On the existence of form-closure configurations on a grid

This paper shows that any polygonal part without parallel edges can be held in form closure by at most four point fingers on two perpendicular lines. The practical implication of this surprising result is that any such part can be held in form closure by at most four point fingers that are constrained to lie on a given regular grid of orthogonal lines-a requirement present in modular fixturing systems. The difficult issue of existence of solutions in modular settings was raised by Zhuang and Goldberg (1996). In contrast to earlier results in this direction, the result in this paper does not require convexity of the part or a lower bound on the part diameter or edge lengths. The paper also considers the extension of the result to parts with parallel edges and to circular fingers with nonzero radius