## 5 Experiments
### Elliptic Trajectory  
The elliptic trajectory was picked to demonstrate the ability of the proposed approach to comply with joint velocity limits and to prevent self-collisions. This trajectory consists of moving the end-effector from an initial pose \(\mathbf{r_{0}}= \begin{bmatrix}\mathbf{P_{0}}&Q_{0}\end{bmatrix} ^{T}\) to a desired final pose \(\mathbf{r_{d}}= \begin{bmatrix}\mathbf{P_{d}}&Q_{d}\end{bmatrix} ^{T}\) using an elliptic path. The trajectory position is defined by  
\[\mathbf{P_{d}}(t)= \begin{bmatrix}A\cos(s(t))+c_{x}\\ B\sin(s(t))+c_{y}\\ m(s(t)-s_{0})+2c_{0}\end{bmatrix} ,\]  
where \(A,B,c_{x},c_{y}\) and \(s_{0}\) are calculated using the \(XY\) coordinates of \(\mathbf{P_{0}}\) and \(\mathbf{P_{d}}\), such that the elliptic path is centered at the closest point to the origin, while it covers a \(90^{\circ}\) angle between the start and end points. The trajectory's \(Z\) coordinate follows a straight line between the points \((z_{0},s_{0})\) and \((z_{d},s_{d})\), where \(z_{0}\) and \(z_{d}\) are the \(Z\) coordinates of \(\mathbf{P_{0}}\) and \(\mathbf{P_{d}}\), respectively; \(s_{0}\) and \(s_{d}\) are the starting and ending angles of the elliptic path, respectively; and \(m\) in the equation above is the slope of this line. A fifth-order polynomial profile [1] was used for the timing variable \(s(t)\). The orientations and rotational velocities along the trajectory were computed using quaternion polynomials [23]. This technique has two main advantages: a smooth velocity profile is obtained, and the rotational velocities and accelerations in the task space are included as boundary constraints. Figure 9: Lissajous trajectory’s comparison of objective functions for manipulability maximization in simulations: (**a**) maximization of the manipulability of the arm; (**b**) maximization of the manipulability of the whole system; (**c**) maximization of a linear combination of the manipulability of the arm and the whole system; and (**d**) maximization of the proposed mobile manipulator manipulability measure. This figure demonstrates the advantages of the manipulability measure for mobile manipulators (panel (**d**)) presented in Section 4.1. See Section 5.4 for a detailed discussion of this figure. The starting configuration of the mobile platform and prismatic joint for this trajectory was set to \(x_{p}=-1.3\) (m), \(y_{p}=0.56\) (m), \(\theta_{p}=0(^{\circ})\), and \(z_{pj}=0.24\) (m). With this configuration, the end-effector's initial pose is given by \(\mathbf{P_{0}}= \begin{bmatrix}-0.781&0.669&1.028\end{bmatrix} \) (m) and \(Q_{0}=\{0,0.7071i-0.7071j+0k\}\). The final pose was selected to have position \(\mathbf{P_{d}}= \begin{bmatrix}1.55&-1.0&0.26\end{bmatrix} \) (m) and orientation \(Q_{d}=\{0.2706,0.6533i+0.6533j-0.2706k\}\), and an execution time \(t_{f}=20(s)\) was chosen. Figure 10a,b show snapshots of the NMM's movement along the elliptic trajectory, in simulations and experiments, respectively. Figure 11 compares the trajectory tracking results between the simulation and the experiment. The small position and orientation errors obtained in the simulation (Figure 11b,c) again demonstrate the good tracking performance of the proposed approach. In the experiment, the position errors are kept below \(1.5\times 10^{-3}\) (m) and the orientation errors are kept below \(1\times 10^{-3}\) as depicted in Figure 11e,f. Once again, the vibrations of the system played a role in the larger errors seen in the experiment. Figure 10: Snapshots of the NMM’s motion for the elliptic trajectory tracking: (a) simulations; and (b) experiments. Figure 12 illustrates the experiment results. The manipulabilities of both the arm and the complete system are again improved at the end of the trajectory compared to the initial configuration, as shown in Figure 12b. However, we notice how both manipulabilities decreased during the time interval \(t=(8,15)(s)\).