Spatial Compliant Motion of a Rigid Body Constrained by a Frictional Contact

In this paper, we study the quasi-static motion of an elastically suspended, unilaterally constrained rigid body. The motion of the rigid body is determined, in part, by the position controlled motion of its support base and by the behavior of the elastic suspension that couples the part to the support. The motion is also determined, in part, by contact with a frictional surface that both couples the rigid body to the unilateral constraint and generates a friction force. The unknown friction force, however, is determined in part by the unknown direction of the rigid-body motion. We derive an analytically solvable set of equations that simultaneously determines both the friction force and the resulting rigid-body motion. We also address the issues of whether a solution to these equations exists and whether the obtained solution is unique. We show that, for any passive compliant system in which the nominal motion imposes contact, a solution to the set of motion equations always exists. We also show that, for any passive system with an upper bounded friction coefficient, the solution is unique. Two sufficient conditions that guarantee the uniqueness of the solution are presented