Abstract
A new approach to modelling pedestrians' avoidance dynamics based on a Fokker–Planck (FP) Nash game framework is presented. In this framework, two interacting pedestrians are considered, whose motion variability is modelled through the corresponding probability density functions (PDFs) governed by FP equations. Based on these equations, a Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals. The existence of Nash equilibria solutions is proved and characterized as a solution to an optimal control problem that is solved numerically. Results of numerical experiments are presented that successfully compare the computed Nash equilibria to the output of real experiments (conducted with humans) for four test cases.
Highlights
Multiple pedestrian motion is a complex social and biological process [1,2], involving psychological and non-deterministic behavioural decisions
A Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals
We introduce a new approach that starts from the framework developed in [21], where optimal control of crowd motion in the framework of stochastic processes and the related Fokker–Planck (FP) equations is discussed, and considers two individuals, who behave with rationality and have enough motion variability to be suitably described by the probability density functions (PDFs) of their motions
Summary
Multiple pedestrian motion is a complex social and biological process [1,2], involving psychological and non-deterministic behavioural decisions. Pedestrian modelling encompasses mathematical approaches ranging from discrete and cellular automata to continuum fluid dynamics, conservation laws and hybrid approaches [1,8,9,10,11,12,13,14] In this framework, avoidance is one of the most important and challenging features in pedestrian motion [15], involving. In our framework, the concept of NE for modelling the decision-making control for collision avoidance plays a central role, and results in the formulation of a partial-differential open-loop Nash game governed by FP equations where the control is included in the drift of the stochastic pedestrian motion, and cost functionals for each pedestrian are defined that include a cost of the control and a collision-penalizing term.
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