# Coupled Dynamic Modeling and Control of Aerial Continuum Manipulation Systems
## 4 Actuation Modeling  
In this section, an actuation mechanism for a general \(n\)-rotor UAV with an attached tendon-driven continuum arm is presented. As discussed in the introduction, 6 DOFs' control of multi-rotors can be achieved using a nonparallel thrust configuration. In such configurations, each propeller is rotated a cant angle, \(\phi_{i}\), around its radius. As shown in Figure 1, \(\phi_{i}\) is the angle between the normal vector of each propeller plane and UAV's vertical plane. These rotors are laid out along the edge of the disk canted tangentially to the edge of the disk [41]. This means that motors with odd and even numbers are rotated toward the left and right side of their radii, respectively. The position of \(n\) motors and their rotations and forces in the horizontal plane of the UAV are defined in Figure 3. In the body reference frame, it is assumed that the \(n^{\text{th}}\) motor is located at the front of the UAV on the \(X_{q}\) axis of the body frame. The other motors are located at angles \(\vartheta\), \(2\vartheta\), \(3\vartheta\), \(\ldots\), \((n_{m}-1)\vartheta\) with respect to the \(X_{q}\) axis. Angle \(\vartheta\) depends on the number of motors (\(n_{m}\)) and is given by \(\vartheta=\frac{2\pi}{n_{m}}\). To control the torque applied to the system, motors with odd numbers are rotating clockwise, while motors with even numbers are rotating counterclockwise [44]. Torque produced by each motor can be expressed as \(\tau_{i}=kT_{i}\), where \(T_{i}\) is the thrust of the \(i\)-th motor, and \(\dot{k}\) is the ratio of the torque to thrust produced by each motor. To compute the net force/torque acting on the UAV from all thrusters, first each motor's thrust and torque is decomposed into components on the body frame as given in Equation (30). Then the total force/torque for motors can be obtained [44]. \[\mathbf{F}_{pos_{i}}=\left[ \begin{array}{c}-\mathit{\Gamma}sin(\vartheta_{i}) sin(\phi_{i})\\ \mathit{\Gamma}cos(\theta_{i})sin(\phi_{i})\\ cos(\phi_{i})\end{array} \right][T_{i}], \tag{30}\]  
where \(\phi_{i}\) is the cant angle from vertical, and \(\vartheta_{i}\) represents the rotor's position in the horizontal UAV frame and equals \(i\vartheta\). In addition, \(\dot{d}\) is the distance from the rotor center to the central axis of UAV (the radius of the \(n\) rotor). Moreover, \(\mathit{\Gamma}\) is a sign parameter that is \(+1\) for even motors and \(-1\) for odd motors. If all cant angles (\(\phi_{i}\)) are equal to zero, it means that all motors are perpendicular to the UAV horizontal plane and the generalized model will reduce to a conventional multi-rotors model. After decomposition of force/torque produced by each motor, the net force and torque of the \(n\)-rotor UAV can be computed as \(F_{pos}=\sum_{i=1}^{n}F_{pos}\) and \(\tau_{att}=\sum_{i=1}^{n}\tau_{att_{i}}\), respectively. The only remaining term is \(u_{tem}\), which is discussed in the following lines. Here, the assumption is that we have three tendons for actuation of the continuum robot. The first tendon's actuator is located on the \(X_{q}\) of the horizontal body reference frame of the multirotor and two others are located with \(\pm\frac{2\pi}{3}\) radians with respect to the first one. To control the end effector, two tendons out of three should be actuated at the same time depending on the \(q_{2}\) angle [31]. Table 2 shows the tendon's actuation policy. The displacement of each tendon with respect to the main backbone can be found by [32]. \[\Delta L_{1}=rq_{1}cosq_{2}, \tag{31}\]  
\[\Delta L_{2}=rq_{1}cos\left(-q_{2}+\frac{2\pi}{3}\right), \tag{32}\]  
\[\Delta L_{3}=rq_{1}cos\left(q_{2}+\frac{2\pi}{3}\right), \tag{33}\]  
where \(r\) is the distance between each tendon actuator and the main backbone.