  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 stiffer faces self-folds
better.
