# An Extended Proxy-Based Sliding Mode Control of Pneumatic Muscle Actuators
## 1 Introduction  
Patients that survive after stroke often suffer from paralysis, and they still want to be productive members of society. Rehabilitation robotics can be considered a specific focus of biomedical engineering and a part of human-robot interaction, increasing the ease of activities in the daily lives of these patients [1, 2]. [PERSON] et al. [3] proposed an interval type-2 fuzzy logic control of a mobile two-wheeled inverted pendulum, which is a promising method for future personal mobilities. [PERSON] et al. [4] and [PERSON] et al. [5] developed a walking intention-based motion control of omnidirectional type cane robot for rehabilitation of the elderly. For most stroke patients undergoing the rehabilitation process, a low-cost home-based rehabilitation solution is of great benefit and therefore in great demand [6, 7, 8]. Previous researchers have used robotics in rehabilitation devices because they can automate to perform traditional therapy methods repetitively. However, the drawbacks of robotics, such as space, price, substantial workspace, and especially noncompliance with human hand movements, still limit the applications of the system. As a result, a small, inexpensive hand and forearm rehabilitation device with human compliance is needed for stroke patients. Compared with motors which are commonly used in rehabilitation robots, pneumatic muscle actuators (PMAs) have several merits including intrinsic compliance, low cost, light weight, very high power-to-weight and power-to-volume ratios, etc. [9, 10, 11]. So far, a large number of rehabilitation robots driven by PMAs can be found in the literature [12, 13, 14, 15, 16]. It should be pointed out that the accurate joint trajectory tracking control of a rehabilitation robot is the basis of a variety of robot-assisted rehabilitation exercises. It can be directly used forpassive training on patients with weak active motor ability. This control demand relies heavily on the accurate control of the actuators, including PMAs, discussed in this study. [PERSON] et al. [17] presented an assist-as-needed control of a PMA-based robotic orthosis for gait rehabilitation. Although PMAs are very good solutions for driving the rehabilitation robots, their control design remains a great challenge. This is because PMAs have characteristics of strong nonlinearity, time-varying parameters, hysteresis, etc. These features jeopardize the control performance of a closed-loop system which is actuated by PMAs. To solve this problem, both model-free and model-based control strategies have been studied for PMAs in recent decades. The PID controller is a typical model-free controller extensively used for PMAs. For stroke rehabilitation, an exoskeleton robotic arm has been built by [PERSON] et al. [16]. A feedback and feedforward PID controller was used to calculate the desired outputs of the robotic arm. [PERSON] et al. established an assistive pneumatic manipulator to help with handling heavy loads [18]. A PID controller is used in this system to correct disturbances and model inaccuracies. Despite the easy implementation, the PID control lacks sound theoretical foundations for convergence analysis. In addition, the coefficients adjustment of the PID controller is also time-consuming. Another kind of model-free controller is realized by soft-computing techniques, e.g., fuzzy sets, artificial neural networks (ANNs), etc. [PERSON] et al. proposed a hybrid fuzzy controller composed of a bang-bang controller and fuzzy controller for a novel curved pneumatic muscle-based rotary actuator for the wearable elbow exoskeleton [19]. [PERSON] et al. [20] proposed an iterative fuzzy controller for PMA-driven rehabilitation robot. [PERSON] [21] showed an adaptive self-organizing fuzzy sliding mode controller (SMC) for a 2-DOF PMA-driven rehabilitation robot. [PERSON] et al. presented the development of a novel adaptive wearable ankle robot for the treatments of ankle sprain through physical rehabilitation [22].