Safe planning and control via L1-adaptation and contraction theory

Autonomous robots that are capable of operating safely in the presence of imperfect model knowledge or external disturbances are vital in safety-critical applications. The research presented in this dissertation aims to enable safe planning and control for nonlinear systems with uncertainties using robust adaptive control theory. To this end we develop methods that (i) certify the collision-risk for the planned trajectories of autonomous robots, (ii) ensure guaranteed tracking performance in the presence of uncertainties, and (iii) learn the uncertainties in the model without sacrificing the transient performance guarantees, and (iv) learn incremental stability certificates parameterized as neural networks. In motion planning problems for autonomous robots, such as self-driving cars, the robot must ensure that its planned path is not in close proximity to obstacles in the environment. However, the problem of evaluating the proximity is generally non-convex and serves as a significant computational bottleneck for motion planning algorithms. In this work, we present methods for a general class of absolutely continuous parametric curves to compute: the minimum separating distance, tolerance verification, and collision detection with respect to obstacles in the environment. A planning algorithm is incomplete if the robot is unable to safely track the planned trajectory. We introduce a feedback motion planning approach using contraction theory-based L1-adaptive (CL1) control to certify that planned trajectories of nonlinear systems with matched uncertainties are tracked with desired performance requirements. We present a planner-agnostic framework to design and certify invariant tubes around planned trajectories that the robot is always guaranteed to remain inside. By leveraging recent results in contraction analysis and L1-adaptive control we present an architecture that induces invariant tubes for nonlinear systems with state and time-varying uncertainties. Uncertainties caused by large modeling errors will significantly hinder the performance of any autonomous system. We adapt the CL1 framework to safely learn the uncertainties while simultaneously providing high-probability bounds on the tracking behavior. Any available data is incorporated into Gaussian process (GP) models of the uncertainties while the error in the learned model is quantified and handled by the CL1 controller to ensure that control objectives are met safely. As learning improves, so does the overall tracking performance of the system. This way, the safe operation of the system is always guaranteed, even during the learning transients. The tracking performance guarantees for nonlinear systems rely on the existence of incremental stability certificates that are prohibitively difficult to search for. We leverage the function approximation capabilities of deep neural networks for learning the certificates and the associated control policies jointly. The incremental stability properties of the closed-loop system are verified using interval arithmetic. The domain of the system is iteratively refined into a collection of intervals that certify the satisfaction of the stability properties over the interval regions. Thus, we avoid entirely rejecting the learned certificates and control policies just because they violate the stability properties in certain parts of the domain. We provide numerical experimentation on an inverted pendulum to validate our proposed methodology