  This paper develops the method of Continuous Pontryagin Differentiable
Programming (Continuous PDP), which enables a robot to learn an objective
function from a few sparsely demonstrated keyframes. The keyframes, labeled
with some time stamps, are the desired task-space outputs, which a robot is
expected to follow sequentially. The time stamps of the keyframes can be
different from the time of the robot's actual execution. The method jointly
finds an objective function and a time-warping function such that the robot's
resulting trajectory sequentially follows the keyframes with minimal
discrepancy loss. The Continuous PDP minimizes the discrepancy loss using
projected gradient descent, by efficiently solving the gradient of the robot
trajectory with respect to the unknown parameters. The method is first
evaluated on a simulated robot arm and then applied to a 6-DoF quadrotor to
learn an objective function for motion planning in unmodeled environments. The
results show the efficiency of the method, its ability to handle time
misalignment between keyframes and robot execution, and the generalization of
objective learning into unseen motion conditions.
