  Models of bipedal locomotion are hybrid, with a continuous component often
generated by a Lagrangian plus actuators, and a discrete component where leg
transfer takes place. The discrete component typically consists of a locally
embedded co-dimension one submanifold in the continuous state space of the
robot, called the switching surface, and a reset map that provides a new
initial condition when a solution of the continuous component intersects the
switching surface. The aim of this paper is to identify a low-dimensional
submanifold of the switching surface, which, when it can be rendered invariant
by the closed-loop dynamics, leads to asymptotically stable periodic gaits. The
paper begins this process by studying the well-known 3D Linear Inverted
Pendulum (LIP) model, where analytical results are much easier to obtain. A key
contribution here is the notion of \textit{self-synchronization}, which refers
to the periods of the pendular motions in the sagittal and frontal planes
tending to a common period. The notion of invariance resulting from the study
of the 3D LIP model is then extended to a 9-DOF 3D biped. A numerical study is
performed to illustrate that asymptotically stable walking may be obtained.
