  Accurate models of robot dynamics are critical for safe and stable control
and generalization to novel operational conditions. Hand-designed models,
however, may be insufficiently accurate, even after careful parameter tuning.
This motivates the use of machine learning techniques to approximate the robot
dynamics over a training set of state-control trajectories. The dynamics of
many robots, including ground, aerial, and underwater vehicles, are described
in terms of their SE(3) pose and generalized velocity, and satisfy conservation
of energy principles. This paper proposes a Hamiltonian formulation over the
SE(3) manifold of the structure of a neural ordinary differential equation
(ODE) network to approximate the dynamics of a rigid body. In contrast to a
black-box ODE network, our formulation guarantees total energy conservation by
construction. We develop energy shaping and damping injection control for the
learned, potentially under-actuated SE(3) Hamiltonian dynamics to enable a
unified approach for stabilization and trajectory tracking with various
platforms, including pendulum, rigid-body, and quadrotor systems.
