There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The theory of affordances

Journal of Statistical Mechanics: theory and experiment

高等学校計算数学学報 = Numerical mathematics

Human mobility: Models and applications

A spatially continuous force-based model for simulating pedestrian dynamics is introduced which includes an elliptical volume exclusion of pedestrians We discuss the phenomena of oscillations and overlapping which occur for certain choices of the forces The main intention of this work is the quantitative description of pedestrian movement in several geometries Measurements of the fundamental diagram in narrow and wide corridors are performed The results of the proposed model show good agreement with empirical data obtained in controlled experiments

/pdf/generalized-centrifugal-force-model-for-pedestrian-dynamics-3zbtgoak8q.pdf

Generalized centrifugal-force model for pedestrian dynamics.

Force-based models describe the interactions of pedestrians in terms
 of physical and social forces. We discuss some intrinsic problems of
 this approach, like penetration of particles, unrealistic
 oscillations and velocities as well as conceptual problems related
 to violations of Newton's laws. We then present the generalized
 centrifugal force model which solves some of these problems.
 Furthermore we discuss the problem of choosing a realistic driving
 force to an exit. We illustrate this problem by investigating
 the behaviour of pedestrians at bottlenecks.

Force-based models of pedestrian dynamics

Experiments with pedestrians could depend strongly on initial conditions. Comparisons of the results of such experiments require to distinguish carefully between transient state and steady state. Thus a modified version of the Cumulative Sum Control Chart algorithm is proposed to robustly detect steady states from density and speed time series of bottleneck experiments. The threshold of the detection parameter in the algorithm is calibrated using an autoregressive model. Comparing the detected steady states with manually selected ones, the modified algorithm gives robust and reproducible results. For the applications, three groups of bottleneck experiments are analysed and the steady states are detected. The results reconfirm that the specific flow is constant as bottleneck width changes. Moreover, we proposed a criterion to judge the difference between the flows in all states and in steady states, which is the ratio of pedestrian number to bottleneck width. The critical value of the ratio is found to be approximately 115 persons/m. This conclusion applies not only for the analysis of existing bottleneck experiments but also for the design of new bottleneck experiments and the validation of evacuation models. Furthermore, the range of steady state in time series of pedestrian characteristics could be effectively controlled by adjusting the value of the ratio.

Measuring the steady state of pedestrian flow in bottleneck experiments

We present an approach for finding the overlap area between two ellipses that does not rely on proxy curves. The Gauss-Green formula is used to determine a segment area between two points on an ellipse. Overlap between two ellipses is calculated by combining the areas of appropriate segments and polygons in each ellipse. For four of the ten possible orientations of two ellipses, the method requires numerical determination of transverse intersection points. Approximate intersection points can be determined by solving the two implicit ellipse equations simultaneously. Alternative approaches for finding transverse intersection points are available using tools from algebraic geometry, e.g., based on solving an Eigen-problem that is related to companion matrices of the two implicit ellipse curves. Implementations in C of several algorithm options are analyzed for accuracy, precision and robustness with a range of input ellipses.

/pdf/calculating-ellipse-overlap-areas-3p1caneddm.pdf

Calculating ellipse overlap areas

We propose in this paper a minimal speed-based pedestrian model for which particle dynamics are intrinsically collision-free. The speed model is an optimal velocity function depending on the agent length (i.e. particle diameter), maximum speed and time gap parameters. The direction model is a weighted sum of exponential repulsion from the neighbours, calibrated by the repulsion rate and distance. The model’s main features like the reproduction of empirical phenomena are analysed by simulation. We point out that phenomena of self-organisation observable in force-based models and field studies can be reproduced by the collision-free model with low computational effort.

Mohcine Chraibi

Papers

Generalized centrifugal-force model for pedestrian dynamics.

Force-based models of pedestrian dynamics

Measuring the steady state of pedestrian flow in bottleneck experiments

Calculating ellipse overlap areas

Collision-free speed model for pedestrian dynamics