# Hysteresis Compensation and Trajectory Tracking Control Model for Pneumatic Artificial Muscles
## 3 Position Control Strategy of PAM
### Feedforward Hysteresis Compensation Control  
Feedforward control based on the inverse hysteresis model of pneumatic artificial muscles is a control strategy designed to accurately compensate for the complex hysteresis effects of PAMs. The hysteresis characteristics of the PAM arise from the nonlinear behavior of its material and the lag between pressure, output force, and displacement, which makes achieving high-precision motion control difficult with traditional control methods. To address this challenge, feedforward control establishes an inverse model of PAM hysteresis, using the model to predict the input pressure required for different target outputs, thereby effectively reducing the tracking errors caused by hysteresis. In practical applications, when the system provides a predetermined target output, the feedforward controller directly calculates the precise input signal based on the inverse hysteresis model, allowing the system to respond quickly and follow the target trajectory. The advantage of this approach is that it eliminates the need to wait for feedback error adjustments, enabling the system to directly compensate for delays caused by hysteresis, thereby improving control accuracy and response speed. This method is particularly effective in high-frequency motion or force control tasks (such as robotics and bionic exoskeletons), where feedforward control based on the inverse hysteresis model performs exceptionally well.  
With the ideal displacement as the inverse model input, the output pressure acts on the PAM. It can effectively compensate for the hysteresis effect in the system, thereby improving control accuracy and response speed. Through feedforward control, the system can adjust the input without relying on feedback, significantly reducing error accumulation. Set \(n=10\) in the formula, the final parameters can then be determined by the PSO algorithm, which not only reduces the complexity of the calculation but enhances the precision of the modeling suitable for real-time control. To verify the performance of feedforward hysteresis compensation, the inverse model was tested through a triangular wave input signal under a 0.05 Hz frequency and 0.5 bar amplitude. Figure 6 displays the fitting results of the feedforward hysteresis loop, real-time fitting results, and real-time fitting errors. Consequently, the maximal error of the inverse model result is 0.03 bar, and the mean error is 0.01 bar. The pressure-displacement hysteresis of the PAM has been adjusted. However, due to the modeling uncertainty and the unmodeled slow driving characteristics, certain errors still exist when taking this inverse model as a PAM output displacement feedforward compensator. In practical applications, combining feedforward control with methods such as incremental PID iterative learning feedback control leverages the fast response capability of feedforward control while using the feedback mechanism to correct errors in real time, thereby enhancing the system's robustness and disturbance rejection ability.  
Figure 6: Fitting results of QUPI inverse model. (**a**) Fitting of length–pressure hysteresis loop. (**b**) Air pressure fitting diagram and real-time error diagram.