An algorithm for stopping a class of underactuated nonlinear mechanical robotic systems

We provide a constructive global discontinuous control law with state dependent switches for a class of under-actuated nonlinear mechanical robotic systems that will drive the system to an arbitrarily small neighborhood of rest from all initial configurations and velocities in arbitrarily small time. Because all physical mobile robotic systems are mechanical in nature, control methodologies which exploit the fact that the system is governed by principles of mechanics which are particularly important for robotic engineers. The philosophy of the approach is that instead of using control algorithms which start with a completely generic dynamical system, we constrain the structure of the system to be one which is a Lagrangian control system. To the extent the structure of the mechanical system can be exploited, stronger control results are possible to obtain, such as the stopping algorithm in this paper. Specifically, for control of general nonlinear systems, there are many unsolved problems for the case when the system is not at an equilibrium, and the results in this paper are an initial contribution to this area. The robot is assumed to be underactuated by one in the configuration space; hence, in the state space it is underactuated by twice the dimension of the configuration space plus two. Our method can easily be extended to construct a global discontinuous control law with state dependent switches that will drive the system to an arbitrarily small neighborhood of any velocity from any initial configuration and velocity in arbitrarily small time.