  In real-world robotics applications, accurate models of robot dynamics are
critical for safe and stable control in rapidly changing operational
conditions. This motivates the use of machine learning techniques to
approximate robot dynamics and their disturbances over a training set of
state-control trajectories. This paper demonstrates that inductive biases
arising from physics laws can be used to improve the data efficiency and
accuracy of the approximated dynamics model. For example, the dynamics of many
robots, including ground, aerial, and underwater vehicles, are described using
their $SE(3)$ pose and satisfy conservation of energy principles. We design a
physically plausible model of the robot dynamics by imposing the structure of
Hamilton's equations of motion in the design of a neural ordinary differential
equation (ODE) network. The Hamiltonian structure guarantees satisfaction of
$SE(3)$ kinematic constraints and energy conservation by construction. It also
allows us to derive an energy-based adaptive controller that achieves
trajectory tracking while compensating for disturbances. Our learning-based
adaptive controller is verified on an under-actuated quadrotor robot.
