Autonomous Driving with Priority-Ordered STL Specifications Under Multimodal Uncertainty

Abstract: Autonomous vehicles must plan trajectories that satisfy a multitude of requirements on safety, passenger comfort, and compliance with traffic rules. However, in safety-critical scenarios, it is not always possible to satisfy all requirements simultaneously, necessitating their prioritization based on importance. At the same time, in these safety-critical scenarios, the uncertainty in trajectory predictions of the surrounding traffic, such as other vehicles and pedestrians, should be explicitly accounted for. In this work, we propose an uncertainty-aware trajectory planning framework that incorporates a predefined lexicographic ordering over Signal Temporal Logic (STL) specifications that stays valid under uncertainty. We implement this formulation with Model Predictive Path Integral (MPPI) control and we demonstrate the effectiveness of our method on simulation scenarios, showing that our framework efficiently handles conflicting objectives under realistic multi-modal uncertainty.

• The paper introduces a novel risk-aware trajectory planning framework using lexicographically prioritized STL specifications to address multimodal uncertainty in autonomous driving.
• It integrates scenario-based CVaR approximation with an MPPI planner, ensuring robust satisfaction of safety-critical objectives in real-time.
• Empirical tests on highway and pedestrian scenarios validate the framework's feasibility, achieving sub-10 ms computation with GPU parallelization.

Lexicographically Ordered STL-Based Trajectory Planning for Autonomous Vehicles under Multimodal Uncertainty

Problem Formulation and Motivations

The paper addresses the challenge of trajectory planning for autonomous vehicles (AVs) subjected to multimodal stochastic uncertainty in the behavior of surrounding traffic agents. It operationalizes safety, comfort, and traffic-rule compliance as Signal Temporal Logic (STL) specifications and develops a planning architecture that enforces a strict lexicographic priority ordering among these rules. The motivation stems from the observation that safety-critical scenarios may not admit simultaneous satisfaction of all requirements, and that conventional deterministic approaches and naïve risk-averaging fail to account for rare but catastrophic violations. The proposed stochastic framework leverages Conditional Value-at-Risk (CVaR) as a risk metric, ensuring robust satisfaction of higher-priority rules even in adverse probabilistic modes, and integrates these constraints into a receding-horizon Model Predictive Path Integral (MPPI) planner capable of navigating the inherent non-smoothness and non-convexity.

Formalism: Multimodal Stochastic Trajectory, STL Rules, and Risk Extensions

The vehicle dynamics are modeled using discrete-time nonlinear state evolution, with the environment (other vehicles, pedestrians) represented as stochastic processes over a probability space encompassing multimodal predictions. STL specifications encode formal requirements, characterized by both qualitative (satisfaction/violation) and quantitative (robustness) semantics. The paper extends deterministic lexicographic ordering, which strictly prioritizes higher-level rules, to risk-aware stochastic settings by defining per-rule loss variables, evaluated via CVaR. This allows the planner to penalize rare but severe violations, overcoming the limitations of expectation-based approaches.

The lexicographic ordering is translated into a rank function and a rank-preserving reward, which quantitatively encode priorities and enable continuous tie-breaking within ranks while strictly segregating satisfaction levels. The resulting planning problem aims to maximize this reward over a receding horizon subject to system dynamics, rule-based constraints, and scenario approximations.

Numerical Solution: Scenario-Based CVaR Approximation and MPPI Planner

Exact computation over the continuous uncertainty distribution is intractable; hence, a scenario tree approximation is adopted, with multimodal traffic predictions instantiated as discrete samples (with weights). Per-sample STL robustness is computed, and CVaR is empirically approximated as a weighted sum over scenarios, converging asymptotically to the true value as sample sizes grow.

The MPPI planner, inherently sampling-based and amenable to highly parallel GPU implementation, is deployed to solve the non-smooth optimization. It propagates perturbed control sequences, computes STL robustness and rank-preserving reward for each rollout, and aggregates results via importance-weighted averaging. This method avoids the pitfalls of non-differentiability and non-convexity present in both STL semantics and the reward structure, efficiently guiding the planner toward feasible and risk-optimal solutions.

Empirical Results: Highway Maneuver and Pedestrian Crossing

Two case studies demonstrate the practical behavior of the framework:

• Highway Take-Over with Multimodal Cut-In: The ego vehicle must choose between maintaining lane discipline or ensuring safety against aggressive cut-in maneuvers by another car. The risk-aware planner decisively follows the top-priority rule: if safety is prioritized, the ego violates lane rules to avoid collision; if lane discipline is prioritized, it stays but risks unsafe proximity. Notably, the rank-preserving reward guarantees that no weighting of lower-priority objectives overrides violations at higher levels.
• Pedestrian Crossing with Discrete Positional Uncertainty: The ego vehicle plans with respect to a stochastic pedestrian position (six candidates, weighted probabilities). Results across varying CVaR levels show the planner steering increasingly conservative avoidance arcs as the CVaR rises, directly quantifying the trade-off between tail risk and task performance. CVaR0.70 presents a regime with minimal violation and moderate conservatism, while CVaR0.90 guarantees safety at the cost of reduced goal attainment and wider trajectory deviation.

Both scenarios validate the computational feasibility (sub-10 ms per step with GPU parallelization) and tunable conservatism of the framework, with the rank-preserving reward enforcing strict priority compliance.

Implications and Future Directions

This research delivers a principled approach for AV trajectory planning with robustly ordered STL specifications under realistic multimodal uncertainty. The practical implications are substantial: the ability to operationalize prioritized safety rules over probabilistic future scenarios aligns with regulatory and ethical requirements, and the integration with MPPI and scenario-based CVaR computation enables real-time deployment.

Theoretically, the work extends deterministic lexicographic reward structures to risk-aware stochastic domains, providing formal guarantees on priority retention, and establishes CVaR as a powerful tool for tail-risk management in safety-critical applications.

Potential future developments include:

• Encoding moral and ethical imperatives directly as STL rules with strict priority ordering.
• Online adaptation of scenario weights and CVaR levels based on real-time intention prediction of traffic participants.
• Validation on closed-loop benchmarks (e.g., nuPlan) with dense multimodal prediction outputs and diverse interaction scenarios.

Conclusion

The presented methodology synthesizes risk-aware receding-horizon trajectory planners with lexicographically ordered STL specifications for AVs, robust to multimodal environmental uncertainty (2606.20336). The combination of scenario-based CVaR approximation and MPPI optimization ensures strict compliance to prioritized objectives. Empirical results confirm its efficacy in practical use cases, and the framework sets a foundation for theoretically rigorous, ethically aligned, and operationally robust trajectory planning in autonomous driving.