Analytic formulation of the 6-3 fully-parallel manipulator’s singularity determination Raffaele Di Gregorio Department of Engineering, University of Ferrara, Via Saragat, 1; 44100 FERRARA (Italy) (Received in Final Form: March 21, 2001) SUMMARY When a parallel manipulator reaches a singular configura- tion (singularity), the end effect (platform) pose cannot be controlled any longer, and infinite active forces must be applied in the actuated joints to balance the loads exerted on the platform. Therefore, these singularities must be avoided during motion. The first step to avoid them is to locate all the platform poses (singularity locus) making the manip- ulator singular. Hence, the availability of a singularity locus equation, explicitly relating the manipulator geometric parameters to the singular platform poses, greatly facilitates the design process of the manipulator. The problem of determining the platform poses, that make the 6-3 fully- parallel manipulator (6-3 FPM) singular, will be addressed. A simple singularity condition will be written. This singularity condition consists in equating to zero the mixed product of three vectors, that are easy to be identified on the manipulator, and it is geometrically interpretable. The presented singularity condition will be transformed into an equation (singularity locus equation) explicitly containing the geometric parameters of the manipulator and the platform pose parameters making the 6-3 FPM singular. Eventually, the singularity locus equation will be reduced to a polynomial equation by using the Rodrigues parameters to parameterize the platform orientation. This polynomial equation is cubic in the platform position parameters and one of sixth degree in the Rodrigues parameters. KEYWORDS: Parallel manipulators; Kinematics; Mobility analy- sis; Singularity locus 1. INTRODUCTION Reduced workspace, high stiffness and positioning preci- sion are required by many industrial applications of the manipulators. These requirements are matched by parallel architectures much more easily than by the serial ones when a specific task manipulator is designed. Parallel manipulators (PM) feature the end-effector (platform) connected to the frame (base) by a number of kinematic chains (legs). A wide family of PMs is the one collecting PMs with platform and base connected to one another by six legs either of type SPS (S and P stand for spherical pair and prismatic pair respectively) or equivalent to the SPS ones from a kinematic point of view, for example, legs of type UPS (U stands for universal joint). The manipulators of this family are named fully-parallel manipulators (FPM). FPMs have six degrees of freedom (dof). Their active joints are the prismatic pairs and their active joint coordinates are the distances (leg lengths) between the spherical pair centers of each leg. FPMs’ architectures are classified 1 according to the number of distinct spherical pairs joining the legs to the base and to the platform. For example, a FPM having m distinct spherical pairs in the base and n distinct spherical pairs in the platform is named m-n FPM. The most general FPM architecture is the one of the 6-6 FPM (Fig. 1), featuring six distinct spherical pairs both in the base and in the platform. All the other architectures can be derived from the 6-6 FPM’s one by making two or more spherical pairs coincide in the base and/or in the platform. Figure 2 shows the 6-3 FPM (Fig. 2). The 6-3 FPM features six spherical pairs in the base and three double spherical pairs in the platform. The 6-3 FPM architecture has been proposed by Stewart. 2 The kinematic analysis of a manipulator consists of three steps: position analysis, velocity analysis and acceleration analysis. In the FPMs, the position analysis is the determi- nation of the relationship between the leg lengths and the platform poses (positions and orientations). The velocity analysis is the determination of the relationship between the first-order time derivatives of the leg lengths and the velocities of the platform points. The acceleration analysis is the determination of the relationship between the second- order time derivatives of the leg lengths and the accelerations of the platform points. Each of these analyses consist of two problem: the direct one and the inverse one. Fig. 1. The 6-6 fully-parallel manipulator. Robotica (2001) volume 19, pp. 663–667. Printed in the United Kingdom © 2001 Cambridge University Press