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 second and first order immobility for a single object—to capture the intuitive requirement for a 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 frictionless point contacts can immobilize any chain of polygons without parallel edges; six contacts can immobilize any chain of three such polygons. Any chain of arbitrary polygons can be immobilized with at most contacts. We also show that "! contacts suffice to robustly immobilize polygons without parallel edges, and that #