# ProjFlow: Projection Sampling with Flow Matching for Zero-Shot Exact Spatial Motion Control

arXiv: https://arxiv.org/abs/2602.22742 · HTML: https://mlsanity.com/paper/2602.22742/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 56 of its 66 references that are in the map.
- First posted on arXiv 26 Feb 2026.

## Placed in

- Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/)

## Builds on (most cited first, 30 of 56)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Adding Conditional Control to Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.05543/) (2023)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [AMASS: Archive of Motion Capture as Surface Shapes](https://mlsanity.com/paper/1904.03278/) (2019)
- [Human Motion Diffusion Model](https://mlsanity.com/paper/2209.14916/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [MotionDiffuse: Text-Driven Human Motion Generation with Diffusion Model](https://mlsanity.com/paper/2208.15001/) (2022)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [AI Choreographer: Music Conditioned 3D Dance Generation with AIST++](https://mlsanity.com/paper/2101.08779/) (2021)
- [Action2Motion: Conditioned Generation of 3D Human Motions](https://mlsanity.com/paper/2007.15240/) (2020)
- [Human Motion Diffusion as a Generative Prior](https://mlsanity.com/paper/2303.01418/) (2023)
- [EDGE: Editable Dance Generation From Music](https://mlsanity.com/paper/2211.10658/) (2022)
- [ReMoDiffuse: Retrieval-Augmented Motion Diffusion Model](https://mlsanity.com/paper/2304.01116/) (2023)
- [Diffusion-based Generation, Optimization, and Planning in 3D Scenes](https://mlsanity.com/paper/2301.06015/) (2023)
- [OmniControl: Control Any Joint at Any Time for Human Motion Generation](https://mlsanity.com/paper/2310.08580/) (2023)
- [Guided Motion Diffusion for Controllable Human Motion Synthesis](https://mlsanity.com/paper/2305.12577/) (2023)
- [MoFusion: A Framework for Denoising-Diffusion-based Motion Synthesis](https://mlsanity.com/paper/2212.04495/) (2022)
- [Bailando: 3D Dance Generation by Actor-Critic GPT with Choreographic Memory](https://mlsanity.com/paper/2203.13055/) (2022)
- [InterGen: Diffusion-based Multi-human Motion Generation under Complex Interactions](https://mlsanity.com/paper/2304.05684/) (2023)
- [Solving Inverse Problems with Latent Diffusion Models via Hard Data Consistency](https://mlsanity.com/paper/2307.08123/) (2023)
- [MotionLCM: Real-time Controllable Motion Generation via Latent Consistency Model](https://mlsanity.com/paper/2404.19759/) (2024)
- [Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing](https://mlsanity.com/paper/2407.01521/) (2024)
- [Solving Linear Inverse Problems Provably via Posterior Sampling with Latent Diffusion Models](https://mlsanity.com/paper/2307.00619/) (2023)
- [Trace and Pace: Controllable Pedestrian Animation via Guided Trajectory Diffusion](https://mlsanity.com/paper/2304.01893/) (2023)
- [CG-HOI: Contact-Guided 3D Human-Object Interaction Generation](https://mlsanity.com/paper/2311.16097/) (2023)

## Built on by (most cited first, 5 of 5)

- [Feed-forward Motion In-betweening for Any 4D](https://mlsanity.com/paper/2606.22131/) (2026)
- [PathGuide: Dynamic Classifier-Free Guidance via On-Policy Transport Alignment](https://mlsanity.com/paper/2608.29107/) (2026)
- [InterCMDM: Block-Causal Diffusion for Autoregressive Human Interaction Generation](https://mlsanity.com/paper/2607.01743/) (2026)
- [MotionSpaceFlow: Representation-Aware Flow Matching in Direct Motion Space](https://mlsanity.com/paper/2609.34190/) (2026)
- [A Benchmark for Spatially Grounded Gesture Generation](https://mlsanity.com/paper/2610.03105/) (2026)

## Abstract (arXiv)

Generating human motion with precise spatial control is a challenging problem. Existing approaches often require task-specific training or slow optimization, and enforcing hard constraints frequently disrupts motion naturalness. Building on the observation that many animation tasks can be formulated as a linear inverse problem, we introduce ProjFlow, a training-free sampler that achieves zero-shot, exact satisfaction of linear spatial constraints while preserving motion realism. Our key advance is a novel kinematics-aware metric that encodes skeletal topology. This metric allows the sampler to enforce hard constraints by distributing corrections coherently across the entire skeleton, avoiding the unnatural artifacts of naive projection. Furthermore, for sparse inputs, such as filling in long gaps between a few keyframes, we introduce a time-varying formulation using pseudo-observations that fade during sampling. Extensive experiments on representative applications, motion inpainting, and 2D-to-3D lifting, demonstrate that ProjFlow achieves exact constraint satisfaction and matches or improves realism over zero-shot baselines, while remaining competitive with training-based controllers.