| This dissertation explores various problems in the control of the rigid body and related dynamical systems with symmetry, utilizing various modeling approaches and control techniques.; We first derive a control law that asymptotically stabilizes an unbalanced top to the sleeping motion. We rewrite the classical Euler-Poisson equations by projecting the phase space onto {dollar}IRsp5.{dollar} The control law is based on the Hamilton-Jacobi-Bellman theory with zero dynamics and partial stability. Lyapunov techniques are used in the analysis.; Next, the control of rotor imbalance with magnetic bearings is considered in the adaptive virtual autobalancing and adaptive autocentering approaches. We derive single-plane and two-plane balancing control algorithms that provide asymptotic estimates of the rotor imbalance, and that guarantee consistent performance under varying spin rate. These algorithms are based on emulation of the mechanical autobalancer. We discuss the theory based on linear analysis, and simulation and experimental results.; We go on to investigate symmetry properties associated with mechanical control systems and certain nonlinear control systems. First, we generalize the classical Serret-Andoyer transformation for the free rigid body to left-invariant, hyperregular Hamiltonian systems on {dollar}Tsp*SO(3),{dollar} employing the notion of symplectic (Marsden-Weinstein) reduction. We then apply this result to the controlled rigid body, and show that for Hamiltonian controls that preserve the rigid body structure, the generalized Serret-Andoyer transformation yields a two dimensional representation of the closed-loop motion in canonical form. Applications to the stability analysis of relative equilibria and numerical integration are also discussed.; Finally, we apply the concept of reduction to certain regulation problems on smooth manifolds. Following the works of Van der Schaft (1981) and Grizzle and Marcus (1985), we show that an output feedback regulation problem possessing certain symmetries can be reduced to one in the orbit space. In this setting, we obtain an extension of the control design of Isidori and Byrnes (1990). Moreover, we show that the adaptive virtual autobalancing and adaptive autocentering controls fit into this general framework. |
