# Motion Planning for a Mobile Humanoid Manipulator Working in an Industrial Environment
## 4 Discussion  
The manner in which a humanoid manipulator completed a task is presented in Figures 5-8 and in the supplementary animation; however, the obtained results require further detailed analysis. In Figure 9, the norm of robot error \(\|\mathbf{e}(t)\|\) and object error \(\|\epsilon(t)\|\) given by the dependencies (15), (25), and (21), (28) are shown. The vertical lines, in the figure, separate individual phases in the completion of the task. As can be seen, according to the properties of the proposed solution, in the first phase of the task, when the robot performed the motion to the given fixed end-effectors positions, the initial error value was large, and it tended to zero, so at the end of this phase, the grip can be made. In the subsequent phases of the motion, the positions of the end-effectors in relation to the contact points \(\mathbf{p}_{C}\) were constant, so the error was equal to zero. The transportation of the object began in the second phase of the task, and the object was moved to the initial point of the path, which was traced in the last phase. In this case, a high precision of positioning at this point was not necessary, so this phase can be completed with a relatively large error (as can be seen in supplementary animation, the beam did not precisely reach the desired orientation). This error was reduced during the third phase of the task, and finally, the beam was placed precisely on the conveyor. To demonstrate the effectiveness of the obstacle avoidance approach, the distances of the robot and the transported beam from the obstacles in the workspace were calculated and are presented in Figures 10 and 11. As can be seen, at the beginning of the first phase, the robot was close to the CNC machine and control panel. After about three seconds, needed in order to bypass the panel, it approached the container in order to grip the beam. This goal was reached at about the 13 th second. The beam was gripped, and the second phase of the task then began. In this phase, the robot lifted the beam, avoiding a collision with the container (see Figure 11), and moved it towards the roller conveyor. This phase ended when the beam was approximately 0.5 m above the conveyor. Finally, in the last phase, the robot placed the beam on the conveyor, so the distances of the robot end-effectors and the object from the conveyor tended to zero (see the third part in Figures 10 and 11). As can be seen, the distances from the obstacles during whole motion were greater than zero, so the trajectory, determined according to the proposed approach, was collision-free. Figure 9: Norms of robot and object errors. The values of the joint angles, obtained for the torso and the left and right arm, were mean normalized, i.e., scaled to a \(-0.5\) to \(0.5\) range, in order to fit in the same range, and are presented in Figures 12-14, respectively. As can be seen, according to the assumed form of mapping \(D_{M}(\mathbf{q})\), all values remained as close as possible to the middle of the ranges permitted. The values close to the limitations resulted from the given initial configuration or the requirements of the task, such as obstacle avoidance or reaching a given position and orientation. Figure 11: The distance of the transported beam from the obstacles. Figure 10: The distance of the robot from the obstacles. Figure 14: Changes of the configuration variables for the right arm. Figure 12: Changes of the configuration variables for the torso. Figure 13: Changes of the configuration variables for the left arm. As was shown in Section 2.4, in addition to the fulfillment of the mechanical and collision avoidance constraints, the maximization of the manipulability measure of the humanoid arms was considered as a secondary objective. Changes of this measure are shown in Figure 15. As can be seen, the manipulability measure at the beginning of the motion was relatively small; however, during the task execution, the effect of the secondary objective term could be seen, and the values of this measure increased.