Single-file motion refers to the movement of agents, such as pedestrians, vehicles, or bicycles, in a single line without overtaking, focusing on the regulation of speed and spacing between them. This behavior is crucial for deriving fundamental diagrams that express the phenomenological relationship between mean speed, flow, and density in traffic engineering.  Single-file traffic flow characteristics can be either smooth (laminar) or exhibit oscillating patterns like stop-and-go waves (accordion traffic, see the experiments by Sugiyama et al., Stern et al. and the simulation modules) which are problematic for traffic safety, performance and the environment and also observed in automated driving system (i.e. adaptive cruise control systems, see the experiments by Gunter et al., OpenACC and the simulation modules) as well as pedestrian and bicycle single-file motions.

Continuous following models, based on ordinary, delayed and stochastic differential equation systems, provide microscopic insights into vehicle and pedestrian behavior, with applications to driving automation and adaptive cruise control (ACC) systems in traffic engineering. We develop isotropic and anisotropic pedestrian and car-following models, including delay, noise, and other heterogeneity mechanisms, analyse their stability and long-term behavior, derive infinite limits and carry out experiment-based simulation and parameter calibration. For instance, the Adaptive Time Gap (ATG) model introduces a safety parameter for time gap regulation that ensure stable uniform dynamics. The Collision-Free Optimal Velocity (CFOV) model is a simplified version of the Optimal Velocity (OV) model by Bando et al. (1995) that mitigates unrealistic collisions and backward movements while describing stop-and-go traffic patterns (see the simulations #1, #2, #3). Derivation of the CFOV microscopic model results in a parabolic convection-diffusion partial differential macroscopic equation while derivation of the ATG models leads to the Lighthill-Whitham-Richards (LWR) macroscopic model.

In the local stability analysis (also called platoon stability), some vehicles follow a leader with assigned speed. The local stability is said to be overdamped when the convergence is oscillation-free. Such a feature allows to ensure, at least partially, the absence of collisions in the dynamics. The global stability analysis (also called string or collective stability) deals with vehicles on an infinite line or with periodic boundary conditions. Both stability properties are expected features of adaptive cruise control systems in traffic engineering. However, string stability conditions, including inductive and convective perturbations that vanish locally, are typically more restrictive than local conditions. Several factors pertubing the stability are identified in our research, including delay and latency in the dynamic, stochastic noise, especially colored and state-dependent noise, as well as individual heterogeneity, and various mechanisms are proposed to offset these effects.