Provably Safe Design Methodology for Automatic Longitudinal Control of Autonomous Vehicles

Formally verified automatic longitudinal control (ALC) of road vehicles in complex driving scenarios, including traffic light and road geometry constraints and adverse weather conditions, with or without vehicle-to-vehicle and vehicle-to-infrastructure communication, remains a challenge. The safety conditions for an ALC system are often time-varying with potential discontinuities, and correct-by-construction approaches, e.g., based on control barrier functions (CBFs), often require smooth approximations of these time-varying specifications. This paper introduces piecewise continuously differentiable CBFs (PCD-CBFs) and presents a systematic and provably correct design methodology for ALC based on PCD-CBFs to address this challenge. Specifications capturing safety, comfort, and traffic rules are represented in terms of forward-invariance conditions on safe sets characterized by PCD-CBFs. A provably safe controller is then synthesized from the specifications via quadratic programming. The methodology is illustrated for a vehicle on a road with complex road geometry and multiple traffic signals. Simulation results show its effectiveness under a broad range of driving environments.