Nonlinear Control and Reduction of Underactuated Mechanical Systems with Symmetry

Dr. Reza Olfati-Saber, CDS, Caltech

In this talk we focus on nonlinear control and reduction of underactuated mechanical systems with kinetic symmetry. Underactuated systems are mechanical control systems with fewer controls than configuration variables. Many real-life control systems in Robotics, Aerospace Vehicles, and Marine Vehicles are examples of underactuated systems. Due to their broad applications, control of underactuated systems is currently an active field of research. The examples of underactuated systems include flexible-link robots, mobile robots, walking robots, cars, snake-type/swimming robots, aircraft, spacecraft, helicopters, satellites, surface vessels, and underwater vehicles. Based on recent surveys, control of general underactuated systems is a major open problem.

One of the main contributions of our work is to reduce control design for (possibly) high-order underactuated systems to control of lower-order nonlinear systems that are in cascade with linear systems. We refer to the process of transformation of an underactuated system to a cascade nonlinear system using a change of coordinates and control as ``reduction''.

Based on three basic properties of mechanical systems, we classify underactuated systems to eight classes. For each class, we address the corresponding reduction problem by giving the required transformation in explicit form. In each case, the obtained normal form after applying the transformation is a cascade nonlinear system with structural properties that simplify nonlinear control design for the original underactuated system. These normal forms are cascade systems in strict feedback form, strict feedforward form, and nontriangular quadratic form.

The triangular structure of the first two types of the normal forms allows effective application of the existing backstepping and forwarding approaches to nonlinear control of complex underactuated systems. Examples of systems with triangular normal forms include a helicopter, the acrobot, the 3D cart-pole system, the VTOL aircraft, the planar ducted fan, the TORA system, and a three-link arm with two controls and a missing actuator at the base.

Stabilization of general nontriangular nonlinear systems is an important open problem. In certain special cases, we have addressed this problem. The proposed controller for the nontriangular normal forms of underactuated systems is in the form of a state feedback that is the solution of a fixed-point equation. Underactuated systems with nontriangular normal forms include flexible-link robots, the model of a helicopter with the coupling effects of the attitude dynamics on the translational dynamics, the rotating pendulum, the pendubot, the beam-and-ball system, and a three-link robot with one missing actuator at the elbow.

For special classes of underactuated systems that are differentially flat, the flat outputs are obtained automatically as a by-product of the reduction process (e.g. the VTOL aircraft and an autonomous helicopter).