A New Look to the Three Axes Theorem

Abstract

This paper analyzes the well known three axes theorem under the light of the Lie algebra se(3) of the Euclidean group, SE(3) and the symmetric bilinear forms that can be defined in this algebra. After a brief historical review of the Aronhold-Kennedy theorem and its spatial generalization, the main hypothesis is that the general version of the Aronhold-Kennedy theorem is basically the application of the Killing and Klein forms to the equation that relates the velocity states of three bodies regardless if they are free to move in the space, independent of each other, or they form part of a kinematic chain. Two representative examples are employed to illustrate the hypothesis, one where the rigid bodies are free to move in the space without any connections among them and other concerning a RCCC spatial mechanism.