# Dynamic Analysis and Trajectory Tracking Control for a Parallel Manipulator with Joint Friction
## 1 Introduction
According to the ISO14242 standard, the high speed and acceleration of the parallel manipulator during the artificial hip joint wear test lead to the need for the servo motor to change speed frequently, so the control strategy based on kinematic parameters is difficult to maintain the accuracy of the parallel manipulator. Therefore, it is important to establish a dynamics model that is close to the actual dynamics characteristics of the parallel manipulator. With an increase in speed, the joint friction effect becomes significant [20, 21], and friction effect plays an important role in controlling and tracking the accuracy of the parallel manipulator [22]. [PERSON]'s [23] experiments with artificial hip joint friction measurements using the Durham friction simulator showed that hip joint friction under high loads cannot be ignored. [PERSON] et al. [24] proposed a parallel manipulator decomposition modeling method, which projected the force of the moving platform in the workspace into the joint space to calculate the normal force of the sliding plane of the driving joint and then obtained the friction force. [PERSON] et al. [25] proposed a joint friction solution method that subtracts the inertial parts from the joint torque, and the reduction of dynamic model accuracy due to the friction model deviation can be avoided. [PERSON] et al. [26] solved the discontinuity problem of the friction model of a 3-DOF parallel manipulator by dividing the self-interval, calculating the friction force in the viscous phase, and enhancing stability. Parallel manipulators are subjected to cyclically varying loads during friction and wear testing, resulting in significant friction that is difficult to eliminate directly through error compensation. Therefore, with regard to the dynamic analysis of parallel mechanisms, the time-varying characteristics of artificial hip joint friction cannot be ignored. In this study, to overcome the shortcomings of series hip joint simulators with regard to their bearing capacity, trajectory simulation, accuracy, etc., and to remedy the lower working efficiency deficiencies using a single moving platform, and considering the cost and dimensions of the assembly, a 2(3 PUS+S) parallel manipulator with two pairs of artificial hip joints and two moving platforms is proposed, which can simultaneously conduct friction and wear tests for two pairs of artificial hip joints. To lay the foundation for further research, the movements and driving forces of the parallel manipulator under the requiredmotion are studied. First, the inverse kinematics for the parallel manipulator is modeled, and the kinematic relationships between the linear module slider and the moving platform are established. Second, the driving torque of each component is analyzed separately. Combined with the Coulomb friction model of artificial joints and thrust ball bearings, a rigid-body dynamics model of a parallel manipulator with joint friction is established. Finally, numerical simulations and experiments are conducted to analyze the kinematic differences between the two moving platforms under the required motion and verify the accuracy of the kinematic and dynamical models.