| An overarching and unifying theme for this document is that viewing mechanical systems through a geometric lens opens up an extensive set of tools that can be brought to bear upon energy, mass, and system---conscious control design for constrained mechanical systems. To demonstrate this thesis we consider the dynamics and control for several mechanical systems.; The moving mass Chaplygin sleigh, a rigid platform and moving mass system with attached blade imposing nonholonomic (velocity) constraints, when viewed through a geometric lens presents a nonholonomic momentum equation. Analysis of this momentum equation reveals natural (uncontrolled) motions of the sleigh system that play a central role in our control design to steer the sleigh to any point in the plane using a moving mass.; We develop a geodesic-based, proportional-derivative (PD) control logic for tracking on a class of Riemannian manifolds. As a specific application of this general control logic, we consider the double gimbal system, a mechanical system comprised of a base, an outer gimbal attached to the base through a revolute joint, and an inner gimbal also attached to the outer gimbal through a revolute joint. Simply stated, our specific application is to develop a control logic for pointing a telescope from an initial pointing direction to a desired pointing direction.; In an n-symplectic (generalized Hamiltonian) setting, kinetic energy dynamics are formulated on the frame bundle of the configuration manifold for a mechanical systems. By adapting the frame bundle dynamics to the constraint distribution (that is, by an appropriate choice of moving frame) a portion of the constrained generalized momenta dynamics are an n-symplectic version of the nonholonomic momentum equation. These general dynamics have been carried out for the simple examples of the vertical rolling hoop and a nonholonomic constrained particle. Preliminary work along the n-symplectic line of thought indicates the possiblity of potential shaping and momenta based control design for nonholonomic mechanical systems. |
