# Motion Planning for a Mobile Humanoid Manipulator Working in an Industrial Environment
## 2 Materials and Methods  
### Model of the System  
In the paper, a holonomic mobile humanoid manipulator, working in an industrial environment, was considered. The robot was inspired by the Rollin' Justin robot [19], developed by the German Aerospace Center (DLR), which is made up of a humanoid dual-arm upper body, mounted on a mobile platform. In contrast to the original design, the robot proposed in this work was equipped with a holonomic \((3,0)\)-type platform with two pairs of Mecanum wheels and the arms with four DOF. The model of the humanoid manipulator on its holonomic platform is shown in Figure 1. As can be seen, the robot consists of four main components:  
* The 3 DOF holonomic platform, which allows any position and orientation to be achieved in the \(XY\)-plane of the workspace;
* The 3 DOF torso with three revolute joints and an additional, passive revolute joint, placed at the end of the kinematic chain, ensuring that the chest is always kept orthogonally to the ground;
* Two arms, each with 4 DOF. In the work here presented, and for the sake of simplicity and in order to increase the readability of the presentation, the orientation of the end-effectors was not taken into account. Such an assumption does not affect the generality of the presented approach because the proposed method did not limit the number of arm joints, so it is possible to complement the arms with additional 3 DOF grippers. The problem with planning a grasping action is beyond the scope of this work; however, it is the subject of ongoing research, and the results presented in such as [23] can be utilized when considering the humanoid manipulator in this paper. It was assumed that the humanoid robot consisted of a series of rigid bodies connected by rigid joints. In this case, the location of each link with respect to a previous link can be easily described by the location, i.e., the position and orientation of the local coordinate frame, attached rigidly to the chosen link, relative to the local coordinate frame attached to the previous link. In order to describe the position of the two end-effectors, i.e., the left and right arms of the robot, in the workspace, modified Denavit-Hartenberg (DH) notation [24] was used, and the parameters of the robot joints and links were determined and are shown in Table 1.  
Figure 1: Model of the humanoid manipulator. \(l_{T_{1}},l_{T_{2}},l_{T_{3}},l_{L_{1}},l_{L_{2}},l_{R_{1}},l_{R_{2}}\) are the lengths of the humanoid links for the torso and the left and right arms, respectively. \(q_{T_{1}},q_{T_{2}},q_{T_{3}}\) are the joints angles of the torso. \(q_{T}^{\star}=pi/2-q_{T_{2}}-q_{T_{3}}\) is the joint angle of the last joint of the torso and is used to compensate the rotations of the two previous joints. \(q_{L_{1}}\dots q_{L_{\
u}},q_{R_{1}}\dots q_{R_{4}}\) are the joint angles of the left and right arms, respectively. The rotation axes of the independent joints of the torso (i.e., the axes for the first, second, and third joints) and all joint axes of left arm are shown in Figure 1; the rotation axes of the right arm are similar. As shown, the arms are located at the end of the torso, and their bases are rotated by an angle of \(\beta\). The grippers are located at distances \(l_{L_{3}}\) and \(l_{R_{3}}\) from the fourth joints of the left and right arm, respectively. The position and orientation of the holonomic platform is uniquely defined by three parameters: \(x\) and \(y\) determine the platform center location in the \(XY\) plane, and \(\theta\) describes the platform orientation.