## 5 Simulations
### Example 1 and Result Discussion  
In the first example, we used two types of mobile robots. The first type of robot was a pusher robot that we developed [25; 26; 27], as shown in Figure 11b. In this robot, we realized the safety manipulation that avoided the robot from falling; we did this by restricting the force that the robot could apply to the environment by using passive points. The pusher robot has a linear actuator. A face contact between the manipulator and an object acted as an active contact to change the mode of the robot-object system. The pusher robot could move to the target position by using wheels, but those wheels were lifted up and not used while the robot manipulated an object. Because of the special mechanism with passive joints, the kinematics of the robot (shown in Figure 9a) is given as explained in the previous section. See the details of the mechanism in our previous studies [25; 26; 27]. Another type of robot is a transporter robot, as shown in Figure 10(c). A transporter robot (see Figure 9b) has a structure similar to the pusher robot, but it does not have a linear actuator and has fewer passive joints (as explained in the previous section). The transporter robot can also move to the target position by using wheels but those wheels are not used until it starts transporting the object. As a result, a transporter robot supports an object passively during manipulation, and a face contact between the robot and an object acted as a passive contact. We assume that a pusher robot and a transporter robot act as an active contact and a passive contact, respectively, and these functions are not changed during manipulation for simplicity. In this manipulation, the task is to load a cuboid up to the transporter robots by using the pusher robots so that the object can be transferred away by the transporter robots. In the initial mode, the cuboid lies against a corner of the two adjacent walls, as shown in Figure 10(a). The target final mode is the object held by the three transporter robots, as shown in Figure 10(b). We consider the interactions between the cuboid and the environment when loading the cuboid up to the transporter robots by using the proposed method. As shown in Figure 12, the faces, the edges, and the vertices of the cuboid are denoted as \(f_{1},f_{2},\ldots,f_{6}\), \(e_{1},e_{2},\ldots,e_{12}\), and \(v_{1},v_{2},\ldots,v_{8}\), respectively. The walls are denoted as \(F_{1}\) and \(F_{2}\), and the floor is denoted as \(F_{3}\). For simplicity, we assume that the frictional coefficient of the walls is small enough to deal with the contacts on them as friction-less contacts. Therefore, the ECFs including \(F_{1}\) and \(F_{2}\) are always dynamic contacts. We used six pusher robots whose geometrical primitives were denoted as \(r_{7},r_{8},r_{9}\), as \(r_{1},r_{2},\ldots,r_{6}\) and three transporter robots whose geometrical primitives were denoted as \(r_{7},r_{8},r_{9}\). For simplicity, the transporter robots and the object always made contact at the bottom of the object. Therefore, \(r_{7},r_{8},r_{9}\) consisted of RCFs only with \(f_{2}\). The initial state is given as \(s_{1}=\{C_{1}^{\rm E},C_{1}^{\rm R}\}\), where \(C_{1}^{\rm E}=\{(f_{6},F_{1})^{\rm e,dn},(f_{2},F_{3})^{\rm e,st},(f_{1},F_{2} )^{\rm e,dn}\}\), and \(C_{1}^{\rm R}=\varnothing\). The final state is given as \(s_{\rm G}=\{C_{\rm G}^{\rm E},C_{\rm G}^{\rm R}\}\), where \(C_{\rm G}^{\rm E}=\varnothing\), and \(C_{\rm G}^{\rm R}=\{(f_{2},r_{7})^{\rm r,ps},(f_{2},r_{8})^{\rm r,ps},(f_{2},r_ {9})^{\rm r,ps}\}\). By applying the proposed method to generate the mode from the viewpoint of statics, 166,553,714 modes were generated, with the type and the corresponding number of robots determined according to the necessary condition to achieve gravity closure. As shown in Table 1, the total number of combinations between the possible ECFs and RCFs was 167,180,587, and we can see that proposed  
Figure 11: Mobile robots used in examples. (**a**) Pusher robot. (**b**) Schematic of a pusher robot.