# Novel Pointing and Stabilizing Manipulator for Optical Space Payloads
## 2 Characteristics and Algorithms for Large-Scale Pointing Adjustment
### Algorithm of Active Vibration Isolation
#### 3.2.2 Stability of Feedforward Controller  
Generally, a feedforward control system would not be at an unstable risk. However, item \(\mathbf{C}(s)\) gives the manipulator feedforward system an internal positive feedback loop (IPFL); thus, the control system would not be stable. Since \(\mathbf{A}(s)\), \(\mathbf{G}_{\mathbf{P}}(s)\), \(\mathbf{C}(s)\), \(\mathbf{D}(s)\), and the nominal optimal feedforward controller are stable, respectively, there is no unstable pole-zero cancellation in the IPFL. The stability is decided by its sensitivity function. According to Formulas (34) and (50), the sensitivity function of the IPFL consisting of the optimal feedforward controller and the nominal plant could be expressed as  
\[ \begin{array}{ll}\mathbf{S}(s)&=\left(\mathbf{E}_{6}-\mathbf{C}(s)\mathbf{K }_{wop}(s)\right)^{-1}\\ &=\left(\mathbf{E}_{6}-\mathbf{C}(s)\mathbf{U}_{12}(s)+\mathbf{C}(s)\mathbf{U} _{11}(s)\mathbf{J}_{\theta f}^{-1}\mathbf{J}_{\theta b}\right)^{-1}\\ &=\left(\mathbf{E}_{6}-\left(-\mathbf{U}_{22}^{-1}(s)\mathbf{U}_{21}(s) \mathbf{G}_{11}(s)\right)\mathbf{U}_{12}(s)+\left(-\mathbf{G}_{22}(s)\mathbf{ U}_{21}(s)\mathbf{U}_{11}^{-1}(s)\right)\mathbf{U}_{11}(s)\mathbf{J}_{\theta f}^{-1} \mathbf{J}_{\theta b}\right)^{-1}\\ &=\left(\left(\mathbf{U}_{22}^{-1}(s)+\mathbf{U}_{22}^{-1}(s)\mathbf{U}_{21}( s)\mathbf{G}_{11}(s)\mathbf{U}_{12}(s)\mathbf{U}_{22}^{-1}(s)\right)\mathbf{U}_{22}(s)- \mathbf{G}_{22}(s)\mathbf{U}_{21}(s)\mathbf{J}_{\theta f}^{-1}\mathbf{J}_{ \theta b}\right)^{-1}\\ &=\left(\mathbf{G}_{22}(s)\mathbf{U}_{22}(s)-\mathbf{G}_{22}(s)\mathbf{U}_{21}( s)\mathbf{J}_{\theta f}^{-1}\mathbf{J}_{\theta b}\right)^{-1}\\ &=\left(\mathbf{U}_{22}(s)-\mathbf{U}_{21}(s)\mathbf{J}_{\theta f}^{-1}\mathbf{ J}_{\theta b}\right)^{-1}\mathbf{G}_{22}^{-1}(s)\\ &=\left(\mathbf{U}_{22}(s)-\mathbf{U}_{21}(s)\mathbf{J}_{\theta f}^{-1} \mathbf{J}_{\theta b}\right)^{-1}\left(\mathbf{U}_{22}(s)-\mathbf{U}_{21}(s) \mathbf{U}_{11}^{-1}(s)\mathbf{U}_{12}(s)\right)\\ &=\left(\mathbf{M}_{eb}+\mathbf{M}_{eb}\mathbf{J}_{\theta f}^{-1}\mathbf{J}_{ \theta b}\right)^{-1}\left(\mathbf{M}_{eb}-s^{2}\mathbf{M}_{ebf}\left(s^{2} \mathbf{M}_{ef}+s\mathbf{C}_{f}+\mathbf{K}_{f}\right)^{-1}\mathbf{M}_{efb} \right)\end{array}  \tag{56}\]  
So, the pole equation of nominal IPFL is  
\[\det\left(s^{2}\mathbf{M}_{ef}+s\mathbf{C}_{f}+\mathbf{K}_{f}\right)=0 \tag{57}\]  
Since \(\mathbf{M}_{ef}\), \(\mathbf{C}_{f}\), \(\mathbf{K}_{f}\) is always symmetric and positive definite, it is certain that all the roots of the pole equation are placed at the left. Therefore, the above sensitivity function is bound to be stable. However, the practical plant would usually differ from the nominal one because of delay, high-frequency modes of the manipulator and payload, feedforward controller approximation, and other uncertainty. All the uncertainty could be first lumped in practical plant \(\mathbf{C}(s)\), and the relationship between the nominal plant \(\mathbf{C}(s)\) and the practical plant \(\mathbf{C}^{\prime}(s)\) could be expressed in a general form:  
\[\mathbf{C}^{\prime}(s)=(\mathbf{E}_{6}+\Delta(s))\mathbf{C}(s) \tag{58}\]  
where \(\Delta\) denotes the model uncertainty and is supposed to be a proper and real rational stable transfer matrix. To overcome the influence of model uncertainty, the sensitivity function of practical IPFL is supposed to be robustly stable, i.e., the loop is stable for any given bounded model uncertainty.