begin{bmatrix}\mathbf{M}_{\alpha}& \mathbf{M}_{\alpha\beta}\\ \mathbf{M}_{\beta\alpha}&\mathbf{M}_{\beta}\end{bmatrix}\begin{pmatrix}\dot{ \vec{\dot{q}}}_{p}\\ \dot{\vec{\dot{q}}}_{b}\end{pmatrix} \] (A16)  
where  
\[\mathbf{M}_{\alpha}=\sum_{i=1}^{3}\mathbf{J}_{pci}^{\mathrm{T}}\Big{(}m_{aci} \mathbf{P}_{ci}^{\perp}\Big{)}\mathbf{J}_{pci}+\sum_{i=1}^{3}\mathbf{J}_{pfi}^{ \mathrm{T}}\Big{(}m_{fui}\mathbf{E}_{3}+m_{afi}\mathbf{P}_{fi}^{\perp}\Big{)} \mathbf{J}_{pfi}\] (A17)  
\[m_{aci}=\frac{\mu_{ci}+m_{ci}\rho_{\alpha}^{2}}{l_{ci}^{2}},\;\;i=1,2,3\] (A18)  
\[m_{afi}=\frac{\mu_{fui}+m_{fui}\rho_{fui}^{2}+\mu_{fdi}+m_{fdi}\rho_{fdi}^{2} }{l_{fi}^{2}}-\frac{2m_{fui}\rho_{fui}}{l_{fi}},\;\;i=1,2,3\] (A19)  
\[\mathbf{M}_{\beta}=\sum_{i=1}^{3}\mathbf{J}_{pci}^{\mathrm{T}}\Big{(}m_{ci} \mathbf{E}_{3}+m_{gi}\mathbf{P}_{ci}^{\perp}\Big{)}\mathbf{J}_{pci}+\sum_{i=1 }^{3}\mathbf{J}_{pfi}^{\mathrm{T}}\Big{(}m_{fdi}\mathbf{E}_{3}+m_{\beta fi} \mathbf{P}_{fi}^{\perp}\Big{)}\mathbf{J}_{bfi}\] (A20)  
\[m_{\beta ci}=\frac{\mu_{ci}+m_{ci}\rho_{ci}^{2}}{l_{ci}^{2}}-\frac{2m_{ci} \rho_{ci}}{l_{ci}^{2}},\;\;i=1,2,3\] (A21)  
\[m_{\beta fi}=\frac{\mu_{fui}+m_{fui}\rho_{fui}^{2}+\mu_{fdi}+m_{fdi}\rho_{fdi} ^{2}}{l_{fi}^{2}}-\frac{2m_{fdi}\rho_{fdi}}{l_{fi}},\;\;i=1,2,3\] (A22)  
\[\mathbf{M}_{\alpha\beta}=\mathbf{M}_{\beta\alpha}^{\mathrm{T}}=-\sum_{i=1}^{ 3}m_{\alpha\beta ci}\mathbf{J}_{pci}^{\mathrm{T}}\mathbf{P}_{ci}^{\perp} \mathbf{J}_{pci}-\sum_{i=1}^{3}m_{\alpha\beta fi}\mathbf{J}_{pfi}^{\mathrm{T}} \mathbf{P}_{fi}^{\perp}\mathbf{J}_{bfi}\] (A23)  
\[m_{\alpha\beta ci}=\frac{\mu_{ci}+m_{ci}\rho_{ci}^{2}}{l_{ci}^{2}}-\frac{m_{ ci}\rho_{ci}}{l_{ci}^{2}},\;\;i=1,2,3\] (A24)  
\[m_{\alpha\beta fi}=\frac{\mu_{fui}+m_{fui}\rho_{fui}^{2}+\mu_{fdi}+m_{fdi}\rho_{ fdi}^{2}}{l_{fi}^{2}}-\frac{m_{fui}\rho_{fui}+m_{fdi}\rho_{fdi}}{l_{fi}},\;\;i=1,2,3\] (A25)