Abstract-We describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game. The algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation. While alternative techniques for computing the reachable set have been proposed, the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters. Because the time-dependent equation's solution is continuous and defined throughout the state space, methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions. A numerical implementation of our formulation is described and has been released on the web. Its correctness is verified through a two vehicle, three dimensional collision avoidance example for which an analytic solution is available.

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Automotive traffic monitoring using probe vehicles with Global Positioning System receivers promises significant improvements in cost, coverage, and accuracy. Current approaches, however, raise privacy concerns because they require participants to reveal their positions to an external traffic monitoring server. To address this challenge, we propose a system based on virtual trip lines and an associated cloaking technique. Virtual trip lines are geographic markers that indicate where vehicles should provide location updates. These markers can be placed to avoid particularly privacy sensitive locations. They also allow aggregating and cloaking several location updates based on trip line identifiers, without knowing the actual geographic locations of these trip lines. Thus they facilitate the design of a distributed architecture, where no single entity has a complete knowledge of probe identities and fine-grained location information. We have implemented the system with GPS smartphone clients and conducted a controlled experiment with 20 phone-equipped drivers circling a highway segment. Results show that even with this low number of probe vehicles, travel time estimates can be provided with less than 15% error, and applying the cloaking techniques reduces travel time estimation accuracy by less than 5% compared to a standard periodic sampling approach.

Hybrid system theory lies at the intersection of the fields of engi-

In this paper, we combine the most complete record of daily mobility, based on large-scale mobile phone data, with detailed Geographic Information System (GIS) data, uncovering previously hidden patterns in urban road usage. We find that the major usage of each road segment can be traced to its own -surprisingly few -driver sources. Based on this finding we propose a network of road usage by defining a bipartite network framework, demonstrating that in contrast to traditional approaches, which define road importance solely by topological measures, the role of a road segment depends on both: its betweeness and its degree in the road usage network. Moreover, our ability to pinpoint the few driver sources contributing to the major traffic flow allows us to create a strategy that achieves a significant reduction of the travel time across the entire road system, compared to a benchmark approach.I n an era of unprecedented global urbanization, society faces a rapidly accelerating demand for mobility, placing immense pressure on urban road networks 1,2 . This demand manifests in the form of severe traffic congestion 3,4 , which decreases the roads' level of service, while at the same time increasing both fuel consumption 5 and traffic-related air pollution 6 . In 2007 alone, congestion forced Americans living in urban areas to travel 4.2 billion hours more, purchase an additional 2.8 billion gallons of fuel, at a total cost of $87.2 billion 3 . To mitigate congestion in urban roads, urban planning 1 , traffic prediction 7-9 and the study of complex networks [10][11][12][13][14][15] have been widely investigated potential influencing factors. However, without comprehensive knowledge of how roads are used dynamically, these studies are conventionally based on expensive and quickly outdated travel surveys or segmented information on traffic flow and travel time 7-9 , which fail to support the researchers with the information needed to cope with modern mobility demand. Up to now our understanding of the origins of the drivers in each road remained limited and not quantitatively solved.In this work we validate for the first time a methodology, which employs comprehensive mobile phone data to detect patterns of road usage and the origins of the drivers. Thus, providing a basis for better informed transportation planning, including targeted strategies to mitigate congestion 3,4 . We formalize the problem by counting the observed number of individuals moving from one location to another, which we put forward as the transient origin destination (t-OD) matrix (Fig. S5, Fig. S11 and Supplementary Information (SI) section II.A). Traditionally, ODs are costly and difficult to obtain, because they are at best based on travel diaries made every few years, which quickly become obsolete and strongly rely on provided reports 7,8 . In contrast, the rapidly increasing penetration rates and massive usage of mobile phones, with towers densely located in urban areas, can provide the most detailed information on daily human mobility [16][17][...

Traffic state estimation (TSE) refers to the process of the inference of traffic state variables (i.e., flow, density, speed and other equivalent variables) on road segments using partially observed traffic data. It is a key component of traffic control and operations, because traffic variables are measured not everywhere due to technological and financial limitations, and their measurement is noisy. Therefore, numerous studies have proposed TSE methods relying on various approaches, traffic flow models, and input data. In this review article, we conduct a survey of highway TSE methods, a topic which has gained great attention in the recent decades. We characterize existing TSE methods based on three fundamental elements: estimation approach, traffic flow model, and input data. Estimation approach encompasses methods that estimate the traffic state, based on partial observation and a priori knowledge (assumptions) on traffic dynamics. Estimation approaches can be roughly classified into three according to their dependency on a priori knowledge and empirical data: model-driven, data-driven, and streaming-data-driven. A traffic flow model usually means a physics-based mathematical model representing traffic dynamics, with various solution methods. Input data can be characterized by using three different properties: collection method (stationary or mobile), data representation (disaggregated or aggregated), and temporal condition (real-time or historical). Based on our proposed characterization, we present the current state of TSE research and proposed future research directions. Some of the findings of this article are summarized as follows. We present model-driven approaches commonly used. We summarize the recent usage of detailed disaggregated mobile data for the purpose of TSE. The use of these models and data will raise a challenging problem due to the fact that conventional macroscopic models are not always consistent with detailed disaggregated data. Therefore, we show two possibilities in order to solve this problem: improvement of theoretical models, and the use of data-driven or streaming-data-driven approaches, which recent studies have begun to consider. Another open problem is explicit consideration of traffic demand and route-choice in a large-scale network; for this problem, emerging data sources and machine learning would be useful.

This article is motivated by the practical problem of highway traffic estimation using velocity measurements from GPS enabled mobile devices such as cell phones. In order to simplify the estimation procedure, a velocity model for highway traffic is constructed, which results in a dynamical system in which the observation operator is linear. This article presents a new scalar hyperbolic partial differential equation (PDE) model for traffic velocity evolution on highways, based on the seminal Lighthill-Whitham-Richards (LWR) PDE for density. Equivalence of the solution of the new velocity PDE and the solution of the LWR PDE is shown for quadratic flux functions. Because this equivalence does not hold for general flux functions, a discretized model of velocity evolution based on the Godunov scheme applied to the LWR PDE is proposed. Using an explicit instantiation of the weak boundary conditions of the PDE, the discrete velocity evolution model is generalized to a network, thus making the model applicable to arbitrary highway networks. The resulting velocity model is a nonlinear and nondifferentiable discrete time dynamical system with a linear observation operator, for which a Monte Carlo based ensemble Kalman filtering data