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/spl ne/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 /spl lceil/(6/5)(p+2)/spl rceil/ contacts suffice to robustly immobilize p polygons without parallel edges, and that /spl lceil/(5/4)(p+2)/spl rceil/ contacts can robustly immobilize p/spl ne/3 arbitrary polygons, and eight contacts can robustly immobilize three polygons.

Authors

• Jae-Sook Cheong
• Ken Goldberg
• Mark H. Overmars
• A. Frank van der Stappen

Keywords

• Fixtures
• Robustness
• Fasteners
• Industrial engineering
• Operations research
• Manufacturing
• Machining
• Resists
• Sufficient conditions
• Planar motors
• Contact Point
• Multiple Edges
• Arbitrary Precision
• Second-order Analysis
• Rigid Parts
• Incident Edges