Numerical Approaches for Constrained and Unconstrained, Static Optimization on the Special Euclidean Group SE(3)

In this paper, rigid body static optimization is investigated on the Riemannian manifold of rigid body motion groups. This manifold, which is also a matrix manifold, provides a framework to formulate translational and rotational motions of the body, while considering any coupling between those motions, and uses members of the special orthogonal group \(\textsf{SO}(3)\) to represent the rotation. Hence, it is called the special Euclidean group \(\textsf{SE}(3)\). Formalism of rigid body motion on \(\textsf{SE}(3)\) does not fall victim to singularity or non-uniqueness issues associated with attitude parameterization sets. Benefiting from Riemannian matrix manifolds and their metrics, a generic framework for unconstrained static optimization and a customizable framework for constrained static optimization are proposed that build a foundation for dynamic optimization of rigid body motions on \(\textsf{SE}(3)\) and its tangent bundle. The study of Riemannian manifolds from the perspective of rigid body motion introduced here provides an accurate tool for optimization of rigid body motions, avoiding any biases that could otherwise occur in rotational motion representation if attitude parameterization sets were used.