Risk-Aware Rulebooks for Multi-Objective Trajectory Evaluation under Uncertainty

Tichakorn Wongpiromsarn

eess.SYMar 4, 2026

Risk-Aware Rulebooks for Multi-Objective Trajectory Evaluation under Uncertainty

Authors: Tichakorn Wongpiromsarn

Organizations: Department of Computer Science, Iowa State University, Ames, IA 50010

Abstract

We present a risk-aware formalism for evaluating system trajectories in the presence of uncertain interactions between the system and its environment. The proposed formalism supports reasoning under uncertainty and systematically handles complex relationships among requirements and objectives, including hierarchical priorities and non-comparability. Rather than treating the environment as exogenous noise, we explicitly model how each system trajectory influences the environment and evaluate trajectories under the resulting distribution of environment responses. We prove that the formalism induces a preorder on the set of system trajectories, ensuring consistency and preventing cyclic preferences. Finally, we illustrate the approach with an autonomous driving example that demonstrates how the formalism enhances explainability by clarifying the rationale behind trajectory selection.

Figures & tables

Panel of Figure 1 — Fig. 1: Illustrative probability density of a random cost variable $f$ , highlighting key risk measures for $\alpha=0.9$ .

Panel of Figure 2 — Fig. 2: AV–pedestrian interaction with their candidate trajectories.

Environment scenario	$\omega_{1}$	$\omega_{2}$	$\omega_{3}$	$\omega_{4}$
$\mathit{Pr}$	0.98	0.001	0.009	0.01
$\mathbf{E}_{\tau_{1}}$	$\xi_{1}$	$\xi_{2}$	$\xi_{1}$	$\xi_{1}$
$\mathbf{E}_{\tau_{2}}$	$\xi_{1}$	$\xi_{2}$	$\xi_{2}$	$\xi_{1}$
$\mathbf{E}_{\tau_{3}}$	$\xi_{1}$	$\xi_{2}$	$\xi_{2}$	$\xi_{2}$
$\mathbf{E}_{\tau_{4}}$	$\xi_{1}$	$\xi_{2}$	$\xi_{1}$	$\xi_{1}$

Fig. 3: The interaction

\mathbf{E}_{\tau_{i}}(\omega_{j})

and probability measure

\mathit{Pr}(\omega_{j})

for each

i,j\in\{1,\ldots,4\}

	$r_{1}(\tau_{i},\xi_{j})$		$r_{2}(\tau_{i},\xi_{j})$		$r_{3}(\tau_{i},\xi_{j})$		$r_{4}(\tau_{i},\xi_{j})$
	$\xi_{1}$	$\xi_{2}$	$\xi_{1}$	$\xi_{2}$	$\xi_{1}$	$\xi_{2}$	$\xi_{1}$	$\xi_{2}$
$\tau_{1}$	0	225	0	0	0	0	0	0
$\tau_{2}$	0	175	0	0	1.77	1.77	0	0
$\tau_{3}$	0	0	0	0	15	15	12.25	12.25
$\tau_{4}$	0	0	1	1	0	0	0	0

Fig. 4: The degree of rule violations for each combination of AV and pedestrian trajectories.

	$r_{2}$	$r_{3}$	$r_{4}$
$\tau_{1}$	0	0	0
$\tau_{2}$	0	1.77	0
$\tau_{3}$	0	15	12.25
$\tau_{4}$	1	0	0

Fig. 5: The degree of rule violations for different AV trajectories. [Left] Probability mass function of rule

r_{1}

values for different AV trajectory

\tau_{i}\in\mathcal{T}

under environmental uncertainty. The x-axis represents the realized value

r_{1}(\tau_{i})

and the y-axis represents the probability of observing that value according to the probability measure

\mathit{Pr}

over the environment scenarios

\Omega

defined in Figure 3 . [Right] The violations for

r_{2}

r_{3}

, and

r_{4}

, where the degree of violation is independent of the environment scenarios.

	Expected	Worst case	$\text{VaR}_{\alpha}$	$\text{CVaR}_{\alpha}$
$\tau_{1}$	0.225	225	$\begin{array}[]{ll}0&\hskip-8.53581pt\text{if }\alpha\leq 0.999\\ 225&\hskip-8.53581pt\text{otherwise}\end{array}$	$\begin{array}[]{ll}\frac{0.225}{1-\alpha}&\hskip-8.53581pt\text{if }\alpha\leq 0.999\\ 225&\hskip-8.53581pt\text{otherwise}\end{array}$
$\tau_{2}$	1.75	175	$\begin{array}[]{ll}0&\hskip-8.53581pt\text{if }\alpha\leq 0.99\\ 175&\hskip-8.53581pt\text{otherwise}\end{array}$	$\begin{array}[]{ll}\frac{1.75}{1-\alpha}&\hskip-8.53581pt\text{if }\alpha\leq 0.99\\ 175&\hskip-8.53581pt\text{otherwise}\end{array}$
$\tau_{3}$	0	0	0	0
$\tau_{4}$	0	0	0	0

Fig. 6: The risk of each AV trajectory with respect to rule

r_{1}

, evaluated using widely used risk measures.

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Autonomous Driving Safety Evaluation

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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.

Abstract

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