Form generation of rigid origami for approximation of a curved surface based on mechanical property of partially rigid frames

Abstract

A method is presented for approximating a curved surface by a developable rigid origami; a polyhedral shape which can be developed to a plane without deformation of its facets. Form generation starts from a triangulated surface, and an optimization problem is solved to obtain a polyhedron which satisfies the geometric conditions for developability. The degrees of freedom of a rigid origami mechanism with only triangular facets are often too large for the engineering application, and therefore, it is sequentially reduced by fixing (removing) some crease lines, along which the rigid facets rotate. However, the crease line that is not fixed often becomes unable to rotate in the process of fixing the crease lines; consequently, the polyhedron cannot be developed to a plane. To avoid such an unfavorable locking situation, selection criteria of the crease line to be fixed are proposed. They are defined based on the eigenvalues and their derivatives of the stiffness matrix of the frame model, which is the numerical model for form generation and mechanism analysis of rigid origami. The performance of the proposed criteria is demonstrated through the examples of surfaces with some patterns of crease lines.