A direct Lyapunov approach to stabilization and tracking of underactuated mechanical systems

Abstract

Mechanical systems play an integral part in our everyday lives. A subset of these systems can be described as underactuated; the defining characteristic of underactuated mechanical systems is that they have fewer control inputs than degrees of freedom. Airplanes, rockets, helicopters, overhead crane loads, surface vessels, and underwater vehicles are all examples of such systems. The control challenges associated with these systems arise from both the underactuation of the control input and the nonlinear nature of the dynamic equations describing the system’s motion. In this work, a control method for stabilization and tracking based on Lyapunov stability theory is presented. The remarkable result of this tracking controller development is that we arrive at three matching equations that are (with the exception of ) identical to matching equations developed for stabilization as shown in White et al. (2006, 2007, 2008). Asymptotic stabilization of the tracking errors (s) is not obtained. However, the norm of s (||s||) will decrease until an ultimate bound is reached, then it will stay within this bound. A lemma is provided for estimating this bound and it is shown that the magnitude of the bound depends upon the eigenvalues and norms of certain matrices in the Lyapunov formulation. Three examples are presented to illustrate the effectiveness of the direct Lyapunov approach. Two examples of holonomic systems are presented. The first is an inverted pendulum cart which is used to illustrate the formulations performance to tracking a desired path on the cart position or actuated axis. The second example is a ball and beam system in which a desired path is tracked by the ball or unactuated axis. The tracking control technique is also applied to an example of a nonholonomic system, a rolling wheel. The control technique is applied in two alternate manners. Finally, the controller is implemented on a laboratory inverted pendulum cart system in hard real time. A desired trajectory for the cart position is tracked and the control law is used to define the desired pendulum trajectory.