On the Commutativity of Sequential Finite Rotations

Abstract

Non-commutativity of finite rotations is a fundamental issue that needs to be addressed in mathematical models dealing with sequential finite rotations. Comprehending rotational motions required to orient a rigid link or associated reference frames in a three-dimensional space can be quite challenging for beginning students. In this paper we present the problem of non-commutativity of finite rotations and a constraint to allow their commutativity. The constraint, in the form of a chain rule, consists of a lemma for forward and backward propagation of the prescribed rotations. The proposed scheme has been implemented using a commercial graphics package for visual demonstration of a rigid body going through three identical finite rotations but in different sequences. The unique final orientation of the body in each of the cases confirms the commutative nature of the constrained rotations. Using this method, sequential finite rotation becomes easy to comprehend and can be used as a tool for learning the concepts associated with finite rotations as well as for kinematical analysis involving sequential rotations.