Workspace and joint space analysis of the 3-RPS parallel robot

The Accurate calculation of the workspace and joint space for 3 RPS parallel robotic manipulator is a highly addressed research work across the world. Researchers have proposed a variety of methods to calculate these parameters. In the present context a cylindrical algebraic decomposition based method is proposed to model the workspace and joint space. It is a well know feature that this robot admits two operation modes. We are able to find out the set in the joint space with a constant number of solutions for the direct kinematic problem and the locus of the cusp points for the both operation mode. The characteristic surfaces are also computed to define the uniqueness domains in the workspace. A simple 3-RPS parallel with similar base and mobile platform is used to illustrate this method. INTRODUCTION The workspace of parallel robots mainly depends upon the actuated joint variables, the range of motion of the joints and the mechanical interferences between the bodies of mechanism. There are different techniques based on geometric, discretization, numerical and algebraic methods which are used to calculate the workspace of parallel robot. The main advantage of the geometric approach is that, it establish the nature of the boundary of the workspace [1]. Also it allows the computation of the surface and volume of the workspace while being very efficient in terms of storage space, but if the rotational motion is included, it becomes more complex. The interval analysis based method can be used to compute the workspace but the computation time depends on the complexity of the robot and the accuracy requested. The ALIAS library is a good implementation for the parallel robots [2]. Discretization methods are usually less complex and take into account all kinematic constraints, but require more space and computation time for higher resolutions. The majority of numerical methods which is used to determine the workspace of parallel manipulators includes the discretization of the pose parameters for the determination of workspace boundaries [3]. Algebraic methods are used in [4–6] to study planar or spatial parallel robots. Two main teps are necessary to perform the workspace and jointspace analysis. First, the discriminant variety is computed to characterize the boundaries of the workspace and jointspace as well as the singularities. Second, the Cylindrical Algebraic Decomposition (CAD) is used to define the connected regions where there exists a constant number of real solutions to the inverse and direct kinematic problem and no parallel or serial singularities [5–7]. For the design or the trajectory planning, the workspace of the parallel manipulator is divided into singularity-free regions [8]. The singularities divide the workspace into aspects and the characteristic surfaces induce a partition of each aspect into a set of regions (the basic regions) [9]. For the parallel robots with several inverse and direct kinematic solutions, the aspects are de1 Copyright c © 2014 by ASME fined as the maximal singularity-free sets in the workspace or the cross-product of the joint space by the workspace. An assembly mode is associated with a solution for the Direct Kinematic Problem (DKP) and a working mode for the Inverse Kinematic Problem (IKP). Practically, a change of assembly mode may occur during the execution of a trajectory between two configurations in the workspace which are not necessarily associated with the same input for a given working mode. The uniqueness domains can be defined as the maximal regions of the workspace where all the displacements of the end-effector can be accomplished without changing of assembly mode and working mode [5]. The main goal of this paper is to generalize the notions of the uniqueness domains for parallel robot with several operation modes and a single working mode. The outline of the paper is as follows. Firstly, the kinematic equations of the 3-RPS parallel robot under study are introduced. The singular configurations are written for both operationmodes. The cylindrical algebraic decomposition algorithm is presented to explain the study of the workspace and the joint space. Then, the characteristic surface is defined for the parallel robots with several operations. Finally, the basic regions and the basic components are computed to define the uniqueness domains for the direct kinematic problem. KINEMATICS OF 3 RPS The robot under study is the 3-RPS parallel robot with three degrees of freedom and has been studied by many researchers [10, 11]. It is the assembly of two equilateral triangles (the base, moving platform) by three identical RPS legs where R is a revolute passive joint, P an actuated prismatic joint and S a passive spherical joint. Thus, the revolute joint is connected to the fixed base and the spherical joint to the mobile platform.