# Design and Trajectory Tracking Control of a Magnetorheological Prosthetic Knee Joint
###### Abstract
Proportional-derivative (PD) control has the characteristics of simple structure and good robustness, can well track the reference swing curve and has become a widely used controller. [PERSON] and [PERSON] [16; 17] applied a PD controller to control MR prostheses, but the parameter settings depend on experience. Moreover, since the MR prosthesis is a strongly non-linear system, especially the multi-axis MR prosthetic knee joint, the non-linear term of the system will greatly interfere with the PD controller. Therefore, [PERSON] and [PERSON] [18; 19] added calculated torque, iterative learning, a neural network, and other nonlinear compensation items based on a PD controller to compensate for the influence of knee joint control swing angle. Aiming at the influence of the hysteresis effect of the MR damper on the swing angle of the MR prosthesis, [PERSON] [20] designed a sliding mode trajectory tracking control (SMTC), and the simulation results illustrate that the SMTC controller had a good control effect and robustness for knee joint swing. [PERSON] [21] introduced adaptive control to the swing control of MR prostheses and proposed a model reference adaptive control (MRAC) algorithm, the principle of which is to design an appropriate adaptive law to estimate the model parameters and adjust the controller output to make it follow the desired trajectory. To solve the problem of parameter uncertainty and strong coupling, [PERSON] et al. [22] devised an adaptive robust force/position control algorithm that makes use of time delay estimation technology, sliding mode control and a fuzzy neural network to achieve finite-time convergence and gait tracking. The simulation results indicate that this strategy has better trajectory tracking characteristics and strong robustness in the presence of unknown external interference. Electromyography (EMG) signal control system is a control method that, by judging the gait based on the electrical muscle signal and then controlling the movement of the prosthesis. [PERSON] et al. ([PERSON] et al., 2019; [PERSON] et al., 2019) used surface electromyography (S-EMG) signals as the basis of intention recognition in continuous walking mode. The results show that the algorithm designed by the support vector machine (SVM) has a high road condition recognition ability, reaching more than 99% in the stance phase. A summary of related research work in the past is shown in Table 1.  
In this paper, the structure of an MR prosthetic knee joint is developed according to the gait data of healthy people, and its movement and dynamics models are also established. Then the MR damper is designed; the forward and reverse mechanical models of the MR damper are derived, and the dynamic performance of the MR damper is obtained through experimental tests. Finally, to solve the uncertainties and the existence of external interference in the MR prosthetic knee joint system, a second-order sliding mode trajectory tracking control method is proposed, and the built MR prosthetic knee joint control system is used to test and analyze its control effect on the swing angle of the knee joint.