Characteristics of the Screw Transformation Matrix and Their Effect on Chasles’ Motion

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 screw transformation matrix in the form of either 3×3 dual number matrix or 6×6 transformation matrix that is executed with rotation and translation. This paper investigates characteristics of the screw transformation matrix, and decomposes the dual part of the transformation matrix into the part with an equivalent translation due to the effect of moving rotation axis and the part resulting from a pure translation. New results are presented and new formulae are generated. The analysis further reveals two new traces of the transformation matrix and presents the relation between the screw transformation matrix and the instantaneous screw, leading to the understanding of Chasles’ motion embedded in a normal body transformation. An algebraic and geometric interpretation of the screw transformation matrix is thus given, presenting an intrinsic property of the screw transformation matrix in relation to the finite screw. The paper ends with a case study to verify the theory and illustrate the principle.