Computing 3-legged equilibrium stances in three-dimensional gravitational environments

Quasistatic multi-legged locomotion consists of a sequence of equilibrium postures where the mechanism supports itself against gravity while moving free limbs to new positions. A posture maintains equilibrium if the contacts can passively support the mechanism against gravity. This paper is concerned with identifying and computing equilibrium postures for mechanisms supported by frictional contacts in a three-dimensional gravitational field. The complex kinematic structure of the mechanism is lumped into a single rigid body B having the same contacts with the environment and a variable center of mass. The identification of equilibrium postures associated with a given set of contacts is reduced to the identification of center-of-mass locations of B that maintain equilibrium stances while all nonlinear frictional constraints are satisfied. Focusing on 3-contact stances, this paper provides an exact formulation for the boundary of the center-of-mass feasible region R as a solution of implicit polynomials. The paper also provides a conservative polyhedral approximation of R by using techniques for projection of convex polytopes. The exact and approximate computations of R are demonstrated with graphical examples