## 4 Controller Design
### Sinusoidal Wave Path-Tracking Experiment with a 2 kg Loading Mass  
The experiment of the sinusoidal path-tracking control are compared with a traditional PID controller and FSMC with a 2 kg loading mass added on the 500 mm length slider. The path-tracking control is set as the sinusoidal wave path with different periods and the amplitude is set to \(\pm\)10 degrees.  
The sinusoidal wave path-tracking control response with frequency of 0.05 Hz and 0.1 Hz for the experiment with a 2 kg loading mass by using the PID controller and FSMC is shown in Figures 22-25, respectively.  
Figure 21: Experimental results for triangular wave path tracking control with a period of 10 s for the 1-DOF robotic lower limb system driven by a single-PAM with a torsion spring with a 2 kg loading mass by using FSMC: (**a**) step response; (**b**) angular tracking error; (**c**) control signal.  
Figure 23: Experimental results for sinusoidal wave path tracking control with frequency in 0.05 Hz for the 1-DOF robotic lower limb system driven by a single-PAM with a torsion spring with a 2 kg loading mass by using the FSMC: (**a**) the angular path response; (**b**) the angular tracking error.  
Figure 24: Experimental results for sinusoidal wave path tracking control with frequency in 0.1 Hz for the 1-DOF robotic lower limb system driven by a single-PAM with a torsion spring with a 2 kg loading mass by using the PID controller: (**a**) step response; (**b**) angular tracking error.  
Figure 22: Experimental results for sinusoidal wave path tracking control with frequency in 0.05 Hz for the 1-DOF robotic lower limb system driven by a single-PAM with a torsion spring with a 2 kg loading mass by using the PID controller: (**a**) the angular path response; (**b**) the angular tracking error.  
According to the experimental results under the load of 2 kg loading mass, the PID controller has an IAE of 100.8331 in a frequency of 0.05 Hz and an IAE of 155.6086 in a frequency of 0.1 Hz. However, the FSMC can reach an IAE of 65.528 in a frequency of 0.05 Hz and an IAE of 93.5461 in frequency of 0.1 Hz. Moreover, the tracking error in both cases is within the range of 0.3 degrees by using FSMC, better than PID control.  
Due to the compressibility of the air pressure and the non-linear elasticity of the rubber, the enthalpy changes are caused. Therefore, it is not easy to achieve good tracking performance for the 1-DOF robotic lower limb system. The traditional PID controller and the FSMC are adopted to overcome the nonlinear behavior of PAMs, and to compare and analyze the performance of the controller.  
The traditional PID controller is applied to a robotic lower-limb system composed of a single-PAM. Compared with the FSMC, the control performance is poor, and the required stabilization time is longer in the step response, and the maximum overshoot is also large. In the sinusoidal wave response, the tracking error are far worse than the FSMC. The FSMC uses the sliding surface to simplify the two-dimensional fuzzy rule base into one-dimensional, which can not only shorten the calculation time, but also can display better robustness against changes in the external environment.  
According to the above various experiments, the proposed novel single-PAM with a torsion spring-driven lower-limb accessory joint mechanism can be verified with satisfactory performance.