The Forced Damped Pendulum: Chaos, Complication and Control

Abstract

This paper will show that a “simple ” differential equation modeling a garden-variety damped forced pendulum can exhibit extraordinarily complicated and unstable behavior. While instability and control might at first glance appear contradictory, we can use the pendulum’s instability to control it. Such results are vital in robotics: the forced pendulum is a basic subsystem of any robot. Most of the mathematical methods used in this paper were initially developed in celestial mechanics, largely by Poincaré. The literature of the field [1, 10] tends to be quite advanced indeed; one object of this paper is to show that computer programs, properly used, can make these advanced topics transparent. All the computer-generated pictures in this paper were produced by the programs Planar Systems and Planar Iterations [6], both written by Ben Hinkle (now at Maple). Some parallels in celestial mechanics When I was a graduate student, I was amazed by the results of Alekseev [1], [10] concerning a system formed by three bodies obeying Newton’s law of gravitation. As shown in Figure 1, two massive bodies of equal mass move in a plane P on ellipses symmetric around a common focus F, and the third body, the satellite, of mass zero, moves on the line L perpendicular to P through F. Once this satellite is launched, its motions are determined uniquely by the gravitational pull of the two massive bodies. The system has a natural unit of time, the “year”—the time it takes the massive bodies to complete a revolution. Choose a time zero, so that it makes sense to speak of the 0th, 1st,...,nth year. Also let x denote the position on the line L, with x = 0 corresponding to F. the massive bodies the satellite (mass 0) Figure 1. Alekseev’s three body system.