# Coupled Dynamic Modeling and Control of Aerial Continuum Manipulation Systems
## 2 Kinematic Modeling  
A tendon driven continuum arm attached to the center of a multi-rotor UAV is considered as the ACMS (Figure 2). The CR arm contains an elastic continuum backbone, a number of tendon spacer disks, and a number of tendons, which can be in the form of cables or wires. The spacer disks are fixed to the elastic backbone and the pinholes on them are required to actuate the tendons. By pulling the tendons, the backbone bends toward the contracted tendon. To describe the ACMS motion, different coordinate frames are introduced (Figure 1), with the inertial reference frame \(\mathcal{F}_{I}:\ \left\{X_{I}\,,\,Y_{I}\,,\,Z_{I}\right\}\) having the origin \(O_{I}\) while \(\mathcal{F}_{q}:\ \left\{X_{q}\,,\,Y_{q}\,,\,Z_{q}\right\}\) possessing the origin \(O_{\text{q}}\) indicates the body frame attached to the center of mass of the multi-rotor. The position of the multi-rotor with respect to \(\mathcal{F}_{I}\) is indicated by \(\mathbf{p}_{q}\), while its attitude is described by Euler angles, \(\mathbf{\phi}_{q}=\left[ \begin{array}{cccc}\mathbf{\varphi}&\theta&\psi\end{array}  \right]^{T}\) containing roll, pitch, and yaw angles, respectively. The relationship between the linear velocity of multi-rotor with respect to \(\mathcal{F}_{I}\), \(\dot{\mathbf{p}}_{q}\), and its linear velocity with respect to \(\mathcal{F}_{q}\), \(\dot{\mathbf{p}}_{q}^{\dot{q}}\), is defined as:  
\[\dot{\mathbf{p}}_{q}=\mathbf{R}_{q}\dot{\mathbf{p}}_{q}^{\dot{q}}, \tag{1}\]  
where \(\mathbf{R}_{q}\) is a rotation matrix of \(\mathcal{F}_{q}\) with respect to \(\mathcal{F}_{I}\). The angular velocity of the multi-rotor with respect to \(\mathcal{F}_{I}\), \(\mathbf{\omega}_{q}\), can be determined based on the derivative of Euler angles, i.e.,  
\[\mathbf{\omega}_{q}=\mathbf{T}_{q}\dot{\mathbf{\phi}}_{q}. \tag{2}\]  
Figure 2: A hexa-rotor ACMS with a tendon-driven CR. Using a rotation matrix and the relationship between angular velocity in the \(\mathcal{F}_{I}\) coordinate system and the derivative of Euler angles given in Equation (2), the angular velocity of the multi-rotor with respect to \(\mathcal{F}_{q}\), \(\mathbf{\omega}_{q}^{q}\), can be found as  
\[\mathbf{\omega}_{q}^{q}=\mathbf{R}_{q}^{T}\mathbf{\omega}_{q}=\mathbf{R}_{q}^{T}\mathbf{T}_{q}\dot{ \mathbf{\phi}}_{q}=\mathbf{Q}\dot{\mathbf{\phi}}_{q}, \tag{3}\]  
where \(\mathbf{Q}=\mathbf{R}_{q}^{T}\mathbf{T}_{q}\). In addition, \(\mathbf{q}=\left[ \begin{array}{cc}q_{1}&q_{2}\end{array} \right]^{T}\) is the angle vector for the arm's end-effector with respect to the UAV. As shown in Figure 1, the end-effector's position can be found using two angles, \(q_{1}\) and \(q_{2}\), where \(q_{2}\) is the angle of the end-effector with respect to the \(X_{\rm q}\) in the UAV's horizontal plane (\(X_{q}Y_{q}\) plane), and \(q_{1}\) denotes the angle of CR's end-effector with respect to the \(Z_{q}\) perpendicular to the UAV's horizontal plane. Thus, the generalized states of the ACMS, \(\mathbf{\xi}\), can be expressed using UAV's position and attitude and CR's end-effector angles as:  
\[\mathbf{\xi}=\left[ \begin{array}{cc}\mathbf{p}_{q}^{T}&\mathbf{\phi}_{q}^{T}&q_{1}&q_{2} \end{array} \right]^{T}. \tag{4}\]  
To calculate the absolute position of each point of the continuum arm at section s in \(\mathcal{F}_{\rm I}\) (where \(s\in\left[ \begin{array}{cc}0&l\end{array} \right]\) denotes the arc length of each section and \(l\) is the length of CR), the following equation holds, indicating that the absolute position of that special point of the arm equals the summation of the multi-rotor's position in \(\mathcal{F}_{I}\) and the relative position of that point of arm with respect to the center of mass of the multi-rotor in \(\mathcal{F}_{I}\),  
\[\mathbf{p}_{s}=\mathbf{p}_{q}+\mathbf{R}_{q}\mathbf{p}_{qs}^{q}, \tag{5}\]  
where \(\mathbf{p}_{qs}^{q}\) is the relative position of each section (point) of arm with respect to \(\mathcal{F}_{q}\), which can be found from forward kinematics of the continuum arm [42].