Failing with Grace: Learning Neural Network Controllers that are Boundedly Unsafe

In this work, we consider the problem of learning a feed-forward neural network (NN) controller to safely steer an arbitrarily shaped planar robot in a compact and obstacle-occluded workspace. Unlike existing methods that depend strongly on the density of data points close to the boundary of the safe state space to train NN controllers with closed-loop safety guarantees, we propose an approach that lifts such assumptions on the data that are hard to satisfy in practice and instead allows for graceful safety violations, i.e., of a bounded magnitude that can be spatially controlled. To do so, we employ reachability analysis methods to encapsulate safety constraints in the training process. Specifically, to obtain a computationally efficient over-approximation of the forward reachable set of the closed-loop system, we partition the robot's state space into cells and adaptively subdivide the cells that contain states which may escape the safe set under the trained control law. To do so, we first design appropriate under- and over-approximations of the robot's footprint to adaptively subdivide the configuration space into cells. Then, using the overlap between each cell's forward reachable set and the set of infeasible robot configurations as a measure for safety violations, we introduce penalty terms into the loss function that penalize this overlap in the training process. As a result, our method can learn a safe vector field for the closed-loop system and, at the same time, provide numerical worst-case bounds on safety violation over the whole configuration space, defined by the overlap between the over-approximation of the forward reachable set of the closed-loop system and the set of unsafe states. Moreover, it can control the tradeoff between computational complexity and tightness of these bounds. Finally, we provide a simulation study that verifies the efficacy of the proposed scheme.