Pith. sign in

REVIEW 4 major objections 6 minor 42 references

Safety Evaluation of Motion Plans Using Trajectory Predictors as Forward Reachable Set Estimators

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read FORCE-OPT extracts calibrated forward reachable sets from Gaussian-mixture trajectory predictors via convex optimization and conformal prediction, adapts them with a Bayesian filter out-of-distribution, and achieves the best balance of…

desk verdict Useful new combination of convex FRS extraction and conformal calibration; the experimental coverage gap needs a clear explanation before the completeness claim can be trusted. read the letter →

arxiv 2507.22389 v1 pith:52BDXNAI submitted 2025-07-30 cs.RO cs.SYeess.SY

Kaustav Chakraborty , Zeyuan Feng , Sushant Veer , Apoorva Sharma , Wenhao Ding , Sever Topan , Boris Ivanovic , Marco Pavone
show 1 more author
Somil Bansal
This is my paper
classification cs.ROcs.SYeess.SY
keywords forwardreachablesetstrajectorypredictionconformalsafetymonitoringautonomousdrivingGaussianmixturemodelsconvexoptimizationout-of-distributionrobustness
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

This paper aims to show that modern trajectory predictors—neural networks trained to forecast where other agents will go—can serve as forward reachable set estimators for safety monitoring, giving tighter sets than worst-case reachability while keeping the monitor complete. The proposed FORCE-OPT algorithm extracts a minimal-volume reachable set from the predictor's Gaussian mixture output by solving a convex program, calibrates that set with conformal prediction so it covers the ground-truth future with a user-set probability, and adds a Bayesian filter that dilates the set when the predictor appears unreliable under distribution shift. On nuScenes, FORCE-OPT reports the lowest balanced error rate—the mean of false positives and false negatives—among calibrated, uncalibrated, and worst-case baselines, both in-distribution and when tested on a different city, with an inference time of about 22 ms per frame. The paper also proves that the probabilistic forward reachable set it optimizes is equivalent, almost everywhere, to the classical worst-case forward reachable set.

What carries the argument

The load-bearing object is the calibrated FRS constructed from the GMM modes of a trajectory predictor. For each mode $i$, FORCE-OPT solves the convex program $\min_{c_i} \sum_i \pi\sqrt{\lambda_{i,1}\lambda_{i,2}}\, c_i$ subject to $\sum_i p_i(1-e^{-c_i/2}) \ge \tau$, where the $c_i$ are Mahalanobis sublevel-set radii and $\lambda_{i,1},\lambda_{i,2}$ are the eigenvalues of each component's covariance; the resulting union of ellipses is the reachable set. Conformal prediction then rescales every covariance by a common factor $\eta$ chosen from the empirical quantile of the non-conformity score $\psi(s,x)=\min_i (V_i(x)/c_i)$, which the invariance property of the convex program makes analytic. A Bayesian filter over a two-point belief adjusts this set further by scaling covariances by $1/\hat{\beta}$ when the predictor's observed performance degrades.

What would settle it

Measure the empirical coverage of FORCE-OPT's reachable sets on a test set with a distribution shift of a different kind than city-level, such as night-time or rainy driving from a different sensor setup; if the empirical coverage falls below the nominal $1-\gamma$ level while the Bayesian belief stays high, then the completeness guarantee fails in exactly the regime the filter was meant to handle.

Watch

Extended reading notes

Core claim

The central discovery is that the probabilistic forward reachable set—the smallest set containing the full probability mass under the agent's state distribution—coincides almost everywhere with the classical worst-case reachable set, which justifies estimating reachable sets from learned predictors. FORCE-OPT turns this into a tractable algorithm: given a GMM trajectory predictor, it solves a convex optimization over the sublevel-set radii of each Gaussian mode to minimize total ellipsoid volume subject to capturing at least a fraction $\tau$ of the probability mass, then applies split conformal prediction by scaling all covariances by a single factor $\eta$, chosen from a calibration set so that the set covers the true future with high probability. The covariance-scaling invariance of the convex program lets the conformal factor be computed analytically from a non-conformity score per scene. On nuScenes, this yields the best balance of false positive and false negative rates across all baselines—5.83% balanced error in-distribution and 9.09% out-of-distribution, dropping to 6.77% with the belief-based worst-case fallback.

Load-bearing premise

The conformal coverage guarantee assumes test scenes are exchangeable with the calibration set, and the Bayesian filter's hand-picked thresholds have no formal coverage guarantee, so if the predictor's error pattern shifts in ways the belief model does not capture, the low false-negative rate can degrade.

Editorial extensions

If this is right

  • FORCE-OPT achieves a balanced error rate of 5.83% in-distribution and 9.09% under city-level out-of-distribution shift, lower than every baseline tested, while running in about 22 ms per frame.
  • Adding the belief-based fallback (FORCE-OPT + pWC-FRS) lowers the out-of-distribution balanced error to 6.77%, showing graceful degradation rather than a hard failure when the predictor becomes unreliable.
  • Increasing the number of GMM modes reduces the conformal inflation factor and improves the false-positive rate, so multi-modality in the predictor translates directly into a less conservative safety monitor.
  • Uncalibrated predictors miss a large share of genuine collisions (false-negative rates up to 55.56% out-of-distribution), while worst-case reachability sets trigger false alarms on roughly 44% of safe scenes, motivating calibration as the middle path.

Reading between the lines

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

  • The same convex extraction could apply to other mixture density estimators besides Gaussians—student-$t$ or normalizing-flow mixtures—by replacing the $\chi^2$ tail in the constraint with the appropriate distribution function, though the analytic conformal-score form would need re-deriving.
  • The two-point Bayesian belief over 'low' and 'high' confidence values is a coarse hedge; a deployment-grade monitor should also track the running empirical coverage of the sets and flag when it deviates from the nominal level, since the conformal guarantee no longer holds once the exchangeability assumption fails.
  • Because the belief thresholds are hand-picked, the Bayesian adaptation could be made testable by learning the belief-transition parameters from logged predictor failures, and one concrete experiment would be measuring whether the resulting balanced error improves over the hand-tuned values.
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.

Referee Report

4 major / 6 minor

Summary. The paper proposes FORCE-OPT, a runtime safety monitor for motion plans. The method treats a multi-modal GMM trajectory predictor as a stochastic model of other agents and extracts a forward reachable set (FRS) by solving a convex program that minimizes the total volume of Gaussian sublevel sets subject to a probability-mass constraint. Split conformal prediction is then used to calibrate the FRS by scaling the GMM covariances, with the non-conformity score defined as the minimal scaling factor that covers the ground-truth trajectory. To handle distribution shift, the authors add a Bayesian filter that adaptively dilates the covariance based on the predictor's observed likelihood, with fallback to parameterized or worst-case reachability when the estimated confidence drops below a threshold. The method is evaluated on nuScenes, using an Autobots predictor trained on Singapore and tested on both Singapore (in-distribution) and Boston (out-of-distribution), with synthetic unsafe scenarios to measure false negatives. The main empirical claim is that FORCE-OPT achieves the best balance of false-positive and false-negative rates, with a BER of 5.83% in-distribution and 6.77% out-of-distribution for the best variant, at roughly 22 ms per safety check.

Significance. If validated, the paper offers a practical and principled route from learned trajectory predictors to calibrated reachability-style safety monitors. The convex formulation in Theorem 2 is correctly derived, and Corollary 1 (invariance of the optimizer under covariance scaling) is true, which makes the conformal non-conformity score analytically tractable. The use of a public dataset, a public predictor, and a synthetic unsafe-scenario construction is a strength; the reported runtime is competitive for online monitoring. The core idea---calibrating FRSs extracted from GMM predictors rather than using raw or worst-case sets---is timely and likely useful for learned autonomy stacks. However, the conformal guarantee and the empirical coverage numbers are currently not reconciled, and the proof of the theoretical FRS equivalence in Theorem 1 has a gap. These issues bear directly on the central claims of completeness and soundness, so they need to be addressed before the contribution can be fully accepted.

major comments (4)
  1. [V-D and Table I] The paper does not disclose whether the 35,220 calibration examples are disjoint from the nuScenes training split used to train Autobots. Split conformal prediction requires exchangeability of calibration and test non-conformity scores; if the calibration data overlap the training split, the scores are in-sample, the calibration factor eta is too small, and the guarantee in Eq. (8) does not transfer to fresh data. This concern is reinforced by Table I: FORCE-OPT's reported ID coverage is 89.95%, which is about five percentage points below the 95% target. With N=35,220 the finite-sample correction term is below 0.7%, so the gap is far too large to be explained by calibration randomness. The authors should disclose the exact split, define precisely what the Cov metric counts (per scene, per agent, or per agent-timestep), and state which empirical quantity corresponds to the conformal coverage object in Eq. (8).
  2. [Appendix, Proof of Theorem 1] The proof of Theorem 1 is not valid as written. It invokes Luzin's N-property to claim that a measurable subset F of F_t with positive volume has a preimage of positive Lebesgue measure under the composed dynamics map. Luzin's N-property concerns forward images of null sets and does not imply this; if the domain of the composed map has lower dimension than X, the image has measure zero, and even in equal dimension the rank conditions are nontrivial. The proof also assumes that the absolutely continuous measure on the control/disturbance space assigns positive probability to every positive-measure subset of its support, which is not stated. Without these assumptions, the support of mu_t can be a proper subset of F_t, in which case the minimal volume set omega*_t need not equal F_t up to a measure-zero set. The authors should either add the required assumptions or weaken Theorem 1 to a statement about the support of the pushforward measure.
  3. [IV-C, Eq. (8)] Equation (8) as printed is not the standard split-conformal finite-sample bound. With delta in (0,1), the term sqrt(-log delta / 2N) is positive, so the displayed inequality states that the miscoverage probability is less than gamma minus a positive term. The standard form from [39, Proposition 2a] is gamma plus sqrt(log(1/delta)/(2N)) (equivalently, with the appropriate quantile-index ceiling). Please correct the sign and state the exact quantile construction used for eta. This matters because the current text presents a formal guarantee that is both nonstandard and inconsistent with the observed 10.05% miscoverage rate in Table I.
  4. [Tables I-II and V-E] All performance numbers are reported as single-point estimates with no confidence intervals, error bars, or number of calibration runs. Because split conformal calibration introduces randomness (and the predictor training may as well), the claimed ordering---e.g., FORCE-OPT's ID BER of 5.83% versus 7.78% for FORCE-OPT + belief, or the OOD ordering among FORCE-OPT variants in Table II---may be within noise. The authors should report means and standard deviations over multiple calibration splits, and provide a sensitivity analysis for the hand-picked Bayesian thresholds (beta_low=0.3, beta_high=1, switch at beta<0.75) since the OOD conclusions depend on those choices.
minor comments (6)
  1. [IV-D, Eq. (13)] The notation \bar{x}_t in Eq. (13) is not defined; presumably it denotes the mean of the t-th GMM component, but it should be introduced explicitly.
  2. [Appendix, Proof of Corollary 1] The proof by contradiction assumes that c*_{i,alpha1} < c*_{i,alpha2} for a single index i and then compares the full sums as if the inequality held componentwise. The invariance statement is true because the covariance scaling factors out of the objective and leaves the constraint unchanged, but the proof as written is not rigorous and should be rewritten.
  3. [V-D] The description of the Nakamura et al. baseline contains a typo ('3% probaility mass') and is ambiguous; it should state clearly whether the control bounds enclose the 99% or 3-sigma region.
  4. [VI] The first sentence of the conclusion contains a stray footnote marker '4' that appears to be a formatting artifact.
  5. [Fig. 1 caption] The caption refers to 'true future positions of the contender (the cyan ellipses do not cover the true ground truth states states shown with red dots)' and later to 'containing the agent's true future path (green dots)'; the color convention for ground truth in panels (b) and (c) is inconsistent and should be reconciled.
  6. [IV-C] The value of delta in the conformal guarantee of Eq. (8) is never specified in the experiments; please state the chosen delta and report the resulting finite-sample correction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the FORCE-OPT derivation is self-contained and the evaluation is against external benchmarks.

full rationale

The paper's central derivation is self-contained. FORCE-OPT extracts forward reachable sets by solving the convex program in Theorem 2, which is derived in the appendix from the standard volume formula for ellipsoids and the chi-squared CDF of the Mahalanobis energy; this is a direct mathematical construction, not a restatement of the target result. Corollary 1 and Lemma 1 are proven in the appendix and support the conformal calibration step, which uses split conformal prediction with an explicit non-conformity score and the standard exchangeability-based coverage bound from Vovk [39]. The Bayesian filter in Section IV-D is adopted from [25], a peer-reviewed ICRA paper by a co-author, but it is used as one component, is also evaluated as an external baseline in Table I, and the paper's central FORCE-OPT claim does not reduce to it; the hand-picked beta thresholds are hyperparameters rather than fitted constants renamed as predictions. The empirical evaluation is conducted on nuScenes test splits against external baselines, and the synthetic unsafe scenarios are generated from test data, so the reported BER, FPR, and FNR are not forced by construction. The reported ID coverage of 89.95% versus the 95% conformal target is a statistical correctness concern about calibration-set exchangeability or metric alignment, not a circularity, and does not affect the circularity score.

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

The central claim rests on the GMM predictor faithfully representing agent uncertainty, on the exchangeability assumption for conformal calibration, and on a heuristic Bayesian filter for out-of-distribution operation. The free parameters (tau, gamma, beta_low, beta_high, switch threshold) are chosen by the authors and are not validated with sensitivity analyses.

free parameters (4)
  • tau (probability mass threshold)
    Appears in equations (4) and (6); must be set close to 1, controls FRS conservativeness but is never reported in the paper.
  • gamma (conformal miscoverage rate) = 0.05
    Desired coverage 0.95, set in Section V-D; determines the calibration quantile.
  • beta_low, beta_high (Bayesian belief bounds) = 0.3, 1.0
    Set in Section V-D for the Bayesian filter; no sensitivity analysis is provided.
  • Belief switch threshold = 0.75
    Triggers fallback to pWC-FRS or WC-FRS in Section V-D; chosen ad hoc without a described tuning procedure.
assumptions (4)
  • domain assumption The output of the trajectory predictor is a GMM whose modes approximate the true push-forward distribution of agent states.
    Core modeling assumption, stated in Section IV-B.
  • domain assumption Calibration and test scenes are drawn exchangeably (IID) from the same distribution for the conformal guarantee.
    Required for split conformal prediction, stated implicitly in Section IV-C.
  • standard math Luzin's N-property can be applied to the preimage of a positive-measure set to conclude the preimage has positive measure.
    Used in the Theorem 1 proof in the Appendix; the direction is valid by contradiction for C^1 maps, but the text does not spell out the full argument.
  • ad hoc to paper The Bayesian belief update from [25] accurately reflects predictor reliability under distribution shift.
    Adopted in Section IV-D without formal guarantees.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Safety Evaluation of Motion Plans Using Trajectory Predictors as Forward Reachable Set Estimators." pith.science (2026). https://pith.science/paper/52BDXNAI

@misc{pith2026250722389,
  author       = {Pith},
  title        = {Pith review of: Safety Evaluation of Motion Plans Using Trajectory Predictors as Forward Reachable Set Estimators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/52BDXNAI}},
  note         = {Machine review of arXiv:2507.22389}
}
read the original abstract

The advent of end-to-end autonomy stacks - often lacking interpretable intermediate modules - has placed an increased burden on ensuring that the final output, i.e., the motion plan, is safe in order to validate the safety of the entire stack. This requires a safety monitor that is both complete (able to detect all unsafe plans) and sound (does not flag safe plans). In this work, we propose a principled safety monitor that leverages modern multi-modal trajectory predictors to approximate forward reachable sets (FRS) of surrounding agents. By formulating a convex program, we efficiently extract these data-driven FRSs directly from the predicted state distributions, conditioned on scene context such as lane topology and agent history. To ensure completeness, we leverage conformal prediction to calibrate the FRS and guarantee coverage of ground-truth trajectories with high probability. To preserve soundness in out-of-distribution (OOD) scenarios or under predictor failure, we introduce a Bayesian filter that dynamically adjusts the FRS conservativeness based on the predictor's observed performance. We then assess the safety of the ego vehicle's motion plan by checking for intersections with these calibrated FRSs, ensuring the plan remains collision-free under plausible future behaviors of others. Extensive experiments on the nuScenes dataset show our approach significantly improves soundness while maintaining completeness, offering a practical and reliable safety monitor for learned autonomy stacks.

Figures

Figures reproduced from arXiv: 2507.22389 by the authors.

Figure 1
Figure 1. In this intersection scenario from the nuScenes dataset, we evaluate the safety of the ego (blue) vehicle’s planned trajectory (black) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Ablation with different number of GMM modes from the trajectory predictor. The performance of FORCE-OPT and its belief-based [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

42 extracted references · 35 canonical work pages

  1. [1]

    Motion planning for autonomous driving: The state of the art and future perspectives,

    S. Teng et al. , “Motion planning for autonomous driving: The state of the art and future perspectives,” TIV, vol. 8, no. 6, pp. 3692–3711, 2023. (a) BER vs. Number of Modes (b) FPR vs. Number of Modes (c) FNR vs. Number of Modes Fig. 2: Ablation with different number of GMM modes from the trajectory predictor. The performance of FORCE-OPT and its belief-...

    work page 2023

  2. [2]

    Planning and decision-making for autonomous vehicles,

    W. Schwarting et al., “Planning and decision-making for autonomous vehicles,” Annual Review of CRAS , vol. 1, no. 1, pp. 187–210, 2018

    work page 2018

  3. [3]

    Trajectron++: Dynamically-feasible trajectory forecasting with heterogeneous data,

    T. Salzmann et al. , “Trajectron++: Dynamically-feasible trajectory forecasting with heterogeneous data,” in ECCV. Springer, 2020, pp. 683–700

    work page 2020

  4. [4]

    Latent variable sequential set transformers for joint multi-agent motion prediction,

    R. Girgis et al. , “Latent variable sequential set transformers for joint multi-agent motion prediction,” arXiv preprint arXiv:2104.00563, 2021

    arXiv 2021

  5. [5]

    A gentle introduction to confor- mal prediction and distribution-free uncertainty quantification,

    A. N. Angelopoulos and S. Bates, “A gentle introduction to confor- mal prediction and distribution-free uncertainty quantification,” arXiv preprint arXiv:2107.07511, 2021

    arXiv 2021

  6. [6]

    A tutorial on conformal prediction

    G. Shafer and V . V ovk, “A tutorial on conformal prediction.” JMLR, vol. 9, no. 3, 2008

    work page 2008

  7. [7]

    nuscenes: A multimodal dataset for autonomous driving,

    H. Caesar et al. , “nuscenes: A multimodal dataset for autonomous driving,” 2020, pp. 11 621–11 631

    work page 2020

  8. [8]

    Formal specification and verification of au- tonomous robotic systems: A survey,

    M. Luckcuck et al. , “Formal specification and verification of au- tonomous robotic systems: A survey,” CSUR, vol. 52, no. 5, pp. 1–41, 2019

    work page 2019

Show all 42 references
  1. [9]

    Formal verification techniques for vision-based autonomous systems–a survey,

    S. Mitra et al. , “Formal verification techniques for vision-based autonomous systems–a survey,” in Principles of Verification: Cycling the Probabilistic Landscape: Essays Dedicated to Joost-Pieter Katoen on the Occasion of His 60th Birthday, Part III . Springer, 2024, pp. 89–108

    2024

  2. [10]

    Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization,

    M. Althoff and J. M. Dolan, “Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization,” CDC, 2011

    2011

  3. [11]

    Controllers for reachability specifications for hybrid systems,

    J. Lygeros et al., “Controllers for reachability specifications for hybrid systems,” Automatica, vol. 35, no. 3, pp. 349–370, 1999

    1999

  4. [12]

    A toolbox of hamilton–jacobi solvers for analysis of nondeterministic continuous and hybrid systems,

    I. M. Mitchell et al., “A toolbox of hamilton–jacobi solvers for analysis of nondeterministic continuous and hybrid systems,” Hybrid Systems: Computation and Control , 2005

    2005

  5. [13]

    Funnel libraries for real-time robust feedback motion planning,

    A. Majumdar and R. Tedrake, “Funnel libraries for real-time robust feedback motion planning,” IJRR, vol. 36, no. 8, pp. 947–982, 2017

    2017

  6. [14]

    Confidence-aware motion prediction for real- time collision avoidance1,

    D. Fridovich-Keil et al., “Confidence-aware motion prediction for real- time collision avoidance1,” IJRR, vol. 39, no. 2-3, pp. 250–265, 2020

    2020

  7. [15]

    Trajectron++: Dynamically-feasible trajectory forecasting with heterogeneous data,

    T. Salzmann et al. , “Trajectron++: Dynamically-feasible trajectory forecasting with heterogeneous data,” in ECCV, 2020, pp. 683–700

    2020

  8. [16]

    Emma: End-to-end multimodal model for au- tonomous driving,

    J.-J. Hwang et al. , “Emma: End-to-end multimodal model for au- tonomous driving,” arXiv preprint arXiv:2410.23262 , 2024

    2024 arXiv

  9. [17]

    Diffstack: A differentiable and modular control stack for autonomous vehicles,

    P. Karkus et al., “Diffstack: A differentiable and modular control stack for autonomous vehicles,” in CoRL. PMLR, 2023, pp. 2170–2180

    2023

  10. [18]

    Interactive joint planning for autonomous vehicles,

    Y . Chen et al. , “Interactive joint planning for autonomous vehicles,” RA-L, 2023

    2023

  11. [19]

    Mp3: A unified model to map, perceive, predict and plan,

    S. Casas et al., “Mp3: A unified model to map, perceive, predict and plan,” in CVPR, 2021, pp. 14 403–14 412

    2021

  12. [20]

    Lookout: Diverse multi-future prediction and planning for self-driving,

    A. Cui et al., “Lookout: Diverse multi-future prediction and planning for self-driving,” in ICCV, 2021, pp. 16 107–16 116

    2021

  13. [21]

    Task-aware risk estimation of perception failures for autonomous vehicles,

    P. Antonante et al., “Task-aware risk estimation of perception failures for autonomous vehicles,” arXiv preprint arXiv:2305.01870 , 2023

    2023 arXiv

  14. [22]

    System-level safety monitoring and recov- ery for perception failures in autonomous vehicles,

    K. Chakraborty et al. , “System-level safety monitoring and recov- ery for perception failures in autonomous vehicles,” arXiv preprint arXiv:2409.17630, 2024

    2024 arXiv

  15. [23]

    Measuring surprise in the wild,

    A. Dinparastdjadid et al. , “Measuring surprise in the wild,” arXiv preprint arXiv:2305.07733, 2023

    2023 arXiv

  16. [24]

    Surprise potential as a measure of interactivity in driving scenarios,

    W. Ding et al. , “Surprise potential as a measure of interactivity in driving scenarios,” arXiv preprint arXiv:2502.05677 , 2025

    2025 arXiv

  17. [25]

    Online update of safety assurances using confidence-based predictions,

    K. Nakamura and S. Bansal, “Online update of safety assurances using confidence-based predictions,” in ICRA. IEEE, 2023, pp. 12 765– 12 771

    2023

  18. [26]

    Safe planning in dynamic environments using conformal prediction,

    L. Lindemann et al. , “Safe planning in dynamic environments using conformal prediction,” RA-L, 2023

    2023

  19. [27]

    Prediction-based reachability for collision avoidance in autonomous driving,

    A. Li et al., “Prediction-based reachability for collision avoidance in autonomous driving,” arXiv preprint arXiv:2011.12406 , 2020

    2011 arXiv

  20. [28]

    Set propagation techniques for reachability analy- sis,

    M. Althoff et al., “Set propagation techniques for reachability analy- sis,” Annual Review of CRAS , vol. 4, no. 1, pp. 369–395, 2021

    2021

  21. [29]

    Zapp! zonotope agreement of prediction and planning for continuous-time collision avoidance with discrete-time dynamics,

    L. Paparusso et al. , “Zapp! zonotope agreement of prediction and planning for continuous-time collision avoidance with discrete-time dynamics,” in ICRA. IEEE, 2024, pp. 9285–9292

    2024

  22. [30]

    Adaptive conformal prediction for motion planning among dynamic agents,

    A. Dixit et al. , “Adaptive conformal prediction for motion planning among dynamic agents,” in L4DC. PMLR, 2023, pp. 300–314

    2023

  23. [31]

    Reactive motion planning with probabilisticsafety guarantees,

    Y . Chen et al. , “Reactive motion planning with probabilisticsafety guarantees,” in CoRL. PMLR, 2021, pp. 1958–1970

    2021

  24. [32]

    Robust, informative human-in-the- loop predictions via empirical reachable sets,

    K. Driggs-Campbell et al. , “Robust, informative human-in-the- loop predictions via empirical reachable sets,” arXiv preprint arXiv:1705.00748, 2017

    2017 arXiv

  25. [33]

    Estimating reachable sets with scenario optimization,

    A. Devonport and M. Arcak, “Estimating reachable sets with scenario optimization,” in L4DC. PMLR, 2020, pp. 75–84

    2020

  26. [34]

    Multi-modal conformal prediction regions by optimiz- ing convex shape templates,

    R. Tumu et al., “Multi-modal conformal prediction regions by optimiz- ing convex shape templates,” in L4DC. PMLR, 2024, pp. 1343–1356

    2024

  27. [35]

    Convex approximation of probabilistic reach- able sets from small samples using self-supervised neural networks,

    J. Xiang and J. Chen, “Convex approximation of probabilistic reach- able sets from small samples using self-supervised neural networks,” arXiv preprint arXiv:2411.14356 , 2024

    2024

  28. [36]

    Data-driven reachability with scenario optimization and the holdout method,

    E. Dietrich et al., “Data-driven reachability with scenario optimization and the holdout method,” arXiv preprint arXiv:2504.06541 , 2025

    2025 arXiv

  29. [37]

    Integrating intuitive driver models in autonomous planning for interactive maneuvers,

    K. Driggs-Campbell et al. , “Integrating intuitive driver models in autonomous planning for interactive maneuvers,” ITS, vol. 18, no. 12, pp. 3461–3472, 2017

    2017

  30. [38]

    Sampling-based reachability analysis: A random set theory approach with adversarial sampling,

    T. Lew and M. Pavone, “Sampling-based reachability analysis: A random set theory approach with adversarial sampling,” in CoRL. PMLR, 2021, pp. 2055–2070

    2021

  31. [39]

    Conditional validity of inductive conformal predictors,

    V . V ovk, “Conditional validity of inductive conformal predictors,” in ACML. PMLR, 2012, pp. 475–490

    2012

  32. [40]

    V olume of n-dimensional ellipsoid,

    A. J. Wilson, “V olume of n-dimensional ellipsoid,” Sciencia Acta Xaveriana, vol. 1, no. 1, pp. 101–6, 2010

    2010

  33. [41]

    The distribution of quadratic forms of gaussian vectors,

    V . Zorin et al. , “The distribution of quadratic forms of gaussian vectors,” Theory of Probability & Its Applications , vol. 33, no. 3, pp. 557–560, 1989. APPENDIX PROOFS Lemma 2: Let A and B be measurable sets. If every measurable subset of A with non-zero measure is a subse...

    1989

  34. [42]

    Using the same argument vol(B \ A) = 0, implying that A = B almost everywhere. Proof: [Proof of Theorem 1 ] We establish this proof by showing that any measurable subset of Ft with non-zero measure is also a subset of ω∗ t , and every measurable subset of ω∗ t with non-zero me...

Pith tools

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