# Hysteresis Compensation and Trajectory Tracking Control Model for Pneumatic Artificial Muscles
## 2 Hysteresis Model and Inverse Model of PAM
### The Inverse Model of QUPI  
In order to address the issue of low-trajectory tracking control accuracy caused by hysteresis characteristics in the operation of pneumatic artificial muscles (PAMs), this paper designs and applies an inverse model based on the QUPI model, aiming to compensate for the hysteresis effect in the control system, thereby improving the system's control accuracy and response speed. In complex dynamic systems, hysteresis often leads to delays and errors between input and output, and traditional control methods show limitations when dealing with such nonlinear effects. The introduction of the inverse model effectively resolves this issue. The core function of the inverse model in the control system lies in its ability to predict the input signal based on known output values, allowing the system to adjust in advance, thus achieving precise feedforward control. This control strategy enables the pre-adjustment of input parameters before system operation, reducing the negative impact of hysteresis, achieving smoother dynamic responses, and higher control accuracy. Additionally, the system can quickly adjust to the correct operating point, significantly improving its response speed. Based on the detailed introduction of the QUPI model earlier, this paper further applies the inverse model of the QUPI operator to system control to  
Figure 3: FN-QUPI model.  
compensate for the hysteresis characteristics of the PAM. The output expression of the QUPI operator is simplified as follows:  
\[\Gamma_{QUPI}[u](k)=q[u](k)+\sum\limits_{j=1}^{n}\mathrm{d}j\mathrm{y}_{\gamma_{ j}}[u](k) \tag{8}\]  
In the equation, \(q\) is a constant, \([u](k)\) represents the input at time \(k\), \(d_{j}\) is the weight coefficient, and \(j\) is the number of QUPI operators.  
Since the relationship between pressure and displacement forms a hysteresis model, its inverse model is also a hysteresis model. The inverse model of the QUPI operator is:  
\[\Gamma_{QUPI}{}^{-1}[u](k)=q^{-1}[u](k)+\sum\limits_{j=1}^{n}g(\overline{r}_ {j})\mathrm{y}_{\gamma_{j}}[u](k) \tag{9}\]  
In the equation, \(\overline{r}_{j}\) represents the threshold of the inverse model, and \(g\) is the density function of the inverse model. The expressions are as follows:  
\[\overline{r}_{j}=qr_{j}+\sum\limits_{i=1}^{j-1}p_{i}(r_{j}-r_{i}) \tag{10}\]  
\[g_{j}=-\frac{p_{j}}{\left(q+\sum\limits_{i=1}^{j}p_{i}\right)\left(q+\sum \limits_{i=1}^{j-1}p_{i}\right)} \tag{11}\]  
In the equation, \(q^{-1}\) is a constant equal to \(1/q\). Therefore, the key to solving the QUPI inverse model lies in identifying the parameter \(q^{-1}\) and the vector. In this paper, particle swarm optimization (PSO) is used for parameter identification to obtain the optimal solution. The QUPI inverse model demonstrates excellent real-time compensation capabilities in practical system operations. By introducing the density function and the recursive update formula for the threshold, the input signal can be dynamically adjusted in real time according to the changes in the system's state.