When a robot performs a task next to a human, physical interaction is inevitable: the human might push, pull, twist, or guide the robot. The state-of-the-art treats these interactions as disturbances that the robot should reject or avoid. At best, these robots respond safely while the human interacts; but after the human lets go, these robots simply return to their original behavior. We recognize that physical human-robot interaction (pHRI) is often intentional -- the human intervenes on purpose because the robot is not doing the task correctly. In this paper, we argue that when pHRI is intentional it is also informative: the robot can leverage interactions to learn how it should complete the rest of its current task even after the person lets go. We formalize pHRI as a dynamical system, where the human has in mind an objective function they want the robot to optimize, but the robot does not get direct access to the parameters of this objective -- they are internal to the human. Within our proposed framework human interactions become observations about the true objective. We introduce approximations to learn from and respond to pHRI in real-time. We recognize that not all human corrections are perfect: often users interact with the robot noisily, and so we improve the efficiency of robot learning from pHRI by reducing unintended learning. Finally, we conduct simulations and user studies on a robotic manipulator to compare our proposed approach to the state-of-the-art. Our results indicate that learning from pHRI leads to better task performance and improved human satisfaction.
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