Kinematic analysis of linkages based in finite elements and the geometric stiffness matrix R. Avile ´s, A. Herna ´ndez * , E. Amezua, O. Altuzarra Escuela Te ´cnica Superior de Ingenierı ´a, Alameda de Urquijo s/n, 48013 Bilbao, Spain Received 8 March 2006; received in revised form 19 July 2007; accepted 21 July 2007 Available online 4 September 2007 Abstract This paper presents a numerical approach to rigid body linkage kinematics, based on a reduced form of the stiffness matrix and in structural analysis concepts. This matrix may be referred to as geometric stiffness matrix, or simply as geo- metric matrix. It is derived from basic nodes and length constraints, and provides full information on the kinematic prop- erties of any linkage, including positions, velocities, accelerations, jerks and singular positions. This approach offers a number of major advantages, especially where simplicity and generality are concerned. The computational cost is also very low, because of the simplicity of the numerical calculations and the reduced dimensions of the matrices involved. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Kinematics; Linkage analysis; Multibody; Numerical methods 1. Introduction Methods for kinematic analysis are most often classified according to the type of procedure used to get a solution, e.g. graphical, analytical, and numerical. Alternatively they can be classified according to the nature of the kinematic problem, as when a distinction is drawn between methods used for position problems and others used in the calculation of velocities and accelerations. While the position problem leads to systems of nonlinear equations with multiple solutions; velocities, acceleration, and jerks analyses are linear problems with a unique solution. Many methods found in bibliography can be defined as grapho-analytic, and are based on a geometric approach that leads to analytic solution procedures. These may be divided into different subgroups. The fore- most subgroup comprises dyadic decomposition methods, described in Refs. [1–3], and are useful for the anal- ysis of low complexity mechanisms. The generalization of these methods are known as interpolation procedures [1,4]. Other methods are useful for mechanisms built up with a series of floating four-bar linkages and are based on inversions of the kinematic chain [2,5]. The modular approach methods are based in decomposing 0094-114X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2007.07.007 * Corresponding author. Tel.: +34 946014222; fax: +34 946014215. E-mail address: a.hernandez@ehu.es (A. Herna ´ndez). Available online at www.sciencedirect.com Mechanism and Machine Theory 43 (2008) 964–983 www.elsevier.com/locate/mechmt Mechanism and Machine Theory