This thesis explores how robots should respond to physical human interactions. From surgical devices to assistive arms, robots are becoming an important aspect of our everyday lives. Unlike earlier robots---which were developed for carefully regulated factory settings---today's robots must work alongside human end-users, and even facilitate physical interactions between the robot and the human. Within the current state-of-the-art, the human's intentionally applied forces are treated as unwanted disturbances that the robot should avoid, reject, or ignore: once the human stops interacting, these robots simply return to their original behavior. By contrast, we recognize that physical interactions are really an implicit form of communication: the human is applying forces and torques to correct the robot's behavior, and teach the robot how it should complete its task. Within this work, we demonstrate that optimally responding to physical human interactions results in robots that learn from these corrections and change their underlying behavior.

We first formalize physical human-robot interaction as a partially observable dynamical system, where the human's applied forces and torques are observations about the objective function that the robot should be optimizing, and, more specifically, the human's preferences for how the robot should behave. Solving this system defines the right way for a robot to respond to physical corrections. We derive three approximate solutions for real-time implementation on robotic hardware: these different approximations assume increasing amounts of structure, and consider cases where the robot is given (a) an arbitrary initial trajectory, (b) a parameterized initial trajectory, or (c) the task-related features. We next extend our approximations to account for noisy and imperfect end-users, who may accidentally correct the robot more or less than they intended. We enable robots to reason over what aspects of the human's interaction were intentional, and which of the human's preferences are still unclear. Our overall approach to physical human-robot interaction provides a theoretical basis for robots that both realize why the human is interacting and personalize their behavior in response to that end-user. The feasibility of our theoretical contributions is demonstrated through simulations and user studies.