Bridging the gap between optimal trajectory planning and safety-critical control with applications to autonomous vehicles

Optimal Control of Connected Automated Vehicles with Event-Triggered Control Barrier Functions

“…This has motivated an approach which combines a solution of the unconstrained problem (8), which can be obtained very fast, with the use of Control Barrier Functions (CBFs) which provide guarantees that (3), ( 4), ( 5) and ( 6) are always satisfied through constraints that are linear in the control, thus rendering solutions to this alternative problem obtainable by solving a sequence of computationally efficient QPs. This approach is termed Optimal Control with Control Barrier Functions (OCBF) [12].…”

“…(iii) This optimal tracking problem is efficiently solved by discretizing time and solving a simple QP at each discrete time step. The significance of CBFs in this approach is twofold: first, their forward invariance property [12] guarantees that all constraints they enforce are satisfied at all times if they are initially satisfied; second, CBFs impose linear constraints on the control which is what enables the efficient solution of the tracking problem through the sequence of QPs in (iii) above.…”

“…The CBF method (details provided in [12]) maps a constraint b q (x(t )) ≥ 0 onto a new constraint which is linear in the control input u i (t ) and takes the general form…”

“…where L f , L g denote the Lie derivatives of b q (x(t )) along f and g respectively and γ(•) stands for any class-K function [12]. It has been established [12] that satisfaction of (9) implies the satisfaction of the original problem constraint b q (x(t )) ≥ 0 because of the forward invariance property.…”

“…An approach combining optimal control solutions with CBFs was recently presented in [12]. In this combined approach (termed OCBF), the solution of an unconstrained optimal control problem is first derived and used as a reference control.…”