# LieSpline-DP: Lie-Group B-Spline Diffusion Policy for Smooth Robot Manipulation

arXiv: https://arxiv.org/abs/2609.15162 · HTML: https://mlsanity.com/paper/2609.15162/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 27 of its 32 references that are in the map.
- First posted on arXiv 14 Sep 2026.

## Placed in

- Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/)
- Offline RL and imitation learning from data / [Robot manipulation imitation learning](https://mlsanity.com/offlinerl/robot-manipulation-imitation-learning/)

## Builds on (most cited first, 27 of 27)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [OpenVLA: An Open-Source Vision-Language-Action Model](https://mlsanity.com/paper/2406.09246/) (2024)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [π0: A Vision-Language-Action Flow Model for General Robot Control](https://mlsanity.com/paper/2410.24164/) (2024)
- [Learning Fine-Grained Bimanual Manipulation with Low-Cost Hardware](https://mlsanity.com/paper/2304.13705/) (2023)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [What Matters in Learning from Offline Human Demonstrations for Robot Manipulation](https://mlsanity.com/paper/2108.03298/) (2021)
- [Real-Time Execution of Action Chunking Flow Policies](https://mlsanity.com/paper/2506.07339/) (2025)
- [SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion](https://mlsanity.com/paper/2209.03855/) (2022)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [EquiBot: SIM(3)-Equivariant Diffusion Policy for Generalizable and Data Efficient Learning](https://mlsanity.com/paper/2407.01479/) (2024)
- [Training-Time Action Conditioning for Efficient Real-Time Chunking](https://mlsanity.com/paper/2512.05964/) (2025)
- [Riemannian Flow Matching Policy for Robot Motion Learning](https://mlsanity.com/paper/2403.10672/) (2024)
- [Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulation](https://mlsanity.com/paper/2309.02685/) (2023)
- [Motion Planning Diffusion: Learning and Adapting Robot Motion Planning with Diffusion Models](https://mlsanity.com/paper/2412.19948/) (2024)
- [Efficient Derivative Computation for Cumulative B-Splines on Lie Groups](https://mlsanity.com/paper/1911.08860/) (2019)
- [BEAST: Efficient Tokenization of B-Splines Encoded Action Sequences for Imitation Learning](https://mlsanity.com/paper/2506.06072/) (2025)
- [ET-SEED: Efficient Trajectory-Level SE(3) Equivariant Diffusion Policy](https://mlsanity.com/paper/2411.03990/) (2024)
- [Learning Native Continuation for Action Chunking Flow Policies](https://mlsanity.com/paper/2602.12978/) (2026)
- [ABPolicy: Asynchronous B-Spline Flow Policy for Real-Time and Smooth Robotic Manipulation](https://mlsanity.com/paper/2602.23901/) (2026)
- [SE(3)-Equivariant Diffusion Policy in Spherical Fourier Space](https://mlsanity.com/paper/2507.01723/) (2025)
- [Spline Policy: A Structured Representation for Robot Policies](https://mlsanity.com/paper/2606.07386/) (2026)
- [B-spline Policy: Accelerating Manipulation Policies via B-spline Action Representations](https://mlsanity.com/paper/2607.09648/) (2026)

## Abstract (arXiv)

Diffusion Policy (DP) is a powerful Learning from Demonstration (LfD) method for robotic manipulation, yet it suffers from discontinuous and non-smooth trajectories. Spline-based action representations promote smooth motion within individual action chunks, but existing spline-based methods neither guarantee cross-chunk C2 continuity nor account for the group structure of SE(3). We therefore propose LieSpline-DP, a Lie-group B-spline diffusion policy that generates end-effector trajectories directly on SE(3) and couples consecutive plans by sharing their boundary control poses, ensuring C2 continuity throughout the entire planned trajectory. Across three real-robot tasks, LieSpline-DP produces lower trajectory jerk and higher task success rates than the DP baseline. The gains are particularly pronounced in real-world tasks involving liquids and flexible objects: in our real-robot experiments, LieSpline-DP achieved a 100% success rate on both pouring and bucket hooking, whereas the DP baseline achieved only 10% and 30%, respectively.