Control of Nonlinear Underactuated Systems

In this paper we introduce a new method to design control laws for non-linear un-deractuated systems. Our method will often work in situations where standard control techniques fail. Even when standard techniques apply, we believe that our approach will achieve better performance. This is because we produce an infinite dimensional family of control laws, whereas most control techniques only produce a finite dimensional family of control laws. We will describe the problem and our solution invariantly, using differential geometry and in local coordinates. We hope that this paper will be useful for both mathematicians and engineers. We include an abstract example of a system which is open-loop unstable and cannot be stabilized using any linear control law, and demonstrate that our method produces a stabilizing control law. We also apply our method to the inverted pendulum cart to compare it with some standard control techniques. An important problem in control theory is how to modify a system of ordinary differential equations so that solutions to the equations satisfy some property. In order to