Computational Geometric Mechanics, Control, and Estimation of Rigid Bodies

Geometric mechanics involves the application of geometric and symmetry techniques to the study of Lagrangian or Hamiltonian mechanics. The goal of computational geometric mechanics is to construct computational algorithms which preserve the geometric properties of mechanical systems [1]. My research is focused on developing computational geometric methods for numerical integration, optimal control, and optimal attitude estimation of rigid bodies. The core idea is constructing computational algorithms from discrete analogues of physical principles, so that the physical properties of the dynamics are preserved naturally in the numerical computations. This is in contrast with the perspective that considers a numerical method as an approximation for a continuous-time equation. In particular, the dynamics of rigid bodies are characterized by Lagrangian or Hamiltonian systems, where the configuration space of each rigid body is a Lie group. I have developed computational methods for rigid bodies, which preserve the underlying geometric structure of rigid body dynamics as well as the Lie group structure of the configuration space. It turns out that the exact geometric properties of the discrete flow not only yield improved qualitative behavior, but also allow for accurate and efficient numerical computations. These geometrically exact but efficient computational methods are applied to optimal control and optimal estimation problems for rigid bodies. The theoretical framework of my research is summarized in the following diagram.