# Dynamic Analysis and Trajectory Tracking Control for a Parallel Manipulator with Joint Friction
## 2 Mechanism and Establishing the Coordinates  
For a parallel manipulator with multiple moving platforms driven by the same set of actuators for achieving the same motion, the positions of the moving platforms should be arranged symmetrically or in parallel to each other or in the same plane. Considering the strengths and weaknesses of the obtained configurations, two 3 PUS+S-type parallel manipulators were designed as the core module of the parallel manipulator, and the two moving platforms were arranged parallel to each other. Theoretically, both the platforms perform the same action. As shown in Figure 1, the 2(3 PUS+S) parallel hip joint simulator comprises two pairs of artificial hip joints, two moving platforms (m\({}_{1}\) and m\({}_{2}\)), one base platform (B), three joint levers, three linear modules (LNi), and one intermediate branch chain. Three active linear modules and one intermediate branch chain were rigidly fixed to the base platform. In this paper, P refers to the prismatic joints composed of the lead screw and nut, U refer to the universal joints installed on the linear module slider, and S refer to the spherical joints installed on the moving platform with a fixed-length joint lever. The figure was shown in our previous paper [27]. Three surrounding branch chain PUS were placed symmetrically at 120\({}^{\circ}\) intervals, with the intermediate branch chain S in the center of the base platform. Two fixed distance sliders were installed on each linear module. The joint lever was designed to connect the slider and moving platform. The motion of the servo motor was transmitted to the moving platform through the linear module and the joint lever. The intermediate branch chain was connected to the moving platform through the spherical joint S.  
The intermediate branch chain, which is divided into two sections to install artificial hip joints, is a passive branch chain. The articular head was assembled on the moving platform, and their rotation centers were coincided. The acetabulum was fixed, and the  
Figure 1: 2(3 PUS+S) parallel manipulator. moving platform could achieve rotational motion with the artificial hip joint as the center. The articular head of joint 2 was assembled on the base platform through a thrust ball bearing and a section of the intermediate branch chain, and the articular head of joint 1 was connected to the acetabulum of joint 2 through a thrust ball bearing and ball spline shaft. The hydraulic system was connected to the acetabulum of joint 1 through a guide rail and pressure sensor. A loading force was applied to artificial joint 1 and then transferred to artificial joint 2 through the ball spline shaft. A spring equal to the gravity of moving platform m\({}_{1}\) was installed in the ball spline, and the height of the artificial joint remained constant during the operation. In Figure 2, \(A_{i}\) (\(i\) = 1, 2, 3) is the intersection of the linear module lead screw and base platform, and \(c_{i}\) is the projection point of \(b_{1i}\) and \(b_{2i}\) on line \(OA_{i}\). Let \(O\) be a center point fixed in the triangle \(A1A2A3\) with a circumscribed circle radius \(E\). The absolute reference frames \(O\)-\(XYZ\) (green) are attached at the point \(O\) with the \(Z\)-axis vertically upward, and the \(X\)-axis is perpendicular to \(A1A2\) and points away from \(OA3\). Similarly, \(a_{ji}\) (\(j\) = 1, 2) and \(b_{ji}\) are the intersection points of the rotation axes of spherical and universal joints, respectively. The relative reference frames \(o_{j}\)-\(x_{j}y_{j}z_{j}\) (red) are attached at the center point \(o_{j}\) of triangle \(a_{j1}a_{j2}a_{j3}\) with a circumscribed circle radius \(e\). The \(y_{j}\)-axis crosses point \(a_{j2}\), the \(z_{j}\)-axis is perpendicular to triangle \(a_{j1}a_{j2}a_{j3}\) and is directed to the hydraulic system. For most serial hip joint simulators, the cantilever structure is predominantly used as the loading part, which causes angle errors in the forced direction of the hip joint.