The Extended Wahba's Problem in Dual and Multi-Dual Algebras

Introduced by Grace Wahba in 1965, the Wahba problem aims to reconstruct the optimal rotation matrix of a spacecraft that minimizes a cost function constructed from a set of measurements of discrete points observed in the frame of a rigid body. This problem is fundamental to various aerospace engineering applications. Over the years, numerous methods have been proposed to solve the generalized forms of the Wahba problem. This paper takes it a step further by extending the Wahba problem to include both pose and the higher-order properties of the general motion of a rigid body. This new problem is closely linked with the parameters that describe the higher-order properties of rigid bodies using dual and multi-dual vectors and tensors. Recent developments highlight the utility of dual and multi-dual tensors as coordinate-free tools for computing the kinematics of higher-order motion parameters. The present research utilizes the homomorphism between the Special Euclidean Lie group, SE (3), and the orthogonal dual and multi-dual tensors Lie groups. This approach enables the development of a novel technique for solving the extended Wahba problem. We present a detailed new procedure for assessing the existence of a solution to this problem. Based on this procedure, we derive closed-form, coordinate-free solutions, highlighting the benefits of working with dual and multi-dual algebra. Applications of this new Wahba problem extend to generalized point-to-point registration (higher-order parameters estimation in motion) and generalized sensor calibration problems in robotics.