## 5 Simulation Setup
### Reachable and Dexterous Workspace of Two-Hybrid Systems  
In the spherical wrist model of the robot arm, three rotational axes represent the pitch, roll, and yaw of the end-effector independently. As shown in Figure 12, these three axes intersect at point \(H\). Point \(P\) is the location of the end-effector, and it is the place where the camera or tool is mounted. For the elbow manipulator mounted with the stereo camera, the camera is placed so that its optical center line coincides with axis C. The location of camera optical center C is determined, and a good estimation of it can be found in this  
Figure 11: The topology of the multi-robotic system for accurate positioning control. paper [36]. For now, assume the optical center is in the middle of \(\overline{PJ}\) (set \(\overline{PC}=17\) mm). For the elbow manipulator mounted with the tool, the screwdriver is attached at point \(P\). Point J indicates the far end of the object (tool or camera) attached to the end-effector. Point H is kept stationary no matter which axis rotates. Therefore, the task of positioning and rotation is decoupled in the spherical wrist robot model. The idea of reachable and dexterous workspaces of an ideal elbow manipulator has been introduced in Ref. [37]. An ideal elbow manipulator is a manipulator whose angles of rotation are free to move in the whole operational range \([0,2\pi]\). However, a realistic elbow manipulator is limited to moving its joints within certain ranges of angle. The workspace of an ideal elbow manipulator (Figure 13) is a sphere centered at joint 1 of the manipulator, denoted as point \(o\). The reachable workspace concerning the center o of a manipulator is the aggregate of all possible locations of point \(J\) attached to the end-effector and denoted as \(W_{o}(J)\). The dexterous workspace to the center o of a manipulator is the aggregate of all possible locations that point \(J\) can reach all possible orientations of the end-effector and is denoted as \(W_{o}^{d}(J)\). The reachable workspace of \(H\) is denoted as \(W_{o}(H)\). It can be shown in the figure:  
\[W_{o}^{d}(J)\subseteq W_{o}(H)\subseteq W_{o}(J). \tag{50}\]  
Figure 12: Spherical wrist model. Figure 13: Workspace of an ideal elbow manipulator. For an ideal elbow manipulator, the radiuses of workspaces are expressed below:  
\[\text{Radius of }W_{o}(H)=L_{2}+\sqrt{{L_{3}}^{2}+{L_{4}}^{2}} \tag{51}\]  
\[\text{Radius of }W_{o}(J)=L_{2}+\sqrt{{L_{3}}^{2}+{L_{4}}^{2}}+L_{t}+\overline{PJ} \tag{52}\]  
\[\text{Radius of}W_{o}^{d}(J)=L_{2}+\sqrt{{L_{3}}^{2}+{L_{4}}^{2}}-L_{t}- \overline{PJ} \tag{53}\]  
where \(L_{2}\), \(L_{3}\), and \(L_{4}\) are link lengths of the manipulator. \(L_{t}\) is the length of the end-effector, and \(\overline{PJ}\) is the length of the object that mounts on the end-effector. As the camera and the tool must be able to rotate in all 3 DoFs when they are at the target position, the dexterous workspace is used as the working space of the visual system and the tool manipulation system. The dexterous workspace of the optical center is used for the visual system, and the dexterous workspace of the far endpoint of the tool is used for the tool manipulator system. \[\text{Radius of }W_{o}^{d}(Visual)=r_{V}=L_{2}+\sqrt{{L_{3}}^{2}+{L_{4}}^{2}}-L_ {t}-\overline{PC}_{camera}=1.716\text{ m} \tag{54}\]  
(From Specification, Appendix A, Table A1.) \[\text{Radius of }W_{o}^{d}(Tool)=r_{T}=L_{2}+\sqrt{{L_{3}}^{2}+{L_{4}}^{2}}-L_{t}- \overline{PJ}_{tool}=1.606\text{ m} \tag{55}\]  
(From Specification, Appendix A, Table A1.) Figure 14 shows the workspace setup for the hybrid system. The dexterous space of the robot arm with the camera is a sphere with a radius of \(r_{V}\), and its sphere center is \(O_{V}\), and the dexterous space of the robot arm with tools is a sphere with a radius \(r_{T}\), and the center of this sphere is \(O_{T}\). Two systems intersect with the ground at an angle of \(\theta_{1}\) and \(\theta_{2}\).