# Aerial Tele-Manipulation with Passive Tool via Parallel Position/Force Control +
## 4 Stability Analysis
### Rotational Stability  
The angular dynamics \(M(\mathbf{\eta})\mathbf{\ddot{\eta}}+C(\mathbf{\eta},\mathbf{\dot{\eta}})\mathbf{\dot{\eta}}+K_{r }\mathbf{\eta}+\mathbf{\tau}_{f}=\mathbf{u}_{\eta}\), where \(\mathbf{\tau}_{f}=-S(\mathbf{d})\mathbf{f}_{\mathbf{t}}\) has the following properties:  
* \(M\) is symmetric positive definite and bounded as \(\mu_{M}\leq\|M\|\leq\gamma_{M}\); where \(\mu_{M},\gamma_{M}\) are its minimum and maximum eigenvalues. Moreover, \(M\) is strictly diagonally dominant (SDD), that is \(M_{ii}>\Sigma_{i\
eq j}|M_{ij}|\);
* \(\dot{M}-2C\) is skew-symmetric, or \(\dot{M}=C+C^{\top}\);
* \(\|\mathcal{C}(\mathbf{\eta},\mathbf{\dot{\eta}})\|\leq\delta_{c}\|\mathbf{\dot{\eta}}\|\), where \(\delta_{c}\leq 2\sigma_{max}(J)\);
* the rotational stiffness matrix is bounded as \(\mu_{k}\leq\|K_{r}\|\leq\gamma_{k}\). **Theorem 1**. : _Applying the rotational part of the control law (5) to the aerial manipulator described by (1), its rotational dynamic is locally asymptotically stable such that \(\mathbf{\dot{\eta}},\mathbf{\dot{\eta}}\to\mathbf{0}\) and \(\int_{0}^{t}\mathbf{\dot{\eta}}(s)ds\to-K_{I}^{-1}(K_{r}\mathbf{\eta}+\mathbf{\tau}_{f})\) as \(t\to\infty\)._  
Proof. : Let \(\mathbf{\zeta}=[\mathbf{\Delta}\mathbf{\eta}^{\top}\mathbf{\dot{\eta}}^{\top}\mathbf{\dot{\eta}}^{ \top}]^{\top}\), where \(\mathbf{\Delta}\mathbf{\eta}=\int_{0}^{t}\mathbf{\dot{\eta}}(s)ds+K_{I}^{-1}(K_{r}\mathbf{\eta} +\mathbf{\tau}_{f})\), and consider the following scalar function:  
\[V(\mathbf{\zeta})=\frac{1}{2}\mathbf{\zeta}^{\top}P\mathbf{\zeta}\, \tag{16}\]  
where the symmetric matrix \(P\) is defined as follows:  
\[P= \begin{bmatrix}K_{I}&K_{12}&\varepsilon M\\ K_{12}&K_{22}&\varepsilon M\\ \varepsilon M&\varepsilon M&M\end{bmatrix} , \tag{17}\]  
with \(\varepsilon=k_{\varepsilon}\frac{U_{M}}{\gamma_{M}}\) and \(0<k_{\varepsilon}<0.5\), \(K_{12}=K_{I}+\varepsilon K_{D}-K_{r}\), and \(K_{22}=K_{P}+\varepsilon K_{D}-K_{12}K_{I}^{-1}K_{r}\). Let \(K_{s}=k_{s}I\) for \(s=\{P,I,D\}\) with \(k_{s}>0\); in a range satisfying inequities,  
\[ \begin{array}{ccc}0<k_{D}<\frac{\mu_{K}}{\varepsilon}-\gamma_{M},&\varepsilon <\frac{\mu_{K}}{\gamma_{M}}\\ k_{P}>k_{I}+\frac{\gamma_{K}}{k_{I}}(\varepsilon k_{D}-\mu_{K})+\varepsilon \gamma_{M}.\end{array}  \tag{18}\]  
\(P\) becomes SDD with positive diagonal elements, and is therefore positive definite. Thus, \(V(\mathbf{\zeta})\) is a positive definite function and hence a Lyapunov function candidate, which is radially unbounded and satisfies the Rayleigh-Ritz inequality [29] as:  
\[\mu_{P}\|\xi\|^{2}\leq V(\xi)\leq\gamma_{P}\|\xi\|^{2}\, \tag{19}\]  
where \(\mu_{P}\) and \(\gamma_{P}\) are the minimum and maximum eigenvalues of \(P\). The time derivative of the Lyapunov candidate (16) is:  
\[\dot{V}=-\mathbf{\zeta}^{\top}Q\mathbf{\zeta}+\mathbf{\zeta}^{\top}B[\mathbf{\tau}_{f}^{\top} \ \mathbf{\dot{\eta}}_{d}^{\top}]^{\top}\, \tag{20}\]with  
\[B = \begin{bmatrix}I&-K_{12}\\ K_{12}&-K_{22}\\ \varepsilon K_{l}^{-1}M&-\varepsilon M\end{bmatrix}  \tag{21}\] \[Q = \begin{bmatrix}\varepsilon K_{l}&\frac{1}{2}((\varepsilon-1)K_{l} +\varepsilon K_{P})&-\frac{1}{2}\varepsilon C^{\top}\\ \frac{1}{2}((\varepsilon-1)K_{l}+\varepsilon K_{P})&\varepsilon K_{P}-(K_{l} +\varepsilon K_{D}-K_{r})&-\frac{1}{2}\varepsilon(C^{\top}+M)\\ -\frac{1}{2}\varepsilon C^{\top}&-\frac{1}{2}\varepsilon(C^{\top}+M)&K_{D}- \varepsilon M(I+K_{l}^{-1}K_{r}),\end{bmatrix}  \tag{22}\]  
In the following we shall show that:  
\[\dot{V}(\xi)\leq-\xi_{1}^{\top}Q_{1}\xi_{1}+\boldsymbol{\zeta}^{\top}B[\hat{ \boldsymbol{\tau}}_{f}^{\top}\ \hat{\boldsymbol{\eta}}_{d}^{\top}]^{\top}\, \tag{23}\]  
where \(\boldsymbol{\zeta}_{1}=[\|\boldsymbol{\alpha}\boldsymbol{\eta}\|\ \|\hat{ \boldsymbol{\eta}}\|\ \|\hat{\boldsymbol{\eta}}\|]^{\top}\) and  
\[Q_{1}=