# Novel Pointing and Stabilizing Manipulator for Optical Space Payloads
## 2 Characteristics and Algorithms for Large-Scale Pointing Adjustment
### Algorithm of Active Vibration Isolation
#### 3.2.1 Active Feedforward Controller  
The feedforward system of the manipulator takes the motion signal of the base as input. As shown in Figure 4, the disturbance from the base would influence the platform motion through two parallel paths, i.e., the primary path (denoted by \(\mathbf{G}_{\text{P}}(s)\)) and the active path (denoted by \(\mathbf{A}(s)\)). The platform would be completely isolated from base disturbance when primary path and active path satisfy that  
\[\mathbf{G}_{\text{P}}(s)\overset{\rightarrow}{\
ot{\mathbf{Q}}}_{b0}(s)=- \mathbf{A}(s)\overset{\rightarrow}{\
ot{\mathbf{F}}}_{f}(s) \tag{47}\]  
Figure 4: Control block diagram of active feedforward vibration isolation. Because of the effect of active isolation force \(\stackrel{{\rightarrow}}{{f}}_{f}\) on the support module, the input of feedforward controller (i.e., \(\stackrel{{\rightarrow}}{{q}}_{b}\)) would differ from that of primary path (i.e., \(\stackrel{{\rightarrow}}{{q}}_{b0}\)) if \(\stackrel{{\rightarrow}}{{f}}_{f}\) were applied. Considering item \(\mathbf{C}(s)\), the relationship between \(\stackrel{{\rightarrow}}{{q}}_{b}\) and \(\stackrel{{\rightarrow}}{{q}}_{b0}\) is  
\[\stackrel{{\rightarrow}}{{Q}}_{b}(s)=\mathbf{S}(s)\stackrel{{ \rightarrow}}{{Q}}_{b0}(s)=(\mathbf{E}_{6}-\mathbf{C}(s)\mathbf{K}_{w}(s))^{- 1}\stackrel{{\rightarrow}}{{Q}}_{b0}(s) \tag{48}\]  
Combined expression (47) and (48), the optimal active feedforward isolation force is  
\[\stackrel{{\rightarrow}}{{F}}_{fopt}(s)=\mathbf{K}_{wcp}(s) \stackrel{{\rightarrow}}{{Q}}_{b}(s)=-\Big{(}\mathbf{E}_{3}- \mathbf{A}^{-1}(s)\mathbf{G}_{\mathrm{P}}(s)\mathbf{C}(s)\Big{)}^{-1}\mathbf{A }^{-1}(s)\mathbf{G}_{\mathrm{P}}(s)\stackrel{{\rightarrow}}{{Q}} _{b}(s) \tag{49}\]  
In addition, the corresponding feedforward controller is  
\[\mathbf{K}_{wcp}(s) =-(\mathbf{A}(s)-\mathbf{G}_{\mathrm{P}}(s)\mathbf{C}(s))^{-1} \mathbf{G}_{\mathrm{P}}(s) \tag{50}\] \[=-\Big{(}\mathbf{J}_{\theta f}\mathbf{G}_{11}(s)+\mathbf{J}_{ \theta b}\mathbf{C}(s)-\mathbf{G}_{\mathrm{P}}(s)\mathbf{C}(s)\Big{)}^{-1} \mathbf{G}_{\mathrm{P}}(s)\] \[=-\Big{(}\mathbf{J}_{\theta f}\mathbf{G}_{11}(s)+\mathbf{J}_{ \theta f}\mathbf{U}_{11}^{-1}(s)\mathbf{U}_{12}(s)\mathbf{C}(s)\Big{)}^{-1} \mathbf{G}_{\mathrm{P}}(s)\] \[=-\Big{(}\mathbf{J}_{\theta f}\Big{(}\mathbf{G}_{11}(s)+\mathbf{U }_{11}^{-1}(s)\mathbf{U}_{12}(s)\Big{(}-\mathbf{G}_{22}^{-1}(s)\mathbf{U}_{21} (s)\mathbf{U}_{11}^{-1}(s)\Big{)}\Big{)}\Big{)}^{-1}\mathbf{G}_{\mathrm{P}}(s)\] \[=-\mathbf{U}_{11}(s)\mathbf{J}_{\theta f}^{-1}\mathbf{G}_{\mathrm{ P}}(s)\] \[=\mathbf{U}_{12}(s)-\mathbf{U}_{11}(s)\mathbf{J}_{\theta f}^{-1} \mathbf{J}_{\theta b}\]  
It is seen that \(\mathbf{K}_{wcp}(s)\) is also immune to the support module modes. Combined expression (26)-(30), (49), and (50), the optimal feedforward force could be expressed as  
Figure 4:0  
where  
Figure 4:1  
Figure 4:2  
Figure 4:3  
Since only static gains are required, the nominal optimal feedforward controller \(\mathbf{K}_{wcp}(s)\) is feasible. Although the nominal optimal feedforward controller is achievable, it would face difficulties when it comes to a practical plant whose dynamic differs from the nominal plant. On the one hand, it is hard to obtain the absolute and multidimensional displacement, velocity, and acceleration of floor disturbance at the same time. On the other hand, the all-pass controller might provoke a strong high-frequency response, considering that the high-frequency characteristics of the practical plant probably differ from that of the nominal system. Given this, a modified feedforward controller is proposed. Firstly, the modified controller takes only the acceleration of floor disturbance as the input signal, and then the integration method is adopted to obtain displacement and velocity signals. Considering the saturation risk of the integrator, the acceleration signal would be filtered by a high-pass filter first.