Learning a Tracking Controller for Rolling $μ$bots

Micron-scale robots ($\mu$bots) have recently shown great promise for emerging medical applications. Accurate controlling $\mu$bots, while critical to their successful deployment, is challenging. In this work, we consider the problem of tracking a reference trajectory using a $\mu$bot in the presence of disturbances and uncertainty. The disturbances primarily come from Brownian motion and other environmental phenomena, while the uncertainty originates from errors in the model parameters. We model the $\mu$bot as an uncertain unicycle that is controlled by a global magnetic field. To compensate for disturbances and uncertainties, we develop a nonlinear mismatch controller. We define the model mismatch error as the difference between our model's predicted velocity and the actual velocity of the $\mu$bot. We employ a Gaussian Process to learn the model mismatch error as a function of the applied control input. Then we use a least-squares minimization to select a control action that minimizes the difference between the actual velocity of the $\mu$bot and a reference velocity. We demonstrate the online performance of our joint learning and control algorithm in simulation, where our approach accurately learns the model mismatch and improves tracking performance. We also validate our approach in an experiment and show that certain error metrics are reduced by up to $40\%$.