# Coupled Dynamic Modeling and Control of Aerial Continuum Manipulation Systems
## 5 Control Design  
The formulated controller is based on the adaptive sliding mode approach, which considers the whole dynamics of ACMS and is robust against modeling uncertainties and external disturbances. In the control design process, the error vector \(\mathbf{e}\), sliding surface \(\mathbf{S}\), and the reference signal rate \(\dot{\mathbf{\zeta}}_{r}\) are considered as:  
\[\mathbf{e}=\mathbf{\zeta}-\mathbf{\zeta}_{d}, \tag{36}\]  
\[\dot{\mathbf{\xi}}_{r}=\dot{\mathbf{\xi}}_{d}-\mathbf{\Lambda}\mathbf{e}, \tag{37}\]  
\[\mathbf{S}=\dot{\mathbf{\xi}}-\dot{\mathbf{\xi}}_{r}=\dot{\mathbf{e}}+\mathbf{\Lambda}\mathbf{e}, \tag{38}\]  
where \(\mathbf{\zeta}_{d}\) and \(\dot{\mathbf{\xi}}_{d}\) are the desired states for ACMS and their desired derivatives, and \(\mathbf{\Lambda}\) is a positive diagonal matrix. With these definitions, the control signal can be proposed as:  
\[\mathbf{u}_{ACMS}=\hat{\mathbf{B}}\,\ddot{\mathbf{\xi}}_{r}+\hat{\mathbf{C}}\,\dot{\mathbf{\xi}}_{ r}+\hat{\mathbf{G}}-\mathbf{k}_{v}\dot{\mathbf{e}}-\mathbf{k}_{p}\mathbf{e}+\hat{\mathbf{\Delta}}\,, \tag{39}\]  
where \(\hat{}\) shows the nominal values of matrices while \(\mathbf{k}_{p}\) and \(\mathbf{k}_{v}\) are positive definite gain matrices. In addition, \(\hat{\mathbf{\Delta}}\) represents the estimated uncertainty, which is calculated using the adaptation law by updating \(\dot{\hat{\mathbf{\Delta}}}\) as:  
\[\dot{\hat{\mathbf{\Delta}}}=-\mathbf{k}_{TI}\mathbf{S}, \tag{40}\]  
where \(\mathbf{k}_{TI}\) is a positive definite gain. Substituting the adaptive sliding mode control law of Equation (39) into the ACMS dynamics Equation (24), the closed loop dynamics of the system can be found as:  
\[\mathbf{\dot{B}}\,\dot{\mathbf{S}}+\hat{\mathbf{C}}\mathbf{S}+\mathbf{k}_{v}\dot{\mathbf{e}}+\mathbf{k}_{ p}\mathbf{e}=\widetilde{\mathbf{\Delta}}\,, \tag{41}\]  
where \(\sim\) is the error on the modeling and uncertainty estimation. Stability analysis of the formulated control for ACMS with coupled dynamics can be conducted using a Lyapunov candidate function as follows. \[L=\frac{1}{2}\Big{\{}\mathbf{S}^{T}\hat{\mathbf{B}}\mathbf{S}+\widetilde{\mathbf{\Delta}}^{T} \mathbf{\Pi}\widetilde{\mathbf{\Delta}}+\mathbf{e}^{T}\big{(}\mathbf{\Pi}\mathbf{k}_{v}+\mathbf{k}_{p} \big{)}\mathbf{e}\Big{\}}>0, \tag{42}\]  
where \(\mathbf{\Pi}=\mathbf{k}_{TI}^{-1}\). The derivative of the Equation (42) can be found as  
\[\dot{L}=\mathbf{S}^{T}\hat{\mathbf{B}}\dot{\mathbf{S}}+\frac{1}{2}\mathbf{S}^{T}\dot{\hat{B}} \mathbf{S}+\widetilde{\mathbf{\Delta}}^{T}\mathbf{\Pi}\dot{\widetilde{\mathbf{\Delta}}}+\mathbf{e} ^{T}\big{(}\mathbf{\Pi}\mathbf{k}_{v}+\mathbf{k}_{p}\big{)}\dot{\mathbf{e}}. \tag{43}\]  
By using the closed loop dynamics of Equation (41) and the skew symmetricity feature of \(\dot{\mathbf{B}}-2\hat{\mathbf{C}}\), the derivative of the Lyapunov function can be written as  
\[\dot{L}=\mathbf{S}^{T}\Big{(}-\mathbf{k}_{v}\dot{\mathbf{e}}-\mathbf{k}_{p}\mathbf{e}+\widetilde{ \mathbf{\Delta}}\Big{)}+\widetilde{\mathbf{\Delta}}^{T}\Pi\dot{\widetilde{\mathbf{\Delta}}} +\mathbf{e}^{T}\big{(}\mathbf{\Pi}\mathbf{k}_{v}+\mathbf{k}_{p}\big{)}\dot{\mathbf{e}}.\]  
Considering the definition of sliding surface, the above equation can be rearranged as  
\[\dot{L}=-\dot{\mathbf{e}}^{T}\mathbf{k}_{v}\dot{\mathbf{e}}-\mathbf{e}^{T}\Pi\mathbf{k}_{p}\mathbf{e}+ \widetilde{\mathbf{\Delta}}^{T}\Big{(}\mathbf{\Pi}\dot{\widetilde{\mathbf{\Delta}}}+\mathbf{S} \Big{)}. \tag{44}\]Substitution of the adaptive law of Equation (40) into the previous Lyapunov derivative with the assumption that \(\Delta\) changes slower than \(\hat{\Delta}\) and consequently, \(\dot{\hat{\Delta}}\approx\dot{\hat{\Delta}}\), one can obtain  
\[\dot{\mathbf{\xi}}_{r}=\dot{\mathbf{\xi}}_{d}-\mathbf{\Lambda}\mathbf{e}, \tag{37}\]0  
Being a positive definite of Equation (42) and a negative semi-definite of Equation (45) means that \(\mathbf{e}\), \(\mathbf{S}\) and \(\mathbf{\tilde{\Delta}}\) are bounded.