The main motivations for driving automation of road vehicles lie in safety aspects. Indeed, most road accidents are due to human errors that could be avoided by using driving automation systems. Yet accident rates by unit of distance traveled in conventional traffic are extremely small quantities. The demonstration of safety enhancement by automation of the driving is currently actively debated. Even basic longitudinal motion planning, i.e. adaptive cruise control (ACC) systems, require rigorous demonstration of their safety. Classical linear speed planners are feedback and relaxation processes for the time gap, based on the distance ahead and the speed of the preceding vehicle. Their active safety is tackled thanks to stability analysis. Generally speaking, stability occurs if the relaxation is sufficiently strong. Unfortunately, the acceleration rates and jerks with stable classical speed planners can exceed the bounds recommended by the ISO standard and lead to unsafe dynamics. We propose a novel nonlinear speed planner for ACC systems. The model provides safe and comfortable speed regulations for various driving situations. Reasons are stability properties of the new planner, that hold for any relaxing order. We discuss applications of the car-following model for ACC systems and the robustness of the stability property to mechanical and computational latencies or noise and measurement errors.