Control Barrier Functions with Circulation Inequalities

Control barrier functions (CBFs) when paired with quadratic programming (QP) offer an increasingly popular framework for control considering critical safety constraints. However, being closely related to artificial potential fields, they suffer from the classical stable spurious equilibrium point problem, in which the controller can fail to drive the system to the goal. The main contribution of this article is showing that this problem can be mitigated by introducing a circulation inequality as a constraint, which forces the system to explicitly circulate obstacles under some conditions. This circulation is introduced in the configuration space and is simple to implement once we have the CBF-constraint, adding a negligible complexity to the resulting optimization problem. Theoretical guarantees are provided for this framework, indicating, under appropriate conditions, the feasibility of the resulting optimization problem, continuity of the control input, characterization of the equilibrium points, a weak form of Lyapunov stability, and uniqueness of the equilibrium points. The provided experimental studies showcase the overall properties and applicability in different scenarios.