Geometric Controls on Nonlinear Manifolds for Complex Aerospace Systems

Many interesting dynamic systems in science and engineering evolve on a nonlinear, or curved, space that cannot be globally identified with a linear vector space. Such nonlinear spaces are referred to as manifolds, and they appear in various mechanical systems such as a pendulum, the attitude dynamics of a rigid body, or complex multibody systems. However, the geometric structures of a nonlinear manifold have not been extensively incorporated in control system engineering. For example, most of the existing nonlinear control system theory is based on nonlinear dynamic systems evolving on a linear space Rn. There are fundamental limitations in applying this traditional nonlinear control theory to dynamic systems evolving on a manifold: stability is guaranteed only near a selected operating condition, or coordinate charts should be switched frequently to avoid singularities. As a result, it is difficult to obtain global stability properties that are uniformly guaranteed for all possible configurations of a dynamic system. Furthermore, an improper characterization of a nonlinear manifold causes undesirable properties, such as ambiguities, sensitivity to measurement errors, or unwinding phenomenon. My research focuses on constructing geometric methods for dynamics and control of mechanical systems evolving on a nonlinear manifold. The central idea is constructing nonlinear control systems directly on nonlinear manifolds, and verifying their performances numerically by using geometric numerical integrators that preserve the underlying geometric properties of a dynamic system. All of these are expressed in a coordinate-free form to avoid any singularities, ambiguities, and complexities inherent to local parameterizations. These allow us to study non-local, large, and aggressive motions of complex dynamic systems globally in an accurate and efficient fashion over a long period. The desirable properties of the proposed geometric approaches have been demonstrated by numerous numerical examples and experiments. More explicitly, the PI has studied the following topics.