Abstract
Generating collision-free and smooth motions remains a central challenge in robotic manipulation, particularly in cluttered environments and narrow passages where feasible regions are highly constrained and fragmented. We propose a trajectory optimization framework that performs geometry-aware updates directly in function space using natural functional gradients. The method optimizes a Gaussian-smoothed surrogate objective that regularizes the optimization landscape through smooth trajectory perturbations while preserving trajectory-level structure. Because the updates are defined intrinsically in function space, trajectory regularity can be controlled independently of a particular time discretization. We derive a practical Monte-Carlo estimator of the natural functional gradient that requires only black-box trajectory evaluations, making the method applicable when analytic gradients are unavailable or unreliable due to collision checking and contact-rich simulation. Experiments on constrained robotic manipulation tasks demonstrate that the proposed method improves trajectory feasibility and produces smoother motions than representative planning and trajectory optimization baselines in environments with narrow geometric clearances.
Method
We model trajectories as elements of a Hilbert space and perform updates directly in function space. This formulation enables trajectory regularity to be controlled through the geometry induced by Gaussian kernels and allows smooth perturbations to be generated independently of a particular trajectory discretization.
Synthetic Experiment
We evaluate the proposed method in a synthetic narrow-passage trajectory optimization problem. NFG achieves the highest success rate among the baselines, consistently discovering feasible trajectories despite the fragmented feasible region.
Simulation Results
In simulation robot evaluation, the robot successfully inserts a cylindrical object into a cabinet while avoiding collisions with surrounding obstacles and cabinet walls. The trajectory objective combines collision avoidance with joint-space motion regularization.
Trajectory Smoothness
The top and bottom rows show the joint positions (rad) and velocities (rad/s) plot of the Franka Research 3 manipulator respectively. The joint trajectories exhibit gradual configuration changes throughout the insertion motion without abrupt transitions near the constrained region.
Additional Results
Example: FR3 - Narrow Cabinet
Example: G1 - Tabletop
Example: Doosan - Shelf
Example: FR3 - Shelf
Qualitative examples on multiple robot platforms and manipulation environments. NFG consistently discovers feasible and smooth trajectories under varying kinematic structures and workspace constraints.
Acknowledgment
We thank RLWRLD for generously supporting our experiments by providing access to the Franka Robotics Research robot platform and the Inspire Hand system.
BibTeX
@inproceedings{park2026nfg,
author = {Kisang Park and Chanwoo Kim and Kyungjae Lee and Sungjoon Choi},
title = {Natural Functional Gradients for Smooth Trajectory Optimization},
booktitle = {Proceedings of Robotics: Science and Systems (RSS)},
year = {2026},
}