D-space and deform closure: a framework for holding deformable parts

We extend the form closure framework for rigid parts to holding a class of deformable parts. In this class, a part is a linearly elastic, frictionless polygon with a finite element mesh and given stiffness matrix. We define the D-space (deformation-space) of a part as the C-space of all its mesh nodes. We define the free space as the intersection of the set of topology preserving mesh configurations with the complement of the union of all D-obstacles that represent collisions of the part with finger bodies. Consider a given set of finger bodies in frictionless contact with a part. When positive work is needed to release the part, we say that it is in deform closure. We present a numerical example and prove two results: (1) if contact set holds a rigid part in form closure, it will hold the equivalent deformable part in deform closure; and (2) deform closure is frame invariant.