Multi-configuration rigidity: Theory for statically determinate structures

This paper introduces the concept of multi-configuration rigidity for kinematically indeterminate structures with elastic springs and unilateral constraints. A simple example is provided by a structure with a single mechanism and a spring that engages two different unilateral constraints. In each of these configurations, the structure can rigidly support loads up to a critical magnitude at which the unilateral constraints become inactive. The general design problem of embedding springs throughout a structure to achieve multi-configuration rigidity, with multiple unilateral constraints and springs, is studied. This problem is cast as a linear program that maximizes the critical loads required to break free from the unilateral constraints, in all target configurations. This problem can be efficiently solved with guarantees of optimality. The formulation is generally applicable to a variety of discrete structures (e.g., linkages, pin-jointed bars, or origami) with unilateral constraints (e.g., contacts or cables).