Finite Displacement Screw Operators With Embedded Chasles’ Motion

Abstract

Rigid body displacement can be presented with Chasles’ motion by rotating about an axis and translating along the axis. This motion can be implemented by a finite displacement screw operator in the form of either a 3 × 3 dual-number matrix or a 6 × 6 matrix that is executed with rotation and translation as an adjoint action of the Lie group. This paper investigates characteristics of this finite displacement screw matrix and decomposes the secondary part that is the off diagonal part of the matrix into the part of an equivalent translation due to the effect of off-setting the rotation axis and the part of an axial translation. The paper hence presents for the first time the axial translation matrix and reveals its property, leading to discovery of new results and new formulae. The analysis further reveals two new traces of the matrix and presents the relationship between the finite displacement screw matrix and the instantaneous screw, leading to the understanding of Chasles’ motion embedded in a rigid body displacement. An algebraic and geometrical interpretation of the finite displacementscrew matrix is thus given, presenting an intrinsic property of the matrix in relation to the finite displacement screw. The paper ends with a case study to verify the theory and illustrate the principle.