Integrating Rotations Using Nonunit Quaternions

Abstract

In this letter, we propose and demonstrate a method for integration of rotations using nonunit quaternions. Unit quaternions are commonly used to represent rotation, in which case the rotation operation involves the conjugate of the unit quaternion. However, a redundant mapping can be defined from all nonzero quaternions to the set of rotation matrices, ${\mathrm{SO}(3)}$ , based on the more general rotation operation involving the quaternion inverse. From this we show that the well-known formula that maps angular velocity to the derivative of a unit quaternion actually represents the minimum-norm solution within a set of solutions for the derivative of a nonunit quaternion. This fact enables efficient, singularity free, numerical integration of rotations over long intervals. The approach inherently preserves the structure of ${\mathrm{SO}(3)}$ during the integration with any standard routine or ordinary differential equation (ODE) solver package without employing specially designed geometric integration schemes, exponential updates, or the many quaternion length enforcement techniques found in the literature. We demonstrate the accuracy of this approach compared to other common methods applied to integrate a known angular velocity function and a classic Lagrange top.