## 5 Experiments  
Experiments were carried out to verify the efficacy of our scheme to solve the motion planning problem for trajectory tracking at the kinematic level. In this section, the results for the tracking of two trajectories, a Lissajous trajectory (see Section 5.4), and an elliptic trajectory (see Section 5.5), are analyzed. These trajectories were picked to demonstrate the ability of our approach to comply with the different constraints introduced in this manuscript while improving the manipulabilities of the whole system and the robot arm. For the Lissajous trajectory, a comparison of different objective functions for manipulability maximization is made to highlight the advantages of the proposed manipulability measure for mobile manipulators. The experiments were performed using a 10-DOF NMM developed by the Industrial Technology Research Institute (ITRI), as shown in Figure 4. This NMM is composed of a differential-drive wheeled mobile platform, a prismatic joint mounted on top of the platform, and a Universal Robots UR5 6-DOF robotic arm attached to the prismatic joint. From this point forward, the UR5 arm's joints are denoted as \(q_{ai}\) with \(i=1,2,\ldots,6\) and the prismatic joint as \(z_{pj}\). The respective Denavit-Hartenberg (DH) parameters are shown in Table 1. The joint limits and joint velocity limits are shown in Table 2.  
Figure 4: Nonholonomic mobile manipulator used for the experiments. The Jacobian of the arm \(J_{a}\) used for calculation of \(\hat{\Omega}_{a}\) in (15) was constructed only using the UR5 arm without the prismatic joint. If the prismatic joint were included, the arm would be allowed to stretch for most tasks since it would always be able to move the end-effector vertically using the prismatic joint, i.e., the manipulability is not affected when the arm is horizontally stretched. Stretching the arm for most tasks is an undesirable behavior, and therefore, the manipulability maximization of the UR5 arm without the prismatic joint is a suitable selection. The Jacobian \(J_{a}\) is calculated with respect to the frame \(X_{a}Y_{a}Z_{a}\) shown in Figure 4.  
Due to the lack of a reliable positioning system, the odometry of the wheels was used in the experiments to compute the position and orientation of the mobile platform, and the forward kinematics were used to compute the end-effector's pose, which in turn was fed back to the motion planning algorithm. Therefore, the errors presented in the experiments are not the real-world errors, but the errors in the trajectory in which it is assumed that a reliable positioning system exists. Furthermore, as mentioned in the introduction section, the scope of this work is the motion planning of NMM for trajectory tracking at the kinematic level; therefore, the objective of the presented experiments is to validate the proposed algorithm at the kinematic level. Problems inherent to the dynamic behavior of the system, including vibrations of the mechanical structure, friction from the ground, etc., have an impact on the real end-effector tracking error, but they are out of scope of this paper and not considered in the algorithm. For these reasons, the simulation results of the position and orientation errors along the trajectory are also shown in this manuscript to highlight the performance of our motion planning algorithm. An important remark is that the system vibrations due to the NMM mechanical structure greatly affected the performance of the motion planning algorithm when the system was close to its joint limits or self-collision constraints. This behavior was not observed when performing the simulations. To address this issue, a moving average filter with a window size of five was used to filter the gradients of the criterion functions for joint limits avoidance and self-collision avoidance. This filter diminished the impact of such vibrations on the motion planning algorithm.