## 5 Simulation and Experimental Verification  
In this section, the designed fuzzy PID controller for the torsional vibration of the flexible-joint robot is validated by simulation and experiment. The simulation model of the flexible-joint robot system is constructed based on MATLAB R2015b/Simulink. In order to further verify the harmony between stability precision and rapidity of the designed fuzzy PID controller in torsional vibration suppression effect of the flexible-joint robot system, the conventional PID controller, whose parameters are determined by the trial and error method through simulation, is used for comparison and analysis. Setting the target speed of joint II as \(2000^{\circ}\)/s, the control effects of the two controllers are shown in Figure 8.  
It is seen from Figure 8b that when the joint flexibility of the joint robot transmission system is not considered, the rotational speed of joint II fluctuates obviously, and it takes a long time to synchronize with the rotational speed of joint I. The rotational speed of joint II can track the target speed without overshoot by the designed fuzzy PID controller, and the torsional vibration of the flexible-joint robot system is effectively suppressed. However, under the same stability time, the conventional PID  
Figure 8: Control effects of the designed fuzzy PID controller and the conventional PID controller on the torsional vibration of the flexible-joint robot system: (**a**) control voltages of the DC motor; (**b**) speed of joint II. controller will lead to a larger overshoot of the rotational speed of joint II, which is not allowed in the actual system. This is because that the parameters of the conventional PID controller are fixed in the control process, and it is difficult to achieve synchronous optimization for the overshoot, the response time and the static error. In addition, the fuzzy PID controller can realize the self-adjustment of the control parameters. In order to satisfy the response speed requirement of the control system in the initial stage, a larger proportional gain is adopted, and when the target speed is approached, the integral gain is increased to reduce the static error. On the other hand, Figure 8a shows the control voltages of the conventional PID controller and the designed fuzzy PID controller for the DC motor. It is found that the control voltage, obtained by the designed fuzzy PID controller, comes to stability faster, which indicates that its adjustment time is shorter. Then, the conclusion that the fuzzy PID controller has a better control effect than the conventional PID controller on the torsional vibration of the flexible-joint robot system is further proved. Finally, the dynamic quality and stability accuracy for the torsional vibration control of the flexible-joint robot are optimized synchronously. Furthermore, a step disturbance is added to the control system in 2 s for comparing the anti-interference performance of the designed controller and the results are shown in Figure 9a. It is known that the two controllers can effectively eliminate the interference and achieve stability. But the conventional PID controller fluctuates greatly and needs about 1.8 s to eliminate the step disturbance, while the fuzzy PID controller can suppress the step disturbance very well. The performances of the fuzzy PID controller are that the disturbance is small and the adjustment time is about 0.6 s.  
On the other hand, the white Gaussian noise, of which the height of the power spectral density is 0.4, is added to the control system in 2 s for further comparing the anti-interference performance of the designed controller and the results are shown in Figure 9b. It is found that the designed fuzzy PID controller have better anti-jamming capability for the white Gaussian noise than that of the conventional PID controller.