The Lie group of rigid body displacements, a fundamental tool for mechanism design

Mathematical tools classically employed in relativistic mechanics are ignor ed by most of the engineers who are involved in the design of mechanical sy stems. The Lie group algebraic structure of the set of rigid-body displacem ents is a cornerstone for the design of mechanical systems. According to Li e's theory of continuous groups, an infinitesimal displacement is represent ed by an operator acting on the affine points of the 3-dimensional Euclidea n space. This operator includes a field of moments which is classically cal led screw or twist. If a set of possible screws (formerly called a screw sy stem) has a Lie-algebraic structure, we are allowed to take the exponential function of these possible screws, thus obtaining a set of operators that represents all possible finite displacements. This last set has the Lie-gro up structure. It is a subgroup of the 6-dimensional displacement group. A c omprehensive list of Lie subalgebras together with the corresponding Lie su bgroups will be presented. A mechanism is a finite set of rigid bodies with material contact at some p airs of body surfaces, which are called kinematic pairs. The essential prob lem in mechanism analysis is to find a mathematical representation of the c onnection between any pair of bodies when all the kinematic pairs are given by the description of the mechanism in a given initial configuration. It c an be shown that the result can be obtained through two operations: the com position and the intersection of mechanical bonds. The first operation corr esponds to a serial arrangement of kinematic pairs, the second to a paralle l arrangement. The scope of this method will be illustrated with examples of new robotic m anipulators which are capable of producing 3-degrees-of-freedom displacemen ts of a platform. Three limbs connect a fixed frame to a moving plate which undergoes pure translation. Each limb generates a subset of possible displ acements which is a Lie subgroup of Schoenflies motions. The intersection s et is the Lie subgroup of spatial translation. The servomotors are fixed an d may be weighty and bulky and therefore very powerful. The three limbs mak e up a kind of deformable truss which is light and stiff. High speed and ac celeration can be produced with accurate positioning. (C) 1998 Elsevier Sci ence Ltd. All rights reserved.