# Fault-Tolerant Control of Multi-Joint Robot Based on Fractional-Order Sliding Mode
## 4 Simulation and Experimental Results
### Simulation Results  
The position command of joint1 is \(\cos(\pi t)\), and for joint2, it is \(\sin(\pi t)\), assuming that the multi-joint manipulator actuator has the constant gain failure, and the degree of constant gain failure (CGF) is 60%, 40%, and 20% of the rated value, respectively. Under the above conditions, the dynamic response curves of tracking error for both joints under different gain states are obtained. The simulation waveforms are shown in Figure 3, where the yellow figure under each figure is the enlarged part corresponding to the original figure. Figure 3 shows the simulation results corresponding to the actuator constant gain fault. The first thing we know from Figure 3a,b is that the fluctuation exists during the tracking process using SMC, and as the CGF decreases from 60 percent to 20 percent, the fluctuation amplitude drops from 0.073 radian to 0.017 radian for both joints. From Figure 3a-f, it is obvious that the control effect under SMC is unsatisfactory, where the fluctuation is its drawback. When the controller is changed to FSMC as proposed in this article, the fluctuation disappears. Another simulation phenomenon could be seen from Figure 3a,b, that there is vibration during the transition stage of starting under the control of FSMC. As it can be seen, when the CGF was 60 percent, the error of joint1 converged to zero rapidly once the simulation was started, and then vibrated for three periods and then tend to be stable. A possible explanation is that the FSMC provides stronger control output, so the response time is reduced, and the manipulator state tends to be stable which is guaranteed by the Lyapunov function. Figure 2: Sliding mode control block diagram. (**a**) FSMC control block. (**b**) SMC control block. Figure 3: _Cont._  
Figure 3: Joint tracking error curves under different CGF. (**a**) Tracking error of joint1 with 60% CGF. (**b**) Tracking error of joint2 with 60% CGF. (**c**) Tracking error of joint1 with 40% CGF. (**d**) Tracking error of joint2 with 40% CGF. (**e**) Tracking error of joint1 with 20% CGF. (**f**) Tracking error of joint2 with 20% CGF. The response time difference of FSMC and SMC under different CGF is not large, as the controller only needs to compensate the gain of the broken actuator. Combining Table 1 and Figure 3, it is obvious that the tracking error of the joints under the control of FSMC is reduced about ten times for joint1 and around five times for joint2. Additionally, the periodic vibration disappears by adopting FSMC, which could be due to the fractional differential operator, where the integration operation reduces the steady error. The result shows the advantage of the FSMC in the joint actuator constant gain fault. The constant deviation fault (CDF) of the joint actuator is also studied separately; the fault degree is from 50 N\(\cdot\)m to 300 N\(\cdot\)m, and the trajectory tracking error curve of the end of the double-joint manipulator is taken out, as shown in Figure 4.  
Figure 4 shows the simulation results corresponding to the actuator constant deviation fault. It is easy to know from Figure 3a,b and Figure 4a,b that the vibration disappears under CDF. It can be interpreted that the constant offset fault does not change during the whole process, which is not similar with CGF, so the effect of CDF at steady state is minor. We also know from Figure 4a-f that the regulation time for both SMC and FSMC under CDF remains unchanged. Additionally, the steady-state error, if given adequate time, reaches zero, but the FSMC has the priority in the rapidity. The priority can be due to the fractional operator, in which the integration calculation can speed up the converging process. The dotted line in Figure 4 is the boundary of five percent error zone, and the corresponding time that errors first go beyond this line is shown in Table 2.