SAD-Flower: Flow Matching for Safe, Admissible, and Dynamically Consistent Planning

TL;DR

SAD-Flower addresses the lack of formal guarantees in data-driven flow-matching planners by integrating a virtual control input into the sampling process. It casts state/action constraints as control barrier functions and dynamic consistency as a control Lyapunov function, employing prescribed-time scheduling to guarantee safety, admissibility, and executability at test time without retraining. The approach solves a sequence of minimum-norm quadratic programs to minimally perturb the learned flow while enforcing constraints, yielding formal guarantees and robust performance across navigation, locomotion, and manipulation tasks. Empirical results show SAD-Flower outperforms several constraint-aware baselines in constraint satisfaction while maintaining competitive task rewards, demonstrating practical deployability in real-world robotics.

Abstract

Flow matching (FM) has shown promising results in data-driven planning. However, it inherently lacks formal guarantees for ensuring state and action constraints, whose satisfaction is a fundamental and crucial requirement for the safety and admissibility of planned trajectories on various systems. Moreover, existing FM planners do not ensure the dynamical consistency, which potentially renders trajectories inexecutable. We address these shortcomings by proposing SAD-Flower, a novel framework for generating Safe, Admissible, and Dynamically consistent trajectories. Our approach relies on an augmentation of the flow with a virtual control input. Thereby, principled guidance can be derived using techniques from nonlinear control theory, providing formal guarantees for state constraints, action constraints, and dynamic consistency. Crucially, SAD-Flower operates without retraining, enabling test-time satisfaction of unseen constraints. Through extensive experiments across several tasks, we demonstrate that SAD-Flower outperforms various generative-model-based baselines in ensuring constraint satisfaction.

Figures (5)

• Figure 1: Algorithm 1: Planning by SAD Flower
• Figure 2: (a) Without enforcing dynamic consistency, applying state and action constraints can result in significant outliers, known as the local trap problem xiao2023safediffuser. (b) Our method satisfies constraints, but using too few numerical integration steps for the ODE introduces jitter in the trajectory. (c) With sufficient integration steps, our method produces dynamically consistent trajectories while satisfying unseen constraints (red ellipses).
• Figure 3: Effect of increasing the number of ODE steps for solving eq. (\ref{['eq:control_ode']}) and the activation time $T_0$ of controller eq. (\ref{['eq:CBF_CLF_control_law']}) on the consistency and performance of trajectories.
• Figure 4: Performance of the proposed SAD-Flower and baselines depending on level of constraint tightening for the Hopper (Med-Expert) benchmark.
• Figure 5: Performance under progressively tightened constraints in the locomotion task. As the allowed torso height decreases, the admissibility constraint becomes stricter, creating a conflict with the task objective of jumping forward.

Theorems & Definitions (7)

• Remark 4.1
• Theorem 5.1
• Theorem B.1
• proof
• Theorem B.2
• proof
• proof : Proof of \ref{['th:CLF-CBF-All']}