# 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
Combining Formulas (56) and (58), the sensitivity function of practical IPFL is  
\[\mathbf{S}^{\prime}(s)=\left(\mathbf{E}_{6}-\mathbf{C}(s)\mathbf{K}_{wop}(s) \right)^{-1}=\mathbf{S}(s)\mathbf{S}_{\Delta}(s) \tag{59}\]  
where  
\[\mathbf{S}_{\Delta}(s)=\left(\mathbf{E}_{6}-\Delta(s)\mathbf{T}(s)\right)^{-1} \tag{60}\]  
\[\mathbf{T}(s)=\mathbf{E}_{6}-\mathbf{S}(s)=-\mathbf{C}(s)\mathbf{K}_{wop}(s) \left(\mathbf{E}_{6}-\mathbf{C}(s)\mathbf{K}_{wop}(s)\right)^{-1} \tag{61}\]  
Since \(\mathbf{S}(s)\) is certainly stable, \(\mathbf{S}^{\prime}(s)\) is robustly stable if, and only if, \(\mathbf{S}_{\Delta}(s)\) is robustly stable. According to small gain theorem, \(\mathbf{S}_{\Delta}(s)\) is stable if  
\[\sigma(\Delta(j\omega))\sigma(\mathbf{T}(j\omega))<1,\ \ \forall\omega\in\mathbb{R} \tag{62}\]where \(\sigma(\cdot)\) represents the maximum singular value of a matrix. According to the dynamic model in the above subsection, the form of \(\mathbf{T}(s)\) could be obtained as  
\[ \begin{split}\mathbf{T}(s)&=\mathbf{E}_{6}-\mathbf{S} (s)\\ &=\mathbf{E}_{6}-\Big{(}\mathbf{U}_{22}(s)-\mathbf{U}_{21}(s) \mathbf{J}_{\theta f}^{-1}\mathbf{J}_{\theta b}\Big{)}^{-1}\Big{(}\mathbf{U}_ {22}(s)-\mathbf{U}_{21}(s)\mathbf{U}_{11}^{-1}(s)\mathbf{U}_{12}(s)\Big{)}\\ &=\Big{(}\mathbf{U}_{22}(s)-\mathbf{U}_{21}(s)\mathbf{J}_{ \theta f}^{-1}\mathbf{J}_{\theta b}\Big{)}^{-1}\Big{(}\Big{(}\mathbf{U}_{22}( s)-\mathbf{U}_{21}(s)\mathbf{J}_{\theta f}^{-1}\mathbf{J}_{\theta b}\Big{)}- \mathbf{G}_{22}^{-1}(s)\Big{)}\\ &=\Big{(}\mathbf{U}_{22}(s)-\mathbf{U}_{21}(s)\mathbf{J}_{ \theta f}^{-1}\mathbf{J}_{\theta b}\Big{)}^{-1}\Big{(}\mathbf{U}_{21}(s) \mathbf{U}_{11}^{-1}(s)\mathbf{U}_{12}(s)-\mathbf{U}_{21}(s)\mathbf{J}_{ \theta f}^{-1}\mathbf{J}_{\theta b}\Big{)}\\ &=\Big{(}\mathbf{U}_{22}(s)-\mathbf{U}_{21}(s)\mathbf{J}_{ \theta f}^{-1}\mathbf{J}_{\theta b}\Big{)}^{-1}\Big{(}-\mathbf{U}_{21}(s) \mathbf{J}_{\theta f}^{-1}\Big{)}\Big{(}\mathbf{J}_{\theta b}-\mathbf{J}_{ \theta f}\mathbf{U}_{11}^{-1}(s)\mathbf{U}_{12}(s)\Big{)}\\ &=\Big{(}\mathbf{I}_{6}-\mathbf{Y}(s)\mathbf{J}_{\theta f}^{-1} \mathbf{J}_{\theta b}\Big{)}^{-1}\Big{(}-\mathbf{Y}(s)\mathbf{J}_{\theta f}^{ -1}\Big{)}\mathbf{G}_{\mathbf{P}}(s)\end{split}  \tag{63}\]  
Then, the maximum singular value response of \(\mathbf{T}(s)\) satisfies that  
\[\sigma(\mathbf{T}(j\omega))\leq\sigma\bigg{(}\Big{(}\mathbf{I}_{6}-\mathbf{Y }(j\omega)\mathbf{J}_{\theta f}^{-1}\mathbf{J}_{\theta b}\Big{)}^{-1}\Big{(} -\mathbf{Y}(j\omega)\mathbf{J}_{\theta f}^{-1}\Big{)}\bigg{)}\sigma(\mathbf{G }_{\mathbf{P}}(j\omega)) \tag{64}\]  
Now, a permissive bound of \(\Delta(s)\) that keeps practical IPFL stable could be drawn according to Formulas (62) and(64). To keep the practical IPFL stable with greater uncertainty, it is expected that \(\sigma(\mathbf{T}(j\omega))\) is as small as possible. Under the condition that \(\upsilon<1\), which is reasonable for the case of large spacecraft and small payload, the smaller the item \(\mathbf{Y}(s)\) and \(\mathbf{G}_{\mathbf{P}}(s)\), the better the robust stability of the IPFL consisting of proposed feedforward controller. In other words, the weaker the'mobile-base' effect and the smaller the passive transmissibility, the better the robust stability. A simple but effective method would be to increase the mode damping ratios of \(\mathbf{G}_{\mathbf{P}}(s)\) or to decrease the ratio of the payload mass to the support module mass.