Nonlinear Dynamics and Chaotic Motions in Feedback-Controlled Two-and Three-Degree-of-Freedom Robots

Abstract

The dynamics of a feedback-controlled rigid robot is most commonly described by a set of nonlinear ordinary differential equations. In this paper we analyze these equations, representing the feedback-controlled motion of twoand three-degrees-of-freedom rigid robots with rnvolute (R) and prismatic (P) joints in the absence of compliance, friction, and potential energy for the possibility of chaotic motions. We first study the unforced or inertial motions of the robots, and show that when the Gaussian or Riemannian curvature of the configuration space of a robot is negative, the robot equations can exhibit chaos. If the curvature is zero or positive, then the robot equations cannot exhibit chaos. We show that among the two-degrees-of-freedom robots, the PP and the PR robot have zer Gaussian curvature while the RP and RR rohots have negative Gaussian curvatures. For the three-degrees-offireedom robots, we analyze the two well-known RRP and RRR configurations of the Stanford arm and the PUMA manipulator, respectively, and derive the conditions for negative curvature and possible chaotic motions. The criteria of negative curvature cannot be usedfor the forced or feedback-controlled motions. For the foired motion, we resort to the well-known numerical techniques and compute chaos maps, Poincard maps, and bifurcation diagrams. Numerical results are presentedfor the two-degreesf-ffreedom RP and RR robots, and we show that these robot equations can exhibit chaos for low controller gains and for large underestimated models. From the bifurcation diagrams, the mute to chaos appears to he through period doubling.