Hermite Curves as Trajectory Priors for Vision-Language-Action Models

Qi Lv1,2,3,Jianming Xing1,Zhao Yang2,Mingyuan Yao2,Yinan Shi2,Yawei Jueluo2,Mike Zheng Shou3*,Xiang Deng1*
1Harbin Institute of Technology (Shenzhen)2Jiangsu Cytoderm Intelligent Technology Co., Ltd.3National University of Singapore*Corresponding authors
PaperArxivGithub

Real-Robot Success and Failure

Representative Task 3 executions on the dual-arm platform. In the baseline rollout, a grasp correction flings an object and prevents task completion; Hermite-VLA Reg maintains stable contact and completes the sequence.

Multi-object placement

Real robot Task 3 / dual arm / multiple objects

Representative physical trials from the same task
pi0.5Failure
A sharp grasp correction flings an object, leaving the sequence incomplete.
Hermite-VLA RegSuccess
Maintains grasp alignment, places every object, and closes the lid.
Readout
Task-level success: 46.7% -> 93.3%
Protocol
Source recordings at 1.0x playback (19.9 s vs 18.8 s)

Abstract

Existing Vision-Language-Action models typically flatten each action chunk into per-timestep controls, leaving smoothness and boundary continuity to be learned implicitly. We instead parameterize the chunk trajectory as a piecewise cubic Hermite curve defined by endpoint positions and velocities.

We instantiate this fixed operator as Hermite Tokens, a Hermite Scaffold, and Hermite Regularization. The training-only regularization variant performs best, improving pi0.5 from 95.9% to 98.7% on LIBERO, 85.7% to 90.9% on LIBERO-plus, and 63.4% to 90.0% across four real-robot tasks without additional inference cost.

Comparison of per-step discretization, sequence tokenization, continuous generative modeling, and the proposed Hermite trajectory prior.
Action-chunk representations. The Hermite prior exposes physically meaningful boundary states instead of treating the trajectory as unrelated samples or generic transform coefficients.

Method

A single fixed Hermite operator supports three integrations into discrete and continuous VLA action modeling.

Architecture of Hermite Tokens, Hermite Scaffold, and Hermite Regularization on top of shared vision-language-action features.
Overview of the three variants. The vision-language backbone is shared; the placement and runtime role of the Hermite operator differ.

What is H? H is a fixed, non-learned interpolation matrix that maps boundary positions and velocities to a dense trajectory. It is not another backbone layer.

Discrete autoregressive

Hermite-VLA DH

Predicts quantized endpoint positions and velocities instead of one token per action timestep.

Training
Boundary variables are supervised as compact token groups.
Inference
Decode the boundary tokens, then apply H once to reconstruct the dense action chunk.

Continuous generative

Hermite-VLA CH

Decomposes each clean-action estimate into a Hermite scaffold and a per-timestep residual.

Training
The clean action is modeled as a0 = H theta + r.
Inference
The scaffold H theta is reconstructed inside every flow-matching solver step.

Training-only regularization

Hermite-VLA Reg

Uses a separate Hermite branch to regularize the shared action features during training.

Training
Optimize the original flow loss together with a trajectory-space auxiliary loss.
Inference
Remove the auxiliary head and H; use the unchanged pi0.5 action head and sampler.

Key finding: the least invasive design performs best. Hermite-VLA Reg shapes learning through an auxiliary trajectory objective while leaving deployment unchanged.

Hermite Curve Explorer

A piecewise cubic curve is determined by endpoint positions and tangents. The resulting speed profile changes immediately as its boundary state changes.

Boundary-state curve
Position profileInteractive piecewise cubic Hermite curveTwo cubic segments controlled by position anchors and tangent handles.interior knot
Speed magnitudeSpeed magnitude of the interactive Hermite curveThe speed profile implied by the position and tangent anchors.chunk time
position tangent

K = 2. Piecewise cubic Hermite trajectory.

Results

Hermite Regularization is strongest across in-distribution simulation, distribution shift, and physical execution while adding no inference-time module.

Evaluationpi0.5Hermite-VLA CHHermite-VLA RegReg gain
LIBERO95.9%97.7%98.7%+2.8 pp
LIBERO-plus85.7%85.1%90.9%+5.2 pp
Real robot63.4%81.7%90.0%+26.6 pp

Success rates on 2,000 LIBERO rollouts, 10,030 LIBERO-plus perturbations, and 60 real-robot trials. Best result in each row is highlighted.

Real-robot tasks

TaskPlatformInstructionpi0.5Hermite-VLA Reg
01Franka / single armOpen the pot, place the wooden block, and close the pot.86.7%100.0%
02Cybopal / single armOpen the pot, place the wooden block, and close the pot.26.7%66.7%
03Cybopal / dual armOpen the lid, place all blocks and balls, then close the pot.46.7%93.3%
04ARX / dual armPick up the towel, fold it neatly, and place it to the side.93.3%100.0%

Trajectory Analysis

Task success alone does not expose how a policy moves. We therefore examine spatial paths and frequency-domain motion directly on physical hardware.

Three-dimensional end-effector paths for pi0.5 and Hermite-VLA Reg on the same real-robot scene, colored by instantaneous jerk.
Executed paths on real-robot Task 3. Thin lines show all successful rollouts; the bold representative path is colored by instantaneous jerk. Hermite-VLA Reg follows a cleaner, lower-jerk path toward the same targets.
Real-robot acceleration power spectral density and cumulative jerk RMS distributions for the three evaluated policies.
Hardware smoothness. Hermite regularization attenuates high-frequency acceleration energy and shifts the episode-level jerk distribution lower.

Citation

Please cite the manuscript when using the method or results.

BibTeX
@misc{lv2026hermitecurvestrajectorypriors,
      title={Hermite Curves as Trajectory Priors for Vision-Language-Action Models},
      author={Qi Lv and Jianming Xing and Zhao Yang and Mingyuan Yao and Yinan Shi and Yawei Jueluo and Mike Zheng Shou and Xiang Deng},
      year={2026},
      eprint={2608.01265},
      archivePrefix={arXiv},
      primaryClass={cs.RO},
      url={https://arxiv.org/abs/2608.01265},
}