# Design and Experimentation of Tensegrity Jumping Robots
## 3 Dynamic Modeling and Simulations Analysis
### Simulations Analysis  
In this section, dynamic simulations of the robot are carried out to verify the feasibility of robot design and performance analysis. The simulation of simultaneous jumping performance for different parameter cases was tested, as shown in Figure 7.  
Figure 6: Dual module step-up jump. (**a**) Jump height and jump distance at four moments. (**b**) Choice of jumping moments; (**c**) Schematic diagram of jumps at 8\({}^{\circ}\) and 16 N/mm.  
The maximum jump height obtained from our simulation is 612.40 mm and the maximum jump distance is 53.94 mm. The comparison shows that the jumping performance of the robot increases as the stiffness of the spring increases for the same case. This is due to the fact that the higher the stiffness, the more energy the robot stores. The higher the tilt angle of the robot, the jumping performance of the robot decreases but the distance traveled to the side increases.  
During the step-up jumping performed by the tensegrity jumping robot, it is theoretically known that differing sequences of energy release and the choice of specific energy release moments will affect the jumping results of the robot. We tested several groups of jumping performance of energy release methods with different stiffness cases. The two jumping methods in each group had the same conditions except for the jumping mode. As shown in Figure 8, the first way is to release the energy of the module below the robot first and then release the energy of the module above the robot after the robot is airborne; as shown in Figure 8, the second way is the opposite. Taking the release time of \(\Delta t_{3}\) as an example, the simulation results are shown in Figure 8. It can be found that the energy utilization of the first way is better than the second one. When the stiffness of elastic cable is 28 N/m, the jumping height of the two ways are 276.15 mm and 297.36 mm, respectively.  
Figure 8: Release order and time. (**a**) Method 1; (**b**) Method 2; (**c**) Release order; (**d**) Release time.  
Figure 7: Simultaneous jump simulation. (**a**) Vertical jump height. (**b**) Horizontal jump distance.  
Secondly, the choice of energy release moment in the second module also affects the robot's jump result parameters. As shown in Figure 8, taking the first way for releasing the energy as an example, eight moments are selected to analyze the efficiency of energy release under different moment choices. We can see that the robot energy utilization efficiency shows a trend of increasing and then decreasing as time moves backward. When the release time is 0.5 s, the jumping height is 73.15 mm, and when the release time is 2.1 s, the jumping height is 150.76 mm.  
On the basis of the above analysis, we tested the jumping height and diagonal jumping distance of the tensegrity jumping robot when the robot is tilted at different angles for two-module step-up jumping within a certain stiffness range. Taking the release time of 1.6s and the first way for releasing the energy as an example, the jumping performance of the robot is shown in Figure 9.  
The maximum jump height obtained from our simulation is 712.16 mm and the maximum jump distance is 63.67 mm. According to Figure 9, we can see that when other conditions are the same, the larger the stiffness is, the robot's jumping height and jumping distance will increase. The larger the tilt angle is, the robot's jumping performance in the vertical direction will be reduced accordingly and the jumping distance in the horizontal direction will be increased.