In this thesis is initiated a more systematic geometric exploration of energy shaping. Most of the previous results have been dealt wih particular cases and neither the existence nor the space of solutions has been discussed with any degree of generality. The geometric theory of partial differential equations originated by Goldschmidt and Spencer in late 1960s is utilized to analyze the partial differential equations in energy shaping. The energy shaping partial differential equations are described as a fibered submanifold of a $ k $-jet bundle of a fibered manifold. By revealing the nature of kinetic energy shaping, similarities are noticed between the problem of kinetic energy shaping and some well-known problems in Riemannian geometry. In particular, there is a strong similarity between kinetic energy shaping and the problem of finding a metric connection initiated by Eisenhart and Veblen. We notice that the necessary conditions for the set of so-called $ \lambda $-equation restricted to the control distribution are related to the Ricci identity, similarly to the Eisenhart and Veblen metric connection problem. Finally, the set of $ \lambda $-equations for kinetic energy shaping are coupled with the integrability results of potential energy shaping. The procedure shows how a poor design of closed-loop metric can make it impossible to achieve any flexibility in the character of the possible closed-loop potential function. The integrability results of this thesis have been used to answer some interesting questions about the energy shaping. In particular, a geometric proof is provided which shows that linear controllability is sufficient for energy shaping of linear simple mechanical systems. Furthermore, it is shown that all linearly controllable mechanical control systems with one degree of underactuation can be stabilized using energy shaping feedback. The result is geometric and completely characterizes the energy shaping problem for these systems. Using the geometric approach of this thesis, some new open problems in energy shaping are formulated. In particular, we give ideas for relating the kinetic energy shaping problem to a problem on holonomy groups. Moreover, we suggest that the so-called Fakras lemma might be used for investigating the stabilization condition of energy shaping. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-09-02 12:12:55.051

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In this thesis we are going to study the energy shaping problem on controlled Lagrangian systems with degree of underactuation less than or equal to two. Energy shaping is a method of stabilization by designing a suitable feedback control force on the given controlled Lagrangian system so that the total energy of the feedback equivalent system has a non-degenerate minimum at the equilibrium. The feedback equivalent system can then be stabilized by a further dissipative force. Finding a feedback equivalent system requires solving a system of PDEs. The existence of solutions for this system of PDEs is guaranteed, under some conditions, in the case of one degree of underactuation. Higher degrees of underactuation, however, requires a more careful study on the system of PDEs, and we apply the formal theory of PDEs to achieve this purpose in the case of two degrees of underactuation. The thesis is divided into four chapters. First, we review the basic notion of energy shaping and state the results for the case of one degree of underactuation. We then devise a general scheme to solve the energy shaping problem with degree of underactuation equal to one, together with some examples to illustrate the general procedure. After that we review the tools from the formal theory of PDEs, as a preparation for solving the problem with two degrees of underactuation. We derive an equivalent involutive system of PDEs from which we can deduce the existence of solutions which suit the energy shaping requirement.

In this thesis we are going to study the energy shaping problem on controlled Lagrangian systems with degree of underactuation less than or equal to two. Energy shaping is a method of stabilization by designing a suitable feedback control force on the given controlled Lagrangian system so that the total energy of the feedback equivalent system has a non-degenerate minimum at the equilibrium. The feedback equivalent system can then be stabilized by a further dissipative force. Finding a feedback equivalent system requires solving a system of PDEs. The existence of solutions for this system of PDEs is guaranteed, under some conditions, in the case of one degree of underactuation. Higher degrees of underactuation, however, requires a more careful study on the system of PDEs, and we apply the formal theory of PDEs to achieve this purpose in the case of two degrees of underactuation. The thesis is divided into four chapters. First, we review the basic notion of energy shaping and state the results for the case of one degree of underactuation. We then devise a general scheme to solve the energy shaping problem with degree of underactuation equal to one, together with some examples to illustrate the general procedure. After that we review the tools from the formal theory of PDEs, as a preparation for solving the problem with two degrees of underactuation. We derive an equivalent involutive system of PDEs from which we can deduce the existence of solutions which suit the energy shaping requirement.

The method of controlled Lagrangians is an energy shaping control technique for underactuated Lagrangian systems. Energy shaping control design methods are appealing as they retain the underlying nonlinear dynamics and can provide stability results that hold over larger domain than can be obtained using linear design and analysis. The objective of this dissertation is to identify the control challenges in applying the method of controlled Lagrangians to practical engineering problems and to suggest ways to enhance the closed-loop performance of the controller. This dissertation describes a procedure for incorporating artificial gyroscopic forces in the method of controlled Lagrangians. Allowing these energy-conserving forces in the closed-loop system provides greater freedom in tuning closed-loop system performance and expands the class of eligible systems. In energy shaping control methods, physical dissipation terms that are neglected in the control design may enter the system in a way that can compromise stability. This is well illustrated through the "ball on a beam" example. The effect of physical dissipation on the closed-loop dynamics is studied in detail and conditions for stability in the presence of natural damping are discussed. The control technique is applied to the classic "inverted pendulum on a cart" system. A nonlinear controller is developed which asymptotically stabilizes the inverted equilibrium at a specific cart position for the conservative dynamic model. The region of attraction contains all states for which the pendulum is elevated above the horizontal plane. Conditions for asymptotic stability in the presence of linear damping are developed. The onlinear controller is validated through experiments. Experimental cart damping is best modeled using static and Coulomb friction. Experiments show that static and Coulomb friction degrades the closed-loop performance and induces limit cycles. A Lyapunov-based switching controller is proposed and successfully implemented to suppress the limit cycle oscillations. The Lyapunov-based controller switches between the energy shaping nonlinear controller, for states away from the equilibrium, and a well-tuned linear controller, for states close to the equilibrium. The method of controlled Lagrangians is applied to vehicle systems with internal moving point mass actuators. Applications of moving mass actuators include certain spacecraft, atmospheric re-entry vehicles, and underwater vehicles. Control design using moving mass actuators is challenging; the system is often underactuated and multibody dynamic models are higher dimensional. We consider two examples to illustrate the application of controlled Lagrangian formulation. The first example is a spinning disk, a simplified, planar version of a spacecraft spin stabilization problem. The second example is a planar, streamlined underwater vehicle. / Ph. D.