Continuity equation and fundamental diagram of pedestrians

Juliane Adrian, Ann Katrin Boomers, Sarah Paetzke, Armin Seyfried
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Abstract

Since the beginning of the century, capturing trajectories of pedestrian streams precisely from video recordings has been possible. To enable measurements at high density, the heads of the pedestrians are marked and tracked, thus providing a complete representation of the phase space. However, classical definitions and local measurements of flow, density, and velocity of pedestrian streams using trajectories are based on different segments in phase space (Lagrangian representation). The flow is defined as an average value over time, while the density is defined as the average value of an area. This leads to inconsistencies in central relations, such as the flow equation or the fundamental diagram. These have a particular effect in inhomogeneous states, such as the stop-and-go waves, where, in addition, the pedestrians do not change their position in the stop phase, but the head of the body moves. In order to obtain a local and spatio-temporally consistent measurement of the quantities flow, density, and velocity while ensuring particle number conservation fields (Euler representation) and the continuity equation could be used. To map trajectories of pedestrians heads parameter free and unambiguously to fields, this article introduces a method based on the Voronoi decomposition. These new definitions of flow, density, speed, and the particle number conserving flow equation are consistent with classical measurements. They are able to scrutinise inconsistencies in the state of the art of pedestrian fundamental diagrams.
行人的连续性方程和基本图示
自本世纪初以来,通过视频记录精确捕捉行人流的轨迹已成为可能。为了能够进行高密度测量,我们对行人的头部进行了标记和跟踪,从而提供了一个完整的相空间表示。然而,使用轨迹对行人流的流量、密度和速度进行经典定义和局部测量,是基于相空间中的不同区段(拉格朗日表示法)。流量被定义为超时的平均值,而密度则被定义为区域的平均值。这就导致了中心关系的不一致,如流动方程或基本图。这在不均匀状态下会产生特别的影响,例如走走停停波,行人在停止阶段不会改变位置,但身体的头部会移动。为了获得局部和时空一致的流量、密度和速度等量的测量值,同时确保粒子数守恒场(欧拉表示法)和连续性方程可以被使用。为了将行人头部的轨迹无参数、明确地映射到场中,本文介绍了一种基于 Voronoi 分解的方法。这些新定义与经典测量结果一致,可用于审查行人基础图中的不一致之处。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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