  The geometric, aesthetic, and mathematical elegance of origami is being
recognized as a powerful pathway to self-assembly of micro and nano-scale
machines with programmable mechanical properties. The typical approach to
designing the mechanical response of an ideal origami machine is to include
mechanisms where mechanical constraints transform applied forces into a desired
motion along a narrow set of degrees of freedom. In fact, to date, most design
approaches focus on building up complex mechanisms from simple ones in ways
that preserve each individual mechanism's degree of freedom (DOF), with
examples ranging from simple robotic arms to homogenous arrays of identical
vertices, such as the well-known Miura-ori. However, such approaches typically
require tight fabrication tolerances, and often suffer from parasitic
compliance. In this work, we demonstrate a technique in which
high-degree-of-freedom mechanisms associated with single vertices are
heterogeneously combined so that the coupled phase spaces of neighboring
vertices are pared down to a controlled range of motions. This approach has the
advantage that it produces mechanisms that retain the DOF at each vertex, are
robust against fabrication tolerances and parasitic compliance, but
nevertheless effectively constrain the range of motion of the entire machine.
We demonstrate the utility of this approach by mapping out the configuration
space for the modified Miura-ori vertex of degree 6, and show that when strung
together, their combined configuration spaces create mechanisms that isolate
deformations, constrain the configuration topology of neighboring vertices, or
lead to sequential bistable folding throughout the entire origami sheet.
