Point-based Jacobian formulation for computational kinematics of manipulators O. Altuzarra, O. Salgado, V. Petuya, A. Herna ´ndez * Department of Mechanical Engineering, Faculty of Engineering of Bilbao, Alameda de Urquijo, s/n. Bilbao 48013, Spain Received 29 December 2005; received in revised form 17 January 2006; accepted 22 January 2006 Available online 20 March 2006 Abstract Computational kinematic analysis of mechanisms is a promising tool for the development of new classes of manipula- tors. In this paper, the authors present a velocity equation to be compiled by general-purpose software and applicable to any mechanism topology. First, the approach to model the mechanism is introduced. The method uses a set of kinematic restrictions applied to characteristic points of the mechanism. The resultant velocity equation is not an input–output equa- tion, but a comprehensive one. In addition, the Jacobian characterizing the equation is dimensionless hence extremely use- ful for singularity and performance indicators. The motion space of the manipulator is obtained from the velocity equation. Angular velocities are compiled out of three non-collinear point velocities. The procedure is applied to a 3- DoF parallel manipulator to illustrate. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Velocity equation; Dimensionless Jacobian; Motion space; Parallel manipulators 1. Introduction The velocity equation is the fundamental tool to solve several problems related to the analysis, design and control of manipulators. It provides velocities and accelerations, manipulability and dexterity indices, singular- ity analysis and force transmission, all of them making use of the Jacobian. This implies that the velocity equa- tion must be formulated so that it could be useful to tackle all these problems globally with maximum efficiency. Formulation of the velocity equation can be done using either conventional or computational methods. First ones are characterized by the fact that their application is highly dependent on the manipulator topology. Among them, the method of screw coordinates and the velocity vector-loop method can be highlighted; both are well described in [1]. The resultant Jacobians are up to 6 · 6 matrices in mixed terms of linear and angular velocities that relate only input and output variables. Regarding computational methods, they are general-pur- pose methods that are applicable to any manipulator topology using matrix computation. Up to now, the 0094-114X/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2006.01.011 * Corresponding author. Tel.: +34 94 6014222; fax: +34 94 6014215. E-mail address: a.hernandez@ehu.es (A. Herna ´ndez). Mechanism and Machine Theory 41 (2006) 1407–1423 www.elsevier.com/locate/mechmt Mechanism and Machine Theory