This paper presents a series of unconstrained Riemannian optimization methods for optimizing over Lie groups describing rigid body attitude and pose, along with novel comparisons between the efficiency and accuracy of those methods across different groups. Requisite operations for Riemannian optimization over the Lie groups of attitude (i.e. the special orthogonal group \(\textsf{SO}(3)\) and the space of quaternions \(\mathbb {H}\) ), and the Lie groups of pose (i.e. the special Euclidean group \(\textsf{SE}(3)\) and the space of dual quaternions \(\mathcal {D}\mathbb {H}\) ) are presented. Cost functions over all these spaces are proposed and analyzed. Riemannian optimization techniques, including unconstrained gradient descent and conjugate gradient algorithms, and several methods of step length determination, are presented and compared against each other. Furthermore, an analysis of the relative efficiency of the Riemannian techniques as compared with Euclidean optimizers constrained to obey structure-preserving criteria is conducted. Case studies, including a classical single shooting optimal controls problem and a pose equilibrium determination scenario, are presented to examine the efficacy of these techniques in a variety of problems. Results suggest that the unconstrained Riemannian optimization methods tend to outperform the conventional, constrained Euclidean optimizers and that certain algorithms may be better suited to specific Lie groups than others.
