  We demonstrate the effectiveness of simple observer-based linear feedback
policies for "pixels-to-torques" control of robotic systems using only a
robot-facing camera. Specifically, we show that the matrices of an image-based
Luenberger observer (linear state estimator) for a "student" output-feedback
policy can be learned from demonstration data provided by a "teacher"
state-feedback policy via simple linear-least-squares regression. The resulting
linear output-feedback controller maps directly from high-dimensional raw
images to torques while being amenable to the rich set of analytical tools from
linear systems theory, allowing us to enforce closed-loop stability constraints
in the learning problem. We also investigate a nonlinear extension of the
method via the Koopman embedding. Finally, we demonstrate the surprising
effectiveness of linear pixels-to-torques policies on a cartpole system, both
in simulation and on real-world hardware. The policy successfully executes both
stabilizing and swing-up trajectory tracking tasks using only camera feedback
while subject to model mismatch, process and sensor noise, perturbations, and
occlusions.
