Instantaneous kinematics and singularity analysis of three-legged parallel manipulators Anjan Kumar Dash†, I-Ming Chen†, Song Huat Yeo† and Guilin Yang‡ (Received in Final Form: July 8, 2003) SUMMARY Instantaneous kinematics and singularity analysis of a class of three-legged, 6-DOF parallel manipulators are addressed in this paper. A generic method of derivation of reciprocal screw and consequently, the instantaneous kinematics model is presented. The advantage of this formulation is that the instantaneous kinematics model possesses well-defined geometric meaning and algebraic structure. Singularity analysis is performed under three categories, namely forward, inverse and combined singularities. A new concept of Passive Joint Plane is introduced to correlate the physical structure of the manipulator and these geometric conditions. In the inverse kinematic analysis, a new approach is introduced. At each leg end point a characteristic parallel- epiped is defined whose sides are the linear velocity components from three main joints of the leg. An inverse singularity occurs when the volume of this parallelepiped becomes zero. Examples are demonstrated using RRRS and RPRS-type parallel manipulators. KEYWORDS: Instantaneous kinematics; Singularity analysis; Parallel manipulators. 1. INTRODUCTION Three-legged parallel manipulators unify the advantages of fully parallel manipulators and serial manipulators. Fully parallel manipulators (of the type Stewart-Gough platform mechanisms) have a high rigidity and accuracy, but have a small workspace and suffer from interference of the six legs. Serial manipulators, on the other hand, have a large workspace though it is less rigid and accurate. Three-legged parallel manipulators, combining the features of serial and fully parallel manipulators, have a fairly large workspace, symmetric actuation scheme, sufficient rigidity and reduced branch interference. However, the design, trajectory plan- ning and application development of these manipulators are challenging because of the closed-loop nature of the mechanism. Therefore, they have received attention from many researchers 1–4 . The class of three-legged parallel manipulators, investi- gated here, comprises of a base and a mobile platform connected by three kinematic chains or legs. Each leg has three 1-degrees of freedom (DOF) joints and a spherical joint at the leg-end connecting to the mobile platform. Hence, each leg has 6 DOFs. To have a symmetric actuation scheme, two joints in each leg are actuated. Two prototypes of this class of manipulators, designed and constructed in our laboratory using modularity concept are shown in Figure 1. One of the main concerns in the design of parallel manipulators is the occurrence of kinematic singularities. Singular configurations are particular poses of the end- effector, in which parallel robots lose their inherent rigidity and the end-effector gains or loses degrees of freedom. In other words, the parallel manipulator becomes uncon- trollable upon encountering a singularity pose. Thus, singularity analysis plays an important role in the design of any parallel manipulator. Singularity analysis is performed by analyzing the two matrices relating the instantaneous velocities of the actuators and the end-effector. Derivation of this instantaneous kinematic relationship is the key to determination of singularity poses. These two important issues, namely study of instantaneous kinematics and singularity analysis of this class of parallel manipulators are the objective of this article. Ma and Angeles 5 classified the kinematic singularities of a closed-loop mechanism into three categories: architecture, configuration and formulation singularities. While archi- tecture singularity is related to a particular architecture of a parallel mechanism, formulation singularity is due to a bad scheme of modeling. Configuration singularity is further classified into three different categories namely forward, inverse and combined singularities. 6 To identify configura- tion singularities, the instantaneous relationship between the vectors of the mobile platform velocity (v) and the active joint velocity ( ˙ q) is established as: A v =B˙ q. (1) The first, second and third kind of configuration singular- ities occur respectively when matrix A is singular, matrix B is singular and both matrices are simultaneously singular. Derivation of the relationship as described in Equation (1) is the stepping stone of any singularity analysis. A number of approaches based on the screw theory and line geometry have been proposed to establish this relationship. Mohamed and Duffy 7 introduced a concept of partial twists for instantaneous kinematic analysis. They proposed that for any parallel device, the twist representing the instantaneous motion of the end-effector is equal to the sum of its partial Corresponding author: I-Ming Chen. E-mail: michen@ntu.edu.sg † School of Mechanical and Production Engineering, Nanyang Technological University, Singapore-639798. ‡ Automation Technology Division, Singapore Institute of Manu- facturing Technology, Singapore-638075. Robotica (2004) volume 22, pp. 189–203. © 2004 Cambridge University Press DOI: 10.1017/S0263574703005496 Printed in the United Kingdom