## 5 Experiments
### Lissajous Trajectory
The remaining joints do not reach their respective limits as shown in Figure 8f-h. Furthermore, as seen in Figure 8d,e,i, the velocity profiles for all the joints are smooth and satisfy the boundary constraints, i.e., the initial and final joint velocities are equal to zero. To demonstrate the advantages of the proposed manipulability measure \(\Omega_{MM}\), a comparison of the manipulability maximization results using different objective functions in simulations of the Lissajous trajectory tracking is shown in Figure 9. Here, a task is considered as failed when none of the constraints are satisfied, and thus the simulation is stopped. These results were obtained by using the same parameters shown in Section 5.3 with the only change being the objective function. The Lissajous trajectory tracking fails when the objective function is set to maximize the manipulability of the arm, i.e., \(F(\mathbf{q})=\widehat{\Omega}_{a}\), as shown in Figure 9a. The manipulabilities of both the arm and the whole system start a fast decay after \(t=44s\). This decrement in the manipulabilities is the result of restricting the homogeneous solution to respect the joint velocity limits. Consequently, the arm manipulability is not maximized anymore and the system moves towards singularity, ultimately failing the task. The results of setting the objective function to the manipulability of the whole system, i.e., \(F(\mathbf{q})=\widehat{\Omega}_{p+a}\), are shown in Figure 9b. In this case, the task succeeds, however, the manipulability of the arm  
Figure 7: Tracking performance comparison between simulations and experiments for the Lissajous trajectory. End-effector’s trajectory in simulations (**a**) and experiments (**d**). Position errors in simulations (**b**) and experiments (**e**). Orientation errors in simulations (**c**) and experiments (**f**). Notice that the position and orientation errors, panels (**b**,**c**), respectively, are small in the simulations. In addition, notice that the trajectories in simulations (**a**) and experiments (**b**) are quite similar. The reason for the larger errors obtained in the experiments (panel (**e**,**f**)) are discussed in Section 5.4.  
de deteriorates, and at the end of the trajectory has a value close to zero. This is the behavior discussed in previous studies [6; 10]. Figure 9c shows the results of maximizing a linear combination of the manipulabilities of the arm and the whole system, i.e., \(F(\mathbf{q})=0.5\hat{\Omega}_{p+a}+0.5\hat{\Omega}_{a}\). Figure 9d shows the results of maximizing the proposed manipulability measure for mobile manipulators, i.e., \(F(\mathbf{q})=\Omega_{MM}\) from Equation (15). Both objective functions preserve the manipulability of the arm. However, it is clear that maximizing the proposed measure has better results overall. First, the manipulability of the arm is higher along the trajectory in Figure 9d compared to the arm manipulability obtained with the linear combination in Figure 9c. Secondly, the obtained final manipulabilities have improved with respect to the initial ones in Figure 9d. On the other hand, the linear combination improves the manipulability of the whole system, but not that of the arm with respect to the values at the start of the trajectory, as shown in Figure 9c. Figure 8: Experiment results for the Lissajous trajectory tracking: (**a**) mobile platform’s trajectory; (**b**) normalized manipulability measures; (**c**) self-collision distances; (**d**) mobile platform’s velocity commands; (**e**) prismatic joint’s position and velocity; (**f**–**h**) arm’s joint positions; and (**i**) arm’s joint velocities. Notice that all the joint limits and velocity limits are respected (panels (**d**–**i**)). Furthermore, the manipulabilities of both the arm and the whole system are improved (panel (**b**)). See Section 5.4 for a detailed discussion of this figure.