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

TL;DR

It is proved that the formalism induces a preorder on the set of system trajectories, ensuring consistency and preventing cyclic preferences, and enhances explainability by clarifying the rationale behind trajectory selection.

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 (6)

• Figure 1: Illustrative probability density of a random cost variable $f$, highlighting key risk measures for $\alpha = 0.9$.
• Figure 2: AV–pedestrian interaction with their candidate trajectories.
• Figure 3: The interaction $\mathbf{E}_{\tau_{i}}(\omega_{j})$ and probability measure $\mathit{Pr}(\omega_{j})$ for each $i, j \in \{1, \ldots, 4\}$.
• Figure 4: The degree of rule violations for each combination of AV and pedestrian trajectories.
• Figure 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 \ref{['fig:ex1:interaction']}. [Right] The violations for $r_{2}$, $r_{3}$, and $r_{4}$, where the degree of violation is independent of the environment scenarios.

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Definition 3
• Remark 1
• Example 1
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8