Optimization on the space of rigid and flexible motions: an alternative manifold optimization approach

In this paper we consider the problem of minimization of a cost function that depends on the location and poses of one or more rigid bodies, or bodies that consist of rigid parts hinged together. We present a unified setting for formulating this problem as an optimization on an appropriately defined manifold for which efficient manifold optimizations can be developed. This setting is based on a Lie group representation of the rigid movements of a body that is different from what is commonly used for this purpose. We illustrate this approach by using the steepest descent algorithm on the manifold of the search space and specify conditions for its convergence.