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Barrier Function Conformal Safety Clearance Certification with CVaR for Driving Trajectory Selection

Pei Yu Chang, Qadeer Ahmed

arXiv:2608.26533v1eess.SYcs.RO

TL;DR

Motion planners rank candidate trajectories without directly certifying the realized clearance of the selected one. The paper combines a differentiable OBB-clearance lower bound with nominal or lower-tail CVaR evaluation and post-selection conformal calibration. On 300 nuPlan sessions, CVaR reduced the correction from 1.43 m to 0.03 m and increased nonnegative certificates from 68.7% to 87.3%, while exact-clearance coverage remained 93.3–96.7%.

  • Problem

    Existing trajectory evaluations do not directly certify the realized physical safety clearance of the selected trajectory.

  • Method

    The framework lower-bounds exact signed OBB clearance with a differentiable separating-axis margin and conformally calibrates the complete prediction, evaluation, and selection procedure.

  • Results

    At 10% target miscoverage, sampled lower-tail CVaR reduced the conformal correction from 1.43 m to 0.03 m and increased nonnegative certificates from 68.7% to 87.3%, with exact-clearance coverage of 93.3–96.7%.

  • Takeaways & Limitations

    The framework provides calibrated lower-clearance certificates for selected trajectories, with CVaR producing tighter and more available certificates while maintaining target coverage.

  • Takeaways & Limitations

    The guarantee is marginal under exchangeability and eligibility assumptions; closed-loop deployment requires calibration of the complete frozen rollout policy.

Abstract

from arXiv · show

Autonomous driving motion planners generate and select candidate trajectories while accounting for interactions with surrounding agents. However, these evaluations do not certify the actual safety clearance of the selected trajectory. The framework evaluates the trajectory selected by ant planners and calibrates the gap between its plan time margin and realized safety clearance. A differentiable separating axis barrier margin deterministically lower bounds exact signed oriented-bounding-box (OBB) safety clearance, connecting the statistical certificate to safety margin. At plan time, the margin is evaluated using either a nominal prediction and sampled lower tail Conditional Value-at-Risk (CVaR), while post-selection conformal calibration over exchangeable drive sessions absorbs prediction and sampling errors. Conformal calibration provides statistical validity independently of predictor correctness. The method is evaluated on a frozen 300 session nuPlan study using native Predictive Driver Model (PDM) Closed loop proposals. At 10% target miscoverage, sampled lower CVaR reduces the conformal correction from 1.43m to 0.03m and increases the rate of nonnegative safety clearance certificates from 68.7% to 87.3%. Across all evaluated statistics, exact-clearance coverage remains above the 90% target at 93.3--96.7%.

I. INTRODUCTION AND RELATED WORK

Existing planners rank and select trajectories but do not directly certify the realized physical clearance of the selected trajectory. The framework combines a geometric lower bound, risk-aware prediction, and post-selection session-level conformal calibration to provide such a certificate.

  • Trajectory-ranking quantities support selection but do not directly certify the realized safety clearance of the selected trajectory.
  • Existing conformal planning primarily covers predicted-agent motion, whereas this framework targets physical clearance for the selected ego trajectory.
  • A differentiable separating-axis margin lower-bounds exact signed OBB clearance, and conformal calibration is applied after prediction, evaluation, and selection.
  • The framework is evaluated on 300 nuPlan sessions with PDM-Closed proposals, producing 93.3–96.7% held-out coverage at 10% target miscoverage.
  • Calibration uses sessions as exchangeability units, aggregates windows within sessions, and permits arbitrary within-session dependence.

III. CERTIFIED GEOMETRIC MARGIN

The certified geometric margin uses separating-axis projections and a smooth construction that deterministically lower-bounds signed OBB safety clearance. The bound remains valid across padding, overlap, time steps, and agent pairs.

  • The two-axis SAT margin takes the maximum of longitudinal and lateral signed projection gaps, with nonnegative safety padding reducing the margin.
  • A smooth margin is constructed from the SAT margin while preserving the lower-bound direction.
  • For disjoint rectangles, each selected-axis projection gap cannot exceed the Euclidean distance between the rectangles.
  • The geometric lemma extends the inequality to overlapping rectangles using the complete set of SAT face-normal axes.
  • Taking minima over evaluated time steps and agents preserves the deterministic lower bound, independently of the predictor.

IV. RISK-AWARE EVALUATION AND CONFORMAL CERTIFICATION

Because future agent motion is unavailable at planning time, the framework evaluates geometric margins under a predictive distribution and calibrates the complete pipeline against observed outcomes.

  • The unknown future agent motion induces uncertainty in each candidate’s geometric margin, so the complete prediction-to-selection pipeline is conformally calibrated.

A. Plan-Time Risk Evaluation

Plan-time evaluation assigns each candidate a margin under predicted futures, using either a nominal statistic or lower-tail CVaR. These statistics are not certificates until post-selection conformal calibration absorbs their errors.

  • Each candidate receives a certified geometric margin under each predicted joint future.
  • The nominal evaluator uses a single predicted future, while predictive uncertainty induces a distribution over candidate margins.
  • Lower-tail CVaR emphasizes adverse predicted futures because smaller safety-clearance values are less safe.
  • In practice, lower-tail CVaR is computed empirically from finite predictor samples.
  • Neither nominal nor CVaR plan-time statistics is itself a realized-clearance certificate; conformal calibration absorbs prediction and sampling errors.

B. Fixed Selection Pipeline for Calibration

The certification target is the trajectory returned by a fixed planning pipeline, including prediction, evaluation, ranking, and fallback choices. Conformal calibration is applied to this selected output rather than to candidates independently.

  • The selection step must be included in calibration because it can depend on predicted motion and the plan-time margin statistic.
  • The fixed procedure generates candidates, evaluates predicted information, and selects one trajectory without using the realized future.Its inputs may include candidate sets, predicted futures, the margin statistic, map and route information, and other plan-time quantities.
  • The calibrated certificate applies directly to the trajectory produced by the fixed planning pipeline.Conformal calibration uses the residual associated with the selected output rather than calibrating candidate trajectories independently.

C. Post-Selection Conformal Calibration

Post-selection split conformal calibration converts the selected trajectory’s plan-time margin into a finite-sample lower certificate for exact OBB safety clearance. Under exchangeability, the guarantee covers the deployment population and applies simultaneously across evaluated windows, but it is marginal rather than conditional on certified trajectories or subgroups.

  • The post-selection residual measures how much the selected trajectory’s plan-time margin overestimates the margin realized under the observed future.Larger residuals correspond to greater overestimation.
  • Exchangeability is assumed across driving sessions, while arbitrary dependence among windows within a session is handled by aggregating window residuals into one unit score.An upper bound on the unit-level score therefore applies to every evaluated window in that session.
  • The split-conformal quantile calibrates post-selection residuals because unit-level scores are computed after applying the fixed selection pipeline.The quantile is the kth order statistic, with k = ⌈(n + 1)(1 −η)⌉; if k > n, it is set to +∞.
  • The calibrated lower safety-clearance certificate is obtained by subtracting the conformal correction from the plan-time margin.This corrected quantity lower-bounds the realized exact OBB safety clearance with finite-sample coverage.
  • Theorem 1 provides coverage of at least 1−η for the selected trajectory’s realized exact safety clearance under Assumptions 1 and 2.The proof combines split-conformal exchangeability with the pointwise geometric relation between smooth margin and exact clearance.
  • The predictor and plan-time statistic affect certificate tightness, whereas conformal calibration supplies statistical validity independently of predictor correctness.The plan-time statistic need not itself be a valid estimate or lower bound of realized safety clearance.
  • A requested buffer ρ ≥0 is certified when the calibrated lower bound meets that buffer, and false certification probability is at most η.The guarantee controls the joint event that certification occurs while realized exact clearance is below the requested buffer.
  • The guarantee is marginal over the deployment population, not conditional on certified trajectories or on city, scenario class, or other subpopulations.Conditional subgroup guarantees require exchangeability to hold for the corresponding population.

A. Experimental Protocol

The evaluation uses a frozen, balanced three-city nuPlan study with PDM-Closed candidate proposals, independent calibration and test sessions, and nominal versus sampled lower-tail CVaR plan-time statistics. It measures conformal correction, exact-clearance coverage, certificate rate, and false-certification rate at target miscoverage levels of 10% and 5%.

  • 300 independent nuPlan drive sessions are drawn equally from Boston, Pittsburgh, and Singapore, with one planning query per session.Each session forms one exchangeability unit, so its unit-level score reduces to a single window residual.
  • Each query uses 15 native PDM-Closed candidate ego trajectories over a 4 s horizon before future annotations are accessed.The predictor uses at most 12 dynamic agents, while retrospective clearance evaluation includes all logged vehicles, pedestrians, and bicycles.
  • The dataset is split into 150 calibration and 150 held-out test sessions with no session overlap, representing an approximately balanced three-city deployment mixture.Only queries with at least one nearby dynamic agent available for plan-time prediction are retained.
  • Two plan-time statistics are evaluated: nominal margin and sampled lower-tail CVaR, with β ∈ {0.30, 0.10} controlling the averaged lower-tail fraction.Each statistic is calibrated independently using the same frozen calibration and test split.
  • Target miscoverage levels are η ∈ {0.10, 0.05}, corresponding to target coverage levels of 0.90 and 0.95.Unless otherwise stated, ρ = 0, so certification requires a nonnegative calibrated lower safety-clearance bound.
  • The primary metrics are conformal correction, exact safety-clearance coverage, certificate rate, and unit-level false-certification rate.Certificate rate counts evaluated windows meeting the calibrated buffer rule, while exact-clearance coverage tests the theorem’s event.

B. Conformal Certification Results

The combined and within-city evaluations show that conformal calibration maintains exact safety-clearance coverage while sampled lower-tail CVaR yields tighter, more available certificates than the nominal statistic.

  • Combined evaluation: At η = 0.10, sampled lower-tail CVaR reduces the conformal correction from 1.43 m to 0.03 m and raises the certificate rate from 68.7% to 87.3%.For β = 0.30 and β = 0.10, exact safety-clearance coverage is 96.0% and 96.7%, respectively.
  • Combined evaluation: At η = 0.05, CVaR requires a 0.24 m correction and retains certificate rates of 84.0–86.0%, versus 3.66 m and 40.7% for nominal selection.Empirical exact safety-clearance coverage remains at or above the corresponding target across settings.
  • Within-city diagnostics: Within-city nominal corrections range from 0.15 m in Pittsburgh to 2.70 m in Singapore, while Singapore’s certificate rate falls to 60%.The larger Singapore correction is attributed to several large nominal residuals in the upper tail of its frozen calibration split.
  • Within-city diagnostics: Within-city CVaR β = 0.10 corrections range from 0.02–0.24 m, with 96–98% exact coverage and 88–90% certificate rates.These results are interpreted as finite-sample calibration diagnostics rather than rankings of city difficulty or additional conditional guarantees.

C. Offline Comparison with PDM Native Selection

The offline comparison measures how often the proposed safety-clearance selection differs from PDM’s native selector on the same proposal bank, while paired scenes illustrate higher proposed clearances.

  • Selection differences: The proposed and native selectors choose different trajectories in 271/300 (90.3%) nominal, 268/300 (89.3%) CVaR .30, and 267/300 (89.0%) CVaR .10 queries.The comparison uses the same 15-proposal bank across the 300-session offline benchmark.
  • Selection differences: The high disagreement reflects different selection rules: a safety-clearance gate followed by progress and effort versus PDM’s native composite score.The comparison is not intended to reproduce PDM’s native composite objective.
  • Paired scenes: In the pedestrian scene, exact OBB safety clearance is 0.73 m for native PDM and 1.12 m for the proposed method at the illustrated frame.The rollout traces also mark contact at 0 m and the 1 m clearance threshold.
  • Paired scenes: In the traffic-light scene, exact OBB safety clearance is 0.53 m for native PDM and 1.02 m for the proposed method at native PDM’s minimum clearance.The proposed trajectory had already diverged following an earlier intervention.
  • Interpretation: The offline comparison measures objective difference, not PDM error or certificate invalidity.This limits how disagreement rates should be interpreted.

D. Closed-Loop Integration Diagnostic

The closed-loop intervention integrates safety-clearance selection with PDM-Closed while usually retaining the native proposal. Its clearance differences are diagnostic rather than evidence of improved closed-loop safety, and the closed-loop variant is not covered by the conformal guarantee.

  • The intervention retains the native proposal when its predicted margin is nonnegative and otherwise selects the highest-score alternative meeting that condition.If no eligible alternative exists, the native proposal is retained and the margin violation is recorded.
  • 93.2% of replanning windows retain the native PDM proposal, with 27 of 50 scenarios requiring no override.The proposed method is described as causing limited disruption to native planner behavior.
  • Illustrated exact OBB safety clearance increases from 0.73 to 1.12 m in the pedestrian scene and from 0.53 to 1.02 m in the traffic-light scene.The pedestrian rollouts retain the same official scenario score; corresponding full-rollout minimum clearances are also reported for the traffic-light example.
  • Mean minimum clearance is 1.238 versus 1.223 m, but bootstrap intervals generally include zero, so the differences remain descriptive integration diagnostics.Below-1 m exposure is 0.2377 versus 0.2497; the study does not treat these differences as evidence of improved closed-loop safety.
  • The closed-loop variant uses an uncalibrated zero-margin gate and therefore has no claimed closed-loop conformal guarantee.Extending the guarantee would require calibrating the complete frozen rollout policy over disjoint drive sessions while accounting for within-session replanning dependence.
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