Robust multiple-contact postures in a two-dimensional gravitational field

We describe a practical method for computing the statically stable postures of a planar mechanism in a two-dimensional gravitational field. Our method lumps the complex kinematic structure of the mechanism into a rigid body B having a variable center of mass, while inertial forces generated by moving parts of the mechanism are lumped into a neighborhood of disturbance wrenches centered at the nominal gravitational wrench. Given this reduction, the statically stable postures of the mechanism correspond to center-of-mass locations of B that guarantee static stability for the neighborhood of wrenches. However, the response of B to an applied wrench involves dynamic ambiguity associated with different reaction modes at the contacts. Hence we compute only those center-of-mass locations where B maintains a non-ambiguous equilibrium for the entire neighborhood of wrenches. These center-of-mass locations are called the robust stability region of the posture. Focusing on two-contact postures, we compute the robust stability region as an arrangement of planar cells. Finally, we sketch an application of the results to robust locomotion planning of a 3-legged mechanism.