# Coupled Dynamic Modeling and Control of Aerial Continuum Manipulation Systems
## 6 Experimental Simulation Results  
In this section, the developed coupled dynamic modeling approach for the considered ACMS configurations (a fully actuated octa-rotor [45] with a tendon driven continuum arm actuated by three tendons [20]) is validated using experimental simulation results to verify dynamics and control methods. In this regard, first, the dynamic model of the coupled ACMS is derived using the MATLAB 2020a symbolic toolbox. Then, the ACMS model and control are implemented in MATLAB 2020a, while sampling time is chosen as 0.01 s and a 5 th order Runge Kutta solver was used. The objective of the experimental simulation scenario is to verify the performance of the control method designed for the coupled system dynamics in free flight motions when there is not any interaction with the environment. For this purpose, similar to [22], we analyzed two scenarios: (1) tracking a helix trajectory while the continuum arm is fixed, and (2) tracking an agile infinity trajectory while the arm moves. In these two control cases, an octa-rotor with the capability of carrying 3 motors required for arm actuation, i.e., Dynamixel MX-28-AT geared servo motor (ROBOTIS, Lake Forest, CA, USA), each with a mass of 91 gr was considered. The selected octa-rotor weight was 3.2 \(kg\)[45] with a cant angle of 25 \(deg\) and the selected continuum arm specifications were the same as [20]. For the first case, we assumed that the arm was fixed in the desired position with respect to the UAV (\(q_{1}=\pi/2\) rad and \(q_{2}=\pi/4\) rad) while the UAV tracks a helical trajectory of \(\left[ \begin{array}{cccc}4 cos\left(\frac{2\pi}{15}t\right)-4&4 sin\left( \frac{2\pi}{15}t\right)&0.2t\end{array} \right]^{T}\)(all in \(m\)) with zero desired Euler angles. Since a fully actuated UAV is selected, it should be able to control its position and attitude independently. That is why desired Euler angles are considered zero. In the second case, UAV was driven to follow an agile infinite path of \(\left[ \begin{array}{cccc}2.5 sin\frac{t}{4}&2.5 sin\frac{t}{2}&2\end{array}  \right]^{T}\)(all in \(m\)) with zero desired Euler angles while the end effector of the arm had a circular motion of (\(q_{1}=\frac{\pi}{2}\)_rad_ and \(q_{2}=\frac{2\pi}{17}t\) rad) with respect to the UAV. The initial condition of the arm was \(q_{1}=\frac{\pi}{18}\) rad and \(q_{2}=\frac{\pi}{18}\) rad in all cases while the initial condition of UAV in the first case was \(\left[ \begin{array}{cccc}0&0&0&0&0&0\end{array} \right]^{T}\)(in \(m\) and \(rad\)) and that of the second case was \(\left[ \begin{array}{cccc}0&0&2&0&0&0\end{array} \right]^{T}\)(in \(m\) and \(rad\)). A minimal pose representation with position and RPY Euler angles was used here. Figure 4 illustrates the trajectories of the drone for these two simulation cases. For the first control case, the helical trajectory with fixed arm angles, the states of ACMS and their desired values are shown in Figure 5. In this figure, ACMS states are shown by blue lines while red lines illustrate desired values of states. As shown in Figure 5, ACMS effectively tracked desired values. Figure 6 presents tracking error for this case, which is around 0.1 \(m\) for position and zero for attitude tracking. Octa-rotor thrusts to track the given helical trajectory and required tension of tendons to reach their fixed desired values are illustrated in Figures 7 and 8, respectively. Figures 7 and 8 show that UAV motor thrusts had an oscillatory behavior around 5 \(N\) to produce the required force and moments  
Figure 6: State’s error for the first control case. Figure 7: UAV motors thrusts in the first control case. Figure 8: CR tensions in the first control case. Figure 9: States of ACMS for the second control case (reference: red). Figure 11: UAV motors thrusts in the second control case. Figure 10: State’s error for the second control case.