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In this work, we consider the problem of learning a feed-forward neuralnetwork controller to safely steer an arbitrarily shaped planar robot in acompact and obstacle-occluded workspace. Unlike existing methods that dependstrongly on the density of data points close to the boundary of the safe statespace to train neural network controllers with closed-loop safety guarantees,here we propose an alternative approach that lifts such strong assumptions onthe data that are hard to satisfy in practice and instead allows for gracefulsafety violations, i.e., of a bounded magnitude that can be spatiallycontrolled. To do so, we employ reachability analysis techniques to encapsulatesafety constraints in the training process. Specifically, to obtain acomputationally efficient over-approximation of the forward reachable set ofthe closed-loop system, we partition the robot's state space into cells andadaptively subdivide the cells that contain states which may escape the safeset under the trained control law. Then, using the overlap between each cell'sforward reachable set and the set of infeasible robot configurations as ameasure for safety violations, we introduce appropriate terms into the lossfunction that penalize this overlap in the training process. As a result, ourmethod can learn a safe vector field for the closed-loop system and, at thesame time, provide worst-case bounds on safety violation over the wholeconfiguration space, defined by the overlap between the over-approximation ofthe forward reachable set of the closed-loop system and the set of unsafestates. Moreover, it can control the tradeoff between computational complexityand tightness of these bounds. Our proposed method is supported by boththeoretical results and simulation studies.

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