# Novel Pointing and Stabilizing Manipulator for Optical Space Payloads
## 2 Characteristics and Algorithms for Large-Scale Pointing Adjustment
### Algorithm of Large-Scale Adjustment  
The kinematics algorithm is central to the large-scale pointing adjustment. The forward kinematic algorithm of typical 6-UPS mechanisms is still effective for the proposed manipulator, so it will not be repeated here. However, the inverse kinematic algorithm would be different. Denote the coordinate transformation matrix from the platform to the base as \(\mathbf{{}_{{}_{B}^{P}R}}\) and represent it in RPY angles:  
\[ \begin{array}{l}\mathbf{{}_{{}_{B}^{P}R}}=\mathbf{{}_{{}_{B}^{P}R}}\left( \overset{\rightarrow}{\boldsymbol{\phi}}\right)\\ \left[\begin{array}{ccc}\cos\phi_{2}\cos\phi_{3}&\cos\phi_{3}\sin\phi_{1}\sin \phi_{2}-\cos\phi_{1}\sin\phi_{3}&\cos\phi_{1}\cos\phi_{3}\sin\phi_{2}+\sin\phi _{1}\sin\phi_{3}\\ \cos\phi_{2}\sin\phi_{3}&\cos\phi_{1}\cos\phi_{3}+\sin\phi_{1}\sin\phi_{2}\sin \phi_{3}&-\cos\phi_{3}\sin\phi_{1}+\cos\phi_{1}\sin\phi_{2}\sin\phi_{3}\\ -\sin\phi_{2}&\cos\phi_{2}\sin\phi_{1}&\cos\phi_{1}\cos\phi_{2}\end{array} \right]\end{array}  \tag{14}\]  
where \(\phi_{1}\), \(\phi_{2}\), and \(\phi_{3}\) represent the roll, pitch, and yaw angle. Then denote  
\[\mathbf{{}_{{}_{B}^{\rightarrow}}}\overset{\rightarrow}{\boldsymbol{t}}_{ \Delta}=\mathbf{{}_{{}_{B}^{P}R}}\mathbf{{}^{T}}\left(\overset{\rightarrow}{ \boldsymbol{t}}_{p}-\overset{\rightarrow}{\boldsymbol{t}}_{b}\right) \tag{15}\]  
where \(\mathbf{{}_{{}_{B}^{P}R}}\) represents the coordinate transformation matrix from the base to the inertial coordinate system. So, Formula (2) could be rewritten in the local coordinate system of the base:  
\[ \begin{array}{l}\left\{d_{fi}=\sqrt{\left(\mathbf{{}_{{}_{B}^{\rightarrow} }}\overset{\rightarrow}{\boldsymbol{t}}_{\Delta}+\mathbf{{}_{{}_{B}^{P}R}} \mathbf{{}_{{}_{pfi}}^{\rightarrow}}-\mathbf{{}_{{}_{pfi}}^{\rightarrow}} \right)^{\top}\left(\mathbf{{}_{{}_{B}^{\rightarrow}}}\overset{\rightarrow} {\boldsymbol{t}}_{\Delta}+\mathbf{{}_{{}_{B}^{P}R}}\mathbf{{}_{{}_{pfi}}^{ \rightarrow}}-\mathbf{{}_{{}_{pfi}}^{\rightarrow}}\right)}-l_{f0}\;\;with\;i=1,2,3 \\ d_{ci}=\sqrt{\left(\mathbf{{}_{{}_{B}^{\rightarrow}}}\overset{\rightarrow}{ \boldsymbol{t}}_{\Delta}+\mathbf{{}_{{}_{B}^{P}R}}\mathbf{{}_{{}_{pfi}}^{ \rightarrow}}-\mathbf{{}_{{}_{pfi}}^{\rightarrow}}\right)^{\top}\left( \mathbf{{}_{{}_{B}^{\rightarrow}}}\overset{\rightarrow}{\boldsymbol{t}}_{ \Delta}+\mathbf{{}_{{}_{B}^{P}R}}\mathbf{{}_{{}_{pfi}}^{\rightarrow}}-\mathbf{{ }_{{}_{pfi}}^{\rightarrow}}\right)}-l_{c0}\;\;with\;i=1,2,3\end{array} \right. \tag{16}\]  
Given the target linear coordinates \(\mathbf{{}_{{}_{B}^{\rightarrow}}}\overset{\rightarrow}{\boldsymbol{t}}_{\Delta}\) and angular coordinates \(\overset{\rightarrow}{\boldsymbol{\phi}}\), the required actuator displacement could be directly solved from Equation (16). However, there are two difficulties with the proposed manipulator. Firstly, only the required angular coordinates \(\overset{\rightarrow}{\boldsymbol{\phi}}\) are knowable when adjusting the LOS. Secondly, the stroke of fine actuators is tiny, and the result \(d_{fi}(i=1,2,3)\) must not exceed permissive stroke.