Safety Uncertainty in Control Barrier Functions using Gaussian Processes

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For a dynamical system, safety is typically guaranteed by constraining the system states within a set defined a priori. A popular approach is to use control barrier functions (CBFs) that encode safety using a smooth function. However, typical constructions of the smooth function do not account for any notion of safety uncertainty for the system inside the safe set. Although, one can formulate uncertainty in the dynamics of the model in a CBF framework, observability of unmodeled dynamics is difficult, particularly in an online setting. Addressing these drawbacks, we present a novel formulation for synthesizing the CBF smooth function by taking into account safety uncertainty using online measurements of the system states. This uncertainty is encoded by computing the posterior variance using Gaussian processes conditioned on past measurement states. Our approach only requires observability of system states rather than the system dynamics. By incorporating safety uncertainty, the safe set can be dynamically expanded or compressed. This is achieved by computing a local safety map online at the present location and identifying samples with minimal safety exceeding the current safety limit. As more data is collected, the safety margin increases. Hence, these minimally safe exploratory samples can be used to expand the current safe set incrementally. We validate our approach experimentally by expanding an initial safe set, along x and y positions independently, for a quadrotor with safety. The experiment video can be seen at: https://youtu.be/9qvOf1UpRPw.