M-Predictive Spliner: Enabling Spatiotemporal Multi-Opponent Overtaking for Autonomous Racing

Edoardo Ghignone; Maurice Brunner; Michele Magno; Nadine Imholz; Nicolas Baumann

REVIEW 4 major objections 5 minor 20 references

M-Predictive Spliner: Enabling Spatiotemporal Multi-Opponent Overtaking for Autonomous Racing

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read A single racing planner can overtake multiple opponents by predicting each opponent's future trajectory and planning against the closest one.

desk verdict Solid multi-opponent extension of PSpliner with real experiments, but the 'arbitrary' claim outruns a design that predicts only the closest opponent's collision region. read the letter →

arxiv 2506.16301 v1 pith:QEA2UXPF submitted 2025-06-19 cs.RO

Nadine Imholz , Maurice Brunner , Nicolas Baumann , Edoardo Ghignone , Michele Magno This is my paper · ORCID

classification cs.RO

keywords autonomousracingmulti-opponentovertakingspatiotemporalplanningGaussianprocessregressionKalmanfiltertrackingre-identificationmotionprediction

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

The paper sets out to remove the single-opponent restriction of Predictive Spliner, a spatiotemporal overtaking planner that uses Gaussian-process regression to predict an opponent's trajectory and computes a future Region of Collision before committing to a pass. It argues that a Kalman-filter tracker with re-identification can keep stable identities for several opponents at once, and that regressing each opponent's spatial path and velocity separately supplies enough predictive information to overtake them in sequence. On a physical 1:10 scale car, the method reports an overtaking success rate up to 91.65 percent against a single opponent and an average 10.13-percentage-point improvement in safety over the prior planner, while the total time to overtake two opponents is nearly identical to the single-opponent time. If the claims hold, predictive, intent-aware overtaking is no longer limited to one rival and can be run on computationally modest hardware.

What carries the argument

The central object is the spatiotemporal Region of Collision (RoC): the planner integrates the closest opponent's Gaussian-process-regressed velocity along its spatial trajectory to predict where and when the paths will overlap, then optimizes the ego trajectory to pass within that region. The supporting mechanism is a multi-opponent tracker built on a Kalman filter with re-identification, which associates LiDAR detections to persistent tracklets and lets the method build a separate Gaussian process for each opponent's spatial path and velocity. A sequential quadratic programming problem then adjusts the ego vehicle's lateral position while respecting track boundaries, curvature limits, and a minimum clearance that keeps other opponents as spatial constraints.

What would settle it

Run the planner with two opponents whose Gaussian-process trajectory predictions overlap at the same corner; if the ego vehicle cuts toward the second opponent's future position or the overtake success rate falls to the spatial-baseline level, the single-Region-of-Collision assumption is falsified.

Watch

Extended reading notes

Core claim

The central claim is that arbitrary multi-opponent overtaking can be handled by a single planner that computes one Region of Collision from the closest opponent's predicted trajectory and treats every other opponent as a spatial obstacle during the pass. To make this possible, the paper extends the perception stack with a Kalman-filter multi-opponent tracker that re-identifies cars across LiDAR detections, so each opponent can be trailed and modeled by its own spatial and velocity Gaussian process even when occluded by another car or track curvature. In experiments on a 1:10 scale autonomous vehicle, the planner matched or beat the single-opponent baseline in one-opponent races, with a 91.65 percent success rate and an average safety improvement of 10.13 percentage points, and it achieved successful overtakes at up to 76.75 percent of the ego vehicle's speed. With two opponents, the average lap time for an overtaking lap was 9.275 seconds versus 9.25 seconds for one opponent, which the paper presents as evidence that the second opponent's model is learned while the first overtake is underway.

Load-bearing premise

The planner assumes that a single predicted collision region computed from the closest opponent is sufficient and that every other opponent can be treated as a static spatial obstacle, so the method can fail if two opponents' predicted regions overlap or if the second opponent is the one that actually threatens the overtake.

Editorial extensions

If this is right

A planner that predicts opponent intent can pass multiple cars in sequence without allocating a separate predicted collision region to each one.
Re-identification lets the system keep modeling an opponent that disappears behind another car or around a bend, so the second overtake can be planned while the first is still being executed.
The reported timing shows the cost of the second overtake is almost invisible: two overtakes took 37.1 seconds total, against 37.0 seconds for two single-overtake races.
The method's CPU usage and latency stay in real-time range on embedded hardware, so multi-opponent prediction does not require a large compute budget.

Reading between the lines

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

The single-RoC approximation is the point most worth probing: the paper justifies it by saying overlapping RoCs were unlikely, so a natural stress test is a track layout where two opponents' predicted futures cross at the same corner.
The one-lap trailing requirement means the method assumes a warm-up period for every opponent; replacing that with a generic prior or online model would make the planner usable in race entries and after tracking failures.
The same architecture, detection, identity-preserving tracking, per-agent trajectory regression, and a single predicted collision region, transfers to any high-speed navigation task with several predictable dynamic agents, not just racing.

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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

Dear colleague,

You should know this paper is a competent engineering extension of PSpliner to multiple opponents, but the 'arbitrary multi-opponent' claim in the abstract is not supported by the design or the experiments. The planner computes a single Region of Collision from the closest opponent's GP and treats every other opponent spatially. The two-opponent experiment has sequential, non-overlapping encounters, so it never stresses the temporal coupling between opponents. That gap is real, and the stress-test note lands.

What's genuinely new and good: the tracker is a solid piece of work. Adapting Norfair to spatial LiDAR tracking with reID, and building GPs for multiple opponents simultaneously even under occlusion, is a useful step. The physical experiments are more than a simulation: they run on a 1:10 car with a real opponent, and the improvement over PSpliner in the tested single-opponent conditions (10.13 pp success rate) is believable as an empirical result. The multi-opponent lap time staying flat is also a nice demonstration.

The soft spots are mostly about overclaiming and evidence thickness. The success rates are point estimates from experiments stopped after five successful overtakes; no confidence intervals. The multi-opponent table averages three trials, no error bars. There's no head-to-head comparison with another multi-opponent planner, so 'previous SotA' really means 'their own single-opponent PSpliner.' The code link is a placeholder. And the computational numbers were measured on an AMD Ryzen, while the vehicle's stated computer is an Intel NUC, which makes the real-time claim less direct.

None of these are fatal. The single-RoC design may be a sensible engineering trade-off, but calling it 'arbitrary multi-opponent' is simply not accurate. The paper would be stronger if it said 'up to two opponents' and explicitly discussed the overlap limitation. I'd want the authors to add error bars, run more multi-opponent trials with overlapping trajectories, and ideally compare against a spatial-only baseline that ignores GP predictions, to isolate the value of the temporal model.

This paper deserves a serious referee. It's a real system with real experiments and an honest comparison against their own prior work. The contribution is incremental but useful for the scaled-racing community and for anyone doing 2D LiDAR-based multi-agent tracking. I'd engage with it, but I'd push for revision before accepting.

Recommendation: send it to peer review with a request for major revision.

Referee Report

4 major / 5 minor

Summary. The paper extends the single-opponent Predictive Spliner (PSpliner) overtaking planner to multiple opponents, naming the result M-PSpliner. The system uses a Kalman-filter-based multi-object tracker with re-identification to maintain opponent identities, fits Gaussian process models of each opponent's spatial trajectory and velocity, and then plans an overtake using a single future Region of Collision (RoC) computed from the closest opponent; other opponents are handled purely spatially. The method is evaluated on a physical 1:10 scale autonomous racing car, with four single-opponent behavior conditions and one two-opponent scenario. The reported results claim an overtaking success rate up to 91.65%, an average 10.13-percentage-point improvement over PSpliner, and nearly equal per-overtake time in the two-opponent case compared with the single-opponent case.

Significance. If the results hold, this is a useful step toward practical multi-opponent autonomous racing: the work is openly built on the ForzaETH stack, provides physical-vehicle validation rather than simulation only, and shows that multi-opponent tracking and simultaneous GP construction are feasible on embedded hardware. The CPU and latency measurements are also a useful comparison point for the community. However, the paper's central claim of 'arbitrary multi-opponent head-to-head racing while considering the opponents' future intent' is not actually implemented, because only the closest opponent receives a temporal RoC; all other opponents are reduced to spatial obstacles. In addition, the quantitative evidence for the headline safety improvement is thin: success rates are point estimates from experiments stopped after five successful overtakes, and the two-opponent experiment consists of three trials with no error bars. The contribution is therefore real but narrower than claimed, and the statistical support needs strengthening before the quantitative headline numbers can be accepted as stated.

major comments (4)

[III-B.2, III-B.3; Abstract] The planner computes exactly one RoC, from the closest opponent, and Section III-B.3 states that all other opponents are accounted for 'spatially if present.' Therefore the abstract's claim of 'arbitrary multi-opponent head-to-head racing while considering the opponents' future intent' overstates the method: only one opponent's future intent enters the spatiotemporal planning. No experiment in Section IV-B includes overlapping RoCs or a case where the non-closest opponent is the limiting constraint; the Table IV scenario has sequential, non-overlapping encounters (Figure 2). Please either compute and plan with multiple RoCs, or restrict the claims to 'closest-opponent spatiotemporal planning with spatial avoidance of other opponents.'
[IV-B.1, Table II] The 10.13-percentage-point improvement is based on experiments stopped after five successful overtakes per condition, and the displayed Rot/c values are coarse counts (e.g., 71.4 = 5/7, 83.3 = 5/6, 100 = 5/5). With these counts, the difference between 71.4% and 83.3% is a single crash, and the four-condition average is compatible with large sampling variability. No confidence intervals, raw crash counts, or significance tests are reported. Please report the raw counts with intervals, or soften the safety-improvement claim to reflect the small-sample nature of the measurement.
[IV-B.2, Table IV] The multi-opponent evaluation uses three trials per condition and reports only means, with no variance, error bars, or individual trial values. Moreover, the test scenario places Opp1 on the shortest path and Opp2 on the centerline, with the ego overtaking them one after another, so it does not exercise simultaneous or overlapping threats. This is insufficient to establish 'arbitrary' multi-opponent capability; additional scenarios that stress the planner with spatially or temporally overlapping opponents, or an explicit statement of this limitation, are needed.
[III-B.2] The decision to use a single RoC is justified by an 'empirical analysis' that is neither described nor referenced. The reader cannot check the claims that computing multiple RoCs is too expensive or that overlapping RoCs are unlikely. Please provide the analysis, report the measured computational cost and overlap statistics, or remove the unsupported justification.

minor comments (5)

[III-B.1] The text says opponents 'must be trailed for one lap' to construct their GPs, but Section IV-B.2 says GP2 is learned while overtaking Opp1. Please clarify whether a full lap of observations per opponent is required or whether partial observations are sufficient.
[Table III and surrounding text] There are typos in the text before Table III ('handle multiple oppnents') and in the table caption ('algorihtm'). These should be corrected.
[IV-B.2, Table IV] The notation 'P T' and 'PPP T' is confusing; please define the total race time symbol cleanly and use it consistently in the table and text.
[V, Open-source availability] The code link github.com/ForzaETH/notavailableyet appears to be a placeholder; this contradicts the claim that 'everything in this project is fully open-sourced.' Please provide the actual repository URL or state clearly that the code will be released later.
[Table V] The CPU utilization is reported per core, which is noted in the table caption, but the high standard deviations (e.g., 30.57 for M-PSpliner vs 31.87 for PSpliner) make the reported CPU difference between M-PSpliner and PSpliner statistically unconvincing; please comment on this variability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: the claimed improvements are empirical hardware results; the single-RoC simplification is a scope limitation, not a self-referential derivation.

full rationale

I find no significant circularity. The central claims rest on physical experiments, not on a derivation that contains its own conclusion. M-PSpliner's GPs are fit to observed opponent laps and then used to integrate a future Region of Collision; the reported overtake success rates and crash counts are independent measurements of the resulting planner behavior, so the 'prediction' is not identical to its fitted inputs by construction. The comparison against PSpliner [4] is an empirical benchmark against the authors' own prior system; although [4] is heavily cited for the RoC computation and trajectory optimization, it is an externally published RA-L system with its own experimental basis, and the present paper adds a new multi-opponent tracker and measures the combined system on hardware. The single-RoC simplification in Section III-B.2 is a real limitation: only the closest opponent receives spatiotemporal treatment and other opponents are handled spatially, which weakens the broad 'arbitrary multi-opponent' headline claim. That is a correctness or scope gap, however, not circularity. No equation is defined in terms of the quantity it is said to predict, no fitted parameter is relabeled as a prediction, and no uniqueness theorem is imported from self-citations. The score of 2 reflects only the self-citation-heavy dependence on [4] and the unpublished 'empirical analysis' justification for using a single RoC; neither of those makes the result circular.

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

The central claim rests on hand-tuned tracker parameters, an unshown empirical choice of a single RoC, and GP hyperparameters inherited from prior work; no new physical entities are introduced.

free parameters (4)

KF measurement noise r = 0.074
Hand-set in Eq. (2)-(3); controls how strongly position detections correct the constant-velocity track, and therefore how smooth the opponent trajectories fed to the GPs are.
KF process noise q = 1.7e-4
Hand-set in Eq. (3); sets how quickly the tracker believes the constant-velocity model is wrong, influencing the velocity estimates used by the planner.
ReID distance threshold tau_ReID = 0.1
Hand-set in Eq. (5); determines whether a detection is re-associated with an existing tracklet, and thus whether opponent identities survive occlusions in multi-opponent scenarios.
Gaussian process kernel and noise hyperparameters = not stated; inherited from PSpliner [4]
The GPR for path and velocity prediction is taken from the authors' prior PSpliner work; no kernel choice or fitted values are given here, so the predictive part of the planner rests on unstated fitted settings.

assumptions (4)

domain assumption Every opponent can be trailed for roughly one lap to build its Gaussian process models before overtaking is attempted.
Section III-B.1 states 'opponents must be trailed for one lap to construct their respective GPs'; the multi-opponent timing claim depends on learning the second opponent's GP while overtaking the first.
ad hoc to paper A single Region of Collision for the closest opponent is sufficient for safe multi-opponent planning.
Section III-B.2 justifies this by 'empirical analysis' that is not shown; non-closest opponents are only treated spatially, so overlapping or more dangerous future RoCs are ignored.
domain assumption The constant-velocity point-mass motion model in the Kalman filter adequately captures opponent motion.
Equations (1)-(3) define the tracker with a constant-velocity model; opponents that brake or accelerate abruptly would violate this model, though the process noise q partially compensates.
domain assumption The four scripted opponent behaviors in Section IV-B.1 are representative of real racing opponents.
Success rates are measured against fixed racing-line, shortest-path, centerline, and reactive gap-following opponents; the method is not tested against learning or model-predictive opponents.

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Cite this review

Pith. "Pith review of M-Predictive Spliner: Enabling Spatiotemporal Multi-Opponent Overtaking for Autonomous Racing." pith.science (2026). https://pith.science/paper/QEA2UXPF

@misc{pith2026250616301,
  author       = {Pith},
  title        = {Pith review of: M-Predictive Spliner: Enabling Spatiotemporal Multi-Opponent Overtaking for Autonomous Racing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QEA2UXPF}},
  note         = {Machine review of arXiv:2506.16301}
}

read the original abstract

Unrestricted multi-agent racing presents a significant research challenge, requiring decision-making at the limits of a robot's operational capabilities. While previous approaches have either ignored spatiotemporal information in the decision-making process or been restricted to single-opponent scenarios, this work enables arbitrary multi-opponent head-to-head racing while considering the opponents' future intent. The proposed method employs a KF-based multi-opponent tracker to effectively perform opponent ReID by associating them across observations. Simultaneously, spatial and velocity GPR is performed on all observed opponent trajectories, providing predictive information to compute the overtaking maneuvers. This approach has been experimentally validated on a physical 1:10 scale autonomous racing car, achieving an overtaking success rate of up to 91.65% and demonstrating an average 10.13%-point improvement in safety at the same speed as the previous SotA. These results highlight its potential for high-performance autonomous racing.

Figures

Figures reproduced from arXiv: 2506.16301 by the authors.

**Figure 2.** Overview of the different timesteps of the overtaking algorithm. At [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Indy autonomous challenge-autonomous race cars at the handling limits,

A. Wischnewski, M. Geisslinger, J. Betz, T. Betz, F. Fent, A. Heilmeier, L. Hermansdorfer, T. Herrmann, S. Huch, P. Karle et al. , “Indy autonomous challenge-autonomous race cars at the handling limits,” in 12th International Munich Chassis Symposium 2021: chassis. tech plus. Springer, 2022, pp. 163–182

work page 2021

[2] [2]

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J. Betz, H. Zheng, A. Liniger, U. Rosolia, P. Karle, M. Behl, V . Krovi, and R. Mangharam, “Autonomous vehicles on the edge: A survey on autonomous vehicle racing,” IEEE Open Journal of Intelligent Transportation Systems, vol. 3, pp. 458–488, 2022

2022

[3] [3]

Forzaeth race stack—scaled autonomous head-to-head racing on fully commer- cial off-the-shelf hardware,

N. Baumann, E. Ghignone, J. K ¨uhne, N. Bastuck, J. Becker, N. Imholz, T. Kr¨anzlin, T. Y . Lim, M. L¨otscher, L. Schwarzenbach et al., “Forzaeth race stack—scaled autonomous head-to-head racing on fully commer- cial off-the-shelf hardware,” Journal of Field Robotics , 2024

work page 2024

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N. Baumann, E. Ghignone, C. Hu, B. Hildisch, T. H ¨ammerle, A. Bet- toni, A. Carron, L. Xie, and M. Magno, “Predictive spliner: Data-driven overtaking in autonomous racing using opponent trajectory prediction,” IEEE Robotics and Automation Letters , 2024

work page 2024

[5] [5]

Optimization-based au- tonomous racing of 1:43 scale rc cars,

A. Liniger, A. Domahidi, and M. Morari, “Optimization-based au- tonomous racing of 1:43 scale rc cars,” Optimal Control Applications and Methods , vol. 36, no. 5, pp. 628–647, 2015

work page 2015

[6] [6]

A novel obstacle avoidance algorithm: “follow the gap method

V . Sezer and M. Gokasan, “A novel obstacle avoidance algorithm: “follow the gap method”,” Robotics and Autonomous Systems , vol. 60, no. 9, pp. 1123–1134, 2012

work page 2012

[7] [7]

Optimal trajectory generation for dynamic street scenarios in a frenet frame,

M. Werling, J. Ziegler, S. Kammel, and S. Thrun, “Optimal trajectory generation for dynamic street scenarios in a frenet frame,” in 2010 IEEE International Conference on Robotics and Automation . IEEE, 2010, pp. 987–993

work page 2010

[8] [8]

Multilayer graph-based trajectory planning for race vehicles in dynamic sce- narios,

T. Stahl, A. Wischnewski, J. Betz, and M. Lienkamp, “Multilayer graph-based trajectory planning for race vehicles in dynamic sce- narios,” in 2019 IEEE Intelligent Transportation Systems Conference (ITSC). IEEE, 2019, pp. 3149–3154

work page 2019