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Authors
Filippos N. Tzortzoglou, Logan E. Beaver, Andreas A. Malikopoulos
Abstract
This website is provided to faciliate the comprehension of our letter entitled "A Feasibility Analysis for Signal Free Intersection". In this letter, we address the problem of improving the feasible domain of the solution of a decentralized control framework for coordinating connected and automated vehicles (CAVs) at signal-free intersections. The framework provides the optimal trajectories of CAVs to cross the intersection safely without stop-and-go driving. However, when traffic volume exceeds a certain level, finding a feasible solution for a CAV may become unattainable. We use concepts of numerical interpolation to identify appropriate polynomials that can serve as alternative trajectories of the CAVs, expanding the domain of the feasible CAV trajectories. We select the alternative polynomials through an optimization problem that aims at minimizing jerk. Finally, we demonstrate the efficacy of our approach through numerical simulations.
The full paper is available in this link.
Finding a solution in Problem 2
As discussed in the manuscript, in order to find a solution to Problem 2 we initialize a feasible range of exit times. We then select the minimum one that satisfies the corresponding constraints. Especially, using Problem 1, we derive the optimal control strategy and check if the solution satisfies all the constraints. If it does, we have found our solution. Otherwise, we increase the exit time by a time step and repeat the process. To understand this better, we can visualize the feasible trajectories for a CAV in the specific scenario depicted in the following figure.
We can even see the process in real-time in the following video:
Note that all the possible trajectories have a specific form which sometimes can be restrictive for defining a feasible trajectory. For that reason in the manuscript (after Section III), we define an approach that significantly increases the feasibility of the controller using ideas from numerical mathematics.
Video simulations comparing the two approaches
Following, we aim to showcase video simulations visualizing the effectiveness of each controller at their limits. Specifically, using the prior approach, we conducted simulations with a traffic flow of 5000 vehicles per hour, while with the proposed approach, simulations were conducted at a traffic flow of 7000 vehicles per hour. The goal of this visualization is to demonstrate the effectiveness of each approach at its limits and to underscore the significance of having a controller with greater feasibility, as it can handle more realistic scenarios.
Prior Approach | Traffic Volume: 5000 veh/h
Proposed Approach | Traffic Volume: 7000 veh/h
A case-study where the proposeed approach circumvents the feasibility issue of the prior approach
Consider the scenario depicted in the following Figure. Let the brown dashes represent the lateral constraints. At t=25 s, a new CAV approaches the intersection. However, this CAV cannot find any trajectory defined by Problem 2 due to both rear-end and lateral constraints. To understand that, we delve into this specific scenario. As a first step let us do not take into account the rear-end constraint. Then, the resulted solution is the red trajectory which respects only the lateral constraint, as illustrated in the zoomed-in frame. However, as it is depicted in the same figure, such a trajectory violates the rear-end constraint (as expected) after t=16 s (note that the repeated black lines after t = 13 s, adjacent to the green trajectory of the CAV that entered at t = 7.2 s, represent the rear-end constraints). Conversely, if we do not take into account the CAV that created this lateral constraint (or in general, if we ignore lateral contraints), we see in the black curve that the new trajectory satisfies the rear-end constraints but violates the lateral constraint (as expected). We confirm the result in the zoomed-in frame, as well. However, if we simultaneously consider both constraints, Problem 2 is not able to give a feasible solution. Finally, the blue curve represents the solution of our proposed approach that simultaneously satisfies both the rear-end and lateral constraints. One might query why the blue trajectory is not achievable from Problem 2. Well, the blue trajectory corresponds to the optimal solution provided by Problem 3 and is equal to
p(t) = 0.0039215t^4 - 0.2592821t^3 + 5.6205011t^2 - 35.53093479t - 0.0000000000002
which is close to a cubic polynomial with a small but non-zero coefficient on the 4th order term . It is evident that this trajectory, being a 4th order polynomial, cannot be derived from Problem 2. Note that the fact the blue trajectory does not change curvature does not imply that it can be achieved from a 3rd order solution of Problem 2, especially given that initial conditions, such as speed, are fixed. This also underscores the significance of the proposed approach to expand the feasibility domain compared to Problem 2.
Visualization of Proposition 1
In the following Figure we can visualize the result provided in Proposition 1 which mandates sufficient distance between vehicles on the same path to ensure safe passage from vehicles on conflicting paths.
In order to better understand how this proposition works, let us examine the position trajectories depicted in the following figure, which correspond to path 4 under a traffic volume of 7000 vehicles per hour for a sampled scenario. Although the vehicles from conflicting paths, which constitute the lateral constraints, sometimes appear to be very close, Proposition 1 ensures that there is always a small time gap allowing a vehicle to safely cross between them. In particular, this can be confirmed by observing the second blue trajectory and the zoomed-in frame associated with the first lateral constraint.
Video simulation in Vissim software
In the following video, we demonstrate how our scenario can be implemented and visualized using a commercial software simulator. We selected a traffic volume of 6,000 vehicles per hour to observe the impact of handling a relatively high traffic volume.
Proposed Approach | Vissim Simulator | 6000 veh/h
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