A Foundation for Analysis of Spherical System Linkages Inspired by Origami and Kinematic Paper Art

Abstract

Origami and its related fields of paper art are known to map to mechanisms, permitting kinematic analysis. Many origami folds have been studied in the context of engineering applications, but a sufficient foundation of principles of the underlying class of mechanism has not been developed. In this work, the mechanisms underlying paper art are identified as “spherical system linkages” and are studied in the context of generic mobility analysis with the goal of establishing a foundation upon which future work can develop.Spherical systems consist of coupled spherical and planar loops, and they motivate a reclassification of mechanisms based on the Chebyshev-Grübler-Kutzbach framework. Spherical systems are capable of complex, closed-loop motion in 3D space despite the mobility calculation treating the links as constrained to a single 2D surface. This property permits generalization of some multi-loop planar mechanisms, such as the Watt mechanism, to a generalized 3D form with equal mobility. A minimal connectivity graph representation of spherical systems is developed, and generic mobility equations are identified. Spherical system linkages are generalized further into spherical/spatial hybrid mechanisms which may have any combination of spherical, planar, and spatial loops. These are represented and analyzed with a polyhedron model. The connectivity graph is modified for this case and appropriate generic mobility equations are identified and adapted.The generic analyses developed for spherical system linkages are sufficient to inform an exhaustive type synthesis process. Generation of all configurations of a paper art inspired mechanism subject to constraints is discussed, and a case study generates all configurations of a spatial chain using specified link types. This design process is enabled by the developed notation and analyses, which are used to identify, depict, and classify kinematic paper art inspired mechanisms