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SSP-DMGTimeNet: Physics-Constrained Learning for Spatiotemporal Trajectory Prediction of Vehicle Platoons

Yuhang Wang, Kailang Ma, Zirui Li, Mingfeng Fan, Kitae Jang, Changju Lee, Heye Huang

arXiv:2609.06961v1cs.AI

TL;DR

Existing car-following predictors often lack explicit platoon-level disturbance-propagation constraints, despite trajectory accuracy not guaranteeing physical consistency. SSP-DMGTimeNet combines delay-aware causal attention, cross-vehicle representations, and time- and frequency-domain string-stability constraints. It achieves low instability on HighD and maintains competitive prediction performance while generalizing to NGSIM datasets.

  • Problem

    Existing prediction methods mainly optimize individual trajectory accuracy and rarely model whether disturbances propagate realistically through a vehicle platoon.

  • Method

    SSP-DMGTimeNet combines multi-scale temporal and cross-vehicle representations with delay-aware causal attention and time- and frequency-domain string-stability constraints.

  • Results

    On HighD, SSP-DMGTimeNet achieves a 0.65% unstable-window rate and 0.898 maximum head-to-tail amplification on the ground-truth excitation subset, with competitive prediction performance and zero-shot NGSIM evaluation.

  • Takeaways & Limitations

    The results support incorporating platoon-level physical constraints into trajectory prediction to balance prediction accuracy with disturbance-propagation stability.

  • Takeaways & Limitations

    The framework’s scope remains bounded by limitations in existing temporal models, which do not guarantee physically meaningful propagation or string-stable predictions without explicit modeling and constraints.

Abstract

from arXiv · show

Existing car-following prediction methods mainly optimize trajectory accuracy, while rarely considering whether predicted disturbances propagate realistically along a vehicle platoon. This limitation may lead to accurate but string-unstable predictions. We propose SSP-DMGTimeNet, a physics-constrained learning framework for spatiotemporal trajectory prediction of vehicle platoons. The model combines multi-scale temporal representations with cross-vehicle interaction features to capture complex and time-varying platoon dynamics. A propagation-delay-aware causal attention mechanism explicitly models upstream-to-downstream disturbance propagation by learning response delays between adjacent vehicles and accumulating them along the platoon. In addition, time- and frequency-domain string-stability losses relieve disturbance amplification across both adjacent vehicles and arbitrary sub-platoons during training. Experiments on HighD show that SSP-DMGTimeNet achieves an unstable-window rate of 0.65\% for five-vehicle platoons and a maximum head-to-tail amplification of 0.898 on the ground-truth excitation subset, while maintaining competitive trajectory prediction performance. In zero-shot evaluation on NGSIM US-101 and I-80, the model achieves velocity MAEs of 1.316~m/s and 1.252~m/s, with unstable-window rates of 3.90\% and 4.10\%, respectively. These results demonstrate that incorporating platoon-level physical constraints can effectively balance trajectory prediction accuracy and disturbance propagation stability.

1. Introduction

Existing car-following predictors often optimize individual trajectory accuracy without explicitly modeling disturbance propagation through vehicle platoons. SSP-DMGTimeNet addresses this gap by combining interaction-aware temporal modeling, delay-aware causal attention, and training-time string-stability constraints.

  • Motivation: Existing models primarily optimize pointwise individual-vehicle accuracy, leaving platoon-level disturbance propagation largely unmodeled.This can produce low MAE or RMSE alongside unrealistic disturbance amplification.
  • Motivation: String stability distinguishes disturbance attenuation from amplification as disturbances propagate downstream through successive vehicles.Amplification can produce larger speed oscillations and degraded traffic safety.
  • Proposed framework: SSP-DMGTimeNet predicts the longitudinal states of consecutive same-lane vehicles from 5 s of history over a 3 s horizon.The framework targets platoon-level rather than isolated vehicle prediction.
  • Proposed framework: The model combines multi-scale temporal modeling, cross-vehicle equilibrium representations, and SP-DACA to capture delayed upstream-to-downstream propagation.SP-DACA learns adjacent-vehicle response delays and accumulates them along the vehicle chain.
  • Contributions: Time- and frequency-domain regularization constrains disturbance amplification across adjacent vehicles and arbitrary sub-platoons during training.This converts string stability from a post-hoc metric into a physical learning constraint.

2. Related Work

Related work has improved trajectory prediction and interaction modeling, but commonly remains focused on vehicle-level errors or control-system stability. The paper motivates explicit delay-aware propagation modeling and string-stability constraints within prediction objectives.

  • Vehicle Platoon Trajectory Prediction: Analytical car-following models offer interpretability and calibratability but use restrictive functional forms for nonlinear and non-stationary driving behavior.Representative models include IDM, OVM, and FVDM.
  • Vehicle Platoon Trajectory Prediction: Data-driven and interaction-aware methods improve trajectory prediction, yet most remain optimized for vehicle-level pointwise errors rather than complete-platoon propagation.Prediction accuracy does not necessarily imply physical consistency or traffic-flow stability.
  • String Stability and Cooperative Adaptive Cruise Control: String stability characterizes whether disturbances attenuate or amplify downstream, and experiments have found amplification in some production ACC systems.CACC addresses this issue through inter-vehicle information sharing and related stability analyses.
  • String Stability and Cooperative Adaptive Cruise Control: Trajectory-prediction studies typically evaluate string stability after generation, if at all, motivating direct incorporation of disturbance amplification into learning objectives.The intended outcome is platoon-level physical consistency in predicted trajectories.
  • Propagation Delay and Causal Attention: Propagation delay reflects driver response, actuation, and communication lags that make downstream vehicles react to earlier upstream states.Delayed car-following models show that response delays can alter stability boundaries and induce traffic oscillations.
  • Propagation Delay and Causal Attention: Conventional causal temporal models block future observations but do not explicitly represent delay accumulation with topological distance or guarantee string-stable predictions.SSP-DMGTimeNet instead combines learnable delays, directional interactions, and string-stability constraints.

3. Problem Definition

The paper formulates platoon prediction as forecasting all vehicles’ longitudinal states from a shared historical window, then evaluates disturbance propagation over finite horizons. Its criteria measure both overall and frequency-selective amplification across adjacent vehicles and arbitrary sub-platoons.

  • Platoon-Level Prediction Task: The task predicts the longitudinal states of an entire same-lane vehicle platoon from T historical frames over P future frames.Vehicles are ordered from leader C1 through follower CN.
  • Platoon-Level Prediction Task: The input uses eight per-vehicle features, including position, velocity, acceleration, spacing, relative motion, and time headway.Relative quantities reduce sensitivity to absolute road-position differences.
  • Platoon-Level Prediction Task: The output predicts four variables: velocity, inter-vehicle spacing, acceleration, and position relative to the leader.These outputs support stability, spacing, comfort and energy-related, and kinematic evaluation.
  • Prediction-Level String-Stability Criterion: Finite-horizon string-stability criteria extend beyond pointwise MAE by testing whether predicted disturbances amplify through the platoon.The detrending operation removes low-frequency trends while retaining disturbance-related fluctuations.
  • Prediction-Level String-Stability Criterion: The time-domain factor Aj→i measures overall head-to-tail amplification for arbitrary sub-platoons, with adjacent-vehicle amplification as a special case.A smaller E indicates weaker cumulative disturbance amplification.
  • Prediction-Level String-Stability Criterion: The frequency-domain gain Gj→i(f) identifies frequency-dependent amplification that the overall time-domain ratio cannot reveal.Together, the measures capture overall and oscillation-specific disturbance propagation.
  • Prediction-Level String-Stability Criterion: A prediction window is δ-string stable when it satisfies the specified tolerance criterion; otherwise, it is classified as unstable.Exceedance area quantifies accumulated time-domain amplification beyond the stability boundary.

4. Method

SSP-DMGTimeNet combines multi-scale temporal modeling, adaptive cross-vehicle features, delay-aware causal attention, and string-stability constraints to predict platoon trajectories while modeling disturbance propagation.

  • Architecture: The framework jointly predicts future longitudinal states from five-vehicle historical platoon states using multi-scale temporal features and cross-vehicle interactions.Its main components include multi-scale temporal token extraction, HGF, CFE, and SP-DACA.
  • Multi-Scale Temporal Token Extraction: Four temporal branches use dilation rates {1, 3, 7, 11}, producing receptive fields from approximately 0.3 to 2.3 s.The branches use depthwise dilated convolutions followed by pointwise channel mixing.
  • Hierarchical Gated Fusion: HGF adaptively weights temporal scales for each vehicle and time step, using vehicle position to support different scale preferences along the platoon.The weights form a normalized convex combination of temporal-scale representations.
  • Cross-Vehicle Cointegration Feature Extraction: Cross-vehicle cointegration residuals characterize instantaneous deviations from long-term equilibrium and are added to the fused representation before attention.The residual uses velocity, spacing, and acceleration channels; bounded equilibrium parameters prevent degeneracy.
  • SP-DACA: SP-DACA models upstream-to-downstream interactions with learned response delays, Gaussian delay biases, and a joint temporal–vehicle causal mask.The delay kernel emphasizes upstream states whose temporal separation matches the expected propagation delay.
  • String-Stability Loss: String-stability losses constrain disturbance amplification across adjacent vehicles, arbitrary sub-platoons, and the frequency domain during training.This converts platoon-level stability from a post-hoc metric into a training-time physical constraint.

5. Experiments

Experiments use HighD for primary training and evaluation, with NGSIM subsets used for zero-shot evaluation, and assess both trajectory accuracy and platoon-level stability.

  • Datasets: HighD recordings 01–45, 46–50, and 51–60 are assigned to training, validation, and testing, respectively.Samples are grouped into consecutive same-lane platoons under continuous-observation and unchanged-order constraints.
  • Datasets: The main setting uses a 5 s history, a 3 s prediction horizon, a 1 s sliding step, and five-vehicle platoons, yielding 62,183 training windows.The corresponding validation and test sets contain 4,672 and 1,847 windows.
  • Baselines and Training: SSP-DMGTimeNet is compared with ten physics-based, data-driven, and hybrid physics–learning baselines under the same five-vehicle sampling setup.Optimization uses AdamW for 80 epochs with batch size 64.
  • Metrics: Evaluation measures trajectory MAE and RMSE alongside unstable-window rate, maximum amplification, exceedance area, tail-vehicle velocity error, and RMS jerk.The metrics cover velocity, spacing, acceleration, downstream accuracy, and motion smoothness.

5.3. Results on HighD

On HighD, SSP-DMGTimeNet maintains competitive trajectory prediction while suppressing disturbance amplification under both per-model and ground-truth-excitation evaluations. Its stability gains coexist with reasonable safety margins and motion smoothness.

  • 0.222 m/s tail-vehicle velocity MAE and 0.347 m/s average platoon velocity MAE indicate no progressive downstream deterioration in prediction error.This trajectory-error pattern does not by itself establish stable disturbance propagation, which is evaluated separately.
  • 0.65% unstable-window rate was achieved under the per-model excitation floor, with valid leader excitation retained in all 1,847 test windows.Other learning-based models retained only 18–41 valid windows, making direct comparison under this criterion potentially biased.
  • 0% unstable-window rate and 0.898 maximum head-to-tail amplification were achieved on the 62-window ground-truth excitation subset.The maximum amplification remained below the string-stability boundary of one, whereas ground-truth trajectories were unstable in 93.55% of these windows.
  • 12.175 s 5th-percentile TTC and 0.116 RMS jerk remained within the learning-based baselines’ range.These results indicate disturbance attenuation without excessively aggressive motion.

5.4. Zero-Shot Generalization on NGSIM

Models trained on HighD were evaluated directly on NGSIM US-101 and I-80 without target-domain fine-tuning or normalization reestimation. SSP-DMGTimeNet remained competitive in velocity prediction and retained substantially lower unstable-window rates than other learning-based models.

  • 1.316 m/s and 1.252 m/s velocity MAEs were achieved on US-101 and I-80, respectively, while remaining competitive among learning-based models.The models were trained exclusively on HighD and evaluated without target-domain fine-tuning or normalization-statistics reestimation.
  • 3.90% and 4.10% unstable-window rates were achieved on US-101 and I-80, respectively, compared with 87.52–97.33% for other learning-based models.On valid ground-truth excitation windows, the corresponding rates were 4.12% and 4.22%, with SSP-DMGTimeNet covering all 3,665 US-101 and 3,485 I-80 excitation windows.

5.5. Sensitivity Analysis on Platoon Length

Stability remains strong for platoons of three to six vehicles but degrades for seven-vehicle platoons, indicating that longer propagation chains are more difficult to stabilize.

  • 0%, 0.65%, and 0.96% unstable-window rates were obtained for platoons with N = 3, 5, and 6 vehicles, respectively.These rates remained below 1% and were substantially lower than those of Int-LSTM and Transformer.

5.6. Ablation Study

Ablation results show that directional delay-aware attention, temporal and cross-vehicle representations, and string-stability losses contribute complementary benefits. The full model balances trajectory accuracy with time- and frequency-domain stability.

  • Architecture and propagation modeling: 0.372 velocity MAE and 2.593 maximum frequency-domain gain resulted from replacing the directional spatial causal mask with full-graph attention.Fixing propagation delay also increased GT-subset maximum amplification from 0.898 to 1.101 and produced a 1.61% unstable-window rate.
  • Feature representations: 1.036 GT-subset maximum amplification and 3.746 maximum frequency-domain gain resulted from removing HGF.Removing CFE increased velocity MAE to 0.385, the largest accuracy degradation among feature-ablation variants.
  • String-stability regularization: 8.06% GT-subset unstable-window rate, 1.103 maximum amplification, and 10.208 maximum frequency-domain gain followed removal of Lfft.The ablation exposes an accuracy–stability trade-off and the role of frequency-domain regularization in suppressing selective disturbance amplification.
  • Overall ablation findings: The full model’s components play complementary roles: SP-DACA models propagation direction and delay, HGF and CFE improve representations, and string-stability losses suppress amplification.Together, these components provide a balanced compromise between trajectory accuracy and time- and frequency-domain stability.

5.7. Interpretability Analysis

The analysis examines whether SSP-DMGTimeNet learns physically plausible propagation delays and preserves disturbance-propagation structure while suppressing amplification in time and frequency domains.

  • Learned propagation delays: Learned propagation delays range from 0.84 to 1.01 s, entirely within the physically plausible interval of 0.8–1.2 s.The values converge below the 1.40 s initialization midpoint, with attention-head differences of approximately 0.17 s.
  • Delay consistency: Predicted propagation delays are compared with ground-truth estimates from cross-correlation peaks over concatenated 8 s trajectories.The analysis retains 66 windows with ground-truth leader disturbance amplitudes above 0.05 m/s and permits delays up to 2.5 s.
  • Time-domain propagation: SSP-DMGTimeNet reproduces the ground-truth diagonal disturbance pattern, preserving arrival order and attenuation without evident amplitude amplification.Int-LSTM and LSTM show less distinct disturbance boundaries, while Transformer deviates for intermediate vehicles.
  • Adjacent-vehicle amplification: SSP-DMGTimeNet has a median adjacent-vehicle amplification factor of approximately 0.6, whereas competing models cluster near or above the stability boundary of one.Values below one indicate string-stable attenuation.
  • Frequency-domain propagation: Across 0.05–0.5 Hz, SSP-DMGTimeNet maintains transfer gains of approximately 0.5–0.9 without frequency-selective amplification.Int-LSTM exceeds two near 0.1 Hz, Transformer increases around 0.35 Hz, and LSTM fluctuates around the stability boundary.

6. Conclusion

SSP-DMGTimeNet is a physics-constrained framework for vehicle-platoon trajectory prediction that models disturbance propagation and constrains amplification. It achieves low instability rates on HighD and remains stable in zero-shot NGSIM evaluations, while longer platoons remain more challenging.

  • Framework: SSP-DMGTimeNet combines propagation-delay-aware causal attention, cross-vehicle representations, and time- and frequency-domain string-stability constraints.The framework supplements conventional trajectory prediction with platoon-level physical constraints.
  • HighD results: 0.65% unstable-window rate and 0.898 maximum head-to-tail amplification are achieved on HighD for five-vehicle platoons and the GT-excitation subset, respectively.The model maintains competitive prediction performance while achieving these stability results.
  • Zero-shot evaluation: 3.90% and 4.10% unstable-window rates are obtained in zero-shot evaluation on NGSIM US-101 and I-80, respectively.The conclusion reports these results as evidence of cross-dataset stability.
  • Platoon length: Disturbance suppression remains effective for moderate platoon sizes but becomes more challenging as the propagation chain increases.The conclusion identifies platoon length as a boundary on suppression performance.
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