Group Theoretical Approaches to Vector Parameterization of Rotations

Abstract

Known parametrizations of rotations are derived from the Lie group theoretical point of view considering the two groups ${\rm SO}(3)$ and ${\rm SU}(2)$. The concept of coordinates of the first and second kind for these groups is used to derive the axis and angle as well as the three-angle description of rotation matrices. With the homomorphism of the two groups the Euler parameter description arises from the axis and angle description of ${\rm SU}(2) $. Due to the topology of $ {\rm SO}(3)$ any three-angle description gives only a local parametrization like Euler angles such that the mapping from their time derivatives to the algebra $\frak{so}(3)$, i.e., to the angular velocity tensor, exhibits singularities. All these parametrizations are based on the generation of the respective group by the $\exp$ map from their algebras. Alternatively the Cayley transformation also maps algebra elements to group elements. This fact is well know on ${\rm SO}(3)$ and yields a representation of rotation matrices in terms for Rodrigues parameter, which is, however, not continuous. Generalizing this transformation to ${\rm SU}(2) $ allows for a singularity-free description of all rotations, which does not contain transcendental functions. While in the considered range the exponential map is of class $ C^\infty $ the $\mathrm{cay}$ map on ${\rm SU}(2) $ is only of class $ C^1$ and on ${\rm SO}(3)$ it is not even continuous. Simulation results exemplify the resultant numerical benefits for the simulation of rigid body dynamics. The problem caused by a lack of a continuous transformation from generalized accelerations to angular accelerations can be avoided for rigid body motions using the Bolzmann-Hamel equations.