Practical Parameterization of Rotations Using the Exponential Map

Abstract

Paramet erizing three degree-of-freedom (DOF) rotat ions is difficult to do well. Many graphics applications demand that we be able to compute and differentiate positions and orientations of articulated figures with respect to their rotational (and other) parameters, as well as integrate differential equations, optimize rotation parameters, and interpolate orientations. Widely used parameterizations such as Euler angles and quaternions are well suited to only a few of these operations. The exponential map maps a vector in R 3 describing the axis and magnitude of a three-DOF rotation to the corresponding rotation. Several graphics researchers have applied it with limited success to interpolation of orientations, but it has been virtually ignored with respect to the other operations mentioned above. In this paper we present formulae for computing, differentiating, and integrating three-DOF rotations with the exponential map. We show that our formulation is numerically stable in the face of machine precision issues, and that for most applications all singularities in the map can be avoided through a simple technique of dynamic reparameterization. We demonstrate how to use the exponential map to solve both the “free ly rotating body” problem, and the important ball-and-socket joint required to accurately model shoulder and hip joints in art iculated figures. Examining several common graphics applications, we explaint he benefits of our formulation of t he exponential map over Euler angles and quaternions, including robustness, sm all state vectors, lack of explicitconst raints, good modeling capabilities, simplicity of solving ordinary differential equations, and good interpolation behavior.