Followerstopper Revisited: Phase-space Lagrangian Controller for Traffic Decongestion

Rahul Bhadani

REVIEW 3 major objections 5 minor 23 references

Followerstopper Revisited: Phase-space Lagrangian Controller for Traffic Decongestion

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper specifies the complete Followerstopper control law, including the previously undocumented nominal speed-smoothing layer, and shows that the commanded phase-space trajectory stays out of the collision-risk zone.

desk verdict Writes down Followerstopper's control law and nominal controller explicitly, but the appendix code contains a 16 m override that contradicts Eq. (8)-(10), so the 'complete reproducible specification' claim fails as stated. read the letter →

arxiv 2506.08036 v2 pith:FINKBMLK submitted 2025-06-03 eess.SY cs.SYnlin.CD

Rahul Bhadani This is my paper

classification eess.SYcs.SYnlin.CD

keywords Followerstopperphase-spacecontrolLagrangiantrafficstop-and-gowavesmixedautonomyautonomousvehiclenominalcontrollercar-following

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Followerstopper is a phase-space controller for a single autonomous vehicle in mixed traffic: it chooses the vehicle's velocity from its relative distance and relative velocity to the car ahead, with the goal of damping stop-and-go waves and avoiding collision. This paper's contribution is the complete, self-contained mathematical specification that the earlier field report did not provide: the switching control law of Eq. (8), the three quadratic phase-space envelopes of Eq. (10) with their numeric parameters, and a previously undocumented nominal controller (Algorithm 1) that smooths the reference speed so the vehicle never receives an unrealistically large velocity jump. If the specification is right, an engineer can reproduce the Arizona ring-road experiment's controller from the paper alone, and the experimentally observed wave damping and collision-avoidance behavior are explained by the phase-space geometry rather than by hidden implementation details.

What carries the argument

The load-bearing object is the quadratic envelope family $d_j(v_{\mathrm{rel}})=\omega_j+\frac{1}{2\alpha_j}\min\{0,v_{\mathrm{rel}}\}^2$, $j=1,2,3$, which creates three mode-switching boundaries in the $(x_{\mathrm{rel}},v_{\mathrm{rel}})$ phase plane. It carries the argument because it turns a kinematic constant-acceleration assumption into explicit safety and smoothness constraints: region $S_1$ inside $d_1$ commands full stop, $S_2$ between $d_1$ and $d_2$ blends toward the leader's speed, $S_3$ between $d_2$ and $d_3$ blends toward the reference speed, and $S_4$ outside $d_3$ follows the reference. A second mechanism is the nominal controller's persistent state $y$, which ramps the commanded reference toward the user's target speed at bounded acceleration and deceleration and then clamps the output to within $[-1,+2]\,\mathrm{m/s}$ of the current vehicle speed, preventing velocity jumps.

What would settle it

Log the commanded velocity and the measured velocity during a field deployment, then plot both trajectories in $(x_{\mathrm{rel}},v_{\mathrm{rel}})$; if the measured trajectory ever crosses the $d_1(v_{\mathrm{rel}})$ boundary while the commanded trajectory does not, the immediate-tracking assumption is violated and the Figure 10 safety conclusion does not cover the actual vehicle.

Watch

Extended reading notes

Core claim

The central claim is that the whole Followerstopper controller reduces to a constant-acceleration phase-space law. Starting from $v_i(t)=v_{0i}+\alpha_i t$ for leader and follower, the paper integrates the relative-velocity equation to obtain $x_{\mathrm{rel}}(t)=\omega+\frac{1}{2\alpha}\min\{0,v_{\mathrm{rel}}(t)\}^2$; the $\min\{0,\cdot\}$ makes the envelope react only when the ego vehicle is closing the gap. This family of parabolas, with three offset and curvature parameter pairs $(\omega_1,\alpha_1)=(4.5,1.5)$, $(\omega_2,\alpha_2)=(5.25,1)$, and $(\omega_3,\alpha_3)=(6.0,0.5)$, divides the phase plane into four switching regions, and the control law of Eq. (8) returns a velocity command ranging from full stop in the innermost region, to matching the leader's speed, to converging to the reference speed $r$. The paper further claims that the nominal controller of Algorithm 1, never before published, is what keeps the reference input from causing abrupt acceleration, and that under the complete law the commanded trajectory in the field experiment stayed outside the collision-risk zone.

Load-bearing premise

The design assumes the car's low-level actuator executes each velocity command from Eq. (8) immediately and exactly; if real-world delays, actuator lag, or state-estimation errors make the actual velocity differ from the commanded one, the actual trajectory could enter the collision-risk zone even though the command does not.

Editorial extensions

If this is right

Reproducibility: Eq. (8), Eq. (10) with the tabulated parameters, and Algorithm 1 are a complete controller implementation; the Simulink model is no longer needed to understand or re-implement the behavior.
Safety claim: for the tested reference speeds (6.5 to 8.0 m/s), the commanded phase-space trajectory remains outside the innermost envelope, that is, outside the collision-risk zone.
Performance: the field data show the strongest wave damping at a 7.5 m/s reference, so reference-speed selection is a usable tuning knob for traffic decongestion.
Deployability: the controller needs only local leader-relative measurements, so a single autonomous vehicle in mixed traffic can act as a Lagrangian actuator without infrastructure or connectivity.
The nominal layer guards against unrealistic acceleration: the reference velocity cannot jump from a low value to a high value in one control step.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the $d_j$ envelopes act as barrier-like safety surfaces, a natural next step—also noted by the author—is to certify safety formally with control barrier certificates under measurement uncertainty; the paper's geometry supplies the candidate barrier.
The same phase-space structure could be extended to uncertain state estimates by inflating each $\omega_j$ and $\alpha_j$ by a margin tied to the estimator's covariance, yielding a tunable robustness knob.
The nominal controller's output clamp $[\mathrm{vel}-1,\mathrm{vel}+2]$ can be viewed as a standalone anti-jerk filter; it could be reused ahead of other reference-tracking controllers without the Followerstopper switching law.
A controlled re-run of the ring experiment at several fixed reference speeds, with both the command and the actual velocity logged, would give a quantitative measure of how much of the wave damping comes from the phase-space law versus the nominal smoothing layer.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper does a useful thing—it writes down the Followerstopper control law and the nominal controller explicitly—but its central claim to be a complete, reproducible specification is undercut by its own appendix code. The m-file contains a 16 m override absent from Eq. (8)-(10).

What is genuinely new: the paper gives the phase-space derivation, the min{0,vrel} modification, the explicit switching law, and the first formal description of the nominal controller. The derivation from Eq. (3) to Eq. (6) is correct, and most of the code matches Eq. (8). Data and code links are provided. That is real, useful material for anyone trying to implement or extend the controller.

The main problem is in Appendix A.1. Lines 30-32 command u_cmd = r whenever dx > 16.0, regardless of closing speed. Eq. (8) only enters S4 at d3(vrel) = 6.0 + vrel^2 (with alpha3 = 0.5), so for vrel = -5 m/s that bound is 31 m. In the 16-31 m region, the published law decelerates; the shipped code runs at reference speed. That is safety-relevant, and the paper never mentions the override. So the claim that Eq. (8)-(10) plus Algorithm 1 is a complete specification of the field controller is not supported by the paper's own appendix. The reader's actuator-tracking concern is real, but this mismatch is more direct: even under perfect tracking, the equation-based controller and the code are different controllers.

Two secondary soft spots. First, Figure 10's safety conclusion is largely circular: the trajectory avoids the collision-risk zone because the commanded velocities are defined by the same parabolas. That is not independent evidence of safety. Second, the wave-dampening claim rests on the earlier self-cited field experiment, with no new data or quantification. These are worth stating clearly, but the 16 m override is the load-bearing issue.

Who should read this: anyone who wants the Followerstopper formulas or the nominal controller details. But they should treat the shipped code as the actual spec, not the equations, until the mismatch is resolved. The paper deserves peer review, not desk rejection, because the formalization of a field-tested controller is genuinely useful and the flaw is fixable. I would want a revised version that either documents the override in the control law and algorithm or removes it and re-runs the safety trace, before trusting it as a specification.

Referee Report

3 major / 5 minor

Summary. This paper revisits the Followerstopper controller, a phase-space-based car-following controller previously field-tested in the Arizona ring-road experiment. The authors derive a parabolic phase-space safety envelope from a constant-relative-acceleration kinematic model, define a switching control law (Eq. 8–10) with four regions, and present a nominal controller (Algorithm 1) for smoothing the reference speed. The manuscript discusses the phase-space design, provides Simulink and MATLAB code, and uses the prior experiment's time-space diagram and phase-space trajectory to claim wave-dampening and safety. The stated contribution is a complete, reproducible mathematical formulation of the controller and the first presentation of the nominal controller.

Significance. The paper addresses a real gap in the literature: the Followerstopper controller has been deployed in a landmark field experiment, but its full mathematical formulation and the nominal speed-smoothing layer were not previously documented. If accurate, the derivation from Eq. (3) to Eq. (6) is correct, and the included code and data links support reproducibility. However, the paper reports no new experimental results, and its central reproducibility claim is compromised by a mismatch between the published control law and the supplied implementation (see Major Comment 1). The phase-space interpretation is instructive and the nominal controller is a useful addition, but the safety and wave-damping conclusions require more careful support than the paper currently provides.

major comments (3)

[Appendix A.1] Lines 30–32 of the Followerstopper m-file contain an unconditional override `if( dx > 16.0 ) u_cmd = r;` that is absent from the control law in Eq. (8)–(10) and from Algorithm 1. For a closing speed of vrel = -5 m/s, d3 = 31 m, so for gaps between 16 m and 31 m the published law commands a speed strictly below the reference r (region S3), whereas the code commands the full reference speed r. This is not a cosmetic discrepancy: it changes the controller's behavior in the approach region that is most relevant for safety. The paper nowhere mentions this override, and its existence undermines the claim that Eq. (8)–(10) plus Algorithm 1 constitutes a complete, reproducible specification of the field controller. The authors must either incorporate this override into the formal control law and discuss its design rationale and safety implications, or remove it from the code and re-verify that Figure 10 still reflects the implemented controller.
[Section 4, Figure 10] The conclusion that the AV's behavior was safe because the phase-space trajectory never enters the Collision Risk Zone is weaker than it appears. The controller is defined so that the commanded velocity keeps the point (xrel, vrel) on or outside the parabolas that separate the risk zone; therefore the observed trajectory staying out of the zone is a partial confirmation of the control law's definition rather than an independent safety certificate. In particular, the claim relies on two unstated assumptions: (i) the vehicle's low-level actuator tracks the velocity command u_cmd without significant delay or error, and (ii) the lead vehicle's deceleration is bounded by the relative-acceleration model used to derive the parabolas. The paper provides no tracking-error data, no robustness analysis, and no formal safety argument (e.g., a control barrier certificate, which the conclusion itself mentions as future work). The authors should either provide such an analysis or explicitly scope the safety claim to the ideal tracking case.
[Section 4, Figure 9] The wave-dampening claim is asserted on the basis of the time-space diagram from the prior field experiment, not on new quantitative evidence in this paper. The text states that stop-and-go waves appeared at t≈79 s and that Followerstopper took over at t≈126 s, with waves dampened after 30–35 s, but no quantitative metrics (e.g., speed variance, wave amplitude, or dissipation rate) are provided, nor is there a comparison against a no-control baseline. Since the paper's title and abstract present wave-dampening as a central achievement, the authors should either reproduce a quantitative analysis (even from the existing dataset, which is available) or clearly label the figure as a reproduction of Stern et al. [15] and defer the quantitative claim to that reference.

minor comments (5)

[Algorithm 1] The pseudo-code is missing comparison operators: the lines `if y max_speed + 1` and `elseif y max_speed − 1` should read `if y > max_speed + 1` and `elseif y < max_speed − 1`. As printed, the algorithm is not well-defined, although the appendix code clarifies the intended logic.
[Section 3.1] There is a typo in the sentence 'so as not to cause any abrupt acceleration or deceleration (or example, at one time-step...' – 'or example' should be 'for example'.
[Equations (8)-(10)] The definition of v(vlead) = min{max{vlead,0}, r} is given as text after Eq. (8); it would be clearer to include it inside the displayed equation or to refer to it explicitly in the notation list, since Eq. (8) uses v(vlead) as a function argument.
[Section 2.1] The derivation of Eq. (6) assumes vrel0 = 0, which is stated, but the use of min{0,vrel} in Eq. (7) and the subsequent control law deserves a brief explanation of how positive relative velocities (increasing gap) are handled, since they collapse the switching boundaries to constants.
[Figure 10] The figure would benefit from a clearer annotation identifying which portions of the trajectory correspond to the different switching regions and to the times when the nominal reference speed is changed, to help the reader connect the phase-space plot to Table 1 and Figure 9.

Circularity Check

1 steps flagged · score 4.0 of 10

Figure 10 safety claim restates the design invariant of Eq. (8)-(10); the appendix's undocumented 16 m override is a non-circular code/equation mismatch.

self definitional [Section 4 (Results), Figure 10, and Eq. (8)-(10)]
"We also provide xrel − vrel phase-space diagram for the AV in Figure 10. We see that the phase-space curve is not seen in the Collision Risk Zone when the autonomous mode was active under Followerstopper control. Thus, we conclude that for the speed regime under which the autonomous vehicle was operating, its behavior was safe."

The switching boundaries d_j in Eq. (10) are parabolas xrel = omega_j + (1/(2 alpha_j)) min{0,vrel}^2, and the control law Eq. (8) commands u=0 inside S1 and a speed below vlead in S2. For vrel<0 the S2/S3 boundary d2 acts as a barrier: as the gap shrinks, the commanded speed ramps down to vlead exactly at d2, so vrel tends to 0 before the state can cross into smaller gaps. Thus, under perfect tracking and state estimation, avoiding the Collision Risk Zone is a direct consequence of the controller's definition, not an independent safety prediction. Figure 10 therefore confirms the design invariant, and the Section 4 conclusion reads the controller's own construction back as an empirical safety result.

full rationale

The paper's central contribution—the explicit phase-space control law (Eq. 8-10) and the nominal smoothing algorithm (Algorithm 1)—is self-contained mathematics and is not derived from its own conclusions. The wave-damping evidence comes from the Arizona ring-road field experiment (Fig. 9, Ref. [15]); despite overlapping authorship, this is external empirical data, not a self-validating citation, so it does not create circularity. The one circular aspect is the safety claim in Section 4: the Collision Risk Zone is the region excluded by the switching parabolas, and Eq. (8) is defined so that the closed-loop vector field cannot enter that region under ideal tracking; Figure 10 therefore illustrates the controller's definition rather than testing an independent prediction. Separately, Appendix A.1 contains an undocumented override (if dx > 16.0, u_cmd = r) that contradicts Eq. (8) in the 16-31 m closing-speed regime; this is a serious reproducibility or specification flaw, but it is an inconsistency, not a circular reduction, so it does not raise the circularity score. Overall score 4 reflects one self-definitional safety conclusion while the main formulation retains independent content.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard kinematics plus several domain assumptions about sensing and actuation that are not formally justified. The controller's safety envelopes and nominal smoothing thresholds are hand-chosen free parameters, which is the main source of circularity burden. No new physical entities are introduced.

free parameters (4)

omega_1, omega_2, omega_3 = 4.5 m, 5.25 m, 6.0 m
Hand-chosen distances that define the switching envelopes in Eq. (10). No tuning procedure or safety derivation is provided.
alpha_1, alpha_2, alpha_3 = 1.5 m/s^2, 1.0 m/s^2, 0.5 m/s^2
Hand-chosen curvature parameters for the safety parabolas in Eq. (10). The paper gives no rule for selecting them.
dx_override = 16.0 m
Hard-coded in the m-file implementation: if dx > 16.0, Followerstopper outputs the reference directly, overriding the switching logic. No justification is given.
Nominal controller constants = dt=0.05 s, bandwidth ±1 m/s, speed floor 1-2 m/s, output clamp [vel-1.0, vel+2.0] m/s
Hard-coded thresholds in Algorithm 1 with no tuning rationale or sensitivity analysis. They shape the reference velocity and hence the controller's behavior.

assumptions (5)

standard math Constant-acceleration kinematics for the two-car system
Used to derive Eq. (4)-(6) in Section 2.1. This is a standard kinematic assumption, though real traffic involves time-varying accelerations.
domain assumption Accurate real-time measurement of xrel and vrel
The controller in Eq. (8) depends on relative distance and relative velocity. The original experiment needed extensive sensor filtering, which is not modeled here.
domain assumption The AV actuator tracks vcmd with negligible error
Section 3.2 describes sending commands to the CAT Vehicle actuator, but no actuator dynamics or delay are included in the analysis. The safety conclusion in Section 4 depends on this assumption.
ad hoc to paper The min{0,vrel} modification and the three chosen parabolas are a valid safety design
Equation (7) introduces min{0,vrel} and Eq. (10) selects specific omega and alpha values. These are design choices, not derived from a formal safety specification.
ad hoc to paper The nominal controller thresholds are appropriate for traffic smoothing
Algorithm 1 uses hard-coded values for time step, speed bands, and output clamps. No derivation or experiment is given to justify these values.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Followerstopper Revisited: Phase-space Lagrangian Controller for Traffic Decongestion." pith.science (2026). https://pith.science/paper/FINKBMLK

@misc{pith2026250608036,
  author       = {Pith},
  title        = {Pith review of: Followerstopper Revisited: Phase-space Lagrangian Controller for Traffic Decongestion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FINKBMLK}},
  note         = {Machine review of arXiv:2506.08036}
}

read the original abstract

This paper revisits Followerstopper, a phase-space-based control system that had demonstrated its ability to mitigate emergent traffic jams due to stop-and-go traffic during rush hour in the mixed-autonomy setting. Followerstopper was deployed on an autonomous vehicle. The controller attenuates the emanant traffic waves by regulating its velocity according to the relative distance and velocity of the leader car. While regulating the velocity, the controller also prevents the collision of the ego vehicle with the lead vehicle within the range specified by the controller's design parameter. The controller design is based on a configurable quadratic curve on relative distance-relative velocity phase-space that allows the transition of the regulated velocity from (i) no modification of input, (ii) decelerating to match the leader's velocity (iii) braking to avoid any imminent collision. In this paper, we explore the phase-space properties of Followerstopper and provide a detailed description of a nonlinear control law that regulates the reference input to Followerstopper within the physics-informed boundaries. We also provide a new discussion on the nominal control law that regulates the reference speed to Followerstopper to avoid unrealistic and unsafe acceleration.

Figures

Figures reproduced from arXiv: 2506.08036 by the authors.

**Figure 2.** Phase-Space Portraits for Three Dynamical Systems Categories for Two-Car-Following Systems [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Phase-Space Portraits for xrel-vrel with a fixed ω = 5 and varying α. Vectors in the region pointing downward/leftward indicate rapid closure of the gap between the leader and the ego vehicle. Quiver plot is shown for α = 1. ing. Trajectories crossing this line indicate transitions between safe/unsafe states. Further, α 0 In the case of a leader-follower scenario, the negative relative velocity of the leader with … view at source ↗

Figures from the paper (6 more)

**Figure 5.** Figure 5: A schematic diagram of Followerstopper controller [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 6.** Figure 6: Switching regions allows a smoother transition from [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

**Figure 7.** Figure 7: Followerstopper Simulink Model. The function block specifies the logic from Equation ( [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 8.** Figure 8: Nominal Controller Implementation in smoothUpParams Simulink Block as specified in Algorithm [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

**Figure 9.** Figure 9: Annotated time-space diagram for Arizona-ring road experiment with Followerstopper’s wave-dampening capability. [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]

**Figure 10.** Figure 10: xrel − vrel evolution overlaid with the switching boundary. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[15]

Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments,

R. E. Stern, S. Cui, M. L. Delle Monache, R. Bhadani, M. Bunting, M. Churchill, N. Hamilton, R. Haulcy, H. Pohlmann, F . Wuet al., “Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments,” Transportation research part C: emerging technologies, vol. 89, pp. 205–221, 2018

work page 2018
[1]

Is there more traffic con- gestion in larger cities?-scaling analysis of the 101 largest us ur- ban centers,

Y. S. Chang, Y. J. Lee, and S. S. B. Choi, “Is there more traffic con- gestion in larger cities?-scaling analysis of the 101 largest us ur- ban centers,” Transport Policy, vol. 59, pp. 54–63, 2017

work page 2017
[2]

A simple contagion process describes spreading of traffic jams in urban networks,

M. Saberi, H. Hamedmoghadam, M. Ashfaq, S. A. Hosseini, Z. Gu, S. Shafiei, D. J. Nair, V . Dixit, L. Gardner, S. T . Walleret al., “ A simple contagion process describes spreading of traffic jams in urban networks,” Nature communications, vol. 11, no. 1, pp. 1–9, 2020

work page 2020
[3]

Understanding con- gested travel in urban areas,

S. Çolak, A. Lima, and M. C. González, “Understanding con- gested travel in urban areas,” Nature communications, vol. 7, no. 1, pp. 1–8, 2016

work page 2016
[4]

Traffic and related self-driven many-particle sys- tems,

D. Helbing, “Traffic and related self-driven many-particle sys- tems,” Reviews of modern physics, vol. 73, no. 4, p. 1067, 2001

work page 2001
[5]

Improving noise assessment at intersections by modeling traffic dynam- ics,

E. Chevallier, A. Can, M. Nadji, and L. Leclercq, “Improving noise assessment at intersections by modeling traffic dynam- ics,” Transportation Research Part D: Transport and Environ- ment, vol. 14, no. 2, pp. 100–110, 2009

work page 2009
[6]

Driver heterogeneity in car following and its impact on modeling traffic dynamics,

S. Ossen and S. P . Hoogendoorn, “Driver heterogeneity in car following and its impact on modeling traffic dynamics,” Trans- portation research record, vol. 1999, no. 1, pp. 95–103, 2007

work page 1999
[7]

Analyzing freeway traffic under con- gestion: Traffic dynamics approach,

D. H. Nam and D. R. Drew, “ Analyzing freeway traffic under con- gestion: Traffic dynamics approach,” Journal of Transportation Engineering, vol. 124, no. 3, pp. 208–212, 1998

work page 1998

Show all 23 references

[8]

Behavioral car-following model for computer simulation,

P . G. Gipps et al., “Behavioral car-following model for computer simulation,” Transport. Res., vol. 15, no. 2, pp. 105–111, 1981

1981
[9]

Stabilization effect of traffic flow in an extended car-following model based on an intelligent transportation system application,

H. Ge, S. Dai, L. Dong, and Y. Xue, “Stabilization effect of traffic flow in an extended car-following model based on an intelligent transportation system application,” Physical Review E , vol. 70, no. 6, p. 066134, 2004

2004
[10]

Carsim: Car-following model for simulation of traffic in normal and stop-and-go conditions,

R. F . Benekohal and J. Treiterer, “Carsim: Car-following model for simulation of traffic in normal and stop-and-go conditions,” Transportation research record, vol. 1194, pp. 99–111, 1988

1988
[11]

A new car-following model accounting for varying road condition,

T . Tang, Y. Wang, X. Yang, and Y. Wu, “ A new car-following model accounting for varying road condition,” Nonlinear Dynamics , vol. 70, no. 2, pp. 1397–1405, 2012

2012
[12]

A review analysis of op- timal velocity models,

H. Lazar, K. Rhoulami, and D. Rahmani, “ A review analysis of op- timal velocity models,” Periodica Polytechnica Transportation Engineering, vol. 44, no. 2, pp. 123–131, 2016

2016
[13]

Traffic jams with- out bottlenecks—experimental evidence for the physical mech- anism of the formation of a jam,

Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S.-i. Tadaki, and S. Yukawa, “Traffic jams with- out bottlenecks—experimental evidence for the physical mech- anism of the formation of a jam,”New journal of physics, vol. 10, no. 3, p. 033001, 2008

2008
[14]

Ivs 05: New developments and research trends for intelligent vehicles,

L. Li, J. Song, F .-Y. Wang, W . Niehsen, and N.-N. Zheng, “Ivs 05: New developments and research trends for intelligent vehicles,” IEEE Intelligent Systems, vol. 20, no. 4, pp. 10–14, 2005

2005
[16]

Controllability analysis and optimal control of mixed traffic flow with human- driven and autonomous vehicles,

J. Wang, Y. Zheng, Q. Xu, J. Wang, and K. Li, “Controllability analysis and optimal control of mixed traffic flow with human- driven and autonomous vehicles,” IEEE Transactions on Intelli- gent Transportation Systems, vol. 22, no. 12, pp. 7445–7459, 2020

2020
[17]

Interaction-aware model predictive control for automated vehicles in mixed-autonomy traffic,

M. Shang, S. Wang, T . Li, and R. Stern, “Interaction-aware model predictive control for automated vehicles in mixed-autonomy traffic,” in 2024 IEEE Intelligent Vehicles Symposium (IV). IEEE, 2024, pp. 317–322

2024
[18]

Mitigating traffic oscillation through control of connected automated vehicles: A cellular au- tomata simulation,

Y. Wang, Y. Jiang, Y. Wu, and Z. Yao, “Mitigating traffic oscillation through control of connected automated vehicles: A cellular au- tomata simulation,” Expert Systems with Applications, vol. 235, p. 121275, 2024

2024
[19]

Benchmarks for rein- forcement learning in mixed-autonomy traffic,

E. Vinitsky, A. Kreidieh, L. Le Flem, N. Kheterpal, K. Jang, C. Wu, F . Wu, R. Liaw, E. Liang, and A. M. Bayen, “Benchmarks for rein- forcement learning in mixed-autonomy traffic,” in Conference on robot learning. PMLR, 2018, pp. 399–409

2018
[20]

Fuel consumption reduction of multi-lane road networks us- ing decentralized mixed-autonomy control,

N. Lichtlé, E. Vinitsky, G. Gunter, A. Velu, and A. M. Bayen, “Fuel consumption reduction of multi-lane road networks us- ing decentralized mixed-autonomy control,” in2021 IEEE Inter- national Intelligent Transportation Systems Conference (ITSC) . IEEE, 2021, pp. 2068–2073

2021
[21]

Learning energy-efficient driving behaviors by imitating experts,

A. R. Kreidieh, Z. Fu, and A. M. Bayen, “Learning energy-efficient driving behaviors by imitating experts,” in 2022 IEEE 25th In- ternational Conference on Intelligent Transportation Systems (ITSC). IEEE, 2022, pp. 2689–2695

2022
[22]

Ros: an open-source robot operat- ing system,

M. Quigley, K. Conley, B. Gerkey, J. Faust, T . Foote, J. Leibs, R. Wheeler, A. Y. Ng et al., “Ros: an open-source robot operat- ing system,” in ICRA workshop on open source software , vol. 3, no. 3.2. Kobe, 2009, p. 5

2009
[23]

Generating a ros/jaus bridge for an autonomous ground vehicle,

P . Morley, A. Warren, E. Rabb, S. Whitsitt, M. Bunting, and J. Sprinkle, “Generating a ros/jaus bridge for an autonomous ground vehicle,” in Proceedings of the 2013 ACM workshop on Domain-specific modeling, 2013, pp. 13–18. 9 A Simulink Models and Code A.1 m-file Implementati...

2013

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [15]

Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments,

R. E. Stern, S. Cui, M. L. Delle Monache, R. Bhadani, M. Bunting, M. Churchill, N. Hamilton, R. Haulcy, H. Pohlmann, F . Wuet al., “Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments,” Transportation research part C: emerging technologies, vol. 89, pp. 205–221, 2018

work page 2018

[2] [1]

Is there more traffic con- gestion in larger cities?-scaling analysis of the 101 largest us ur- ban centers,

Y. S. Chang, Y. J. Lee, and S. S. B. Choi, “Is there more traffic con- gestion in larger cities?-scaling analysis of the 101 largest us ur- ban centers,” Transport Policy, vol. 59, pp. 54–63, 2017

work page 2017

[3] [2]

A simple contagion process describes spreading of traffic jams in urban networks,

M. Saberi, H. Hamedmoghadam, M. Ashfaq, S. A. Hosseini, Z. Gu, S. Shafiei, D. J. Nair, V . Dixit, L. Gardner, S. T . Walleret al., “ A simple contagion process describes spreading of traffic jams in urban networks,” Nature communications, vol. 11, no. 1, pp. 1–9, 2020

work page 2020

[4] [3]

Understanding con- gested travel in urban areas,

S. Çolak, A. Lima, and M. C. González, “Understanding con- gested travel in urban areas,” Nature communications, vol. 7, no. 1, pp. 1–8, 2016

work page 2016

[5] [4]

Traffic and related self-driven many-particle sys- tems,

D. Helbing, “Traffic and related self-driven many-particle sys- tems,” Reviews of modern physics, vol. 73, no. 4, p. 1067, 2001

work page 2001

[6] [5]

Improving noise assessment at intersections by modeling traffic dynam- ics,

E. Chevallier, A. Can, M. Nadji, and L. Leclercq, “Improving noise assessment at intersections by modeling traffic dynam- ics,” Transportation Research Part D: Transport and Environ- ment, vol. 14, no. 2, pp. 100–110, 2009

work page 2009

[7] [6]

Driver heterogeneity in car following and its impact on modeling traffic dynamics,

S. Ossen and S. P . Hoogendoorn, “Driver heterogeneity in car following and its impact on modeling traffic dynamics,” Trans- portation research record, vol. 1999, no. 1, pp. 95–103, 2007

work page 1999

[8] [7]

Analyzing freeway traffic under con- gestion: Traffic dynamics approach,

D. H. Nam and D. R. Drew, “ Analyzing freeway traffic under con- gestion: Traffic dynamics approach,” Journal of Transportation Engineering, vol. 124, no. 3, pp. 208–212, 1998

work page 1998