# Dynamic Analysis and Trajectory Tracking Control for a Parallel Manipulator with Joint Friction
## 1 Introduction  
Hip replacement surgery is commonly used to treat pain and disability due to cartilage injury, synovial fluid reduction, and osteoarthritis [1, 2]. However, debris from friction on joint contact surfaces enters circulation, and many patients require revision surgery after 15-20 years [3]. Therefore, it is necessary to examine the friction and wear characteristics of artificial hip joints using in vitro simulation methods. Simulators of the artificial hip joint must simulate the motion and load changes of the human hip joint simultaneously, and most existing simulators often use a serial manipulator as the core module [4]. [PERSON] et al. [3] developed a pendulum friction simulator that enables the acetabulum and femoral head to maintain the correct center of rotation and evaluated the efficacy of the simulation system by testing the natural porcine hip joint and artificial hip joint. [PERSON] et al. [5] developed a serial manipulator to test the biomechanics of hip joint specimens, and reduced the hip joint clamping error by installing force sensors. [PERSON] et al. [6] used a pendulum simulator to analyze the effect of the texturing of the ultrahigh-molecular-weight polyethylene acetabular cup on tribological properties, and the results showed that surface texturing could improve the stability of the friction of the artificial hip joints. Although serial manipulators can partially simulate the kinematic and dynamic performance of the human hip joint, they are deficient in terms of repeatability,comparative accuracy, complex trajectory simulation, and dynamic loading [7, 8]. Moreover, a serial hip joint simulator can test only a single type of joint motion. Generally, parallel manipulators with a closed-chain structure consisting of multiple kinematic chains have the advantages of high precision [9] and high load capacity [10], are widely used in medicine engineering [11] and precision support [12], and are being developed to achieve high speeds and more reconfigurability [13]. The application of parallel manipulators to artificial hip joint testing provides precise trajectories and facilitates carrying higher loads. [PERSON] et al. [14, 15] designed a 3 SPS+1 PS parallel manipulator with satisfactory performance in simulating the kinematics, dynamics, and load changes of the hip joint. In the ISO 14242-1:2014(E) standard, a complete test requires the completion of 5 \(\times\) 10\({}^{6}\) cycles. At present, most of the existing hip joint simulators have only one moving platform, resulting in lower efficiency in many practical applications. Therefore, it is a great prospect to design a parallel manipulator that can be used for the friction and wear test of various size of hip joints and can perform the test on two hip joints simultaneously. Generally, kinematic analysis and dynamic modeling are the basis for structural optimization, performance analysis, and controller design of a parallel manipulator. Scholars have performed many studies on lower-mobility parallel mechanisms and achieved significant results. [PERSON] et al. [16] proposed a 1T2R parallel manipulator dynamic performance and acceleration capability evaluation method that discussed the dynamic isotropic performance and area of the feasible acceleration region under the influence of velocity, external load, and gravity. [PERSON] et al. [17] established a dynamics model of a 2 DOF overconstrained tracking parallel manipulator based on the principle of virtual work, and the effects of inertia and friction are considered in the identification to achieve a balance between accuracy and efficiency. [PERSON] [18, 19] proposed a kinematic modeling method for parallel manipulators with a simple topology to flexibly establish motion equations in task space coordinates and end-effector coordinates.