Analytic Characterization of a Class of Three-contact Frictional Equilibrium Postures in Three-dimensional Gravitational Environments

Quasistatic legged locomotion over uneven terrains requires characterization of the mechanism’s feasible equilibrium postures. This paper characterizes the feasible equilibrium postures of mechanisms supported by three frictional point contacts in a three-dimensional gravitational environment, for a subclass of contact arrangements, called tame , for which the friction cones lie above the plane spanned by the contacts. The kinematic structure of the mechanism is lumped into a single rigid body Β having the same contacts with the environment and a variable center of mass. The equilibrium postures associated with a given set of contacts become the center-of-mass locations of Β that maintain a feasible equilibrium with respect to gravity. The paper establishes the relations between the feasible equilibrium region and the classical support polygon principle. For tame 3-contact arrangements, the paper identifies and characterizes geometrically three types of boundary curves of the feasible equilibrium region, where two of them are obtained is closed-from, and the third is given implicitly as a solution of a set of nonlinear equations, which can be traced numerically. The three types of boundary curves are then associated with the onset of three different modes of non-static contact motions. Finally, the paper reports on experimental results that verify the theoretical predictions by using a 3-legged prototype.