Actual control of most humanoid robots is based on the 3D linear inverted pendulum and assumes an horizontal displacement of the center of mass of the robot while obviously the center of mass in human walking is characterized by vertical oscillations. The objective of the paper is to show that these oscillations have a crucial role for the high level control of the walk. Based on a controlled length inverted pendulum model of the walker, it will be shown that vertical oscillations induce a self stabilization of the walk while this self stabilization is not observed in the case of a horizontal motion of the center of mass. The results are essentially based on the evolution of the angular momentum throughout the walk. The decrease of the angular momentum during the change of support is determinant to introduce a dependence between the path of the center of mass and the walking velocity. For a large set of walking characteristics (stride, velocity ...) a self synchronization of the motion in the sagittal and frontal planes appears that allows a low level control to produce stable cyclic gaits.