Fixturing hinged polygons

We study the problem of fixturing a chain of hinged objects in a given placement with frictionless point contacts. We define the notions of immobility and robust immobility - which are comparable to the second and first order immobility for a single object - to capture the intuitive requirement for the fixture of a chain of hinged objects. Robust immobility differs from immobility in that it additionally requires insensitivity to small perturbations of contacts. We show that (p+2) frictionless point contacts can immobilize any chain of p≠3 polygons without parallel edges; six contacts can immobilize any chain of three such polygons. Any chain of p arbitrary polygons can be immobilized with at most (p+4) contacts. We also show that ┌(6/5)(p+2)┐ contacts suffice to robustly immobilize p polygons without parallel edges, and that ┌(5/4)(p+2)┐ contacts can robustly immobilize p≠3 arbitrary polygons, and eight contacts can robustly immobilize three polygons.