# Jumping with Expandable Trunk of a Metamorphic Quadruped Robot--The Origaker II
## 5 Contrast Simulations of Jumping Motion of the Origaker II with and without Expandable Trunk  
A CAD model was developed for the Origaker II and a simulation was carried out in Matlab and Adams with this model. In Adams, all the 15 degrees of freedom (DOF) of the trunk and four legs are under control by the motion splines designed in Section 3. The friction between foot tips of the robot and the ground is set to Coulomb friction. The static and dynamic coefficients are 0.2 and 0.1, respectively. Figure 8 shows the simulation in Adams, indicating that the Origaker II can successfully perform jumping motion. There is a simulation video of the jumping process in the Supplementary Materials. Figure 7: The change of the trunk work with the robot trunk angle. These pictures show the process of jumping motion designed in Section 3. Firstly, the Origaker II contracts its trunk and legs into a ready posture. Then, this robot expands its front legs and trunk to adjust the direction of jumping. Next, the robot completely expands its trunk and hind legs, and jumps into the air. The system is non-holonomic in the flying mode but stable enough with friction performing little impacts on the results. Finally, it drops to the ground and slides a short distance. The motion after the robot drops on the ground is not designed. So the robot stays the posture in the air and slides a short distance to consume the remaining kinetic energy. This robot can correctly achieve the jumping motion of the frog. The contact forces between the ground and four legs are shown in Figure 9. The robot is over the ground at first. Then the robot lands on the ground firstly emerged that the peaks of the contact forces before the time \(t=0.5\) s. Because of the jumping motion of the robot, there are other peaks near to the time \(0.5\) s. The robot jumps into the air at those peaks. Finally, the robot lands on the ground and this brings the peaks next to the time \(1\) s. This figure illustrates the process of the contact forces between the ground and legs. Figure 8: The simulation of jumping motion in Adams. (**a**) The Origaker II is in the initial posture; (**b**) The front legs kick the ground; (**c**) The hind legs and the trunk expand; (**d**) The robot jumps into the air; (**e**) The robot falls on the ground. Figure 9: The contact forces between the ground and legs. (**a**) The contact force of the left front leg; (**b**) The contact force of the right front leg; (**c**) The contact force of the left hind leg; (**d**) The contact force of the right hind leg. To show more explicitly, a CAD model of the rigid trunk robot which is obtained by locking the joints of the expandable trunk has been developed. The robot without expandable trunk goes through the same footholds as the Origaker II. We also simulate its jumping motion in Adams, presented in Figure 10. This robot jumps in a similar way of the Origaker II. Nevertheless, there are different performances between the two robots. In order to reveal these differences, the positions of COG in \(y\)-axis and \(z\)-axis are investigated firstly. The coordinates along \(y\)-axis denote the jumping distance and the coordinates along \(z\)-axis represent the jumping height. Figure 11 points out the positions and velocities of the COG of the two robots from the PostProcessor in Adams. The red and blue lines indicate the positions of COG of the Origaker II and the rigid trunk robot (without the expandable trunk) respectively. The jumping performances of the two robots are similar in general tendency. As can be seen in Figure 10a, at the time \(t=1\) s when the robot lands on the ground, the jumping distance (excluding the sliding distance) of the Origaker II with expandable trunk is about 150 mm larger than that of the rigid trunk robot. It is further inferred that the Origaker II has a larger kinetic energy profited from the expandable trunk. This confirms that the trunk motion increases the jumping distance.