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Machine Theory
Mechanism and Machine Theory 43 (2008) 1519–1529
ate/mechmt
Discrete kinematic systems: Basic relations
binatorial kinematics
Ge´za Karsai *
Budapest University of Technology and Economics, Institute of Machine Design, 1111 Budapest, Hungary
Received 21 December 2007; received in revised form 15 January 2008; accepted 18 January 2008
Available online 24 March 2008
Abstract
The study examines the properties of rigid body systems and generally discrete kinematic systems with group -
work theory methods. It formulates the loop law – the constraint equation of closed cyclic subsystems – for discretionary
motion groups, and deduces therefrom a plete collection of formulas for structure, degree of freedom and
degree of constraint. The binatorial kinematics is proposed for this chapter of kinematics, based on correlations
and involved in structural analysis and synthesis, without limiting its scope to specific system types.
Ó 2008 Elsevier Ltd. All rights reserved.
Keywords: Kinematic system; Structure; Combinatorial kinematics; Degree of freedom
1. Introduction
A discrete kinematic system is the collection of a finite number or many but ponents, named
bodies, which are in a kinematic relation with each other. Prototypes include rigid body systems, that is,
mobile mechanisms and stable trusses. This study attempts to formulate characteristically, review, and sum-
marize the basic principles of kinematics, then the structural binatorial correlations of discrete sys-
tems, without being strictly limited to rigid body systems or specific system types.
The theory of mechanisms designates the formula of kinematic degree of freedom (DOF) as a basic relation
of the subject. This formula is known in several forms. The literature review of Gogu [1], for example, pub-
lishes dozens of versions including a critical analysis for each. Essentially, however, there are only two basic
versions exist. The f