A Practical Parameterization of 2 and 3 Degree of Freedom Rotations

This paper presents the RQ (Reparameterized Quaternion) parameterization for three degree of freedom (DOF) rotations as an alternative to quaternions for the applications of differential control and dynamic simulation. Because it requires just three Euclidean parameters instead of the four non-Euclidean parameters of quaternions, RQ is more efficient, less difficult to incorporate into large systems, and able to make use of Euclidean interpolants, such as cubic splines. While RQ does possess singularities in its parameter space, we show how they can be easily avoided for the intended applications. After discussing RQ’s properties and limitations when applied to interpolation and spacetime optimizations, we derive an extension to two DOF rotations, and show how it can be used to model a ball-and-socket joint to a very good approximation.