The parametrization of a rigid-body rotation is a classical subject in rigid-body 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 frame-invariant. The full 3 × 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 singularity-free, and thus, ideal to study rigid-body kinematics. However, quaternions, comprising four components, are subject to one scalar constraint, which has to be included in the mathematical model of rigid-body 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 well-conditional region than Euler angles offer. Moreover, the two parametrizations exhibit an outstanding complementarity: where the conditioning of one degenerates, the other comes to rescue.
A Reparametrization of the Rotation Matrix in Rigid-Body Dynamics
and Control Engineering,
Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 22, 2014; final manuscript received March 6, 2015; published online March 31, 2015. Assoc. Editor: Alexander F. Vakakis.
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Zhu, X., and Angeles, J. (May 1, 2015). "A Reparametrization of the Rotation Matrix in Rigid-Body Dynamics." ASME. J. Appl. Mech. May 2015; 82(5): 051003. https://doi.org/10.1115/1.4030006
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