# Passivity-Based Control for a Micro Air Vehicle Using Unit Quaternions
## 5 Numerical Validation
### Simulation Results  
A simulated experiment was designed in which a position reference is given to the quad-rotor. A sequence of five set-points are defined in a time lapse of 25 seconds, thus following a 1 meter square at 1 meter height as illustrated in Figure 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Characteristic** & \multicolumn{6}{c}{**References**} \\ \hline  & [17] & [18] & [19] & [20] & [21] & Our work \\ \hline The algorithm avoids complicated PDEs 1  & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\checkmark\) & \(\checkmark\) \\ \multicolumn{1}{c}{} & Quad-rotor full dynamics control & \(\checkmark\) & \(\checkmark\) & \(\times\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) \\ \multicolumn{1}{c}{} & There is not change in the UAV mechanical structure 2 & \(\times\) & \(\times\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) \\ \multicolumn{1}{c}{} & Simple real-time experiments & \(\times\) & \(\times\) & \(\times\) & \(\checkmark\) & \(\times\) & \(\checkmark\) \\ \multicolumn{1}{c}{} & Three-dimensional maneuvers & \(\checkmark\) & \(\checkmark\) & \(\times\) & \(\times\) & \(\times\) & \(\checkmark\) \\ \multicolumn{1}{c}{} & The algorithm has no issues with singularities & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\checkmark\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Qualitative comparison of control algorithms based on passivity applied to a quad-rotor. \[p_{d}=\left\{ \begin{array}{ccccc}\left[0&0&1\right]^{T}&,&t&<5\\ \left[1&0&1\right]^{T}&,&5<=&t&<10\\ \left[1&1&1\right]^{T}&,&10<=&t&<15\\ \left[0&1&1\right]^{T}&,&15<=&t&<20\\ \left[0&0&1\right]^{T}&,&20<=&t\end{array} \right. \tag{35}\]  
In order to proof in simulation the robustness to unknown bounded perturbations, additive uniformly distributed noise was considered with bounds empirically chosen between \(\pm 0.5\) m for the position, and between \(\pm 5\%\) for the orientation, and translational and rotational velocities. The feedback translational signals plus the additive noise close the loop to calculate the required force (see Figure 4a) to move the vehicle in the 3D (3-dimensional) space. Note in this figure that the force in the \(z\) axis is stabilized around \(mg=(0.4306)(9.81)=4.22\) N in order to maintain the vehicle's altitude. The control torques are calculated such that the vehicle's orientation \(q\) follows the trajectory \(q_{d^{\prime}}\), see Figure 4b. The force in the inertial frame is then used to compute the orientation trajectory which will point the thrust force to the required direction in the inertial frame. Figure 5 illustrates the trajectory tracking  
Figure 4: Control inputs. (**a**) Control Forces; (**b**) Control Torques. Figure 3: Quad-rotor’s translation in 3D (3-dimensional) space. for the vehicle's attitude, illustrated in axis-angle representation for a better understanding, here, \(\bar{\theta}=2\ln\mathbf{q}\), and \(\bar{\theta}_{d}=2\ln\mathbf{q}_{d}\). Finally, Figure 6 represents the translational stabilization. In the first one, the position for each axis is stabilized according to the reference. The quality indicator IAE (Integral of Absolute Error) is chosen to compare the behavior and efficiency of the proposed controller with other control techniques for a quad-rotor using a quaternion representation of the attitude. PD (Proportional Derivative), LQR (Linear Quadratic Regulator) and backstepping control strategies presented in [12] are selected to compare them with the PBC strategy. The results of IAE are shown in Table 2.  
We can observe that the quality indicators IAE for the four strategies are similar, i.e., they exhibit a very similar behavior. The two controller with better performance are PD and PBC for the positions \(x\) and \(y\), Backstepping and PBC for the position \(z\). Figure 5: Vehicle’s attitude and reference quaternions expressed in axis-angle representation.