  From the complex motions of robots to the oxygen binding of hemoglobin, the
function of many mechanical systems depends on large, coordinated movements of
their components. Such movements arise from a network of physical interactions
in the form of links that transmit forces between constituent elements.
However, the principled design of specific movements is made difficult by the
number and nonlinearity of interactions. Here, we model mechanical systems as
linkages of rigid bonds (edges) connected by joints (nodes), and formulate a
simple but powerful framework for designing full nonlinear coordinated motions
using concepts from dynamical systems theory. We begin with principles for
designing finite and infinitesimal motions in small modules, and show that each
module is a one-dimensional map between distances across pairs of nodes. Next,
we represent the act of combining modules as an iteration of this map, and
design networks whose geometries reflect the map's fixed points, limit cycles,
and chaos. We use this representation to design different folding sequences
from a deployable network and a soliton, to a branched network acting as a
mechanical AND gate. Finally, we design large changes in curvature of the
entire network, and construct physical networks from laser-cut acrylic,
origami, and 3D printed material to demonstrate the framework's potential and
versatility for designing the full conformational trajectory of morphing
metamaterials and structures.
