Solution for Direct Kinematics of 3-PRS Parallel Manipulator Using Sylvester Dialytic Elimination Method

Abstract

This paper presents the direct kinematics of a 3-PRS parallel manipulator. Through analysis position, the direct kinematics model is established, which is a nonlinear system of three equations in three unknown. The equations are derived using recursively the Sylvester dialytic elimination method. Finally, a numerical example is provided. Therefore, the direct kinematics is minimal.. Introduction Parallel manipulator is a mechanism composed of a moving platform connected to a fixed base by means of at least two limbs [1]. Parallel manipulators have constituted a very active field of research over the last 20 years. Compared to serial manipulators, parallel manipulators essentially have two well-known advantages, namely greater precision in positioning and increased rigidity with respect to the relationship between size and workload limit. Many parallel manipulators with less than six Degrees Of Freedom (DOF) have been introduced, such as the famous DELTA robot with three translational DOF [2]. One of the challenges in studying parallel manipulators consists of the difficulty in solving their direct kinematics problems, which leads to systems of polynomial equations [3]. For direct kinematics, the input joint variables are given and all possible moving platform positions that would result from the given input values need to be found. Direct kinematic analysis is an essential component of the design, programming and control of any mechanism. Solution approaches for such a problem can be divided into two classes: numerical methods and analytic techniques [4]. To find all possible solutions of the direct kinematics problems of parallel manipulators, analytical techniques are found in the literature [5, 6]. Three common approaches to solving these systems of polynomial equations are the Dialytic Elimination, Polynomial Continuation and Grobner bases. The purpose of this paper is to solve the direct kinematics problem of a 3-PRS parallel manipulator. Deriving the equations governing the problem leads to three couple equations. Then the equations are derived using recursively the Sylvester dialytic elimination method. Finally, a numerical example is provided. Description of the 3-PRS Parallel Manipulator The schematic of the 3-PRS parallel manipulator is shown in Fig. 1. The architecture of the manipulator is composed of a moving platform, a fixed base and three PRS type active limbs with the linear actuators fixed the base. Each PRS type active limb connects the moving platform to the base by a prismatic joint, P at Bi, a revolute joint at Ci, and a spherical joint at Mi. The prismatic joint is actuated and the other joints are passive. A reference frame O-xyz is attached to the base at point O, located at the center of the base. The z axes is perpendicular to the base. Point Bi is assumed to lie at a radial distance of rB from the point O. BiCi is the direction of the prismatic joint movement, which is parallel to z axes. The axes of the revolute joint is perpendicular to the BiO and BiCi. Point OM is the center of the moving platform. And point Mi is the center of the spherical joint, which is assumed to lie at a radial distance of rM from the point OM. The distance from Ci to Mi is L. The angle φi is defined from BiCi to CiMi. And the angle θBi is defined from x axes to OBi. The drive parameters qi is the distance from Bi to Ci. International Conference on Mechanics and Civil Engineering (ICMCE 2014) © 2014. The authors Published by Atlantis Press 151 Fig. 1 Schematic of the 3-PRS parallel manipulator Direct Kinematics The position of Mi in the O-xyz coordinate frame for the i limb can be expressed as xMi, yMi and zMi ( ) ( ) ( ) M B B