An eigenproblem approach to classical screw theory Sandipan Bandyopadhyay a, * , Ashitava Ghosal b a Department of Engineering Design, Indian Institute Technology – Madras, Chennai 600 036, India b Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India article info Article history: Received 6 December 2007 Received in revised form 25 July 2008 Accepted 26 July 2008 Available online 16 October 2008 abstract This paper presents a novel algebraic formulation of the central problem of screw theory, namely the determination of the principal screws of a given system. Using the algebra of dual numbers, it shows that the principal screws can be determined via the solution of a generalised eigenproblem of two real, symmetric matrices. This approach allows the study of the principal screws of the general two-, three-systems associated with a manipulator of arbitrary geometry in terms of closed-form expressions of its architecture and configura- tion parameters. We also present novel methods for the determination of the principal screws for four-, five-systems which do not require the explicit computation of the recipro- cal systems. Principal screws of the systems of different orders are identified from one uni- form criterion, namely that the pitches of the principal screws are the extreme values of the pitch. The classical results of screw theory, namely the equations for the cylindroid and the pitch-hyperboloid associated with the two- and three-systems, respectively have been derived within the proposed framework. Algebraic conditions have been derived for some of the special screw systems. The formulation is also illustrated with several examples including two spatial manipulators of serial and parallel architecture, respectively. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction The theory of screws has been used to analyse finite and instantaneous motions of rigid bodies over the past few centu- ries. One of the major applications of screw theory has been in describing the instantaneous motion of a rigid body in terms of the principal screws, which form a canonical basis of the motion space. The elements of screw theory emerged from the works of Mozzi [1] (1763 AD), Cauchy, and Chasles [2] (1830 AD). However, in 1900, Ball [3] established formally the theory of screws and applied it to the analysis of rigid-body motions of multiple degrees-of-freedom. The fundamental concepts of classical screw theory, including the principal screws of a screw system, the cylindroid, the pitch-hyperboloid, and the rec- iprocity of screws were introduced in his treatise. In 1976, Hunt [4] rejuvenated screw theory from geometric considerations and applied it to the analysis and synthesis of mechanisms. The determination of the principal screws of a given system has attracted a significant amount of research ever since Hunt’s contribution. It is well known that for the general two-degrees-of-freedom rigid-body motion, the instantaneous screw axis (ISA) lies on a ruled surface (known as the cylindroid) generated by the two principal screws of the system. For general three-degrees-of-freedom rigid-body motion, the three principal screws meet orthogonally at a point. The points on the resultant ISA having the same pitch lie on a quadratic surface, known as the pitch-hyperboloid. Hunt [4], Gibson and Hunt [5] have also discussed the special cases of various screw systems and proposed a classification of various screw systems. 0094-114X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2008.07.006 * Corresponding author. Tel.: +91 4422574733. E-mail address: sandipan@iitm.ac.in (S. Bandyopadhyay). Mechanism and Machine Theory 44 (2009) 1256–1269 Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt