Controlling the configuration space topology of mechanical structures.

Linkages are mechanical devices constructed from rigid bars and freely rotating joints studied both for their utility in engineering and as mathematical idealizations in a number of physical systems. Recently, there has been a resurgence of interest in designing linkages in the physics community due to the concurrent developments of mechanical metamaterials, topological mechanics, and the discovery of anomalous rigidity in fiber networks and vertex models. These developments raise a natural question: to what extent can the motion of a linkage or mechanical structure be designed? Here, we describe a method to design the topology of the configuration space of a linkage by first identifying the manifold of critical points, then perturbing around such critical configurations. Unlike other methods, our methods are tractable and provide a simple visual toolkit for mechanism design. We demonstrate our procedure by designing a mechanism to gate the propagation of a soliton in a Kane-Lubensky chain of interconnected rotors.

Robust folding of elastic origami.

Self-folding origami, structures that are engineered flat to fold into targeted, three-dimensional shapes, have many potential engineering applications. Though significant effort in recent years has been devoted to designing fold patterns that can achieve a variety of target shapes, recent work has also made clear that many origami structures exhibit multiple folding pathways, with a proliferation of geometric folding pathways as the origami structure becomes complex. The competition between these pathways can lead to structures that are programmed for one shape, yet fold incorrectly. To disentangle the features that lead to misfolding, we introduce a model of self-folding origami that accounts for the finite stretching rigidity of the origami faces and allows the computation of energy landscapes that lead to misfolding. We find that, in addition to the geometrical features of the origami, the finite elasticity of the nearly-flat origami configurations regulates the proliferation of potential misfolded states through a series of saddle-node bifurcations. We apply our model to one of the most common origami motifs, the symmetric "bird's foot," a single vertex with four folds. We show that though even a small error in programmed fold angles induces metastability in rigid origami, elasticity allows one to tune resilience to misfolding. In a more complex design, the "Randlett flapping bird," which has thousands of potential competing states, we further show that the number of actual observed minima is strongly determined by the structure's elasticity. In general, we show that elastic origami with both stiffer folds and less bendable faces self-folds better.

Topological transitions in the configuration space of non-Euclidean origami.

Origami structures have been proposed as a means of creating three-dimensional structures from the micro- to the macroscale and as a means of fabricating mechanical metamaterials. The design of such structures requires a deep understanding of the kinematics of origami fold patterns. Here we study the configurations of non-Euclidean origami, folding structures with Gaussian curvature concentrated on the vertices, for arbitrary origami fold patterns. The kinematics of such structures depends crucially on the sign of the Gaussian curvature. As an application of our general results, we show that the configuration space of nonintersecting, oriented vertices with positive Gaussian curvature decomposes into disconnected subspaces; there is no pathway between them without tearing the origami. In contrast, the configuration space of negative Gaussian curvature vertices remains connected. This provides a new, and only partially explored, mechanism by which the mechanics and folding of an origami structure could be controlled.