Nonlinear Dynamics of Computer Controlled Machines

Abstract

Position and force-control is a standard mechanical controlling problem, for example in robotics. When unstable equilibria of mechanical systems are stabilized by means of some control, the task can be considered as a position or force control of a system with negative stiffness. Analytical investigation of simple models with 1–2 DOF have a central role in understanding technical phenomena and in forming the common sense in design work. Basic text books often call the attention for the destabilizing effect of sampling in these systems. In this paper, simple closed form formulae for stability limits are presented in basic parameter space of the sampling time, control gains and mechanical parameters. However, the experimentally verified performances of these systems are often worse than expected after the stability analysis of the digitally controlled system. Another digital effect, the quantization have a strong effect on the system behavior. This nonlinearity is very strong locally, and often results in some small amplitude stochastic oscillation of the robot around its desired end position. The paper will show the chaotic nature of these oscillations and some ways to characterize them.