# Hysteresis Compensation and Trajectory Tracking Control Model for Pneumatic Artificial Muscles
## 2 Hysteresis Model and Inverse Model of PAM
### The FN-QUPI Model  
Currently, the classic PI (Prandtl-Ishlinskii) model is widely used in the field of pneumatic artificial muscle hysteresis modeling. Its advantages lie in having a specific inverse model that supports analytical solutions while maintaining low structural complexity and ease of application. However, this model struggles to describe the asymmetric hysteresis of the PAM. The shortcomings of the classic PI model can be addressed through appropriate optimization methods, such as introducing a nonlinear envelope function within the play operator. This enhancement improves the model's adaptability to different hysteresis phenomena, providing higher robustness when handling complex inputs. A representative example is the CPI model [18], where the descending edge of the CPI (classical Prandtl-Ishlinskii) operator is multiplied by a weight coefficient (greater than zero) to yield the UPI (unparallel Prandtl-Ishlinskii) operator [19], making it easier to apply in the modeling of complex hysteresis phenomena. The UPI operator can be expressed as:  
\[ \begin{cases}F_{r,\alpha_{j}}[u](t)=\max\Bigl{\{}u(t)-r,\min\Bigl{\{}\alpha_{j }(u(t)+r),F_{r,\alpha_{j}}(t^{-})\Bigr{\}}\Bigr{\}}\\ \qquad\qquad t^{-}=\lim\limits_{\delta\to 0}(t-\delta);\alpha_{j}>0,r\geq 0 \end{cases}  \tag{1}\]  
In the equation, \(r\) represents the dead-zone threshold, \(u(t)\) represents the input signal at time \(t\), and \(\alpha_{j}\) represents the slope of the descending edge of the UPI operator. Considering that the quadratic function and the hysteresis curve of the pneumatic artificial muscle have similar shapes, and given its basic properties of being invertible, bounded, and continuous, a quadratic function is adopted as the envelope function for the UPI model in this study. This model is named the QUPI model. By integrating this function with the UPI operator, the descending and ascending edges of complex hysteresis phenomena can be obtained. The schematic diagram of the QUPI operator is shown in Figure 1 (the horizontal axis represents the pneumatic muscle pressure \(p\), and the vertical axis represents the pneumatic muscle displacement \(y\)). The envelope function of the basic operator in the QUPI model consists of two quadratic parabolas, with the axis of symmetry aligned with the starting points of the ascending and descending branches of the hysteresis curve, making it more consistent with hysteresis characteristics. Based on the analytical expression of a quadratic function, the analytical expressions for a single QUPI operator can be derived when the pressure increases and decreases, as shown in Equations (2) and (3). Subsequently, the final QUPI model is formed by stacking Na \(\times\) Nb small QUPI operators, and the structure of the QUPI model is shown in Figure 2. The computational formula of the QUPI model is given in Equation (4). When \(\dot{p}>0\)  
\[y=\left\{ \begin{array}{ll}y_{0}&p-p_{0}\leq r_{1}\\ \frac{y_{0}p_{n}^{2}-2y_{0}p_{n}p_{0}-2y_{0}p_{n}r_{1}+y_{n}p_{n}^{2}+2y_{0}p_ {0}r_{1}+y_{n}r_{1}^{2}}{p_{n}^{2}-2p_{n}p_{0}-2p_{n}r_{1}+p_{0}^{2}+2p_{0}r_{1 }+r_{1}^{2}}&\\ +\frac{p^{2}(y_{n}-y_{0})}{p_{n}^{2}-2p_{n}p_{0}-2p_{n}r_{1}r_{0}^{2}+2p_{0}r_ {1}+r_{1}^{2}}&\\ \frac{2p(p_{1}+r_{1})(y_{n}-y_{0})}{p_{n}^{2}-2p_{n}p_{0}-2p_{n}r_{1}+p_{0}^{2 }+2p_{0}r_{1}+r_{1}^{2}}&\\ \end{array} \right. \tag{2}\]  
When \(\dot{p}<0\),  
\[y=\left\{ \begin{array}{ll}\frac{y_{0}p_{n}^{2}-2y_{0}p_{n}p_{0}-2y_{0}p_{n}r _{1}+y_{n}p_{n}^{2}+2y_{0}p_{0}r_{1}+y_{n}r_{1}^{2}}{p_{n}^{2}-2p_{n}p_{0}-2p_{ n}r_{2}+p_{0}^{2}+2p_{0}r_{1}+r_{1}^{2}}&\\ +\frac{p^{2}(y_{n}-y_{0})}{p_{n}^{2}-2p_{n}p_{0}-2p_{n}r_{1}r_{0}^{2}+2p_{0}r_ {1}+r_{1}^{2}}&\\ \frac{2p(p_{1}+r_{1})(y_{n}-y_{0})}{p_{n}^{2}-2p_{n}p_{0}-2p_{n}r_{1}+p_{0}^{ 2}+2p_{0}r_{1}+r_{1}^{2}}&\\ \end{array} \right. \tag{3}\]  
\[\Gamma_{\textrm{QUPI}}[u](t)=\sum_{i=1}^{Na}\sum_{j=1}^{Nb}d_{ij}y_{r_{1}\
u _{2}}[u](t) \tag{4}\]  
In the equation, \(p\) represents the pneumatic pressure of the pneumatic muscle, and \(y\) represents the displacement of the pneumatic muscle.