  The hybrid zero dynamics (HZD) approach has become a powerful tool for the
gait planning and control of bipedal robots. This paper aims to extend the HZD
methods to address walking, ambling and trotting behaviors on a quadrupedal
robot. We present a framework that systematically generates a wide range of
optimal trajectories and then provably stabilizes them for the full-order,
nonlinear and hybrid dynamical models of quadrupedal locomotion. The gait
planning is addressed through a scalable nonlinear programming using direct
collocation and HZD. The controller synthesis for the exponential stability is
then achieved through the Poincar\'e sections analysis. In particular, we
employ an iterative optimization algorithm involving linear and bilinear matrix
inequalities (LMIs and BMIs) to design HZD-based controllers that guarantee the
exponential stability of the fixed points for the Poincar\'e return map. The
power of the framework is demonstrated through gait generation and HZD-based
controller synthesis for an advanced quadruped robot, ---Vision 60, with 36
state variables and 12 control inputs. The numerical simulations as well as
real world experiments confirm the validity of the proposed framework.
