High-Order Control Barrier Functions

We approach the problem of stabilizing a dynamical system while optimizing a cost and satisfying safety constraints and control limitations. For (nonlinear) affine control systems and quadratic costs, it has been shown that control barrier functions (CBFs) guaranteeing safety and control Lyapunov functions (CLFs) enforcing convergence can be used to (conservatively) reduce the optimal control problem to a sequence of quadratic programs (QPs). Existing works in this category have two main limitations. First, with one exception, they are based on the assumption that the relative degree of the system with respect to a function enforcing a safety constraint is one. Second, the QPs can easily become infeasible, in particular for problems with many safety constraints and tight control limitations. We propose high-order CBFs (HOCBFs), which can accommodate systems of arbitrary relative degrees. For each safety constraint, by using Lyapunov-like conditions, we construct a set of controls that renders the intersection of a set of sets forward invariant, which implies the satisfaction of the original constraint. We formulate optimal control problems with constraints given by HOCBF and CLF, and propose two methods—the penalty method and the parameterization method—to address the feasibility problem. Finally, we show how our methodology can be extended for safe navigation in unknown environments with long-term feasibility. We illustrate the proposed framework on adaptive cruise control and robot control problems.