Model Predictive Robustness of Signal Temporal Logic Predicates

TL;DR

This work addresses the challenge of robustly evaluating STL predicates for autonomous driving by introducing model predictive robustness (MPR), which accounts for system dynamics via predictive models. It formalizes MPR as a normalized probability-based robustness $ ho^{\text{MP}}_p$ computed over a finite horizon, using Monte Carlo trajectory predictions coupled with an STL monitor. Gaussian process regression learns the predicate robustness online from a rich feature set that captures rule relevance and vehicle interactions, yielding smooth predictions with uncertainty quantification. Empirical results on highD and German interstate rules show that MPR provides higher fidelity robustness estimates than model-free approaches and that incorporating MPR into trajectory planning yields more rule-compliant behavior, often surpassing human drivers in the dataset. The approach offers a scalable, data-driven pathway to integrate formal safety specifications into real-time planning for autonomous systems.

Abstract

The robustness of signal temporal logic not only assesses whether a signal adheres to a specification but also provides a measure of how much a formula is fulfilled or violated. The calculation of robustness is based on evaluating the robustness of underlying predicates. However, the robustness of predicates is usually defined in a model-free way, i.e., without including the system dynamics. Moreover, it is often nontrivial to define the robustness of complicated predicates precisely. To address these issues, we propose a notion of model predictive robustness, which provides a more systematic way of evaluating robustness compared to previous approaches by considering model-based predictions. In particular, we use Gaussian process regression to learn the robustness based on precomputed predictions so that robustness values can be efficiently computed online. We evaluate our approach for the use case of autonomous driving with predicates used in formalized traffic rules on a recorded dataset, which highlights the advantage of our approach compared to traditional approaches in terms of precision. By incorporating our robustness definitions into a trajectory planner, autonomous vehicles obey traffic rules more robustly than human drivers in the dataset.

Figures (5)

• Figure 1: Scheme of model predictive robustness computation for the predicate $\mathrm{in\_same\_lane}$. The prediction model generates a finite set of trajectories for all rule-relevant vehicles within a certain time period, of which the rule compliance is checked by an STL monitor. The robustness is calculated based on the future probability of satisfying the predicate.
• Figure 2: A curvilinear coordinate system aligned with the reference path $\Gamma$.
• Figure 3: Comparison of different sampling strategies for the ego vehicle.
• Figure 4: Evaluation results of the predicate $\mathrm{in\_same\_lane}$. (a) shows the distribution of its model-free and model predictive robustness with $1,000$ randomly selected data points, while (b) analyzes the sensitivity of the feature variables, where the values are scaled so that the most relevant feature variable has a relevance of one.
• Figure 5: Robustness-aware trajectory planning. The trajectories in (a) are color-coded according to the robustness, which increases from red with negative to blue with positive values. (b) and (c) depict the evaluation results from using model-free and model predictive robustness, respectively. The shaded regions in (b) denote the $2$-$\sigma$ model uncertainty corresponding to a 95.4% confidence level. In the right-hand panels of (b) and (c), the blue dots to the right of the yellow dotted line indicate that the robustness of the optimal trajectory is higher than that of the human trajectory. The red dots to the left of the line indicate a lower robustness.

Theorems & Definitions (3)

• Definition 1: Characteristic Function donze2010robust
• Definition 2: Model Predictive Robustness
• Theorem 1