# Design and Trajectory Tracking Control of a Magnetorheological Prosthetic Knee Joint
###### Abstract  
This paper developed a new magnetorheological (MR) prosthetic knee joint using an MR damper as the brake. According to the gait data of healthy people walking on flat ground, the structure of a MR prosthetic knee joint was expounded in detail, and its motion and dynamic model was also established. In addition, an MR damper was developed according to the specific needs of an MR prosthesis. The forward and reverse mechanical models of the MR damper were established, and its damping performance was obtained through experimental tests. In addition, to solve the problems of uncertainty and external interference in the MR prosthetic knee joint system, a second-order sliding mode controller was proposed. The experimental test results show the maximum positive error of the knee joint swing trajectory is 9.4\({}^{\circ}\), which effectively tracks the reference swing trajectory. magnetorheological prosthetic knee joint; MR damper; second-order sliding mode controller; trajectory tracking control +
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Among them, the used MR damper is a new type of smart device that uses smart material MR fluid as the transmission medium, giving it the advantages of low energy consumption, wide dynamic range, fast response and continuously adjustable damping. For a typical MR damper used in the prosthesis, the maximum average energy consumption is about 5 W; the dynamic range can reach 9, and the response time is about 10 ms [8; 9]. Prosthetic knee joints based on the MR damper can be divided into single-axis prosthetic knee joints and multi-axis prosthetic knee joints. The single-axis structure is common and can offer a stable pace and speed. [PERSON] et al. [10] integrated a rotary MR damper into a prosthetic knee joint and designed a single-axis prosthesis. Compared with a single-axis MR damper prosthesis, the multi-axis MR damper prosthesis can better simulate the trajectory of the instantaneous center of rotation, so that it has better bionics and a more natural gait. [PERSON] et al. [11] combined the MR damper and the four-bar linkage to design a four-axis MR damper prosthesis, wherein the MR damper occupies a large installation space so that the MR damper and the four-bar linkage swing each other when the knee joint is moving, which reduces gait quality. [PERSON] et al. [12] designed a prosthesis that not only reduces the space and mass of the prosthesis but also effectively solves the problem of relative movement between the four-bar mechanism and the MR damper during walking, using an integrated four-bar structure and a double-bar MR damper. [PERSON] et al. [13] developed a four-bar MR prosthetic knee joint, which utilizes a four-bar linkage mechanism to simulate the rotation of the center of the knee joint and track the gait of the prosthesis; an MR damper is used to control the torque of the prosthetic knee joint, and the upper and lower ends of the knee joint are connected to the thigh and the lower leg, respectively. [PERSON] et al. [14] linked a four-bar linkage with a commercial MR damper of the type RD-8040-1 to create an MR prosthetic knee joint. However, the used MR damper occupied a large installation space, which seriously affects the gait of the prosthetic knee joint. The control algorithm of MR prostheses is another research hotspot. According to the repetitive motion of the human body during normal walking, [PERSON] et al. [15] introduced a finite state machine (FSM) to control the bending angle of an MR prosthetic knee joint by planning typical gait states. The method adopted open-loop control to command the MR damper to provide the required knee joint torque for the gait, thereby controlling the leg swing trajectory of the lower limb prosthesis. However, the FSM control can not track the reference swing angle of the leg of the lower limb prosthesis, and the open-loop control method has weak suitability to the environment and can not adjust the parameters for different sports and users, resulting in poor adaptability.