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Model based string stability of adaptive cruise control systems using field data

George Gunter, Caroline Janssen, William Barbour, Raphael E. Stern, Daniel B. Work

arXiv:1902.04983v1eess.SYcs.SY

TL;DR

Commercial ACC is increasingly available, but empirical evidence on its traffic stability is limited. The authors collect car-following data and calibrate models for two following settings. Both calibrated models are string unstable, although some disturbances are dampened in moderate-sized platoons.

  • Problem

    Empirical evidence on the performance and traffic stability of commercially available ACC systems is limited, despite their growing availability.

    Source: §I · pp. 2–3

  • Method

    The authors conduct car-following experiments with a commercial ACC vehicle and calibrate an optimal-velocity relative-velocity model for minimum and maximum following settings.

    Source: §I · p. 3, A. ACC model · p. 5

  • Results

    Both calibrated ACC models are string unstable, while practical disturbances can be dampened in moderate-sized platoons before eventually amplifying in longer platoons.

    Source: C. String stability of calibrated models · p. 14, §VI · p. 17

  • Takeaways & Limitations

    The tested system can amplify some disturbances, but its string instability does not prevent damping of disturbances in moderate-sized platoons.

    Source: §VI · p. 17

    The results do not establish whether ACC vehicles perform better or worse than human drivers.

    Source: §I · p. 3

Abstract

from arXiv · show

This article is motivated by the lack of empirical data on the performance of commercially available Society of Automotive Engineers level one automated driving systems. To address this, a set of car following experiments are conducted to collect data from a 2015 luxury electric vehicle equipped with a commercial adaptive cruise control (ACC) system. Velocity, relative velocity, and spacing data collected during the experiments are used to calibrate an optimal velocity relative velocity car following model for both the minimum and maximum following settings. The string stability of both calibrated models is assessed, and it is determined that the best-fit models are string unstable, indicating they are not able to prevent all traffic disturbances from amplifying into phantom jams. Based on the calibrated models, we identify the consequences of the string unstable ACC system on synthetic and empirical lead vehicle disturbances, highlighting that some disturbances can be dampened even with string unstable commercial ACC platoons of moderate size.

I. INTRODUCTION

This work tests whether a commercially available ACC system is string unstable and examines its effects on traffic disturbances. The question matters as ACC becomes common, while prior evidence about commercial systems and their stability remains limited.

I. INTRODUCTION

  • String stability means small equilibrium perturbations dissipate as they travel from vehicle to vehicle; under homogeneous following dynamics, one vehicle pair suffices to assess a platoon.Phantom jams can arise from traffic oscillations that amplify against the flow, and can increase fuel consumption and emissions,.Source: §I · pp. 1–2
  • Prior work found that human driving alone can trigger phantom jams, while connected platoons have been shown to achieve stability in theory and experiments,,, –.A single autonomous vehicle in a stream of 20 human-driven vehicles stabilized traffic and dampened stop-and-go waves.Source: §I · pp. 1–2
  • By the second quarter of 2018, 16 of the 20 best-selling US cars were available with ACC, and several included it as standard equipment.The paper argues this makes understanding the traffic-stability implications of commercial ACC important.Source: §I · p. 2
  • Earlier tests also found string-unstable commercial ACC: a 2003 study reported substantial overshoots, and a 2014 platoon test found instability,.The present work builds on these efforts to assess recent commercial systems.Source: §I · p. 3
  • The tested commercial ACC system is string unstable, so some disturbances are amplified; the study also examines consequences for synthetic and empirical traffic disturbances.The findings do not establish whether ACC vehicles perform better or worse than human drivers.Source: §I · p. 3
  • The study estimates a model from experimental car-following data, then analyzes its string stability, avoiding the need for a platoon of at least three vehicles.The paper notes that its results do not determine whether ACC vehicles perform better or worse than human drivers, who may also have string-unstable dynamics.Source: §I · p. 3

II. ACC DYNAMICAL MODEL AND STABILITY ANALYSIS

The paper uses an optimal-velocity relative-velocity car-following model to represent ACC dynamics and analyze string stability. Its constant effective time-gap formulation is designed to be straightforward to analyze and fit to field data.

A. ACC model

  • The model treats vehicle dynamics and ACC as a single system, simplifying analysis and calibration to field data when commercial controller logic is difficult to replicate.The authors report good reconstruction of observed ACC behavior in field tests.Source: A. ACC model · p. 4
  • The model builds on optimal-velocity dynamics, in which follower speed relaxes toward a desired speed prescribed by the current spacing.The paper introduces the relative-velocity extension as an additional relaxation toward the leader's velocity.Source: A. ACC model · p. 4
  • The model combines spacing-based optimal-velocity following with a term that relaxes follower speed toward leader speed; α and β control the trade-off between these behaviors.The paper adopts this optimal-velocity relative-velocity model with a constant effective time-gap term for ACC vehicles.Source: A. ACC model · p. 4
  • The constant effective time-gap formulation uses jam distance η and desired effective time-gap τe, with gains k1 and k2 on the spacing and relative-velocity terms.It uses the linear optimal-velocity function V(s) := (s − η)/τe and α := k1τe.Source: A. ACC model · pp. 4–5

B. Stability analysis

The section analyzes string stability using a car-following model and shows how effective time-gap affects stability and vehicle responses. Larger time-gaps can support stability, but reduce throughput.

B. Stability analysis

  • String stability means small equilibrium-flow perturbations dissipate along the platoon; unstable perturbations amplify as they propagate.The analysis evaluates a single-vehicle model using a sufficient stability condition based on its parameter derivatives; jam-spacing η does not affect stability.Source: B. Stability analysis · pp. 5–6
  • For small k1 and k2, a larger effective time-gap is needed for stability; increasing τe can first reduce λ2 and then increase it.At large τe the system is stable, but λ2 approaches 0 and throughput falls as headway increases.Source: B. Stability analysis · p. 6, Figure 1 · p. 6
  • A nine-vehicle simulation shows overshoot with τe = 0.75 s, while τe = 3.2 s produces stable, smoother responses without overshoot.The comparison uses k1 = 0.5, k2 = 0.5, and η = 8.Source: B. Stability analysis · p. 7, Figure 2 · p. 7

III. EXPERIMENTAL OVERVIEW AND TEST VEHICLE DESCRIPTION

The field experiments measure an ACC-equipped follower tracking a lead vehicle's prescribed velocity profile. Eighteen tests cover nine lead-vehicle profiles and the minimum and maximum ACC following settings.

III. EXPERIMENTAL OVERVIEW AND TEST VEHICLE DESCRIPTION

  • Each test pairs a lead vehicle executing a prescribed velocity profile with an ACC-engaged following vehicle.The lead uses cruise control where available; profiles below 28 mph are driven manually.Source: §III · pp. 7–8
  • The 18 tests combine nine lead-vehicle velocity profiles with the ACC's minimum or maximum following setting.The desired ACC velocity is set 5 mph above the profile's maximum lead-vehicle velocity, keeping the follower in gap-closing mode; tests use flat roadways.Source: §III · pp. 7–8
  • Profiles A and B use 5 mph velocity steps held for at least 60 seconds, spanning 5–30 mph and 35–55 mph, respectively.Both profiles repeat the step sequence while decreasing speed.Source: §III · p. 8
  • Profiles C–E cover a high-speed step test from 60 to 70 mph in 5-mph increments with 60-second holds, plus low- and medium-speed oscillations at 20/30 and 45/50 mph, each held 30 seconds.Source: §III · pp. 8–9
  • Profile I applies rapid 5, 10, 15, and 20 mph decreases from 50 mph, holding each decrease for 5 seconds before returning to 50 mph.Vehicle positions and velocities are logged by GPS evaluation kits connected to laptops, at up to 10 Hz.Source: §III · pp. 7, 9

A. Calibration of the ACC following dynamics

The ACC model is calibrated by minimizing velocity RMSE between simulated and measured follower trajectories. Parameter estimates are constrained by the model, initial conditions, and rational driving constraints.

A. Calibration of the ACC following dynamics

  • The calibration selects model parameters to minimize the RMSE between simulated and measured follower velocities.Measured lead-vehicle velocity evolves spacing in the simulation, while initial spacing and follower velocity are set to their measured values.Source: A. Calibration of the ACC following… · pp. 9–10
  • A constrained interior-point search estimates nonnegative k1, k2, τe, and η, with rational driving constraints enforcing physically realistic dynamics.The trajectory is simulated with explicit Euler steps at the 10 Hz measurement rate; a Runge-Kutta alternative was slower and yielded parameters not substantially different from the optimal values.Source: A. Calibration of the ACC following… · p. 9
  • Because the optimization is nonlinear and may have local minima, 100 randomly initialized runs are performed and the parameter set with the lowest RMSE is retained.The state variables depend on the parameters and are also decision variables.Source: A. Calibration of the ACC following… · pp. 9–10

V. RESULTS

The results section begins with GPS measurement accuracy, then compares calibrated models with measured trajectories, and finally assesses calibrated-model stability and its consequences.

V. RESULTS

  • The results cover GPS accuracy, model calibration and trajectory comparison, then stability and its consequences for both following settings.The section overview does not provide the accuracy or stability outcomes.Source: §V · p. 10

A. Validation of GPS measurements

The GPS units were checked by comparing two receivers mounted a known distance apart on the same vehicle. Their position and velocity measurements agreed sufficiently for recording vehicle position and velocity.

A. Validation of GPS measurements

  • The receivers’ mean absolute velocity difference was 0.06 m/s, less than 3% of the lowest test velocity.Relative position accuracy was 0.43 m, or 1–3% of typical following distances of 15–60 m.Source: A. Validation of GPS measurements · p. 10
  • The mean recorded distance between the antennas was 1.37 m, compared with an actual distance of 0.94 m.The distance was computed using the Haversine formula.Source: A. Validation of GPS measurements · p. 10
  • The U-blox EVK-M8T was deemed suitable for recording position and velocity data.Figure 4 shows the distributions of position-measurement error and instantaneous velocity difference between the two receivers.Source: A. Validation of GPS measurements · p. 11, Figure 4 · p. 11

B. Model calibration and validation

A single model for each following setting was calibrated across the velocity profiles, using the first half of each profile for training and the second half for holdout testing. The calibrated models fit the data well, with similar training and test errors, and their velocity–space-gap relationships closely matched the empirical data.

B. Model calibration and validation

  • For each following setting, one model was calibrated across all velocity profiles, with each profile split in half for training and holdout testing.Velocity RMSE selected the best-fitting parameters; space-gap RMSE was also reported.Source: B. Model calibration and validation · p. 11
  • For minimum following, test RMSEs were 0.22 m/s for velocity and 1.37 m for space-gap, versus training RMSEs of 0.23 m/s and 1.51 m.The test errors were slightly lower than training errors; profile F’s velocity and space-gap test errors were 0.25 m/s and 0.84 m, respectively.Source: B. Model calibration and validation · pp. 11, 14, Figure 5 · p. 12
  • For maximum following, training RMSEs were 0.28 m/s for velocity and 3.00 m for space-gap; test RMSEs were 0.30 m/s and 2.77 m.Profile E’s velocity and space-gap test RMSEs were 0.26 m/s and 2.48 m, respectively; training and test errors were similar.Source: Figure 6 · p. 12, B. Model calibration and validation · p. 14
  • The maximum-following model’s overall fit was slightly worse than the minimum-following model’s.For the maximum setting, the reported training and test velocity and space-gap errors were similar.Source: B. Model calibration and validation · p. 14
  • The calibrated velocity–space-gap relationships for both settings closely agreed with experimental data, including the jam-distance y-intercept.The slopes correspond to effective time-gaps of 0.5 s for minimum following and 1.7 s for maximum following.Source: B. Model calibration and validation · p. 14, Figure 7 · p. 14

C. String stability of calibrated models

Both calibrated ACC settings are string unstable, although their responses depend on disturbance frequency. The Bode analysis and sinusoidal simulation show that a string-unstable setting can still dissipate some disturbances.

C. String stability of calibrated models

  • λ2 is 70.7 for the minimum setting and 8.36 for the maximum setting; both calibrated models are string unstable.Because λ2 is non-negative for both models, perturbations may grow as they propagate through the platoon.Source: C. String stability of calibrated models · p. 14
  • Positive transfer-function amplitude in Figure 8 marks frequencies where disturbances grow, and both calibrated models have such frequencies.String stability requires amplitude below 0 dB at all frequencies.Source: C. String stability of calibrated models · pp. 14–15, Figure 8 · p. 15
  • The minimum setting amplifies disturbances below 0.358 rad/s, while the maximum setting does so below 0.118 rad/s.Both settings dissipate larger-frequency disturbances; their largest amplitudes are 1.25 dB at ω = 0.204 rad/s and 0.386 dB at ω = 0.062 rad/s, respectively.Source: C. String stability of calibrated models · p. 15
  • At ω = 0.204 rad/s, a 10-vehicle simulation shows the minimum setting amplifying a sinusoidal disturbance while the maximum setting dissipates it.This frequency is where the minimum-setting transfer function has its largest amplitude.Source: C. String stability of calibrated models · p. 15

D. String unstable platoons following empirical lead vehicle velocity profiles

A recorded slowdown from test I is used to examine empirical lead-vehicle disturbances in calibrated ACC platoons. The minimum setting amplifies the disturbance, while the maximum setting initially dampens it in moderate platoons, although amplification returns in longer ones.

D. String unstable platoons following empirical lead vehicle velocity profiles

  • The test I lead-vehicle profile represents a sudden slowdown and a realistic braking event that could trigger a phantom jam.The simulation applies the calibrated minimum- and maximum-setting parameters to the platoon.Source: D. String unstable platoons following… · pp. 15–16
  • For the recorded test I slowdown, the minimum-setting 15-vehicle platoon amplifies the disturbance, while the maximum-setting platoon initially dampens it.With the maximum setting, velocity perturbations eventually grow again in platoons longer than about 30 vehicles, but the initial decay keeps their overall magnitude small in moderate platoons.Source: D. String unstable platoons following… · pp. 15–16, Figure 10 · p. 16

VI. CONCLUSIONS

  • The conclusion reports that the commercially available ACC system is string unstable under both settings, yet can dampen practical slowdowns in moderate platoons before disturbances amplify in longer ones.The instability finding is described as consistent with a distinct commercial ACC system reported by Milanés and Shladover.Source: §VI · p. 17
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