LieSpline-DP
Lie-Group B-Spline Diffusion Policy for Smooth Robot Manipulation
1Shanghai Jiao Tong University
TL;DR: Generate SE(3) spline trajectories and reuse boundary control poses to keep robot motion smooth across predictions.
Abstract
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.
Highlights
LieSpline-DP achieves substantially higher success rates than DP baseline on real-world tasks: pouring, bucket hooking, and cube stacking . We observed that smooth actions produced by LieSpline-DP are especially beneficial when manipulating liquids and flexible objects. Cube stacking further demonstrates LieSpline-DP's capability for precise manipulation in the real world. The videos below show example executions of LieSpline-DP and the DP baseline on these tasks.
LieSpline-DP
Diffusion Policy (baseline)
To test reactiveness, we introduce human disturbances during stacking.
All videos play at their original speed.
Method
Asynchronous Execution and Spline Coupling
LieSpline-DP represents motion as a cubic B-spline on SE(3), shaped by a sequence of control poses. Inference runs asynchronously: before the current execution segment ends, the policy uses the latest observations to prepare the next plan while the robot keeps moving. To connect the plans, the policy fixes three boundary control poses inherited from the current plan and generates only the future controls. With a shared, uniformly spaced spline grid, the two plans join as a single C2-continuous trajectory, preserving reference pose, velocity, and acceleration across the handover.
The animation below uses a planar quadratic B-spline to provide an intuitive analogy for how our method couples consecutive plans. Hover over the curve to inspect the contributing control points and their basis weights.
The animation illustrates quadratic splines (C¹); the method uses cubic SE(3) splines with three shared boundary controls (C²).
Policy and model architecture
An observation encoder provides visual and robot-state features to a Transformer diffusion model. Block-causal self-attention lets future control tokens attend to the fixed boundary prefix, while cross-attention supplies the observation context. The model denoises local control-pose coordinates, which are mapped to SE(3) and decoded by a Lie-group B-spline into end-effector waypoints.
Spline dataset construction
First, we fit dense demonstrations with SE(3) splines, prioritizing key events such as grasp and release. Second, we convert the fitted trajectories into training windows of local control-pose coordinates: observations and boundary controls provide the context, and future controls are the prediction target.
Experiments
We evaluate LieSpline-DP on three real-robot tasks—pouring, bucket hooking, and cube stacking—using a UR5e with a Unitree Dex1 gripper. We compare task success and commanded-trajectory smoothness with Diffusion Policy baseline.
Task success
LieSpline-DP achieves higher success rates on all three tasks, including 100% success on pouring and bucket hooking.
| Method | Pouring ↑ | Bucket hooking ↑ | Cube stacking ↑ |
|---|---|---|---|
| DP | 10 | 30 | 55 |
| LieSpline-DP | 100 | 100 | 80 |
20 trials per method and task, without deliberate human perturbations.
Trajectory smoothness
LieSpline-DP reduces pooled p95 commanded jerk by more than 10× across all three tasks.
