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The Lie group of rigid body displacements, a fundamental tool for mechanism design. (English) Zbl 1049.70506

Summary: Mathematical tools classically employed in relativistic mechanics are ignored by most of the engineers who are involved in the design of mechanical systems. The Lie group algebraic structure of the set of rigid-body displacements is a cornerstone for the design of mechanical systems. According to Lie’s theory of continuous groups, an infinitesimal displacement is represented by an operator acting on the affine points of the 3-dimensional Euclidean space. This operator includes a field of moments which is classically called screw or twist. If a set of possible screws (formerly called a screw system) 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-group structure. It is a subgroup of the 6-dimensional displacement group. A comprehensive list of Lie subalgebras together with the corresponding Lie subgroups will be presented.
A mechanism is a finite set of rigid bodies with material contact at some pairs of body surfaces, which are called kinematic pairs. The essential problem in mechanism analysis is to find a mathematical representation of the connection between any pair of bodies when all the kinematic pairs are given by the description of the mechanism in a given initial configuration. It can be shown that the result can be obtained through two operations: the composition and the intersection of mechanical bonds. The first operation corresponds to a serial arrangement of kinematic pairs, the second to a parallel arrangement.
The scope of this method will be illustrated with examples of new robotic manipulators which are capable of producing 3-degrees-of-freedom displacements of a platform. Three limbs connect a fixed frame to a moving plate which undergoes pure translation. Each limb generates a subset of possible displacements which is a Lie subgroup of Schoenflies motions. The intersection set is the Lie subgroup of spatial translation. The servomotors are fixed and may be weighty and bulky and therefore very powerful. The three limbs make up a kind of deformable truss which is light and stiff. High speed and acceleration can be produced with accurate positioning.

MSC:

70B10 Kinematics of a rigid body
70B15 Kinematics of mechanisms and robots
22E70 Applications of Lie groups to the sciences; explicit representations
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