A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games

Kuang Huang; Xuan Di; Qiang Du; Xi Chen

doi:10.3934/dcdsb.2020131

Article Contents

2020, Volume 25, Issue 12: 4869-4903. Doi: 10.3934/dcdsb.2020131

This issue Previous Article Using automatic differentiation to compute periodic orbits of delay differential equations Next Article Numerical investigation of ensemble methods with block iterative solvers for evolution problems

A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games

1.
Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, United States
2.
Department of Civil Engineering and Engineering Mechanics and Data Science Institute, Columbia University, New York, NY 10027, United States
3.
Department of Applied Physics and Applied Mathematics and Data Science Institute, Columbia University, New York, NY 10027, United States
4.
Department of Computer Science, Columbia University, New York, NY 10027, United States

^* Corresponding author: Xuan Di
^* Corresponding author: Xuan Di

Received: October 31, 2019

Revised: December 31, 2019

Early access: April 2020

Published: December 2020

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Abstract

This paper proposes an efficient computational framework for longitudinal velocity control of a large number of autonomous vehicles (AVs) and develops a traffic flow theory for AVs. Instead of hypothesizing explicitly how AVs drive, our goal is to design future AVs as rational, utility-optimizing agents that continuously select optimal velocity over a period of planning horizon. With a large number of interacting AVs, this design problem can become computationally intractable. This paper aims to tackle such a challenge by employing mean field approximation and deriving a mean field game (MFG) as the limiting differential game with an infinite number of agents. The proposed micro-macro model allows one to define individuals on a microscopic level as utility-optimizing agents while translating rich microscopic behaviors to macroscopic models. Different from existing studies on the application of MFG to traffic flow models, the present study offers a systematic framework to apply MFG to autonomous vehicle velocity control. The MFG-based AV controller is shown to mitigate traffic jam faster than the LWR-based controller. MFG also embodies classical traffic flow models with behavioral interpretation, thereby providing a new traffic flow theory for AVs.

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Mathematics Subject Classification: Primary: 49N90, 90B20; Secondary: 35Q91.

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Figure 1. From Micro to Macroscopic Traffic Flow Models

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Figure 2. From an $ N $-car differential game to MFG (adapted from [39])

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Figure 3. Connections between MFG and LWR

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Figure 4. [MFG-LWR]

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Figure 5. Density Evolution of [MFG-NonSeparable] and [MFG-Separable]

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Figure 6. Fundamental diagram of [MFG-NonSeparable]

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Figure 7. Density, speed and optimal cost profiles for [MFG-NonSeparable] and [MFG-Separable] at $ t = 0 $ and $ t = 1.5 $

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Figure 8. Convergence of solution algorithm in $ L^1 $ norm

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Figure 9. $ N = 21 $ cars' trajectories integrated from the MFE solution of [MFG-NonSeparable]

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Figure 10. MFE-constructed control cost v.s. best response strategy cost, $ N = 21 $ cars

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Figure 11. Accuracy v.s. Number of cars

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Table 1. Classification of macroscopic traffic flow models

	Speed	Acceleration rate
Traditional	First-order (e.g., LWR)	Higher-order (e.g., PW/ARZ)
Game-theoretic	First-order MFGs	Higher-order MFGs

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