Inverse design of surfaces by deployable origami

Origami has attracted broad interest as a tool to achieve complex surfaces through careful patterning and folding of sheets. The rules which govern origami design are purely geometric, so they can be applied across scales from medical stents to robotics and solar sails. Yet, the rules are also delicate and non-linear, so it is challenging to design crease patterns a priori that lead to targeted surfaces on demand. This is the focus of the current work. We develop a sequential two-stage optimization framework for origami design with a marching algorithm that explicitly parameterizes the geometry and kinematics of all possible rigidly and flat-foldable quadrilateral mesh origami. The optimized origami sheets can fold by a single degree-of-freedom motion from an easily manufactured flat state to a compact fully-folded state, attaining targeted shapes along the path of their deployment. The attainable surfaces include those with modest but diverse curvatures and unprecedented ones with sharp ridges. The framework provides a general, efficient and versatile design strategy for shape-morphing with origami.