A Tracking Algorithm for Car Paths on Road Networks

Abstract

Abstract. In this paper we introduce a computation algorithm to trace car paths on road networks, whose load evolution is modeled by conservation laws. This algorithm is composed by two parts: computation of solutions to conservation equations on each road and localization of car position resulting by interactions with waves produced on roads. Some applications and examples to describe the behavior of a driver traveling in a road network are showed. Moreover a convergence result for wave front tracking approximate solutions, with BV initial data on a single road is established.

Scalar conservation law with discontinuity arising in pedestrian modeling

“…In paper [2], Bretti and Piccoli build the trajectory of a single vehicle in traffic flow. This algorithm will be extended to our pedestrian simulation with a discontinuous flux function.…”

“…Section 2 shows the approximation of the solution with finite volume methods, without any specific attention to the treatment of the separation point. In the last section, we complete the finite volume methods with the tracking of a single pedestrian, using the algorithm developed by Bretti and Piccoli [2]. The plot of the trajectory displays the change of direction and gives a better insight on the turning phenomena which can happen during a simulation.…”

“…and the unique viscosity solution to the Dirichlet problem (2), (4) is given by the value function of the corresponding control problem with discontinuous coefficient c(ρ), i.e.…”

Optimal Paths on Urban Networks Using Travelling Times Prevision

“…where x in is the initial position at the initial time t in (see [9]). For finding the optimal route on the network in a generic time instant t 0 , the CKSPa individuates, among all possible arcs exiting from the origin o, the arc I ψ with minimal travel time t oψ (t 0 ) connecting o with the node ψ.…”

“…The latter are the weights of a Constrained KShortest Paths algorithm (CKSPa). Using a further procedure (analyzed in [9]), that computes the trajectory of a car on a network, we compute the time t 1 at which a car reaches the successive node s 1 . At t 1 , if the current density conditions are different from the forecasted ones, average density values on roads are computed invoking again the simulator with new initial conditions and the new densities are used as weights (travel times) that the CKSPa needs to select the optimal arc among those outgoing from s 1 .…”