# An Extended Proxy-Based Sliding Mode Control of Pneumatic Muscle Actuators
## 1 Introduction
In their study, an adaptive fuzzy logic controller based on Mamdani inference has been developed and appended with the fuzzy-based disturbance observer (FBDO) to compensate for the transient nature of the PMA. [PERSON] et al. proposed several model predictive control (MPC) methods for PMAs based on ANNs [23; 24]. In their work, the uncertain PMA model was approximated by the echo state networks (ESNs) so that the MPC control strategy obtained very high accurate control performance. Although the soft-computing techniques are proved to be capable of approximating arbitrary nonlinear characteristics, the requirement of large storage space and high computation ability confines the real-time and compact PMA control applications. In the case of the model-based controller, [PERSON] et al. applied the dynamic surface control (DSC) characterized by convenient design and good transient performance for realizing pneumatic muscle (PM) tracking control [25]. In [26], a Maxwell-slip model used as a lumped-parametric quasi-static model is proposed to capture the force/length hysteresis of a PMA. The obtained model is simple but physically meaningful and easy to handle in terms of control. The main challenge of model-based controllers for PMAs is that the identified model parameters are highly uncertain. This demands strong robustness of the controllers for PMAs. The SMC might be a suitable solution due to its high robustness to both internal system uncertainties and external disturbances. [PERSON] developed a fully nonlinear model that encompasses all the major nonlinearities in a PMA-actuated servo system [27]. Based on this model, a SMC is developed to obtain robust control in the existence of model uncertainties and disturbances. [PERSON] et al. used the sliding mode approach to the tracking control problem of a planar arm manipulator system driven by a new type of actuator, which comprises a PMA and a torsion spring [28]. [PERSON] et al. [29] applied a SMC for PMA-based gait rehabilitation exoskeleton for verifying the compliance of the exoskeleton. [PERSON] and [PERSON] presented a cascaded SMC scheme for a new pneumatic linear axis which could be seen as an alternative to an electric direct linear drive [30]. On the other hand, it should be noted that the SMC has a \"chattering\" problem which may bring damages to practical actuators. Recently, a proxy-based sliding mode control (PSMC) approach has been proposed as a safe extension of PID control and is the simplest type of SMC. The PSMC can produce overdamped resuming motion from large positional errors without sacrificing tracking accuracy during normal operation. Also, the \"chattering\" phenomenon in conventional SMC is avoided by introducingthe so-called \"proxy\". Due to these advantages, the PSMC is widely applied not only in PMA applications [31] but also many other control fields [32]. However, the stability analysis of original PSMC is based on a strict conjecture of local passivity of well-tuned PID controller [32]. There is much room to improve in the view of the point of the general theoretical framework. Also, the basic idea of designing PID-type virtual coupling between the proxy and physical object can be generally extended in our opinion. In this study, we propose a novel PSMC with elaborately designed couple force between the proxy and object. Based on this technique, the Lyapunov stability of proposed PSMC can be analyzed straightforwardly by using the passivity approach. Furthermore, we successfully applied the approach in a real PMA control platform. By using this new PSMC, better performance can be achieved compared with some conventional control approaches. Since the proposed method presents well in the PMA tracking control, it has a good prospect of becoming an excellent pneumatic rehabilitation robot control approach. This will result in a safe and quantitative passive training implemented by the rehabilitation robot. The rest of paper is organized as follows. Section 2 gives the dynamic model of PMA.