On the angular velocity of a rigid body: Matrix and vector representations

The angular velocity is an important property associated with the velocity state of moving rigid bodies. Unlike the velocity vector of a point, angular velocity vector is not in general equal to the time derivative of any single vector. Hence a unified, simple and comprehensible treatment of the subject may benefit the velocity analysis of complex multibody systems. This paper contributes with a new point of view of matrix and vector representations of angular velocity from the very foundations of classical kinematics of rigid bodies. This contribution was given a systematic, integrated and unified treatment, thus allowing the derivations to be based upon quantities which are expressed in terms of geometric objects (vectors) and geometric operations (vector addition, dot, and cross product). As a result, the approach leads naturally to simple and particularly useful expressions for the angular velocity vector, which allow a readily extension to three important representations involving the position and velocity of three noncollinear points pertaining to a moving rigid body.