# Hysteresis Compensation and Trajectory Tracking Control Model for Pneumatic Artificial Muscles
## 1 Introduction  
Largely comprised of a fiber woven mesh, a hollow rubber tube, and multiple metal connectors, the PAM is an extensive bionic actuator. Compared to motor-powered or hydraulic robots, PAM-driven ones are more flexible. More specifically, the PAM is characterized by a higher power-to-mass percentage, complying with bodies. Thus, it is broadly used in the rehabilitation and medical domains. Moreover, the PAM is applied to the exoskeleton mechanism, with one terminal sliding mode control utilized for achieving more elevated trajectory tracking precision [1]. Though excellent usage prospects, the intrinsic nonlinear hysteresis feature is one of the dominant barriers impacting the accurate trajectory tracking control of the PAM-powered robot. At present, hysteresis compensation and modeling have slowly become key research areas [2]. From a mathematical perspective, the dynamic models of the PAM are capable of being simply classified into physics-based models and phenomenon-based models [3]. The physics-based model primarily construes the geometric correlation and material attributes of the PAM. As an illustration, [PERSON] et al. [4] reflected on the thickness of rubber tubes while creating a static model of the PAM using the force balance method. Due to the intricate math derivation and numerous model parameters in physics-based models, it brings large difficulties to the controller design while increasing the significant computing cost. Or, phenomenon-based models are formulated via experimental data and simplified model descriptions. As found by [PERSON] et al. [5], the PAM is used as a structureconsisting of a spring unit, a damping unit, and a Coulomb friction unit. Prior research has indicated that numerous phenomenological hysteresis models were placed forward and used in the PAMs. Pervasive hysteresis models are capable of being simply divided into integral models and differential models. Pervasive integrated hysteresis models encompass Krasnosel'skii-Pokrovskii, Maxwell [6], Preisach [7; 8], and Prandtl-Ishlinskii models. A precise one is indispensable, whilst improvement in model precision frequently causes a growth in model intricacy. Pervasive differential hysteresis models include the Bouc-Wen, Duhem, Dahl, [PERSON], and Leuven models [9; 10; 11; 12], wherein the Bouc-Wen model is capable of offering receivable modeling precision with fewer parameters. The appeal of differential models is attributed to their formulaic ability to facilitate optimization, modeling, and controller design. In differential hysteresis models, nonlinear differential equations are employed to describe and depict hysteresis. Integral hysteresis models are also known as operator hysteresis models. For nonlinear systems with hysteresis characteristics, the control strategy typically involves establishing an inverse hysteresis model and serially connecting an inverse compensator to the system to counteract the effects of hysteresis, known as inverse model control. This approach generally includes the following: (1) The feedforward control, where the inverse model of the hysteretic nonlinear system is developed and used as a feedforward compensator for open-loop control. The hysteresis nonlinearity in the control channel is partially or fully compensated by the inverse model. [PERSON] [13] employed a polynomial approach and the least-squares method to fit the hysteresis characteristics of pneumatic artificial muscles as accurately as possible, minimizing the impact of hysteresis on the system's control precision. The external control loop was transformed into a PID controller and a feedforward controller, significantly improving the model's control accuracy. However, its improvements to the nonlinearity of the closed-loop main control loop were limited, with inadequate disturbance rejection and robustness. (2) The feedforward plus feedback control, where the inverse model serves as the feedforward controller in a closed-loop control system, thus predicting the desired control signal before tracking errors occur.