A Reparametrization of the Rotation Matrix in Rigid Body Dynamics

Abstract

The parametrization of a rigidbody rotation is a classical subject in rigidbody dynamics. Euler angles, the rotation matrix and quaternions are the most common representations. However, Euler angles are known to be prone to singularities, besides not being frameinvariant. The full 3 أ— 3 rotation matrix conveys all the motion information, but poses the problem of an excessive number of parameters, nine, to represent a transformation that entails only three independent parameters. Quaternions are singularityfree, and thus, ideal to study rigidbody kinematics. However, quaternions, comprising four components, are subject to one scalar constraint, which has to be included in the mathematical model of rigidbody dynamics. The outcome is that the use of quaternions imposes one algebraic constraint, even in the case of mechanically unconstrained systems. An alternative parametrization is proposed here, that (a) comprises only three independent parameters; (b) is fairly robust to representation singularities; and (c) satisfies the quaternion scalar constraint intrinsically. To illustrate the concept, a simple, yet nontrivial case study is included. This is a mechanical system composed of a rigid, toroidal wheel rolling without slipping or skidding on a horizontal surface. The simulation algorithm based on the proposed parametrization and fundamentally on quaternions, together with the invariant relations between the quaternion rate of change and the angular velocity, is capable of reproducing the falling of the wheel under deterministic initial conditions and a random disturbance acting on the tilting axis. Finally, a comparative study is included, on the numerical conditioning of the parametrization proposed here and that based on Euler angles. Ours shows as broader wellconditional region than Euler angles offer. Moreover, the two parametrizations exhibit an outstanding complementarity: where the conditioning of one degenerates, the other comes to rescue.