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

cs.ROJun 3, 2025

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

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

Organizations: Chair of Robotics and Systems Intelligence (RSI), Munich Institute of Robotics and Machine Intelligence (MIRMI), Technical University of Munich (TUM), 80992 Munich, Germany · Chair of Information-oriented Control (ITR), School of Computation, Information and Technology (CIT), Technical University of Munich (TUM), 80333 Munich, Germany · Chair of AI Planning in Dynamic Environments, Munich Institute of Robotics and Machine Intelligence (MIRMI), Technical University of Munich (TUM), 80992 Munich, Germany · School of Mechanical Engineering, Translational Research Center, Tongji University, Shanghai 201804, China, affiliated with Shanghai Yangzhi Rehabilitation Hospital and also with Shanghai Key Laboratory of Wearable Robotics and Human-Machine Interaction ·

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.

Figures & tables

Method	Obstacle Avoidance	Safety Guarantee	Dynamics Consistency	Admissible Action	Long-Horizon Planning	Dataset Creation	Real-world Experiments
Diffuser [ 7 ]	✗	✗	✗	✗	✗	✓	✗
Decision Diffuser [ 24 ]	✓	✗	✓	✗	✗	✓	✗
Diffusion-based Policy [ 8 , 12 , 13 ]	✗	✗	⚫	✗	✗	✓	✓
ChainedDiffuser [ 11 ]	✗	✗	⚫	⚫	✓	✓	✓
DPCA [ 34 ]	✓	✗	✗	✗	✗	✗	✗
DPCC [ 35 ]	✓	✗	✓	✓	✗	✓	✗

TABLE I : Comparison of representative diffusion- and flow-based motion planners across key properties. Long-Horizon Planning: the method generates a complete task-level trajectory rather than a single short-horizon action chunk. Dataset Creation: the method provides an explicit data generation pipeline or a publicly released dataset. Real-World Experiments: the method is validated on a physical robotic platform. ✓ denotes that the property is explicitly addressed in the respective work, ✗ that it is not, and ⚫ that it is not applicable.

Fig. 1 : Function $r(t)$ over time with $c_{r}=0$ and $\gamma_{r}(t,r)=\underline{\gamma}_{r}=r$ in Definition 5 .

Fig. 2 : Function $g(t)$ over time with $c_{r}=0$ , $\gamma_{r}(t,r)=\underline{\gamma}_{r}=r$ in Definition 5 and $g(0)=1$ , $\bar{g}(0)=2$ in Theorem 1 with PTZF (a) and with PT function (b). Green area denotes the free generation process.

Fig. 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.

Fig. 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).

Fig. 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).

Fig. 6 : Terminal refinement process for the car racing scenario. The window-based segmentation strategy preserves the global trajectory shape (left). Refine the safe segment where kinodynamic consistency may be violated, due to numerical errors (middle). The final refined trajectory achieves safety, admissibility, and kinodynamic consistency (right).

Fig. 7 : Performance heatmap for car racing scenarios in forward and reverse driving cases. All values are normalized to $[0,1]$ with inversion applied to KC, Cost, and Time such that higher scores uniformly represent better performance.

Fig. 8 : Visualization of state and action rollout trajectories generated by UniConFlow versus optimization-based methods.

Fig. 9 : The experimental setup for the manipulator. The obstacle is modeled as an ellipsoid for collision avoidance purposes, with its geometric center defined by the attached marker. Marker positions are measured via an NDI Polaris Vega XT optical tracker visualized in Cigna-6D, while the robot state is visualized in RViz.

Fig. 10 : The datasets (left) are obtained either by expert demonstration with Franka arm or optimal control in a simulation environment. In the sampling process (middle), the rectangle filled with diagonal lines signifies the obstacles. The green path represents the collision-free trajectory navigating between red obstacles within the bounded workspace after PTZF-based guided sampling, while the blue path before PTZF-based guided sampling. The terminal refinement (right) is the same process in the car racing case in Fig. 6 .

Fig. 11 : Success rate in 3D spatial motion scenario.

Fig. 12 : Values of the RMSE of kinodynamic consistency, cost, and inference time (ms) in 3D spatial motion scenario.

Pipeline Component	Pendulum		Raceline		Pseudo-2D Manip.		3D Manip.
	Time(ms)	(%)	Time(ms)	(%)	Time(ms)	(%)	Time(ms)	(%)
Generative Model ODE	1.95	78.0	2.00	76.9	22.50	71.0	81.50	77.6
PTZF Constraints & QP	0.35	14.0	0.22	8.5	3.20	10.1	15.50	14.8
Terminal Refinement	0.20	8.0	0.38	14.6	6.00	18.9	8.00	7.6
Total Inference	2.50	100	2.60	100	31.70	100	105.00	100

TABLE II : Computational Cost Ablation of Inference Components

Task	Stage 1 (PTZF Output)		Stage 2 (Terminal Refinement)
	Certified Rate	State RMSE	State Corr. $\ell_{2}$ Dist.	State Fréchet Distance	Action Corr. $\ell_{2}$ Dist.		Action Fréchet Distance
Pendulum	$88.4\%$	$0.03$	$0.07$	$0.38$	$0.31$		$1.13$
Raceline	$98.1\%$	$0.07$	$0.06$	$0.52$	$0.31$	$0.003$	$0.59$	$0.009$
Pseudo-2D Manip.	$89.2\%$	$29.84$	$7.12$	$13.49$	$4.18$		$9.21$
3D Manipulator	$86.1\%$	$17.18$	$12.01$	$15.79$	$6.73$		$11.17$

TABLE III : Inference Components Impact Ablation of Terminal Enforcement on Generative Output

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Fig. 13 : Visualization of the motion of the tip of the inverted pendulum over time.

Method	SR-S (%) ( $\uparrow$ )	SR-A (%) ( $\uparrow$ )	AR (%) ( $\uparrow$ )	TSR (%) ( $\uparrow$ )	KC-F ( $\downarrow$ )	KC-I ( $\downarrow$ )	Cost ( $\downarrow$ )	Time (ms) ( $\downarrow$ )
Diffuser	86.10	86.40	95.53	83.66	0.007 $\pm$ 0.000	1.18 $\pm$ 0.50	535.30	1.9 $\pm$ 0.001
Diffuser-D	86.10	74.60	76.27	69.06	0.000 $\pm$ 0.000	0.0000 $\pm$ 0.0000	601.52 $\pm$ 202.28	2.9 $\pm$ 0.001
Trunc	86.10	86.10	95.92	83.66	0.007 $\pm$ 0.000	1.18 $\pm$ 0.51	535.29	2.2 $\pm$ 0.0001
CG	87.35	86.70	91.85	83.92	0.087 $\pm$ 0.004	1.48 $\pm$ 0.69	536.40	133.3 $\pm$ 0.001
RoS	87.62	86.30	94.28	84.19	0.033 $\pm$ 0.008	1.68 $\pm$ 0.51	533.67	139.9 $\pm$ 0.001
ReS	87.62	85.80	98.49	85.19	0.033 $\pm$ 0.008	1.20 $\pm$ 0.54	535.68	137.2 $\pm$ 0.001

Appendix

TABLE IV : Results comparison on different metrics against baselines in the double inverted pendulum scenario.

Fig. 14 : Visualization of the motion of the car’s states over time in the forward driving scenario.

Fig. 15 : Visualization of the motion of the car’s states over time in the reverse driving scenario.

Method	SR-S (%)	SR-A (%) $\uparrow$	AR (%) $\uparrow$	TSR (%) $\uparrow$	KC-F $\downarrow$	KC-I $\downarrow$	Cost $\downarrow$	Time (ms) $\downarrow$
Diffuser	0.00	0.00	100.00	0.00	40.48 $\pm$ 16.23	5.30 $\pm$ 1.09	99.62 $\pm$ 30.52	2.10 $\pm$ 0.0001
Diffuser-D	0.00	0.00	93.72	0.00	0.00 $\pm$ 0.00	0.00 $\pm$ 0.00	423.86 $\pm$ 51.21	2.30 $\pm$ 0.0001
Trunc	0.00	0.00	100.00	0.00	41.16 $\pm$ 17.01	4.96 $\pm$ 2.79	103.27 $\pm$ 28.08	2.10 $\pm$ 0.0001
CG	0.00	0.00	100.00	0.00	53.90 $\pm$ 9.14	4.45 $\pm$ 1.57	123.67 $\pm$ 34.57	286.40 $\pm$ 0.0001
RoS	7.27	0.00	100.00	0.00	54.70 $\pm$ 14.22	4.27 $\pm$ 1.59	130.71 $\pm$ 41.50	312.10 $\pm$ 0.0001
ReS	14.55	0.00	100.00	0.00	52.63 $\pm$ 14.25	4.73 $\pm$ 1.74	113.59 $\pm$ 42.13	23.90 $\pm$ 0.0001

Appendix

TABLE V : Results comparison on different metrics against baselines for the car racing scenario (forward driving).

Method	SR-S (%)	SR-A (%) $\uparrow$	AR (%) $\uparrow$	TSR (%) $\uparrow$	KC-F $\downarrow$	KC-I $\downarrow$	Cost $\downarrow$	Time (ms) $\downarrow$
Diffuser	0.00	0.00	100.00	0.00	26.29 $\pm$ 12.323	4.53 $\pm$ 2.84	126.59 $\pm$ 27.34	2.90 $\pm$ 0.00
Diffuser-D	0.00	0.00	91.22	0.00	0.00 $\pm$ 0.00	0.00 $\pm$ 0.00	348.22 $\pm$ 79.85	3.00 $\pm$ 0.000
Trunc	0.91	0.00	100.00	0.00	24.74 $\pm$ 17.17	4.55 $\pm$ 2.76	129.77 $\pm$ 33.11	2.90 $\pm$ 0.00
CG	0.00	0.00	100.00	0.00	23.95 $\pm$ 6.29	4.05 $\pm$ 1.57	132.74 $\pm$ 28.57	178.40 $\pm$ 0.00
RoS	0.00	0.00	100.00	0.00	24.50 $\pm$ 17.73	4.03 $\pm$ 1.59	137.03 $\pm$ 27.88	177.30 $\pm$ 0.00
ReS	0.91	0.00	100.00	0.00	26.58 $\pm$ 11.34	3.97 $\pm$ 1.74	133.89 $\pm$ 28.39	18.90 $\pm$ 0.00

Appendix

TABLE VI : Results comparison on different metrics against baselines for the car racing scenario (reverse driving).

Fig. 16 : Visualization of the spatial motion of the manipulator’s end-effector states over time in the pseudo-2D scenario.

Fig. 17 : Visualization of the spatial motion of the manipulator’s end-effector states over time in the 3D scenario.

Method	SR-S (%)	SR-A (%) $\uparrow$	AR (%) $\uparrow$	TSR (%) $\uparrow$	KC-F $\downarrow$	KC-I $\downarrow$	Cost $\downarrow$	Time (ms) $\downarrow$
Diffuser	86.29	0.00	100.00	0.00	2057.94 $\pm$ 59.74	37.09 $\pm$ 2.91	$\approx 1.66\times 10^{11}$	27.00 $\pm$ 0.02
Diffuser-D	86.36	0.00	100.00	0.00	2057.89 $\pm$ 51.97	37.08 $\pm$ 7.24	$\approx 1.66\times 10^{11}$	28.00 $\pm$ 0.08
Trunc	100.00	0.00	100.00	0.00	1228.73 $\pm$ 36.21	35.36 $\pm$ 8.17	$\approx 1.25\times 10^{10}$	59.91 $\pm$ 1.76
CG	95.84	0.00	100.00	0.00	2059.33 $\pm$ 32.49	37.08 $\pm$ 26.42	$\approx 3.62\times 10^{10}$	4439.57 $\pm$ 0.02
RoS	94.51	0.00	100.00	0.00	2059.94 $\pm$ 36.84	37.08 $\pm$ 2.55	$\approx 3.62\times 10^{10}$	4435.32 $\pm$ 0.04
ReS	97.78	0.00	100.00	0.00	2059.05 $\pm$ 29.66	37.08 $\pm$ 4.75	$\approx 3.62\times 10^{10}$	4462.81 $\pm$ 0.03

Appendix

TABLE VII : Results comparison on different metrics against baselines for the manipulator pseudo-2D case.

Method	SR-S (%)	SR-A (%) $\uparrow$	AR (%) $\uparrow$	TSR (%) $\uparrow$	KC-F $\downarrow$	KC-I $\downarrow$	Cost $\downarrow$	Time (ms) $\downarrow$
Diffuser	0.00	0.00	100.00	0.00	692.44 $\pm$ 2.43	0.06 $\pm$ 0.05	$\!\approx\!1.17\!\times\!10^{7}\pm$ 4.40 $\!\times\!10^{5}$	103.3 $\pm$ 0.7
Diffuser-D	0.00	0.00	100.00	0.00	0.00 $\pm$ 0.00	0.00 $\pm$ 0.00	$\!\approx\!1.18\!\times\!10^{6}\pm$ 4.67 $\!\times\!10^{5}$	103.4 $\pm$ 0.9
Trunc	100.0	7.02	100.00	7.02	55.93 $\pm$ 28.76	0.08 $\pm$ 0.10	$\!\approx\!2.37\!\times\!10^{7}\pm$ 7.17 $\!\times\!10^{5}$	24.5 $\pm$ 7.4
CG	12.30	0.00	100.00	0.00	692.37 $\pm$ 2.64	0.06 $\pm$ 0.05	$\!\approx\!1.19\!\times\!10^{7}\pm$ 5.12 $\!\times\!10^{5}$	102.9 $\pm$ 0.5
RoS	7.02	0.00	100.00	0.00	677.34 $\pm$ 2.43	0.06 $\pm$ 0.03	$\!\approx\!1.17\!\times\!10^{7}\pm$ 4.15 $\!\times\!10^{5}$	107.1 $\pm$ 0.2
ReS	3.95	0.00	100.00	0.00	689.28 $\pm$ 2.54	0.05 $\pm$ 0.04	$\!\approx\!1.16\!\times\!10^{7}\pm$ 5.03 $\!\times\!10^{5}$	107.6 $\pm$ 0.1

Appendix

TABLE VIII : Results comparison on different metrics against baselines for the manipulator 3D case.

Fig. 18 : Percentage breakdown of the computational cost for the UniConFlow inference pipeline.

Experiment	$\bar{D}_{\mathrm{raw}}$	$D_{\mathrm{raw}}^{\max}$	$t_{\max}$
Raceline	$1.0049$	$1.7129$	$0.9411$
Pendulum	$1.7209\times 10^{-2}$	$3.7317\times 10^{-2}$	$0.8813$
2D manipulator	$1.1258\times 10^{-1}$	$5.7431\times 10^{-1}$	$0.7913$
3D manipulator	$1.6400\times 10^{-1}$	$2.7364\times 10^{-1}$	$0.5597$

Appendix

TABLE IX : Slack magnitude analysis across tasks.

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ConFlow: Constraints-Guided Learning with Flow Matching for Motion Generation

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Constrained Generative Modeling Flow Matching Trajectory Generation Robot Motion Generation Flow-Based Generative Modeling

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In recent years Flow Matching has become a prominent method for generative modeling robot motion generation. In its generic form Flow Matching is an ODE-based neural sampler that is trained by regressing empirical flow fields associated with motion samples as data. However, in robot motion generation we often have additional constraints that might not be present in the collected data. The majority of current approaches train the flow on the available data and use inference-time guidance to enforce task-specific constraints. To address this mismatch, we propose ConFlow, a constraint-guided flow matching framework that incorporates constraint information directly into the training objective via differentiable barrier or cost functions. To address design specifications such as smoothness and boundary conditions, we propose replacing the standard Gaussian source distribution used in flow matching training with a conditional Gaussian Process. Our approach also uses infeasible demonstrations as negative supervision, improving constraint satisfaction without requiring additional expert data. Experiments on a two-robot navigation task demonstrate that ConFlow achieves lower collision rates and higher trajectory quality than standard flow matching baselines, with or without inference-time guidance. These results validate training-time constraint integration as an effective approach to closing the training--inference gap in generative motion models.

Abstract

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