  To overcome the obstructions imposed by high-dimensional bipedal models, we
embed a stable walking motion in an attractive low-dimensional surface of the
system's state space. The process begins with trajectory optimization to design
an open-loop periodic walking motion of the high-dimensional model and then
adding to this solution, a carefully selected set of additional open-loop
trajectories of the model that steer toward the nominal motion. A drawback of
trajectories is that they provide little information on how to respond to a
disturbance. To address this shortcoming, Supervised Machine Learning is used
to extract a low-dimensional state-variable realization of the open-loop
trajectories. The periodic orbit is now an attractor of the low-dimensional
state-variable model but is not attractive in the full-order system. We then
use the special structure of mechanical models associated with bipedal robots
to embed the low-dimensional model in the original model in such a manner that
the desired walking motions are locally exponentially stable. The design
procedure is first developed for ordinary differential equations and
illustrated on a simple model. The methods are subsequently extended to a class
of hybrid models and then realized experimentally on an Atrias-series 3D
bipedal robot.
