## 5 Simulations
### Example 1 and Result Discussion
(**c**) Schematic of a transporter robot. (**d**) Mobile robot with a six-axis manipulator. Figure 12: (**a**) Initial mode and (**b**) Target final mode. The targeted object for manipulation is a cuboid. There are two walls and the floor as the environment. We used six pusher robots and three transporter robots. protocol reduced the number of modes. The reduction in the total number of modes was not dramatical because by considering all the possible contact between the robots and the object's geometrical primitives, there existed a huge amount of RCFs. For each ECF, our method eliminated the unavailable robot combinations. However, for most of the ECFs, the environmental constraints enabled most of the robot combinations to provide sufficient constraints to achieve the gravity closure, so the reduction of the candidates of robot combination was not really much. Further, after considering the possible geometrical primitives of the object that made contact with the robots, this reduction became less significant. The mode transition graph was generated from the above modes by using the rules defined in the previous section. Also, as shown in Table 1, the resultant number of transitions was \(8.6\times 10^{9}\), where we can see that the proposed rules significantly reduced the candidates of transition between modes, which originally would be \(1.4\times 10^{16}\). We can see a dramatic reduction in the number of mode transitions here because one contact states can only change to another one under the constraints of geometrical boundary relationships and under the rules that the robot contact changes one by one. With such geometrical boundary relationships and the rules to change the robot contacts considered, a mode has only very few or even no transitable modes, whereas all the other modes will be its transitable modes and a large number of meaningless transitions will be caused if our method is not adopted. Finally, by searching the mode graph, a total of 27,216 paths which minimized the cost were obtained in the graph. Table 2 shows one of the obtained paths with the lowest cost in the mode graph. As mentioned above, the paths in the modes were decided without planning the exact configuration of robots. Therefore, the configurations of the object and the robots shown in Table 2 are examples of the modes. In the path shown in Table 2, the object was pushed by a pusher robot and rotated about the object's edge \(e_{5}\) alongside the wall \(F_{1}\). A transporter robot was then inserted underneath it. The pusher robot then moved the object so that only the object's vertex \(v_{3}\) was in contact with the floor \(F_{3}\), and another transporter robot was inserted underneath the object. Finally, with two transporter robots at the bottom, the object lost contact with the floor \(F_{3}\) through contact with the pusher robot, which allowed the third transporter robot to enter underneath the object. Thus, the object was finally loaded onto three transporter robots. We can see that the obtained path is reasonable. From the viewpoint of statics and geometry, it was also reasonable to include the paths in which only the combinations of geometrical primitives were different. The resultant paths also contained those paths that were impossible to realize from the viewpoint of the force balance determined by the configurations of the object and the robots. Those paths will be omitted in the later planning phase when deciding the object-robot system's exact configurations based on the transitions of the modes; they will not be dealt with in the current phase where only the transitions of the modes are decided. The important result is that the candidates of the mode transposition were narrowed down, as shown Table 1, and this facilitated the planning to determine the configurations.