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This paper extends the gap-based navigation technique Potential Gap with safety guarantees at the local planning level for a kinematic planar nonholonomic robot model, leading to Safer Gap . It relies on a subset of navigable free space from the robot to a gap, denoted the keyhole region. The region is defined by the union of the largest collision-free disc centered on the robot and a collision-free trapezoidal region directed through the gap. Safer Gap first generates Bézier-based collision-free paths within the keyhole regions. The keyhole region of the top scoring path is encoded by a shallow neural network-based zeroing barrier function (ZBF) synthesized in real-time. Nonlinear Model Predictive Control (NMPC) with Keyhole ZBF constraints and output tracking of the Bézier path, synthesizes a safe kinematically feasible trajectory. The Potential Gap projection operator serves as a last action to enforce safety if the NMPC optimization fails to converge to a solution within the prescribed time. Simulation and experimental validation of Safer Gap confirm its collision-free navigation properties. 

Barrier function-based inequality constraints are a means to enforce safety specifications for control systems. When used in conjunction with a convex optimization program, they provide a computationally efficient method to enforce safety for the general class of control-affine systems. One of the main assumptions when taking this approach is the a priori knowledge of the barrier function itself, i.e., knowledge of the safe set. In the context of navigation through unknown environments where the locally safe set evolves with time, such knowledge does not exist. This manuscript focuses on the synthesis of a zeroing barrier function characterizing the safe set based on safe and unsafe sample measurements, e.g., from perception data in navigation applications. Prior work formulated a supervised machine learning algorithm whose solution guaranteed the construction of a zeroing barrier function with specific level-set properties. However, it did not explore the geometry of the neural network design used for the synthesis process. This manuscript describes the specific geometry of the neural network used for zeroing barrier function synthesis, and shows how the network provides the necessary representation for splitting the state space into safe and unsafe regions.