# Novel Pointing and Stabilizing Manipulator for Optical Space Payloads
## 2 Characteristics and Algorithms for Large-Scale Pointing Adjustment
### Algorithm of Large-Scale Adjustment
For the convenience of dealing with complex constraints, the Lagrange method is adopted to establish the dynamics model, as shown in Figure 3. The elastic potential energy of the dynamic system could be expressed as  
the elastic potential energy of all the fine legs,  
\[V_{lf}=\frac{1}{2}\sum_{i=1}^{3}k_{fi}\Big{(}\delta d_{fi}\Big{)}^{2}=\frac{1}{2} \delta\overset{\rightarrow}{d}_{f}^{\text{T}}\mathbf{K}_{f}\delta\overset{ \rightarrow}{d}_{f} \tag{18}\]  
the elastic potential energy of the support module,  
\[V_{bm}=\frac{1}{2}\overset{\rightarrow}{\mathbf{x}}_{m}^{\text{T}}\mathbf{K}_{bm} \overset{\rightarrow}{\mathbf{x}}_{m} \tag{19}\]  
where \(\overset{\rightarrow}{\mathbf{x}}_{m}\) represents the modal coordinates that describe the vibration deformation, as shown in Figure 3. Furthermore, the kinematic energy of the idealized dynamic system includes  
the kinematic energy of the payload,  
\[T_{p}=\frac{1}{2}\overset{\rightarrow}{\mathbf{q}}_{p}^{\text{T}}\mathbf{M}_{p} \overset{\rightarrow}{\mathbf{q}}_{p}=\frac{1}{2}\overset{\rightarrow}{\mathbf{q}}_ {p}^{\text{T}} \begin{bmatrix}m_{p}\mathbf{E}_{3}&-m_{p}\widetilde{\mathbf{p}}_{p} \\ m_{p}\widetilde{\mathbf{p}}_{p}&\mathbf{I}_{p}\end{bmatrix} \overset{\rightarrow}{ \mathbf{q}}_{p} \tag{20}\]  
and the kinematic energy of the manipulator's legs  
\[T_{ffu}+T_{lfd}+T_{lc}=\frac{1}{2} \begin{pmatrix}\overset{\rightarrow}{\mathbf{q }}_{p}^{\text{T}}&\overset{\rightarrow}{\mathbf{q}}_{b}^{\text{T}}\end{pmatrix} \begin{bmatrix}\mathbf{M}_{a}&\mathbf{M}_{\beta b}\\ \mathbf{M}_{\beta a}&\mathbf{M}_{\beta}\end{bmatrix}\begin{pmatrix}\overset{ \rightarrow}{\mathbf{q}}_{p}\\ \overset{\rightarrow}{\mathbf{q}}_{b}\end{pmatrix}  \tag{21}\]  
A detailed derivation of Formula (21) can be found in Appendix A. The kinematic energy of the support module origin from two kinds of motion, i.e., the bulk motion and the deformation motion, so the kinematic energy could be expressed as  
\[T_{b}=\frac{1}{2}\overset{\rightarrow}{\mathbf{q}}_{b}^{\text{T}}\mathbf{M}_{b} \overset{\rightarrow}{\mathbf{q}}_{b}= \begin{pmatrix}\overset{\rightarrow}{\mathbf{q }}_{b}&\overset{\rightarrow}{\mathbf{x}}_{m}^{\text{T}}\end{pmatrix} \begin{bmatrix} \mathbf{M}_{b}&\mathbf{M}_{b12}\\ \mathbf{M}_{b12}&\mathbf{M}_{b22}\end{pmatrix} \begin{pmatrix}\overset{ \rightarrow}{\mathbf{q}}_{b}\\ \overset{\rightarrow}{\mathbf{x}}_{m}\end{pmatrix}  \tag{22}\]  
where the mass matrix associated with bulk motion is  
\[\mathbf{M}_{b}= \begin{bmatrix}m_{b}\mathbf{E}_{3}&-m_{b}\widetilde{\mathbf{p}}_{b }\\ m_{b}\widetilde{\mathbf{p}}_{b}&\mathbf{I}_{b}\end{bmatrix}  \tag{23}\]  
Figure 3: Schematic diagram of the simplified dynamic system. As for non-conservative forces, the floor disturbance force \(\overset{\rightarrow}{\boldsymbol{f}}_{b}\), the fine leg active force \(f_{fi}\) (vector form is \(\overset{\rightarrow}{\boldsymbol{f}}_{f}\)), and the force arising from fine actuator electromagnetic damping \(c_{f_{i}}\) (matrix form is \(\mathbf{C}_{f}\)) are considered. Note that coarse actuators are regarded as immobile during active isolation because they are too slow. On this basis, the dynamic differential equations could be obtained according to Euler-Lagrange equations. Then, according to Formula (11), it has the following variational relationship:  
\[\delta \begin{pmatrix}\overset{\rightarrow}{\boldsymbol{\theta}}_{p}\\ \overset{\rightarrow}{\boldsymbol{q}}_{b}\\ \overset{\rightarrow}{\boldsymbol{x}}_{m}\end{pmatrix} = \begin{bmatrix} \mathbf{J}_{\theta f}&\mathbf{J}_{\theta b}&\mathbf{O}\\ \mathbf{O}&\mathbf{E}_{6}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{E}\end{bmatrix} \delta \begin{pmatrix}\overset{ \rightarrow}{\boldsymbol{d}}_{f}\\ \overset{\rightarrow}{\boldsymbol{q}}_{b}\\ \overset{\rightarrow}{\boldsymbol{x}}_{m}\end{pmatrix}  \tag{24}\]  
so the dynamic equations could be rewritten as  
\[