  We present an approach for approximately solving discrete-time stochastic
optimal-control problems by combining direct trajectory optimization,
deterministic sampling, and policy optimization. Our feedback motion-planning
algorithm uses a quasi-Newton method to simultaneously optimize a reference
trajectory, a set of deterministically chosen sample trajectories, and a
parameterized policy. We demonstrate that this approach exactly recovers LQR
policies in the case of linear dynamics, quadratic objective, and Gaussian
disturbances. We also demonstrate the algorithm on several nonlinear,
underactuated robotic systems to highlight its performance and ability to
handle control limits, safely avoid obstacles, and generate robust plans in the
presence of unmodeled dynamics.
