  Safety-critical applications require controllers/policies that can guarantee
safety with high confidence. The control barrier function is a useful tool to
guarantee safety if we have access to the ground-truth system dynamics. In
practice, we have inaccurate knowledge of the system dynamics, which can lead
to unsafe behaviors due to unmodeled residual dynamics. Learning the residual
dynamics with deterministic machine learning models can prevent the unsafe
behavior but can fail when the predictions are imperfect. In this situation, a
probabilistic learning method that reasons about the uncertainty of its
predictions can help provide robust safety margins. In this work, we use a
Gaussian process to model the projection of the residual dynamics onto a
control barrier function. We propose a novel optimization procedure to generate
safe controls that can guarantee safety with high probability. The safety
filter is provided with the ability to reason about the uncertainty of the
predictions from the GP. We show the efficacy of this method through
experiments on Segway and Quadrotor simulations. Our proposed probabilistic
approach is able to reduce the number of safety violations significantly as
compared to the deterministic approach with a neural network.
