## 5 Simulations
### Example 2 and Result Discussion  
In the second example, a cuboid is loaded onto a step by using mobile robots, as shown in Figure 13. 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 floor is denoted as \(F_{1}\) and the faces consisting of the step are denoted as \(F_{2}\) and \(F_{3}\). \begin{table}
\begin{tabular}{l l} \hline \hline
**Image** & **ECF and RCF in the Mode** \\ \hline \multirow{4}{*}{\(C_{1}^{\mathrm{E}}=\varnothing\)} & \(C_{1}^{\mathrm{E}}=\{(f_{6},F_{1})^{\mathrm{e,dn}},(f_{1},F_{2})^{\mathrm{e,dn} },(f_{2},F_{3})^{\mathrm{e,st}}\}\) \\  & \(C_{1}^{\mathrm{E}}=\varnothing\) \\  & \(\Downarrow\quad l(s_{1},s_{2})=1\) \\  & \(C_{2}^{\mathrm{E}}=\{(f_{6},F_{1})^{\mathrm{e,dn}},(f_{1},F_{2})^{\mathrm{e,dn }},(f_{2},F_{3})^{\mathrm{e,st}}\}\) \\  & \(\Downarrow\quad l(s_{2},s_{3})=1\) \\  & \(C_{3}^{\mathrm{E}}=\{(f_{1},F_{2})^{\mathrm{e,dn}},(e_{5},F_{3})^{\mathrm{e,st }}\}\) \\  & \(C_{3}^{\mathrm{E}}=\{(f_{3},F_{1})^{\mathrm{x,ac}}\}\) \\  & \(\Downarrow\quad l(s_{3},s_{4})=1\) \\  & \(C_{4}^{\mathrm{E}}=\{(f_{1},F_{2})^{\mathrm{e,dn}},(e_{5},F_{3})^{\mathrm{e, st}}\}\) \\  & \(\Downarrow\quad l(s_{4},s_{5})=1\) \\  & \(\Downarrow\quad l(s_{4},s_{5})=1\) \\  & \(\Downarrow\quad l(s_{4},s_{5})=1\) \\  & \(\Downarrow\quad l(s_{4},s_{5})=1\) \\  & \(\Downarrow\quad l(s_{4},s_{5})=1\) \\  & \(\Downarrow\quad c_{5}^{\mathrm{E}}=\{(v_{3},F_{3})^{\mathrm{e,st}}\}\) \\  & \(C_{5}^{\mathrm{E}}=\{(f_{3},F_{1})^{\mathrm{x,ac}},(f_{2},\tau_{7})^{\mathrm{ eps}}\}\) \\  & \(\Downarrow\quad l(s_{5},s_{6})=1\) \\  & \(\Downarrow\quad l(s_{5},s_{6})=1\) \\  & \(\Downarrow\quad l(s_{6},s_{7})=1\) \\  & \(\Downarrow\quad l(s_{6},s_{7})=1\) \\  & \(\Downarrow\quad l(s_{6},s_{7})=\varnothing\) \\  & \(C_{7}^{\mathrm{E}}=\varnothing\) \\  & \(C_{7}^{\mathrm{E}}=\{(f_{3},r_{1})^{\mathrm{x,ac}},(f_{2},\tau_{7})^{\mathrm{ eps}},(f_{2},\kappa)^{\mathrm{EPS}}\}\) \\  & \(\Downarrow\quad l(s_{7},s_{8})=1\) \\  & \(\Downarrow\quad l(s_{7},s_{8})=1\) \\  & \(C_{8}^{\mathrm{E}}=\varnothing\) \\  & \(C_{8}^{\mathrm{E}}=\{(f_{3},r_{1})^{\mathrm{x,ac}},(f_{2},\tau_{7})^{\mathrm{ eps}},(f_{2},\kappa)^{\mathrm{EPS}},\\ \(f_{2},r_{9})^{\mathrm{eps}}\}\) \\  & \(\Downarrow\quad l(s_{8},s_{\mathrm{C}})=1\) \\  & \(\Downarrow\quad c_{6}^{\mathrm{E}}=\varnothing\) \\  & \(C_{6}^{\mathrm{E}}=\{(f_{2},r_{7})^{\mathrm{EPS}},(f_{2},r_{8})^{\mathrm{EPS}},( f_{2},r_{9})^{\mathrm{EPS}}\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: One of the manipulation paths that minimized the cost. Figure 13: (**a**) Initial mode and (**b**) target final mode. Targeted object for manipulation is cuboid. There is a step in the environment. We used six pusher robots to bring the cuboid on the step. A mobile robot with a standard six-axis manipulator is shown in Figure 11d. The robot generates a frictional point of contact with the object. We used three robots and denoted their geometrical primitives as \(r_{1},r_{2}\), and \(r_{3}\). There were no obstacles around the step; therefore, the manipulation could be performed by either using or not using the environment. By using the proposed method, we generated the modes with the necessary number of robots to achieve gravity closure. Then, in the created mode graph, we took the initial state as \(s_{1}=\{\{(f_{2},F_{1})^{\mathrm{e,st}}\},\varnothing\}\) and the final state as \(s_{\mathrm{G}}=\{\{(f_{2},F_{3})^{\mathrm{e,st}}\},\varnothing\}\) and searched for the manipulation paths with the minimum total cost. Using the procedure given in the previous example, we generated 292,415 modes and \(2.9\times 10^{6}\) transitions. The total number of combinations of the ECFs and RCFs was 299,967, and the total number of combinations of the generated modes was \(4.5\times 10^{10}\), as shown in Table 3. For the same reason we analyzed in Example 1, our method also narrowed down the number of possible modes, and especially, dramatically decreased the transitions between them.