# Fault-Tolerant Control of Multi-Joint Robot Based on Fractional-Order Sliding Mode
## 4 Simulation and Experimental Results
### Simulation Results
\begin{table}
\begin{tabular}{l l l l l} \hline \hline  & & **CGF 60\%** & **CGF 40\%** & **CGF 20\%** \\ \hline \multirow{2}{*}{SMC} & Joint1 & 0.073 & 0.031 & 0.017 \\  & Joint2 & 0.074 & 0.033 & 0.018 \\ \hline \multirow{2}{*}{FSMC} & Joint1 & 0.007 & 0.002 & 0.001 \\  & Joint2 & 0.015 & 0.006 & 0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Maximum errors (rad) at steady state. Figure 4: _Cont._  
From Table 2, it is very easy to see that the FSMC is about three times faster than the SMC in terms of joint position response time, but the steady-state error of the conventional sliding mode is close to that of the fractional-order sliding mode. Combining Figure 4 and Table 2, we can conclude that the actuator constant offset fault has nearly no effect on the response time and steady-state error. Furthermore, from Figures 3 and 4, we can know that the offset fault of the actuator has more influence than the gain error in steady-state error, from which it could be interpreted that the gain fault of the actuator could be compensated by the gain design in FSMC and SMC controller, while the offset fault is eliminated by increasing the regulation time. According to the simulation results under constant gain and constant deviation situations for the double-joint manipulator, the fractional-order sliding mode has obvious advantages for the gain or deviation fault in a wide range of fault amplitude. In particular, it has obvious advantages in improving the response speed of the multi-joint manipulator and reducing the steady-state error. In particular, the greater the degree of failure is, the more prominent the advantages of FSMC can be. A hardware experimental platform is built to investigate the control laws of FSMC and SMC. The platform is shown in Figure 5. \begin{table}
\begin{tabular}{l l l l l l} \hline  & & **CDF** 50 **N**\(\cdot\)**m** & **CDF** 100 **N**\(\cdot\)**m** & **CDF** 200 **N**\(\cdot\)**m** & **CDF** 300 **N**\(\cdot\)**m** \\ \hline \multirow{2}{*}{SMC} & Joint1 & 0.11 & 0.11 & 0.11 & 0.11 \\  & Joint2 & 0.12 & 0.12 & 0.12 & 0.12 \\ \hline \multirow{2}{*}{FSMC} & Joint1 & 0.04 & 0.04 & 0.03 & 0.03 \\  & Joint2 & 0.04 & 0.04 & 0.04 & 0.04 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison table of maximum time (s) of dynamic error in different CDF. Figure 4: Position tracking error curves of double-joint under different CDF. **(a)** Tracking error of joint1 with 50 N\(\cdot\)m CDF. **(b)** Tracking error of joint2 with 50 N\(\cdot\)m CDF. **(c)** Tracking error of joint1 with 100 N\(\cdot\)m CDF. **(d)** Tracking error of joint2 with 100 N\(\cdot\)m CDF. **(e)** Tracking error of joint1 with 200 N\(\cdot\)m CDF. **(f)** Tracking error of joint2 with 200 N\(\cdot\)m CDF. **(g)** Tracking error of joint1 with 300 N\(\cdot\)m CDF. **(h)** Tracking error of joint2 with 300 N\(\cdot\)m CDF. Two experiments under extreme situations are investigated here. The first one is the constant gain failure with 80% output torque, and the other one is constant deviation fault where 300 N\(\cdot\)m of the loss is included. The oscilloscope is used to capture the experiment data. Figure 6 shows the output voltage waveform of the double-joint manipulator using SMC and FSMC where the output torque of the joint actuator is only 80% of the rated value. Figure 6 shows that under the condition of large actuator constant gain fault, the output voltage amplitudes of the conventional sliding mode and fractional-order sliding mode controllers are similar in steady state. However, the output voltage of the fractional sliding mode has a fluctuation of about 10 V in amplitude. Furthermore, the voltage waveforms of the FSMC and SMC when the joint actuator has a constant deviation fault are shown in Figure 7.  
Figure 5: Robotic experimental platform. Figure 6: Output voltage of FSMC and SMC under constant gain fault (80%).