Trajectory Tracking Runtime Assurance for Systems with Partially Unknown Dynamics

We consider the problem of tracking a reference trajectory for dynamical systems subject to a priori unknown state-dependent disturbance behavior. We propose a formulation that embeds the uncertain system into a higher dimensional deterministic system that accounts for worst case disturbances. Our main insight is that a single controlled trajectory of this embedding system corresponds to a controlled forward invariant interval tube around the reference trajectory. By taking observations of the system, we then propose to estimate the state-dependent uncertainty with Gaussian Process regression, which improves the accuracy of the forward invariant tube as data is collected. Given a safety objective, we also provide conditions on when an additional observation of the unknown disturbance behavior needs to be collected to maintain safety. We demonstrate our formulation on a case study of a planar multirotor attempting a safe landing in an unknown wind field.