# Design and Analysis of a Novel Variable Stiffness Joint Based on Leaf Springs
## 4 Dynamics Modeling and Simulation Analysis
### Results of Dynamic Modeling  
In order to examine the stiffness response performance and position tracking performance of the designed VSJ, step and sine tracking signals were designed to carry out motion simulation, based on the above controller (Figure 15). The step response of the output position of the main drive module, at the stiffness level of 200 Nm/rad, is illustrated in Figure 16. In the step response, the no-load tracking response is relatively fast, while, as the load increases, the settling time is significantly longer, having been extended by 0.25 s, whereas the overshoot also increases by 5%. In the large stroke step response, the settling time of the main drive module output is slightly increased, extended by 0.16 s, and the overshoot is also increased by 15%. As the load increases, the overshoot and settling time also increase significantly, indicating that more advanced control algorithms need to be investigated, allowing the VSJ to cope with load changes in such working conditions.  
The response curve of the lever position in the VSJ is illustrated in Figure 17. Under both no-load and load conditions, the response speed of the VSJ remains almost the same. In the large-stroke step response, as the load increases, the overshoot of the lever position increases slightly, meaning it can be ignored, indicating that its elastic deflection angle has almost no effect on the lever position. This further indicates that the coupling between the elastic deflection and actuator stiffness is slight, which is consistent with the above simulation results shown in Figure 3.  
The synchronous tracking performance of the output frame under a sine signal with a frequency of 1 Hz and an amplitude of 30\({}^{\circ}\) is demonstrated in Figure 18, where the stiffness of the VSJ is fixed at 200 Nm/rad. The output position tracking performance and tracking error are shown in Figure 18a,b, respectively. The maximum tracking error under load and no-load conditions occurs in the initial motion stage, which is about 13\({}^{\circ}\) and 11\({}^{\circ}\), respectively; this is largely due to the serial elastic stage in the VSJ. After the system gradually stabilizes, the tracking error under no-load and load conditions converges to 6\({}^{\circ}\) and 4\({}^{\circ}\), respectively. Clearly, the load has a certain deterioration effect on the tracking performance. In addition, in order to verify the anti-interference performance of the synchronous variable stiffness under the no-load condition, the VSM also tracks a sine signal with a frequency of 1 Hz and an amplitude of 200 Nm/rad, as shown in Figure 18c.  
Figure 16: Step response of the main drive module output position.  
Figure 15: PID controller block diagram.  
The maximum tracking error is about 10 Nm/rad. Under the load condition, the maximum tracking error is about 16 Nm/rad. The results emphasize that the load has a very small impact on the joint stiffness.  
Figure 17: Step response of the lever position.  
Figure 18: Trajectory tracking performance: (**a**) Output position; (**b**) Position tracking error; (**c**) Stiffness tracking results.