Quasi-Static Motions of a Three-Body Mechanism Along a Plane

Abstract

Quasi-static motion of a three-body system along a horizontal plane in the presence of dry friction is considered. The control forces are due to pairwise interaction between the bodies. For the quasi-static motion, the control forces should be chosen so that the equilibrium conditions hold for each body and, hence, for the entire system. The quasi-static motions, for which one of the bodies is moving, while the other two bodies are in a state of rest, are described. It is shown that if the products of the weight of each body by the corresponding friction coefficient satisfy the triangle inequalities, then each body can be quasi-statically moved to any prescribed position in the plane, whereas the other two bodies are at rest. Thus, quasi-static controllability of the system, subject to the aforementioned assumptions about the parameters, is proved. An optimal control problem for the moving body is solved, and the shortest (minimizing the work against friction) trajectory is shown to be a two-link broken line or a straight line segment. An algorithm for transferring the system to a given state is presented. The results obtained can be used for designing control strategies for mobile robotic systems.