  In this paper, we consider the problem of using a robot to explore an
environment with an unknown, state-dependent disturbance function while
avoiding some forbidden areas. The goal of the robot is to safely collect
observations of the disturbance and construct an accurate estimate of the
underlying disturbance function. We use Gaussian Process (GP) to get an
estimate of the disturbance from data with a high-confidence bound on the
regression error. Furthermore, we use neural Contraction Metrics to derive a
tracking controller and the corresponding high-confidence uncertainty tube
around the nominal trajectory planned for the robot, based on the estimate of
the disturbance. From the robustness of the Contraction Metric, error bound can
be pre-computed and used by the motion planner such that the actual trajectory
is guaranteed to be safe. As the robot collects more and more observations
along its trajectory, the estimate of the disturbance becomes more and more
accurate, which in turn improves the performance of the tracking controller and
enlarges the free space that the robot can safely explore. We evaluate the
proposed method using a carefully designed environment with a ground vehicle.
Results show that with the proposed method the robot can thoroughly explore the
environment safely and quickly.
