A Computational Method for the Synthesis of Rigid Origami Crease Patterns

Abstract

While origami is an ancient art form, its application in engineering science has only been popularized in recent decades when the scientific community recognized its numerous benefits. The benefits and the widespread applicability of origami are accompanied by a set of geometric and kinematic challenges involving rigid foldability, an exponential number of Rigid Body Modes (RBMs), as well as complex relations between the kinematic determinacy, the number of Degrees-Of-Freedom (DOF), and symmetry. These challenges complicate the adoption of origami principles for scientific purposes, which led to the development of various computational methods in related works that support the application of origami in engineering design tasks. However, most of these methods isolate and address specific challenges and focus on the adaptation of existing crease patterns rather than the design of novel crease patterns, leading to today’s design process that is tedious, time-consuming, and limited to a handful of experienced scientists. This gap motivates the present thesis and defines its objective as the development of a computational method for the synthesis of rigidly foldable crease patterns to support the application of origami in engineering design tasks. The first approach to the computational method is a numerical approach that introduces a new kinematic simulation method with which a manually adapted flasher pattern is analyzed. This analysis contributes by visualizing the search space and by revealing the existence of rigidly foldable regions in the search space of rigid foldability, based on which the flasher pattern is optimized using a stochastic search method. While successful for a single crease pattern topology, the numerical approach is too time-intensive to be scaled, leading to a deeper investigation into analytical kinematics. This investigation yields the Principle of Three Units (PTU) stating that the kinematic behavior of a single vertex is only dependent on its vertex triangle. By applying the triangle inequality, the PTU results in the conditions for the rigid and flat foldability of single degree-n vertices. The corresponding kinematic model enables the assessment of different RBMs and offers an active selection of RBMs to be modeled. In addition, the PTU leads to two guidelines for the generation of kinematically determinate and acyclic crease pattern graphs. The guidelines and conditions arising from the PTU are then embedded within a graph grammar whose rule set consists of two rules. The rule application is automated, leading to a new approach for the computational method that enables the enumeration of the vast search space of origami, the synthesis of novel crease patterns including grippers and robotic arms, and yields the potential to apply the origami principle to yet uncharted territories in engineering design.