###### Abstract
[10] proposed formulating the motion planning problem as an optimization problem where the manipulability is maximized and the joint limits and joint velocity limits are included as constraints. This optimization problem is reformulated as a quadratic programming problem and converted into a linear variational inequality problem, that can be solved by different numerical methods. This approach is effective but does not consider boundary constraints for joint velocities. These constraints are also relevant because, for a given task, zero joint velocities are expected at the start and end of the trajectory. Additionally, NMMs are not only subject to physical limits but also to self-collisions, especially between the manipulator and the mobile platform, which are not included in their work. An important remark is that maximizing the manipulability of the whole system might result in poor manipulability for the arm alone, even though the manipulability for the whole system is preserved or improved [6; 10]. Additionally, it is important that both the robot arm subsystem and the whole mobile manipulator system maintain a certain level of manipulability once a coordinated task is completed. After completing an arbitrary path, a subsequent task might only require the arm to manipulate an object. If the arm is in a configuration with low manipulability, executing such a task might not be feasible. For these reasons, in this work, we propose a new manipulability measure for mobile manipulators that, when maximized, and as demonstrated in our experiments,intrinsically improves the manipulability of the robot arm as well as the manipulability of the whole system. The solution to the motion planning of NMMs for trajectory tracking presented in this work includes joint constraints (range and maximum velocities), self-collision avoidance and manipulation capability preservation. Figure 1 summarizes how all these constraints are included in our solution. Both the particular and homogeneous solutions of our proposed scheme are weighted to avoid joint limits and self-collisions while the trajectory is tracked. The homogeneous solution is used to maximize the manipulability by exploiting the redundancy of the system. To satisfy joint velocity boundary constraints, the step size for searching the maximum manipulability is varied. The joint velocity limits are satisfied by restraining the maximum step size based on the allowable self-motion. Experimental results demonstrate that our method can successfully solve the motion planning problem of NMMs under all the mentioned constraints. This work focuses on task space trajectory tracking at kinematic level. In other words, the outputs of the motion planning algorithm are the joint positions and velocities that will be fed to a joint space dynamic controller for motion control. Then, the motion controller is responsible for suppressing the model uncertainties and external disturbances to guarantee that the actual joint positions follow the ones output by the motion planning algorithm. In the experiments shown in this paper, we use the built-in motion controller of the commercial manipulator and leave the design of our own motion controller for future research. The contributions of this work are detailed as follows:  
* A motion planning solution for NMMs that allows to include joint physical constraints and the execution of a secondary task is presented. * Multiple constraints required for the practical implementation of task space trajectory tracking are included in a unified solution. These constraints include joint limits, joint velocity limits, joint velocity boundary constraints and self-collision avoidance. * A new manipulability measure for mobile manipulators is presented. It is demonstrated that the maximization of this measure simultaneously improves the manipulabilities of the whole system and the robot arm. This paper is organized as follows. Section 2 describes the kinematic modeling of NMMs.