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UniConFlow: A Unified Constrained Flow-Matching Framework for Certified Motion Planning

Zewen Yang, Xiaobing Dai, Dian Yu, Zhijun Li, Majid Khadiv, Sandra Hirche, Sami Haddadin

TL;DR

UniConFlow presents a unified, constraint-aware flow-matching framework for certified motion planning that enforces both equality and inequality constraints at inference via a prescribed-time zeroing function (PTZF). By encoding kinodynamic consistency as Lyapunov certificates and safety/feasibility as barrier certificates, and solving a minimal-impact QP-guided correction, the method yields training-free, certified trajectories without retraining. Two practical strategies—violation-segment extraction and trajectory compression—address long-horizon, high-dimensional planning and torque-limited manipulation, enabling real-time performance. Across a toy double inverted pendulum, real-to-sim car racing, and sim-to-real robotic manipulation tasks, UniConFlow outperforms state-of-the-art generative planners and conventional optimization baselines on safety, kinodynamic consistency, and action feasibility, while maintaining lower compute times and higher sample fidelity.

Abstract

Generative models have become increasingly powerful tools for robot motion generation, enabling flexible and multimodal trajectory generation across various tasks. Yet, most existing approaches remain limited in handling multiple types of constraints, such as collision avoidance, actuation limits, and dynamic consistency, which are typically addressed individually or heuristically. In this work, we propose UniConFlow, a unified constrained flow matching-based framework for trajectory generation that systematically incorporates both equality and inequality constraints. Moreover, UniConFlow introduces a novel prescribed-time zeroing function that shapes a time-varying guidance field during inference, allowing the generation process to adapt to varying system models and task requirements. Furthermore, to further address the computational challenges of long-horizon and high-dimensional trajectory generation, we propose two practical strategies for the terminal constraint enforcement and inference process: a violation-segment extraction protocol that precisely localizes and refines only the constraint-violating portions of trajectories, and a trajectory compression method that accelerates optimization in a reduced-dimensional space while preserving high-fidelity reconstruction after decoding. Empirical validation across three experiments, including a double inverted pendulum, a real-to-sim car racing task, and a sim-to-real manipulation task, demonstrates that UniConFlow outperforms state-of-the-art generative planners and conventional optimization baselines, achieving superior performance on certified motion planning metrics such as safety, kinodynamic consistency, and action feasibility. Project page is available at: https://uniconflow.github.io.

UniConFlow: A Unified Constrained Flow-Matching Framework for Certified Motion Planning

TL;DR

UniConFlow presents a unified, constraint-aware flow-matching framework for certified motion planning that enforces both equality and inequality constraints at inference via a prescribed-time zeroing function (PTZF). By encoding kinodynamic consistency as Lyapunov certificates and safety/feasibility as barrier certificates, and solving a minimal-impact QP-guided correction, the method yields training-free, certified trajectories without retraining. Two practical strategies—violation-segment extraction and trajectory compression—address long-horizon, high-dimensional planning and torque-limited manipulation, enabling real-time performance. Across a toy double inverted pendulum, real-to-sim car racing, and sim-to-real robotic manipulation tasks, UniConFlow outperforms state-of-the-art generative planners and conventional optimization baselines on safety, kinodynamic consistency, and action feasibility, while maintaining lower compute times and higher sample fidelity.

Abstract

Generative models have become increasingly powerful tools for robot motion generation, enabling flexible and multimodal trajectory generation across various tasks. Yet, most existing approaches remain limited in handling multiple types of constraints, such as collision avoidance, actuation limits, and dynamic consistency, which are typically addressed individually or heuristically. In this work, we propose UniConFlow, a unified constrained flow matching-based framework for trajectory generation that systematically incorporates both equality and inequality constraints. Moreover, UniConFlow introduces a novel prescribed-time zeroing function that shapes a time-varying guidance field during inference, allowing the generation process to adapt to varying system models and task requirements. Furthermore, to further address the computational challenges of long-horizon and high-dimensional trajectory generation, we propose two practical strategies for the terminal constraint enforcement and inference process: a violation-segment extraction protocol that precisely localizes and refines only the constraint-violating portions of trajectories, and a trajectory compression method that accelerates optimization in a reduced-dimensional space while preserving high-fidelity reconstruction after decoding. Empirical validation across three experiments, including a double inverted pendulum, a real-to-sim car racing task, and a sim-to-real manipulation task, demonstrates that UniConFlow outperforms state-of-the-art generative planners and conventional optimization baselines, achieving superior performance on certified motion planning metrics such as safety, kinodynamic consistency, and action feasibility. Project page is available at: https://uniconflow.github.io.

Paper Structure

This paper contains 67 sections, 7 theorems, 157 equations, 22 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Flow Matching Model
  4. Lyapunov Certification with Equality Constraint
  5. Barrier Certification with Inequality Constraints
  6. Problem Setting
  7. Flow Matching-based Constrained Generation
  8. Certified Robot Motion Planning
  9. Unified Constrained Generation Framework
  10. Equality Constraint
  11. Inequality Constraint
  12. UniConFlow with Quadratic Programming
  13. Comparison to Conventional Prescribed Time Approaches
  14. Certified Robot Motion Planning Strategy
  15. Certified Motion Planning as Constrained Generation
  16. ...and 52 more sections

Key Result

Lemma 1

Given $N_g$ equality constraints $g_i(\cdot)=0$ for $i = 1, \cdots, N_g$ and any initial value $\bm{\mathcal{T}}(0)$. Let the update rate for $\bm{\mathcal{T}}$ satisfy for all $i = 1, \cdots, N_g$, then all equality constraints are satisfied asymptotically, i.e., $\lim_{t \to \infty} g_i(\bm{\mathcal{T}}(t)) = 0$.

Figures (22)

  • Figure 1: Function $r(t)$ over time with $c_r = 0$ and $\gamma_r(t,r) = \underline{\gamma}_r = r$ in \ref{['definition_PTZF']}.
  • Figure 2: Function $g(t)$ over time with $c_r = 0$, $\gamma_r(t,r) = \underline{\gamma}_r = r$ in \ref{['definition_PTZF']} and $g(0) = 1$, $\bar{g}(0) = 2$ in \ref{['theorem_equality_constraint']} with PTZF (a) and with PT function (b). Green area denotes the free generation process.
  • Figure 3: Visualization of the motion of the double inverted pendulum. State sequences (top row) and action rollouts (bottom row) for the stabilization task across different methods: Diffuser-D, TVS, SafeFlow, PCFM, and UniConFlow.
  • Figure 4: Performance comparison across normalized metrics for the double inverted pendulum scenario. SR-S, SR-A, AR, and TSR represent percentage-based success metrics, while KC-F, KC-I, Cost, and Time are inverted min-max normalized metrics (higher is better).
  • Figure 5: Dataset creation pipeline for generative models in car racing scenarios. Geometric raw data including centerline, track boundaries, and raceline for the Nürburgring Nordschleife (left); extracted and rasterized track segment (middle); training data is generated through optimal control problems with random initial conditions for both forward and reverse driving (right).
  • ...and 17 more figures

Theorems & Definitions (19)

  • Definition 1: Class-$\mathcal{K}$ Function
  • Lemma 1: Lyapunov Certificate for Equality Constraint
  • Definition 2: Extended Class-$\mathcal{K}$ Function
  • Lemma 2: Barrier Certificate for Inequality Constraint
  • Definition 3: Certified Sample
  • Example 1
  • Definition 4: Certified Motion Trajectory
  • Definition 5
  • Definition 6
  • Example 2
  • ...and 9 more