## 6 Discussion  
A scheme was proposed to solve the motion planning of NMMs considering joint limits, joint velocity limits, joint velocity boundary constraints, and self-collision avoidance while maximizing the manipulabilities of both the robot arm and the whole system. The proposed solution uses a weighted input velocity vector and a weighted Jacobian to penalize the movement of joints that get close to a position constraint. A proposed quadratic cost function is minimized when solving the motion planning problem for redundant NMMs. This cost function includes a secondary task to be satisfied that is also weighted to comply with the position constraints. The maximization of an introduced manipulability measure for mobile manipulators is used as the secondary task to push the system away from singularities. In the experiments section, it is demonstrated that maximizing this new measure simultaneously improves the manipulability of the whole system and that of the manipulator alone. This work focuses on the motion planning for trajectory tracking at the kinematic level, which must not only comply with joint position constraints, but must also respect joint velocity constraints and joint velocity boundary constraints. Joint velocity boundary  
Figure 12: Experiment results for the elliptic trajectory tracking: (**a**) mobile platform’s trajectory; (**b**) normalized manipulability measures; (**c**) self-collision distance; (**d**) mobile platform’s velocity commands; (**e**) prismatic joint’s position and velocity; (**f**-**h**) arm’s joint positions; (**i**) arm’s joint velocities. Notice that all the joint limits and velocity limits are respected (panels (**d**-**i**)). The potential self-collisions described in Section 5.2 are prevented (panel (**c**)). Furthermore, the manipulabilities of both the arm and the whole system are improved (panel (**b**)). See Section 5.5 for a detailed discussion of this figure. constraints are satisfied by varying the magnitude of the homogeneous solution at the start and end of the trajectory. The manipulability maximization at these points is not necessarily zero; hence, using an adequate variation in the magnitude of self-motion is needed. Joint velocity limits are satisfied by evaluating the maximum allowable self-motion for each joint. Using this information, the step size of the gradient ascent/descent is limited when required, and consequently, the joint velocity limits are not exceeded. The experiments for 6-DOF trajectories were conducted to verify the efficacy of the proposed scheme. The results demonstrate that the proposed approach can solve the motion planning problem for NMMs to perform trajectory tracking at the kinematic level while considering the constraints required for real implementation including manipulation capability preservation or improvement. The experiments designed in Section 5 consider an open environment without obstacles, because this is the scope of our manuscript. However, the proposed solution can be extended to prevent collisions with obstacles by including collision pairs between the robot arm and these obstacles, using the same definitions as in Section 4.2.2. This will penalize the movement of the arm's joints that get closer to an obstacle in the environment. Nevertheless, in the case of the platform, stopping it is not an efficient approach. In this case, an additional task can be added to the solution to push the platform away from the obstacles. One way to achieve this is by using a task priority scheme [24] using the nullspace of the motion planning algorithm. Even though our work focuses on NMMs, our approach can be effortlessly adapted for use with holonomic mobile manipulators. Future work will focus on dynamic modeling and controller design to deal with system vibrations and tire slip such that the tracking errors of the system can be reduced. Conceptualization, methodology, J.L. and T.H. ; software, validation, visualization, writing--original draft preparation, J.L.