Asymptotic Stability and Chaotic Motions in Trajectory Following Feedback Controlled Robots

Abstract

A feedback controlled robot manipulator with positive controller gains is known to be asymptotically stable at a set point and for trajectory following in the sense of Lyapunov. However, when the end-effector of a robot or its joints are made to follow a time-dependent trajectory, the nonlinear dynamical equations modeling the feedback controlled robot can also exhibit chaotic motions and as a result cannot follow a desired trajectory. In this paper, using the example of a simple two-degree-of-freedom robot with two rotary (R) joints, we take a relook at the asymptotic stability of a 2R robot following a desired time-dependent trajectory under a proportional plus derivative (PD) and a model-based computed torque control. We demonstrate that the condition of positive controller gains is not enough and the gains must be large for chaos not to occur and for the robot to asymptotically follow a desired trajectory. We apply the method of multiple scales (MMS) to the two nonlinear second-order ordinary differential equations (ODEs), which describes the dynamics of the feedback controlled 2R robot, and derive a set of four first-order slow flow equations. At a fixed point, the Routh–Hurwitz criterion is used to obtain values of proportional and derivative gains at which the controller is asymptotically stable or indeterminate. For the model-based control, a parameter representing model mismatch is used and the controller gains for a chosen mismatch parameter value are obtained. From numerical simulations with controller gain values in the indeterminate region, it is shown that for some values, the nonlinear dynamical equations are chaotic, and hence, the 2R robot cannot follow the desired trajectory and be asymptotically stable.