QuateRA: The Quaternion Regression Algorithm

This work proposes a batch solution to the problem of estimating fixed angular velocity using orientation measurements. Provided that the angular velocity remains constant with time, it is shown that the orientation quaternion belongs to a constant plane of rotation as time evolves. Motivated by this fundamental property, the angular velocity's direction can be determined by estimating the quaternion plane of rotation. Under the small angle assumption on the attitude measurement noise, the plane of rotation is estimated by minimizing a constrained total least-squares cost function, and the proposed algorithm produces a unique optimizing solution through a batch approach (no need for iterations). The angular velocity magnitude is estimated by projecting the measured quaternions onto the estimated plane of rotation, and then computing the least-squares evolution of the quaternion angle in the plane. A Monte Carlo analysis of the proposed algorithm is performed, validating the method and comparing it with a multiplicative extended Kalman Filter, which is a traditional method in the literature.