Many interesting dynamic systems in science and engineering evolve on a nonlinear, or curved space that cannot be globally identified with a linear space. Such nonlinear spaces are referred to as manifolds, and they appear in various mechanical systems, such as a planar pendulum or a complex robotic system. However, geometric structures of a nonlinear manifold have not been carefully incorporated into control system engineering. Conventional nonlinear control systems based on local coordinates of a manifold suffer from singularities, ambiguities, and complexities, thereby severely restricting control performance.
This talk summarizes recent advances in geometric approaches for four major topics in nonlinear dynamics and controls, namely computational mechanics, optimization, feedback control, and estimation. It is shown that aggressive maneuvers of complex dynamic systems can be achieved in an intrinsic and elegant fashion, by constructing control systems directly on a manifold. The desirable properties of geometric mechanics and controls are illustrated by both computational and experimental results of several aerospace and robotic systems, including binary asteroid, formation reconfiguration of satellites, tethered spacecraft, and quadrotor unmanned aerial vehicles.