# Aerial Tele-Manipulation with Passive Tool via Parallel Position/Force Control +
## 4 Stability Analysis
### Rotational Stability
\begin{bmatrix}\varepsilon k_{l}&\frac{1}{2}((\varepsilon-1)k_{l}+ \varepsilon k_{P})&\frac{1}{2}\varepsilon\delta_{\mathbb{C}}\|\hat{\eta}\| \\ \frac{1}{2}((\varepsilon-1)k_{l}+\varepsilon k_{P})&\varepsilon(k_{P}-k_{D})+ \mu_{K}-k_{l}&\frac{1}{2}\varepsilon(\gamma_{M}+\delta_{\mathbb{C}}\|\hat{ \eta}\|)\\ \frac{1}{2}\varepsilon\delta_{\mathbb{C}}\|\hat{\eta}\|&\frac{1}{2} \varepsilon(\gamma_{M}+\delta_{\mathbb{C}}\|\hat{\eta}\|)&k_{D}-\varepsilon \gamma_{M}(1+\frac{\gamma_{K}}{k_{l}})\end{bmatrix} . \tag{24}\]  
To obtain (23), we choose \(k_{P},k_{l},k_{D}\) such that:  
\[ \begin{array}{c}\varepsilon(2\mu_{K}+\gamma_{M})<k_{l}<\mu_{K}\\ (\frac{1}{\varepsilon}-\varepsilon)k_{l}<k_{P}<2\mu_{K}+\gamma_{M}\\ \varepsilon\gamma_{M}(\frac{3}{2}+\frac{\gamma_{K}}{k_{l}})<k_{D}<\mu_{K}(1- \varepsilon)\end{array} . \tag{25}\]  
Consequently, \(\varepsilon\) must be chosen such that  
\[\varepsilon<min\Bigg{\{}\frac{\mu_{K}-k_{I}}{\mu_{K}-\frac{k_{P}}{2}+\frac{ \gamma_{M}}{2}},\frac{2}{3}\frac{\mu_{K}(2\mu_{K}+\gamma_{M})-\gamma_{K}\gamma _{M}}{\gamma_{M}(2\mu_{K}+\gamma_{M})},\frac{\mu_{M}}{2\gamma_{M}},\frac{\mu_ {K}}{\gamma_{M}}\Bigg{\}}. \tag{26}\]  
Then, it is also assumed that the spherical joint mechanical stiffness is chosen properly in accordance with the system moment of inertia such that \(\mu_{K}>\frac{\gamma_{M}}{2}(1+\frac{\gamma_{M}}{\mu_{K}})\), which can be simplified as \(\sigma_{min}(K_{r})>\gamma_{M}\), which simply means that the higher the inertia, the stiffer the spring that must be chosen. Let \(\mu_{v}=min\{k_{1}-\varepsilon k_{P},\ \varepsilon k_{P}+2(\mu_{K}- \varepsilon k_{D})-(\varepsilon+1)k_{1}-\varepsilon\gamma_{M},\ \ k_{D}-\frac{3}{2} \varepsilon\gamma_{M}-\frac{\varepsilon\gamma_{M}\gamma_{K}}{k_{l}}\}\), for trajectories bounded by:  
\[\|\hat{\boldsymbol{\eta}}\|\leq\frac{1}{\delta_{\mathbb{C}}}\mu_{v}. \tag{27}\]  
\(Q_{1}\) will be SDD with positive diagonal elements, and therefore positive definite. Thus, we can conclude:  
\[\dot{V}(t,\boldsymbol{\zeta})\leq-\mu\|\boldsymbol{\zeta}\|^{2}+\kappa\| \boldsymbol{\zeta}\|\, \tag{28}\]  
where \(\mu=\sigma_{min}(Q_{1})\) and \(\kappa=\delta_{B1}\delta_{f}+\delta_{B2}\delta_{v}\) with \(\delta_{B1}=max\{1,\ k_{1}+\varepsilon k_{D}-\mu_{K},\ k_{P}+\varepsilon\gamma _{M}/k_{I}\}\), \(\delta_{B2}=max\{k_{I}+\varepsilon k_{D}-\mu_{K},\ \ k_{P}+\varepsilon k_{D}-\mu_{K}(k_{I}+ \varepsilon k_{D}-\mu_{K})/k_{I},\ \ \varepsilon\gamma_{M}\}\), and \(\|\hat{\boldsymbol{\eta}}_{d}\|\leq\delta_{v}\) is the upper bound on the norm of \(\hat{\boldsymbol{\eta}}_{d}\). \(\|\hat{\boldsymbol{\tau}}_{f}\|\leq\delta_{f}\) is the upper bound on the norm of the \(\hat{\boldsymbol{\tau}}_{f}\), which is a reasonable assumption considering that the rotational dynamics is faster than translational [30]. Let \(W\) be the positive root of \(V=W^{2}\), as in [31], considering (19) we can state \(\dot{W}\leq-\frac{\mu}{2\gamma_{P}}W+\frac{\kappa}{2\sqrt{\mu_{P}}}\), which means:  
\[W(t)\leq W(0)e^{-\frac{\mu}{2\gamma_{P}}t}-\frac{\kappa}{2\sqrt{\mu_{P}}}\int_{0 }^{t}e^{-\frac{\mu}{2\gamma_{P}}(t-s)}ds\leq W(0)e^{-\frac{\mu}{2\gamma_{P}}t }+\frac{\kappa\gamma_{P}}{\sqrt{\mu_{P}}\mu}. \tag{29}\]Considering (19), we can ensure that (27) is satisfied if \(W(0)+\frac{\kappa_{TP}}{\sqrt{k_{P}}p_{t}}\leq\frac{1}{\lambda_{C}}\mu_{v}\). Therefore, choosing sufficiently large \(k_{P}\), and sufficiently small \(k_{I}\) and \(k_{D}\) such that (18) and (25) are satisfied, the solution of the error system converges to zero asymptotically.