Degenerate mechanical systems for tracking control and prevention of vehicle roll-over and lane departure

Mechanical systems, from robot manipulators to automobiles, share a common structure that has been recognized and studied for more than a century. One useful feature of such systems is the asymptotic stability of the minimum of the potential energy of a fully damped mechanical system, which guarantees that a marble placed in a bowl will always come to rest at the bottom. This behavior is closely related to geometric structures arising from a Riemannian metric defined by the kinetic energy. Many strategies for set point regulation have been developed to exploit this behavior by using the control inputs to make the closed loop system a mechanical system whose potential energy is at a minimum at the desired set point. This thesis describes a strategy for tracking a desired trajectory for fully actuated systems by using the control inputs to make the closed loop system behave like a mechanical system. The main difficulty that arises is the need for the closed loop "kinetic energy" to be zero when the actual trajectory is tracking the desired trajectory, even if the velocity is non-zero. To accomodate this need, a framework is developed of "degenerate mechanical systems" -- systems which have the important structures of mechanical systems, but whose kinetic energy is degenerate. Stability results analogous to the well-known stability results of mechanical systems are developed for these "degenerate mechanical systems". This leads to a control strategy of which many existing tracking control techniques are special cases. Most of the existing control techniques, however, depend on a bound on the velocity of the desired trajectory to guarantee convergence. This requirement is fairly innocuous because it is unlikely for a desired trajectory to have unbounded velocities, but the guarantees for the convergence rate and disturbance rejection performance would depend on the bounds on the velocity. The framework used here leads to new insights about how this dependence can be eliminated. The framework of "degenerate mechanical systems" can also be used to develop velocity estimators using position feedback, which can be combined with full state feedback tracking controllers to produce tracking controllers requiring only position feedback. The same framework can be used in designing controllers for mechanical systems for which motion in certain directions should not be penalized. An example of such a situation is the problem of preventing vehicles from roll-over, which presents the additional challenge of being underactuated. Some propositions are developed concerning the possibility of implementing the desired closed loop dynamics when the system is underactuated, and are applied to the task of preventing vehicle roll-over and road departure.