Design and Transformation Control of Triangulated Origami Tessellation: A Network Perspective

Origami is known as a traditional art of paper folding. It has attracted extensive attention due to its self-folding mechanism, shape-morphing capability, and deployable structures. This article develops network-based methods for designing and controlling a three-dimensional (3D) triangulated origami tessellation to approximate multiple surfaces. The desired surfaces are represented by sets of discrete nodes and the origami tessellation to be designed is composed of triangles. Then, the tessellation design problem is formulated as an optimization problem of minimizing the distance between the origami triangle vertices and the discrete nodes subject to developability and rigid-foldability constraints. Solving the resulting optimization problem leads to an origami tessellation with folding states associated with each target surface. To achieve transformation between different shapes, we first leverage graph rigidity theory to define every 3D origami shape uniquely up to translations and rotations. Next, in order to minimize the control efforts, the shape transformation control problem is formulated as an optimal control problem subject to the derived rigidity conditions, whose feasibility is guaranteed by transformability of the origami and controllability of dynamic vertices. Finally, simulation examples for surface approximation are provided to verify the effectiveness of the network-based design and control methods.