Which Trajectory Is Safer? Combining Rule and Learning for Safety Evaluation

Xuewen Luo; Linlian Jiang; Bo Yu; Fan Ding; Xinxin Zuo; Yang Wang

cs.ROOct 13, 2026

Which Trajectory Is Safer? Combining Rule and Learning for Safety Evaluation

Authors: Xuewen Luo, Linlian Jiang, Bo Yu, Fan Ding, Xinxin Zuo, Yang Wang

Organizations: Xuewen Luo and Bo Yu are with the University of Utah. · Linlian Jiang, Xinxin Zuo, and Yang Wang are with Concordia University. · Fan Ding is with Monash University.

Abstract

Collision rate remains a dominant metric for evaluating autonomous driving trajectory safety, but its binary nature obscures substantial safety differences among collision-free trajectories. Existing surrogate-safety metrics provide finer-grained assessments, yet often trade off physical interpretability against adaptivity to interaction context. We introduce FutureRisk, a continuous metric that quantifies the latent future risk of a trajectory by how closely and persistently its realized interactions approach unsafe states. At its core, our Context-Adaptive Risk Model (CARM) combines a context-dependent safety-time distribution with an independently supervised critical-event-time distribution. Its safety criterion is bounded around a closed-form physical prior, while a bounded residual provides additional adaptation without abandoning this physical anchor. Experiments on DeepAccident and Bench2Drive show that FutureRisk achieves stronger agreement with native collision labels than existing surrogate-safety measures, discriminates risk among collision-free trajectories, and transfers zero-shot across datasets. Comparisons across four planners further show that FutureRisk reveals safety differences hidden by binary collision outcomes. Together, these results establish FutureRisk as a fine-grained, interpretable, and context-adaptive metric for trajectory safety evaluation. Code is available at https://github.com/Xuewen2025/safety-evaluation.

Figures & tables

Fig. 1: Comparison of trajectory-safety metrics. (A) Rule-based metrics are physically interpretable but rely on fixed criteria. (B) Learning-based metrics adapt to context but lack an explicit physical anchor. (C) FutureRisk combines physical interpretability with bounded, context-adaptive learning.

Fig. 2: Overview of FutureRisk. A shared encoding $\bm{h}^{j}$ feeds a safety-time head bounded around the physics prior $\tau_{\mathrm{phy}}^{j}$ and an independent event-timing head, composed into $\widetilde{R}_{\mathrm{CARM}}^{j}$ . Blue is trained in Stage 1, the red residual in Stage 2 with the rest frozen; noisy-OR gives $R$ .

Rule-based
Method	AUROC $\uparrow$	AUPRC $\uparrow$
$y_{\mathrm{analytic}}$	0.648	0.372
raw 3D-TTC	0.646	0.350
Forward TTC (2-D) [ 19 ]	0.685	0.425
Forward TTC (1-D) [ 19 ]	0.684	0.424
PET [ 15 ]	0.667	0.303

Table I: Instance-level discrimination against the binary collision label, DeepAccident.

Rule-based
	$\rho$ vs. $a(t)$		$\rho$ vs. $m$
Method	in	cross	in	cross
$y_{\mathrm{analytic}}$	0.007	– (const.)	1.000 (circ.)	n/a
raw 3D-TTC	0.006	– (const.)	– (const.)	n/a
Forward TTC (2-D) [ 19 ]	0.274	0.283	– (const.)	n/a
Forward TTC (1-D)	0.265	0.291	– (const.)	n/a

Table II: Fine-grained discrimination on collision-free trajectories. In-domain:

n=6630/286

for

a(t)/m

; cross-domain:

n=3333

zero-shot. Cross-domain

m

is undefined (

m=+\infty

throughout).

Model	(in) $\uparrow$	(in) $\uparrow$	(cross) $\uparrow$	(cross) $\uparrow$	$(R,a_{\max})$ $\uparrow$
	AUROC	AUPRC	AUROC	AUPRC	$\rho$
w/o $\mathcal{L}_{\mathrm{evt}}$	0.821	0.502	0.829	0.357	0.082
Full CARM	0.793	0.490	0.827	0.354	0.276

Table III: Ablation of independent critical-event supervision. The event head is retained when removing

\mathcal{L}_{\mathrm{evt}}

Model	AUROC (in) $\uparrow$	AUPRC (in) $\uparrow$	AUROC (cross) $\uparrow$	AUPRC (cross) $\uparrow$
raw safety-time prior	0.731	0.442	0.799	0.344
$+\,\Delta_{\theta}$ only	0.753	0.483	0.812	0.341
$+\,\sigma_{s}$ only	0.718	0.417	0.802	0.331
Stage-1 only (both, $g_{r}\equiv 0$ )	0.764	0.459	0.818	0.333
Full CARM	0.793	0.490	0.827	0.354

Table IV: Ablation of safety-time and residual terms; the learned critical-event branch is retained throughout.

Method	Anchored $\uparrow$	AUROC (in) $\uparrow$	AUROC (cross) $\uparrow$	$\rho$ vs. $a_{\max}$ $\uparrow$
RSS [ 18 ]	$100\%$	0.529	0.662	0.363
Lu et al. [ 7 ]	$0\%$	0.662	0.990 (circ.)	$-$ 0.001
GSSM [ 8 ]	n/a	0.567	0.817	0.070
FutureRisk (CARM)	$70.1\%$	0.793	0.827	0.276

Table V: Interpretability–adaptivity trade-off. Anchoring is structural for RSS and Lu et al. , empirical for FutureRisk, and undefined for GSSM’s non-decomposable output.

Fig. 3: Per-agent anchored-fraction distributions on DeepAccident and Bench2Drive, grouped by collision and critical-event labels. The dotted line marks the $70.1\%$ in-domain mean; the display is clipped to $[-1,1]$ , while all values remain in the reported statistics.

Fig. 4: Planner ranking by mean $R$ (left) and geometric collision rate (right) over available clips ( $n=13$ for TCP; $n=15$ otherwise); safer planners are plotted higher. Parallel lines (TCP, AD-MLP) indicate agreement; crossing lines (UniAD, VAD) indicate disagreement.

Fig. 5: Mean and median $R$ by quintile of the evasive-severity proxy $a_{\max}$ on pooled native-label collision-free DeepAccident instances ( $n=6630$ ). Both show an overall increasing trend with severity; Spearman $\rho=0.276$ .

Fig. 6: One AccidentTwoWays scenario (Route 1105) under UniAD (red) and VAD (blue). UniAD collides at $t=0.6$ s ( $R=0.962$ ), while VAD avoids overlap within 4 s but closes from $57.8$ m to $3.4$ m at about $4$ m/s ( $R=0.975$ ). (a) Closed-form risk $r(t)$ : nonzero for UniAD but zero for VAD. (b1)/(b2) Bird’s-eye trajectories; light boxes mark $t=0$ and solid boxes the moment of interest. (c) Critical-event hazard $q_{k}$ for the relevant agent.

Fig. 7: Per-frame risk for two DeepAccident instances (rows) under four methods (columns): a near-miss (top; native label $0$ ) and a collision missed by geometric overlap (bottom; native label $1$ , contact at $t=4.0$ s). Path color indicates risk (green low, red high); GSSM is normalized to its own range per row, while the others share one $[0,1]$ scale. Gray dashed: interacting agent; X : contact; o : closest approach.

Fig. 8: $R$ , 3D-TTC score, and PET score as the ego path of one DeepAccident collision instance is shifted laterally by $\delta$ (top: min–max normalized; bottom: raw scores). The rule-based curves change abruptly near their thresholds, while $R$ changes smoothly.

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