# 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{M}_{ef}&\mathbf{M}_{efb}&\mathbf{O}\\ \mathbf{M}_{\mathit{cbf}}&\mathbf{M}_{\mathit{eb}}&\mathbf{M}_{b12}\\ \mathbf{O}&\mathbf{M}_{b21}&\mathbf{M}_{b22}\end{bmatrix}\begin{pmatrix} \overset{\rightarrow}{\boldsymbol{d}}_{f}\\ \overset{\rightarrow}{\boldsymbol{q}}_{b}\\ \overset{\rightarrow}{\boldsymbol{x}}_{m}\end{pmatrix} +\begin{bmatrix} \mathbf{C}_{f}&\mathbf{O}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{K}_{bm}\end{pmatrix} \begin{pmatrix}\overset{ \rightarrow}{\boldsymbol{d}}_{f}\\ \overset{\rightarrow}{\boldsymbol{q}}_{b}\\ \overset{\rightarrow}{\boldsymbol{x}}_{m}\end{pmatrix} = \begin{pmatrix} \overset{\rightarrow}{\boldsymbol{f}}_{f}\\ \overset{\rightarrow}{\boldsymbol{f}}_{b}\\ \overset{\rightarrow}{\boldsymbol{x}}_{m}\end{pmatrix}  \tag{25}\]  
where  
\[\mathbf{M}_{ef}=\mathbf{J}_{pf}^{\mathrm{T}}\big{(}\mathbf{M}_{p}+\mathbf{M}_{ \alpha}\big{)}\mathbf{J}_{pf} \tag{26}\]  
\[\mathbf{M}_{\mathit{cbf}}=\mathbf{M}_{\mathit{efb}}+\mathbf{M}_{\beta}+\mathbf{ J}_{\mathit{pb}}^{\mathrm{T}}\big{(}\mathbf{M}_{p}+\mathbf{M}_{\alpha}\big{)} \mathbf{J}_{\mathit{pb}}+\mathbf{J}_{\mathit{pb}}^{\mathrm{T}}\mathbf{M}_{ \alpha\beta}+\mathbf{M}_{\beta\alpha}\mathbf{J}_{\mathit{pb}} \tag{27}\]  
Note that nonlinear terms corresponding to the centrifugal force and the Coriolis force in the above dynamic equation were neglected. Apply Laplace Transform to the dynamic equations, and the vibration deformation \(\overset{\rightarrow}{\boldsymbol{x}}_{m}\) could be derived from \(\overset{\rightarrow}{\boldsymbol{q}}_{b}\) the support module bulk motion:  
\[\overset{\rightarrow}{\boldsymbol{X}}_{m}(s)=\Big{(}s^{2}\mathbf{M}_{b22}+ \mathbf{K}_{bm}\Big{)}^{-1}s^{2}\mathbf{M}_{b21}\overset{\rightarrow}{ \boldsymbol{Q}}_{b}(s) \tag{29}\]  
where \(\overset{\rightarrow}{\boldsymbol{X}}_{m}(s)\) and \(\overset{\rightarrow}{\boldsymbol{Q}}_{b}(s)\), respectively, represent the Laplace transform of \(\overset{\rightarrow}{\boldsymbol{x}}_{m}\) and \(\overset{\rightarrow}{\boldsymbol{q}}_{b}\). Then define the notation:  
\[ \begin{bmatrix}\mathbf{U}_{11}(s)&\mathbf{U}_{12}(s)\\ \mathbf{U}_{21}(s)&\mathbf{U}_{22}(s)\end{bmatrix} = \begin{bmatrix}s^{2} \mathbf{M}_{ef}+s\mathbf{C}_{f}+\mathbf{K}_{f}&s^{2}\mathbf{M}_{\mathit{cbf}} \\ s^{2}\mathbf{M}_{\mathit{ebf}}&s^{2}\Big{(}\mathbf{M}_{\mathit{eb}}+\mathbf{ M}_{b12}(s^{2}\mathbf{M}_{b22}+\mathbf{K}_{bm})^{-1}s^{2}\mathbf{M}_{b21}\Big{)} \end{bmatrix}  \tag{30}\]  
Furthermore, the transfer function matrix of \(\overset{\rightarrow}{\boldsymbol{d}}_{f}\) and \(\overset{\rightarrow}{\boldsymbol{q}}_{b}\) could be obtained:  
\[ \begin{pmatrix}\overset{\rightarrow}{\boldsymbol{D}}_{f}(s)\\ \overset{\rightarrow}{\boldsymbol{Q}}_{b}(s)\end{pmatrix} = \begin{bmatrix} \mathbf{U}_{11}(s)&\mathbf{U}_{12}(s)\\ \mathbf{U}_{21}(s)&\mathbf{U}_{22}(s)\end{bmatrix} ^{-1} \begin{pmatrix} \overset{\rightarrow}{\boldsymbol{F}}_{f}(s)\\ \overset{\rightarrow}{\boldsymbol{F}}_{b}(s)\end{pmatrix} = \begin{bmatrix} \mathbf{G}_{11}(s)&\mathbf{G}_{12}(s)\\ \mathbf{G}_{21}(s)&\mathbf{G}_{22}(s)\end{bmatrix}\begin{pmatrix}\overset{ \rightarrow}{\boldsymbol{F}}_{f}(s)\\ \overset{\rightarrow}{\boldsymbol{F}}_{b}(s)\end{pmatrix}  \tag{31}\]  
where \(\overset{\rightarrow}{\boldsymbol{D}}_{f}(s)\), \(\overset{\rightarrow}{\boldsymbol{F}}_{f}(s)\) and \(\overset{\rightarrow}{\boldsymbol{F}}_{b}(s)\), respectively, represent the Laplace transform of \(\overset{\rightarrow}{\boldsymbol{d}}_{f}\), \(\overset{\rightarrow}{\boldsymbol{f}}_{f}\), and \(\overset{\rightarrow}{\boldsymbol{f}}_{b}\). Combining Formulas (24) and (31), the transfer function matrix of \(\overset{\rightarrow}{\boldsymbol{\theta}}_{p}\) and \(\overset{\rightarrow}{\boldsymbol{q}}_{b}\) could be obtained:  
\[ \begin{pmatrix}\overset{\rightarrow}{\boldsymbol{\Theta}}_{p}(s)\\ \overset{\rightarrow}{\boldsymbol{Q}}_{b}(s)\end{pmatrix} =