Robust Control Barrier–Value Functions for Safety-Critical Control

This paper works towards unifying two popular approaches in the safety control community: Hamilton-Jacobi (HJ) reachability and Control Barrier Functions (CBFs). HJ Reachability has methods for direct construction of value functions that provide safety guarantees and safe controllers, however the online implementation can be overly conservative and/or rely on chattering bang-bang control. The CBF community has methods for safe-guarding controllers in the form of point-wise optimization using quadratic programs (CBF-QP), where the CBF-based safety certificate is used as a constraint. However, finding a valid CBF for a general dynamical system is challenging. This paper unifies these two methods by introducing a new reachability formulation inspired by the structure of CBFs to construct a Control Barrier-Value Function (CBVF). We verify that CBVF is a viscosity solution to a novel Hamilton-Jacobi-Isaacs Variational Inequality and preserves the same safety guarantee as the original reachability formulation. Finally, inspired by the CBF-QP, we propose a QP-based online control synthesis for systems affine in control and disturbance, whose solution is always the CBVF's optimal control signal robust to bounded disturbance. We demonstrate the benefit of using the CBVFs for double-integrator and Dubins car systems by comparing it to previous methods.