[Paper Review] Rule-based Optimal Control for Autonomous Driving

Wei Xiao, Noushin Mehdipour+5 more|arXiv (Cornell University)|Jan 14, 2021

Formal Methods in Verification25 references4 citations

TL;DR

This paper proposes a rule-based optimal control framework for autonomous vehicles that prioritizes traffic laws and driving behavior rules using a hierarchical structure. By integrating Control Lyapunov Functions (CLFs) for stability and Control Barrier Functions (CBFs) for safety, the method recursively relaxes lower-priority rules to resolve conflicts, enabling real-time feasible control while minimizing violation of high-priority rules, with validation through multi-scenario case studies showing successful trajectory generation and pass/fail evaluation.

ABSTRACT

We develop optimal control strategies for Autonomous Vehicles (AVs) that are required to meet complex specifications imposed by traffic laws and cultural expectations of reasonable driving behavior. We formulate these specifications as rules, and specify their priorities by constructing a priority structure. We propose a recursive framework, in which the satisfaction of the rules in the priority structure are iteratively relaxed based on their priorities. Central to this framework is an optimal control problem, where convergence to desired states is achieved using Control Lyapunov Functions (CLFs), and safety is enforced through Control Barrier Functions (CBFs). We also show how the proposed framework can be used for after-the-fact, pass / fail evaluation of trajectories - a given trajectory is rejected if we can find a controller producing a trajectory that leads to less violation of the rule priority structure. We present case studies with multiple driving scenarios to demonstrate the effectiveness of the proposed framework.

Motivation & Objective

• To develop a control framework that enables autonomous vehicles to satisfy complex, prioritized traffic rules and behavioral expectations.
• To resolve conflicts among competing traffic rules by introducing a priority-structured relaxation mechanism.
• To ensure safety and stability using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) in a real-time optimal control setting.
• To enable post-hoc evaluation of vehicle trajectories via a pass/fail criterion based on rule priority structure.
• To demonstrate the framework’s effectiveness on realistic vehicle dynamics with conflicting rule scenarios.

Proposed method

• Formalize traffic and behavioral rules as a priority-ordered set using a pre-order structure.
• Model the control problem as a sequence of quadratic programs (QPs) that enforce CBFs for safety and CLFs for convergence.
• Implement iterative rule relaxation based on priority, where lower-priority rules are relaxed only if higher-priority constraints are satisfied.
• Use violation scores to quantify rule deviations and compare trajectories for pass/fail evaluation.
• Apply the framework recursively: solve QP at each iteration with relaxed rules until a feasible solution is found.
• Leverage the priority structure to determine whether a candidate trajectory is outperformed by a generated one, thus rejecting it.

Experimental results

Research questions

• RQ1 How can conflicting traffic rules with varying priorities be systematically resolved in autonomous vehicle control?
• RQ2 Can a recursive optimal control framework based on CLFs and CBFs ensure safety and stability while minimizing violation of lower-priority rules?
• RQ3 How can a pass/fail evaluation mechanism be designed to objectively assess vehicle trajectories based on rule priority?
• RQ4 To what extent can this framework generate feasible, safe, and behaviorally reasonable trajectories in complex driving scenarios?
• RQ5 Can the framework be applied to realistic vehicle dynamics with hard constraints on control inputs and state variables?

Key findings

• In Scenario 1, the framework generated a feasible trajectory that violated only medium- and low-priority rules (r3 and r5), with violation scores of 0.124 and 0.111, respectively, while satisfying the highest-priority collision avoidance rule (r1).
• For Scenario 2, the optimal control produced a trajectory with lower violation scores (0.111 for r5 vs. 0.646 in the candidate), leading to rejection of the candidate due to better compliance with the priority structure.
• In Scenario 3, the framework successfully generated a trajectory that satisfied all high-priority rules (including pedestrian safety) while relaxing lower-priority rules, with violation scores of 0.058 for r3 and 0.359 for r5.
• The pass/fail evaluation correctly rejected candidate trajectories that violated higher-priority rules, even if they had lower violation scores on lower-priority rules, confirming the priority-based ranking logic.
• The recursive relaxation process ensured feasibility by progressively relaxing rules starting from the lowest priority, enabling convergence to a viable control solution.
• The framework demonstrated robustness in handling conflicting constraints, such as avoiding pedestrians, staying in lane, and maintaining speed, by prioritizing safety over comfort and speed limits.