Designing Control Barrier Functions Using a Dynamic Backup Policy

This paper presents a systematic approach to construct control barrier functions for nonlinear control affine systems subject to arbitrary state and input constraints. Taking inspiration from the reference governor literature, the proposed method defines a family of backup policies, parametrized by the equilibrium manifold of the system. The control barrier function is defined on the augmented state-and-reference space: given a state-reference pair, the approach quantifies the distance to constraint violation at any time in the future, should the current backup policy reference remain constant. Sensitivity analysis is then used to compute the (possibly nonsmooth) Jacobian with respect to the augmented state vector. To showcase its simple yet general nature, the proposed method is applied to an inverted pendulum on cart.