# Control of an Omnidirectional UAV for Transportation and Manipulation Tasks
## 7 Simulation Results
The performance of the internal control loop is measured by the roll-pitch joint tracking errors (\((\alpha_{des}-\alpha)\) and \((\beta_{des}-\beta)\)), reported in Figure 5. Furthermore, in this case, the peak error is below 5 deg, and the oscillation magnitude is about 2 degrees. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Gain** & \multicolumn{8}{c}{**Value**} \\ \hline \(\mathbf{K}_{o}\) & diag\([1000\) & \(1000\) & \(1000\) & \(400\) & \(400\) & \(35\) & \(200\) & \(200\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Momentum observer gain. Figure 3: First case study: desired position and orientation trajectories for the platform. The second simulation case study is aimed to prove the controller robustness in the presence of an external disturbance. In this case, the control outputs (18), (28), and (29) are replaced by (48). A task involving grasping and transportation of an object has been considered: the ODQuad moves to the grasping point, grasps the object, and then rigidly holds it during its motion. Figure 6 shows some snapshots of the simulation. Figure 4: First case study: position and orientation tracking errors. Figure 5: First case study: roll-pitch joint errors. The position tracking error, reported in Figure 7a, increases along the \(z\) axis during the grasping phase (at about 10 s), as expected, and comes back to the initial value after 3 s. In other words, after an initial transition due to the first contact with the object, the wrench estimator allows to correctly compensate the effect of the additional load attached to the vehicle. Figure 6: Second case study: screenshots of Matlab’s Mechanics Explore. (**1**) ODQuad stays in hovering position. (**2**) ODQuad grasps the object, (**3**) ODQuad performs a position and orientation trajectory. (**4**) ODQuad comes to the initial configuration moving the grasped object. Figure 7: Second case study: platform position and orientation tracking errors. Then, during the transportation, an error is experienced along the \(y\) axis and in the platform orientation (with a peak of about 6 deg), probably due to the dynamics of the gimbal mechanism. Figure 7(a) reports the roll-pitch joint tracking errors: the roll and pitch errors during the transportation oscillate between \(-5\) and \(5\) deg. Figure 7(b) reports the actuation torques of the roll-pitch joints: it can be noticed that, even in the presence of an external load, the required effort can be easily obtained via off-the-shelf motors. Finally, the estimated interaction forces and torques, computed via the observer (52), are reported in Figure 8(a),b. In detail, the force along \(z\) axis correctly converges to the load weight (Figure 8(a)), while small forces along \(x\) and \(y\) axes are experienced during the transportation, due to the object inertia and the unmodeled ODQuad's dynamics (Coriolis and friction terms are neglected both in controller and in observer). Regarding the estimated torques (Figure 8(b)), small torques are experienced around the \(x\) and \(y\) axes due to the misalignment during the motion between the object and the platform CoM. Figure 8: Second case study: roll-pitch joint tracking error (**a**) and torques (**b**).