# Coupled Dynamic Modeling and Control of Aerial Continuum Manipulation Systems
## 3 Dynamic Modeling  
To derive the coupled dynamic model of ACMS, the Euler-Lagrange formulation must be considered. The Lagrangian, \(L\), is defined based on the total kinetic energy, \(K\), and potential energies, \(U\), of the system as [43]  
\[L=K-U. \tag{11}\]  
Since the damping effect in continuum arm modelling is not negligible, the modified Euler-Lagrange equation [43] is applied as:  
\[\frac{d}{dt}\frac{\partial L}{\partial\dot{\xi}_{i}}-\frac{\partial L}{ \partial\dot{\xi}_{i}}+\frac{\partial D}{\partial\dot{\xi}_{i}}=u_{i}, \tag{12}\]  
in which \(i=1,\ \dots,\ 8\) and \(\mathbf{\xi}_{i}\) is the \(i\)-th generalized coordinate, while \(u_{i}\) is the \(i\)-th generalized force. In addition, \(D\) is the viscous damper, which is assumed linearly as \(D=\frac{1}{2}\mathbf{\mu_{\xi}^{2}}\), where \(\mathbf{\mu}=\left[ \begin{array}{cccccccc}0&0&0&0&0&0&\mu_{1}&\mu_{2}\\ \end{array} \right]\). Here, \(\mu_{1}\) and \(\mu_{2}\) are CR's damping coefficients. Total kinetic energy of the ACMS, \(K\), is composed of the contribution of three kinetic energy components related to the multi-rotor, \(K_{q}\), main backbone, \(K_{m,b}\), and disks, \(K_{d}\). The total kinetic energy and its components are calculated in Equations (13)-(16). In Equation (14), the kinetic energy of the multi-rotor is computed, while in Equation (15), that of the main backbone for each section is presented. Therefore, total kinetic energy for the backbone is determined by integration of Equation (15) along the length of the continuum arm. From Equation (16), the kinetic energy for the \(j\)-th disk of the continuum arm is obtained. As a result, the total kinetic energy for whole disks is the superposition of energy for each disk. The equations in Equation (17) indicate that position, rotational velocity and rotation matrix of each disk are the same as those of each section of the continuum arm where disks are fixed to the main backbone. Finally, total kinetic energy of the whole system is determined in Equation (18). \[K=K_{q}+K_{m,b}+K_{d}, \tag{13}\]  
\[K_{q}=\frac{1}{2}m_{q}\dot{p}_{q}^{T}\dot{p}_{q}+\frac{1}{2}\mathbf{\omega}_{q}^{T }\mathbf{R}_{q}\mathbf{I}_{q}\mathbf{R}_{q}^{T}\mathbf{\omega}_{q}=\frac{1}{2}m_{q}\dot{p}_{q} ^{T}\dot{p}_{q}+\frac{1}{2}\mathbf{\dot{\phi}}_{q}^{T}\mathbf{Q}^{T}\mathbf{I}_{q}\mathbf{Q} \dot{\mathbf{\phi}}_{q}, \tag{14}\]  
\[K_{m,b}(s)=\frac{1}{2}\rho_{s}A_{s}\dot{p}_{s}^{T}\dot{p}_{s}+\frac{1}{2}\mathbf{ \omega}_{s}^{T}\mathbf{R}_{q}\mathbf{R}_{s}^{d}\mathbf{I}_{s}\mathbf{R}_{s}^{d}\mathbf{I}_{q}^{T} \mathbf{\omega}_{s}, \tag{15}\]  
\[K_{d}(j)=\frac{1}{2}m_{d}\dot{p}_{d}^{T}\dot{p}_{d}+\frac{1}{2}\mathbf{\omega}_{d} ^{T}\mathbf{R}_{q}\mathbf{R}_{d}^{d}\mathbf{I}_{d}\mathbf{R}_{d}^{d}\mathbf{R}_{d}^{T}\mathbf{R}_{q}^{ T}\mathbf{\omega}_{d}, \tag{16}\]  
\[\mathbf{p}_{d}=\mathbf{p}_{s}|_{s=jh^{\prime}},\ \ j=1,2,3,\dots,\ n\]  
\[\mathbf{\omega}_{d}=\left.\mathbf{\omega}_{s}\right|_{s=jh^{\prime}}, \tag{17}\]  
\[\mathbf{R}_{d}^{d}=\left.\mathbf{R}_{s}^{d}\right|_{s=jh^{\prime}},\]  
\[K=K_{q}+\int_{0}^{l}K_{m,b}(s)ds+\sum_{j=1}^{n}K_{d}(j). \tag{18}\]  
Here, \(m_{q}\) is the mass of the multi-rotor, while \(m_{d}\) denotes the mass of each disk. In addition, \(\mathbf{\omega}_{d}\), \(\mathbf{p}_{d}\) and \(\mathbf{R}_{d}^{q}\) are angular velocity, position and rotation matrix of each disk, respectively. Here, \(\mathbf{I}_{s}\) and \(\mathbf{I}_{d}\) present the moment of inertia for the main backbone at each section and that for each disk, respectively, while \(\mathbf{I}_{q}\) denotes the moment of inertia for the multi-rotor. Moreover, \(\rho_{s}\) denotes the density of the main backbone at section \(s\), while \(A_{s}\) is the cross-section area for each section of the continuum arm. Furthermore, \(n\) is the number of disks and \(h\) is the distance between disks. Here, the assumption is that the distance between all disks is equal.