Least Restrictive Hyperplane Control Barrier Functions

Control Barrier Functions (CBFs) can provide provable safety guarantees for dynamic systems. However, finding a valid CBF for a system of interest is often non-trivial, especially if the shape of the unsafe region is complex and the CBFs are of higher order. A common solution to this problem is to make a conservative approximation of the unsafe region in the form of a line/hyperplane, and use the corresponding conservative Hyperplane-CBF when deciding on safe control actions. In this letter, we note that conservative constraints are only a problem if they prevent us from doing what we want. Thus, instead of first choosing a CBF and then choosing a safe control with respect to the CBF, we optimize over a combination of CBFs and safe controls to get as close as possible to our desired control, while still having the safety guarantee provided by the CBF. We call the corresponding CBF the least restrictive Hyperplane-CBF. Finally, we also provide a way of creating a smooth parameterization of the CBF-family for the optimization, and illustrate the approach on a double integrator dynamical system with acceleration constraints, moving through a group of arbitrarily shaped static and moving obstacles.