## 5 Experiments
### Lissajous Trajectory  
The Lissajous trajectory was picked to demonstrate the ability of the proposed approach to track a trajectory for which the nonholonomic constraints greatly affect the movement of the system. The proposed Lissajous trajectory is defined by  
\[\mathbf{r_{d}}(t)= \begin{bmatrix}\mathbf{P_{d}}(t)\\ \mathcal{Q}_{d}(t)\end{bmatrix} = \begin{bmatrix}x_{0}\\ y_{0}\\ z_{0}\\ Q_{0}\end{bmatrix} + \begin{bmatrix}A\cos(s(t)+\pi/2)\\ B\cos(2(s(t)+\pi/2)+\pi/2)\\ C\cos(2s(t))-C\\ 0\end{bmatrix} ,\]  
where \([x_{0},y_{0},z_{0}]^{T}\) is the end-effector's starting position, \(s(t)\) is the trajectory timing variable, and \(A,B\) and \(C\) define the size of the path. The orientation is set to be the same for the entire  
Figure 5: NMM’s self-collision distances. trajectory, i.e., \(Q(t)=Q_{0}\). Notice that this is a 6-DOF trajectory because the orientation is constrained for the entire trajectory. The starting configurations of the mobile platform and prismatic joint for this trajectory were set to \(x_{p}=-0.1\) (m), \(y_{p}=-0.13\) (m), \(\theta_{p}=-90(^{\circ})\) and \(z_{pj}=0.2\) (m). With this configuration, the end-effector's initial pose is given by \(P_{\mathbf{0}}=\left[-0.009\quad-0.6491\quad 0.9884\right]\) (m) and \(Q_{0}=\{0,0i+1j+0k\}\). A trapezoidal velocity profile [1] was used for the derivative of the timing variable \(s(t)\). The parameters for the path size were set to \(A=1.3\) (m), \(B=1.3\) (m) and \(C=0.27\) (m), and an execution time \(t_{f}=64(s)\) was chosen. Figure 5(a),b show snapshots of the NMM's movement along the Lissajous trajectory, in simulations and experiments, respectively. Figure 7 compares the trajectory tracking results between the simulation and the experiment. In the simulation, the position and orientation errors are kept small during the whole trajectory, as shown in Figure 6(b),c. This demonstrates the good tracking performance at the kinematic level of our proposed motion planning algorithm. The observed larger errors in the experiments, as exhibited in Figure 6(e),f, are due to the vibrations of the system during the experiments. Furthermore, as mentioned before, the control of the dynamic behavior of the system is beyond the scope of this work. Nevertheless, the position errors in the experiments are kept below \(2\times 10^{-3}\) (m) and the orientation errors below \(1.5\times 10^{-3}\). Likewise, we observe that the obtained trajectories in the simulation (Figure 6(a)) and the experiment (Figure 6(d)) are fairly similar. Figure 8 illustrates the experiment results. Note that the negative joint velocity limit \(-u_{i}^{+}\) is denoted as \(u_{i}^{-}\) in the pertinent figures. The manipulabilities of both the arm and the complete system are kept high during the execution of the trajectory, and their final values are higher than at the start of the trajectory, as shown in Figure 7(b). It is important to remark that there are no potential self-collisions during the execution of this trajectory. Even though \(d_{\mathit{wrist}}\) is negative at the beginning of the trajectory, as shown in Figure 7(c), the wrist is above the top of the mobile platform and therefore the joint movements were not restricted, as explained in Section 5.2. Figure 6: Snapshots of the NMM’s motion for the Lissajous trajectory tracking: (**a**) simulations; and (**b**) experiments. Figure 8d-i demonstrate the fulfillment of the joint limits and joint velocity boundary constraints. All the joints are kept within their corresponding limits. Notice that the movement of \(q_{a1}\), Figure 8f, is restricted in the time interval \(t=(29,32)(s)\) to respect its lower limit. This time interval corresponds to the trajectory section between snapshots number three and four in both Figure 6a,b. For this section of the trajectory, the movement of the end-effector in the XY plane is taken care of by the platform due to the restriction imposed to \(q_{a1}\). We observe in Figure 8e that the velocity limits of the prismatic joint are respected.