IJRRAS 7 (1) ● April 2011 www.arpapress.com/Volumes/Vol7Issue1/IJRRAS_7_1_13.pdf 88 DIRECT POSITION KINEMATICS OF A THREE REVOLUTE- PRISMATIC-SPHERICAL PARALLEL MANIPULATOR Soheil Zarkandi* & Mohammad Reza Esmaili Department of Mechanical Engineering, Babol University of Technology, P.O. Box 484, Babol, Iran * E-mail: zarkandi@gmail.com ABSTRACT In this work, the direct position kinematics of a 3 degree-of-freedom parallel manipulator with three identical limbs, type revolute-prismatic-spherical (RPS), is analyzed. In contrast to the previous studies on this class of manipulators, the revolute joints of the proposed manipulator are actuated rather than the prismatic joints. Direct position kinematics of the manipulator leads to a system of three nonlinear equations in three unknowns that are reduced to a univariate polynomial of degree eight and two quadratic equations in sequence using Sylvester dialytic elimination method. In addition, to show the efficiency of the presented method a numerical example is provided. Keywords: Parallel Manipulators; Direct Kinematics; Sylvester Dialytic Elimination Method; Analytical Solution. 1. INTRODUCTION Parallel manipulators are the mechanisms composed of a moving platform connected to a fixed base by at least two kinematic chains (legs). The most studied type of parallel manipulator is without doubt the so-called general Gough– Stewart platform, a fully parallel manipulator introduced by Gough as a universal tire-testing machine almost five decades ago [1,2] and proposed as a flight simulator by Stewart in 1965 [3]. However, in many industrial applications, such as some assembly operations, parallel manipulators with fewer degrees of freedom than six can be successfully used instead of the general Gough–Stewart platform. The Delta robot, invented by Clavel, is a typical example of such applications. With this in mind, a significant amount of research has been devoted to the study of parallel manipulators with fewer than six degrees of freedom; see for instance [4–9]. In fact, this class of parallel manipulators offers significant advantages such as the simpler mechanical assembly, a larger workspace than a general Gough–Stewart platform and a simpler direct position kinematics. Direct position kinematics of parallel manipulators lead to systems of nonlinear equations which are very difficult to solve. Solution approaches for such equations can be divided into two classes: numerical (iterative) methods and analytic methods. There are different numerical methods that can deal with simultaneous non-linear equations such as Newton or conjugate gradient method commonly used as iterative methods [10], Homotopy continuation method as an improved iterative method [11], and other new numerical methods [12,13]. On the other hand, there are some analytical methods that have effectively used to solve the systems of nonlinear equations. Two well known analytical techniques which are used for solving polynomial systems are Bezout’s elimination method and Sylvester’s dialytic elimination method. In these methods a set of polynomials of multiple variables are reduced into a polynomial of only one variable. Many scholars have used these methods for solving direct position kinematics of parallel manipulators, e.g. [14-16]. The 3-RPS parallel manipulators constitute a class of parallel manipulators with fewer degrees of freedom than six. A 3-RPS parallel manipulator, see Fig. 1, is a mechanism where the moving platform is connected to the fixed platform by means of three limbs. Each limb is composed by a lower body and an upper body connected to each other by means of a prismatic joint. The moving platform is connected at the upper bodies via three distinct spherical joints while the lower bodies are connected to the fixed platform by means of three distinct revolute joints. The 3-RPS parallel manipulator, in which the prismatic joints are actuated, was introduced by Hunt [17] and has been the motive of an exhaustive research field where a great number of contributions, encompassing a wide range of topics, such as kinematic and dynamic analyses, synthesis, singularity analysis, extensions to hyper-redundant manipulators, etc., see for instance [18–21]. But, by actuating the revolute joints, a new 3-R PS parallel manipulator is achieved while R denotes the actuated revolute joint. To the best knowledge of the author, no study has been done for this type of 3-R PS manipulators. This paper analyzes the direct position kinematics of such a manipulator in an analytical form using the Sylvester dialytic elimination method.