A Geometric PID Control Framework for Mechanical Systems

TL;DR

This paper develops a geometric PID control framework tailored for various mechanical systems, extending classical PID control to non-Euclidean spaces and demonstrating its effectiveness through simulations and experiments in robotics.

Contribution

It introduces a geometric PID controller for fully and under actuated mechanical systems, including feedback regularization and applications to systems on Lie groups.

Findings

Robust almost-global stability for multi-rotor aerial vehicles

Semi-almost-global exponential tracking of spherical robots

Effective control of nonholonomic systems on Lie groups

Abstract

These lectures demonstrate the development of a PID control framework for mechanical systems. Based on the observation that mechanical systems are essentially double integrator systems, we generalize the linear PID controller to mechanical systems that have a non-Euclidean configuration space. Specifically we start by presenting the development of the geometric PID controller for fully actuated mechanical systems and then extend it to a class of under actuated interconnected mechanical systems of practical significance by introducing the notion of feedback regularization. We show that feedback regularization is the mechanical system equivalent to partial feedback linearization. We apply these results for trajectory tracking for several systems of interest in the field of robotics. First, we demonstrate the robust almost-global stability properties of the geometric PID controller developed for fully actuated mechanical systems using simulations and experiments on a multi-rotor-aerial-vehicle. The extension to the class of under actuated interconnected systems allow one to ensure the semi-almost-global locally exponential tracking of the geometric center of a spherical robot on an inclined plane of unknown angle of inclination. The results are demonstrated using simulations for a hoop rolling on an inclined plane and then for a sphere rolling on an inclined plane. The final extension that we present here is that of geometric PID control for holonomically or non-holonomically constrained mechanical systems on Lie groups. The results are demonstrated by ensuring the robust almost global locally exponential tracking of a nontrivial spherical pendulum.