# Novel Pointing and Stabilizing Manipulator for Optical Space Payloads
## 2 Characteristics and Algorithms for Large-Scale Pointing Adjustment
### Algorithm of Large-Scale Adjustment
\begin{bmatrix} \mathbf{J}_{\theta f}&\mathbf{J}_{\theta b}\\ \mathbf{O}_{6}&\mathbf{E}_{6}\end{bmatrix}\begin{bmatrix}\mathbf{G}_{11}(s)& \mathbf{G}_{12}(s)\\ \mathbf{G}_{21}(s)&\mathbf{G}_{22}(s)\end{bmatrix}\begin{pmatrix}\overset{ \rightarrow}{\boldsymbol{F}}_{f}(s)\\ \overset{\rightarrow}{\boldsymbol{F}}_{b}(s)\end{pmatrix} = \begin{bmatrix} \mathbf{A}(s)&\mathbf{B}(s)\\ \mathbf{C}(s)&\mathbf{D}(s)\end{bmatrix}\begin{pmatrix}\overset{ \rightarrow}{\boldsymbol{F}}_{f}(s)\\ \overset{\rightarrow}{\boldsymbol{F}}_{b}(s)\end{pmatrix}  \tag{32}\]Using Schur complement, a detailed expression of each sub-matrix could be obtained as  
\[ \begin{array}{rl}\mathbf{D}(s)&=\mathbf{G}_{22}(s)=\left(\mathbf{U}_{22}(s)- \mathbf{U}_{21}(s)\mathbf{U}_{11}^{-1}(s)\mathbf{U}_{12}(s)\right)^{-1}\\ &=\mathbf{U}_{22}^{-1}(s)+\mathbf{U}_{22}^{-1}(s)\mathbf{U}_{21}(s)\mathbf{G}_{ 11}(s)\mathbf{U}_{12}(s)\mathbf{U}_{22}^{-1}(s)\end{array}  \tag{33}\]  
\[ \begin{array}{rl}\mathbf{C}(s)=\mathbf{G}_{21}(s)&=-\mathbf{U}_{22}^{-1}(s) \mathbf{U}_{21}(s)\mathbf{G}_{11}(s)=-\mathbf{G}_{22}(s)\mathbf{U}_{21}(s) \mathbf{U}_{11}^{-1}(s)\end{array}  \tag{34}\]  
\[ \begin{array}{rl}\mathbf{B}(s)&=\mathbf{J}_{\theta f}\mathbf{G}_{12}(s)+ \mathbf{J}_{\theta b}\mathbf{G}_{22}(s)\\ &=\mathbf{J}_{\theta f}\left(-\mathbf{U}_{11}^{-1}(s)\mathbf{U}_{12}(s) \mathbf{G}_{22}(s)\right)+\mathbf{J}_{\theta b}\mathbf{G}_{22}(s)\\ &=\left(\mathbf{J}_{\theta b}-\mathbf{J}_{\theta f}\mathbf{U}_{11}^{-1}(s) \mathbf{U}_{12}(s)\right)\mathbf{D}(s)\\ \mathbf{A}(s)&=\mathbf{J}_{\theta f}\mathbf{G}_{11}(s)+\mathbf{J}_{\theta b} \mathbf{G}_{21}(s)\\ &=\mathbf{J}_{\theta f}\left(\mathbf{U}_{11}(s)-\mathbf{U}_{12}(s) \mathbf{U}_{22}^{-1}(s)\mathbf{U}_{21}(s)\right)^{-1}+\mathbf{J}_{\theta b} \mathbf{C}(s)\end{array}  \tag{36}\]  
Set the active isolation force \(\overset{\rightarrow}{f}\) to zeros, and the disturbance transmissibility of the manipulator could be obtained as  
\[\overset{\rightarrow}{\mathbf{\Theta}}_{p}(s)=\mathbf{B}(s)\overset{ \rightarrow}{F}_{b}(s)=\mathbf{B}(s)\left(\mathbf{D}^{-1}(s)\overset{ \rightarrow}{Q}_{b}(s)\right)=\mathbf{G}_{\mathrm{P}}(s)\overset{ \rightarrow}{Q}_{b}(s) \tag{37}\]  
According to Formulas (33) and (35), the detail expression could be obtained as  
\[\mathbf{G}_{\mathrm{P}}(s)=\mathbf{B}(s)\mathbf{D}^{-1}(s)=\left(\mathbf{J}_{ \theta b}-\mathbf{J}_{\theta f}\mathbf{U}_{11}^{-1}(s)\mathbf{U}_{12}(s)\right) \tag{38}\]  
It is first seen that the disturbance transmissibility of the manipulator is immune to the support module modes. Then, since the DC gain of item \(\mathbf{U}_{11}^{-1}(s)\mathbf{U}_{12}(s)\) is zero, low-frequency disturbance transmissibility is \(\mathbf{J}_{\theta b}\). Considering that  
\[\mathbf{J}_{\theta b}=\left[ \begin{array}{cc}\mathbf{O}_{3}&\mathbf{P}_{ \mathrm{B}}\mathbf{R}\end{array} \right] \tag{39}\]  
the singular value frequency response of the disturbance transmissibility would always begin with 0 dB. In addition, the form of \(\mathbf{G}_{\mathrm{P}}(s)\) implies a non-zeros nominal feedthrough gain:  
\[ \begin{array}{rl}\mathbf{A}_{ft}&=\lim_{s\rightarrow\infty}\mathbf{G}_{ \mathrm{P}}(s)=\mathbf{J}_{\theta b}-\mathbf{J}_{\theta f}\mathbf{M}_{ef}{}^{ -1}\mathbf{M}_{efb}\\ &=\mathbf{J}_{\theta b}-\mathbf{J}_{\theta f}\left(\mathbf{J}_{pf}^{\mathrm{ T}}\left(\mathbf{M}_{p}+\mathbf{M}_{\alpha}\right)\mathbf{J}_{pf}\right)^{-1} \left(\mathbf{J}_{pf}^{\mathrm{T}}\mathbf{M}_{\alpha\beta}+\mathbf{J}_{pf}^{ \mathrm{T}}\left(\mathbf{M}_{p}+\mathbf{M}_{\alpha}\right)\mathbf{J}_{pb} \right)\end{array}  \tag{40}\]  
Considering that large feedthrough gain would damage the suppression of high-frequency disturbance, Formula (40) implies that the passive vibration isolation capability of the manipulator is likely to be poor. According to Formula (36), the plant dynamics \(\mathbf{A}(s)\) would be influenced by item \(\mathbf{C}(s)\), i.e., the'mobile-base' effect. Item \(\mathbf{C}(s)\) describes the response characteristics of the support module under the excitation of fine actuator force.