## 6 Simulation Results
### Stability Analysis  
As mentioned, the designed fuzzy controller is a Mamdani type fuzzy controller, in which the development of fuzzy rule is based on the rules of the PD controller. In general, such a controller has good stability except perturbations. The input signals of the controller include the joint position errors e(t) and the joint velocity errors \(\dot{\rm e}(\rm t)\), and the output signal of the controller is the adjustment amounts of the joint driving forces/torques u(t) at each machining instant. A good fuzzy control rule system including the above-mentioned rules  
Figure 16: The simulation result of the fuzzy controller with initial trajectory error. will ensure that the calculation of the output value is consistent with the change of input signal and there is less possibility of overshoot. The numerical simulation results show this fact. On the other hand, the input signal that can greatly change the output quantity is the physical value domain of driving force. Numerical experiments show that when the physical value domain of the driving force is selected at a smaller level than the value to be reached, the control simulation results are not good, with large errors, which are understandable. When the domain of physical values is expanded, the errors decrease, and when the physical value domain is large enough to cover all the necessary values of the driving force, the control accuracy will be good, as shown by the simulation results. The next question will be whether increasing the input signal by expanding the domain of physical values without control, i.e., when the value domain is selected as too large, the value of the output driving force may greatly increase? The fuzzy Inference and Defuzzification show that the output physical value of a variable depends on the input physical value domain and the membership function value of such variable. As the physical value domain increases, the output value increases but non-linearly. Numerical experiments in the direction of gradually expanding the domain of input physical values so that the physical value of the output u(t) increases gradually to the appropriate value, we realize: The error of e(t), e(t) reduction; the membership value of linguistic variables corresponding to the large value of the output u(t) decreases; membership values of linguistic variables with small values increase. In the fuzzy set [NB, NS, Z, PS, PB], the membership value of u(t) corresponding to Z is fast progressing to 1, other values decrease and even the NB, PB values almost go to 0 in the first steps. Based on this and based on the fuzzy inference formula, it is easy to see that the growth rate of the physical value of the output u(t) decreases and the slope of the graph decreases. The slope of the graph gradually reaches 0, the graph is horizontal, then u(t) does not increase. Above that limit, if the physical value domain of the input u(t) continues to rise too high, the physical output value u(t) does not continue to increase, but the controller error will increase again. Figure 17 shows the membership function values corresponding to [NB, NS, Z, PS, PB] of the fuzzy set u(t), with the physical value domain of u(t) selected as in Table 6. The membership function value corresponding to NB, PB is almost zero, with Z close to 1. The membership function values corresponding to NS and PS of the physical value u(t) are opposite the sign and adjust the physical output value u(t). An increase of the physical value domain of u(t) by 3 times obtains the results shown in Figure 18. Membership function values corresponding to NS, PS are about 3 times smaller than the values shown in Figure 17.  
Figure 17: Membership function value of the fuzzy set u(t) corresponding to the physical value domain of u(t) is selected as in Table 6.  
Therefore, thanks to the integration of expert knowledge, the fuzzy rules are chosen, then there will be a reasonable upper bound value for the physical value domain of control output signal.