  This paper studies robustness in planar grasping from a geometric
perspective. By treating grasping as a process that shapes the free-space of an
object over time, we can define three types of certificates to guarantee
success of a grasp: (a) invariance under an initial set, (b) convergence
towards a goal grasp, and (c) observability over the final object pose. We
develop convex-combinatorial models for each of these certificates, which can
be expressed as simple semi-algebraic relations under mild-modeling
assumptions. By leveraging these models to synthesize certificates, we optimize
certifiable grasps of arbitrary planar objects composed as a union of convex
polygons, using manipulators described as point-fingers. We validate this
approach with simulations and real robot experiments, by grasping random
polygons, comparing against other standard grasp planning algorithms, and
performing sensorless grasps over different objects.
