# Novel Pointing and Stabilizing Manipulator for Optical Space Payloads
## 2 Characteristics and Algorithms for Large-Scale Pointing Adjustment
### Mechanism Singularity  
Integrating into a parallel configuration, the characteristic differences between the two kinds of actuators would introduce additional constraints. From the perspective of coarse actuators, the strokes of fine actuators are so small that they could be deemed immobile. Similarly, from the perspective of fine actuators, coarse actuators are so slow that they  
Figure 1: Schematic diagram and the virtual assembly of the pointing and stabilizing manipulator. could be deemed not to move. Since the constraint couplings of parallel mechanisms, the additional constraint might cause the mutual obstruction of the actuators and, thus, the failure of the division of action. Therefore, the mechanism characteristic will be analyzed in this section to guarantee the theoretical feasibility of the proposed parallel solution. To ensure that the division of labor still succeeds under extreme conditions, it is desired that the fine (or coarse actuators) are still able to control the pointing motion (i.e., the angular motion) of the payload independently under the condition that the coarse (or fine actuators) are completely immobile. From the kinematics view, it is a mechanism singularity problem. As shown in Figure 2, there are vector loops in the proposed manipulator. On this basis, the vector variation relation is  
\[ \begin{cases}\delta\overrightarrow{\mathbf{t}}_{b}+\delta\overrightarrow{\mathbf{r} }_{bf}+\delta\overrightarrow{\mathbf{I}}_{fi}=\delta\overrightarrow{\mathbf{t}}_{p} +\delta\overrightarrow{\mathbf{r}}_{pfi}\text{ }with\text{ }i=1,2,3\\ \delta\overrightarrow{\mathbf{t}}_{b}+\delta\overrightarrow{\mathbf{r}}_{bci}+ \delta\overrightarrow{\mathbf{I}}_{ci}=\delta\overrightarrow{\mathbf{t}}_{p}+ \delta\overrightarrow{\mathbf{r}}_{pci}\text{ }with\text{ }i=1,2,3\end{cases}  \tag{1}\]  
and the displacement of each actuator could be expressed as  
\[ \begin{cases}d_{fi}=\sqrt{\left(\overrightarrow{\mathbf{t}}_{p}-\overrightarrow {\mathbf{t}}_{b}+\overrightarrow{\mathbf{r}}_{pfi}-\overrightarrow{\mathbf{r}}_{bf} \right)\mathbb{T}\left(\overrightarrow{\mathbf{t}}_{p}-\overrightarrow{\mathbf{t}}_{ b}+\overrightarrow{\mathbf{r}}_{pfi}-\overrightarrow{\mathbf{r}}_{bf} \right)}-l_{f0}\text{ }with\text{ }i=1,2,3\\ d_{ci}=\sqrt{\left(\overrightarrow{\mathbf{t}}_{p}-\overrightarrow{\mathbf{t}}_{b}+ \overrightarrow{\mathbf{r}}_{pci}-\overrightarrow{\mathbf{r}}_{bci}\right)\mathbb{T} \left(\overrightarrow{\mathbf{t}}_{p}-\overrightarrow{\mathbf{t}}_{b}+ \overrightarrow{\mathbf{r}}_{pci}-\overrightarrow{\mathbf{r}}_{bci}\right)}-l_{c0} \text{ }with\text{ }i=1,2,3\end{cases}  \tag{2}\]  
where \(d_{fi}\) and \(d_{ci}\) represent fine actuator displacement and coarse actuator displacement, respectively, and \(l_{f0}\) and \(l_{c0}\) denote the original fine leg length and coarse leg length, respectively. Combining Equations (1) and (2), the variation relation could be rewritten as  
\[\delta\overrightarrow{\mathbf{d}}=\mathbf{J}_{p}\delta\overrightarrow{\mathbf{q}}_{p} -\mathbf{J}_{b}\delta\overrightarrow{\mathbf{q}}_{b} \tag{3}\]  
Figure 2: Vector loops in the pointing and stabilizing manipulator. coarse actuators could completely and independently control the angular motion and be free from the interference of fine actuators if matrix \(\mathbf{J}_{\theta c}\) is nonsingular and frees from \(\delta\overset{\rightarrow}{\boldsymbol{d}}_{f}\). Assuming matrix \(\mathbf{N}_{c}\) is nonsingular, i.e., the direction of each coarse actuator is linearly independent of the other coarse actuators, according to Formula (6) and Schur lemma, there has  
\[\left|\mathbf{J}_{lp}\right|=\left|\mathbf{N}_{c}\right|\left|\mathbf{M}_{pf}- \mathbf{N}_{f}\mathbf{N}_{c}^{-1}\mathbf{M}_{pc}\right| \tag{12}\]  
and  
\[\mathbf{J}_{\theta f}=\left(\mathbf{M}_{pf}-\mathbf{N}_{f}\mathbf{N}_{c}^{-1 }\mathbf{M}_{pc}\right)^{-1} \tag{13}\]  
It could be seen that matrix \(\mathbf{J}_{\theta f}\) is nonsingular if, and only if, \(\mathbf{J}_{lp}\) is nonsingular.