Learning Smooth SE(3) Trajectories under Left-Invariant Riemannian Metrics

2026-08-02

AI Summary

This work addresses the challenge of generating smooth SE(3) trajectories that satisfy arbitrary boundary conditions while coupling rotation and translation, a task for which conventional numerical optimization methods are computationally prohibitive for real-time applications. The authors propose a learning-based framework that parameterizes body twist trajectories using high-order polynomials, where a neural network predicts a subset of coefficients and the trajectory duration, while the remaining coefficients are analytically determined from the boundary conditions. During training, the model incorporates Euler–Lagrange conditions, a metric-weighted smoothness objective, and feasibility constraints. This approach introduces, for the first time, metric-conditioned learning into SE(3) trajectory generation, supporting arbitrary left-invariant Riemannian metrics and boundary conditions. It achieves geometric consistency and millisecond-level inference speeds while closely approximating numerically optimized solutions, demonstrating success in real-time motion primitive generation and dynamic trajectory optimization for quadrotors.

Abstract

Optimal trajectory generation for rigid-body motions on Lie groups can be formulated as a variational problem that minimizes energy functionals defined by Riemannian metrics. While closed-form solutions exist for special cases such as product metrics and rest-to-rest boundary conditions, solving the general problem with arbitrary boundary states and coupled rotational-translational metrics often requires computationally expensive numerical boundary value solvers. These limitations restrict the use of geometrically consistent trajectory generation in real-time robotic planning and control. This paper presents a learning-based framework for approximating higher-order smooth trajectories on SE(3) under general left-invariant Riemannian metrics. The method parameterizes body-twist trajectories using high-order polynomials and relies on a neural network to learn a subset of the polynomial coefficients and the trajectory duration. The remaining coefficients are analytically determined to enforce the boundary conditions. The training of the network is guided by losses derived from Euler-Lagrange optimality conditions, metric-weighted smoothness objectives, and feasibility constraints. The metric-conditioned framework enables generalization across diverse metric structures and motion conditions. Extensive numerical experiments demonstrate that the proposed approach generates smooth trajectories that closely approximate solutions from numerical optimization while achieving millisecond-level inference times. We demonstrate two practical applications of the proposed framework: real-time generation of diverse motion primitives with waypoint traversal, and refinement for quadrotor flight under dynamic conditions. These results suggest that learning-based motions with geometric structure can provide an efficient alternative to conventional optimization-based methods for trajectory generation on SE(3).

Problem

• SE(3)
• trajectory generation
• Riemannian metrics
• smoothness
• real-time planning

Innovation

• SE(3) trajectory generation
• left-invariant Riemannian metrics
• learning-based motion planning
• geometrically consistent trajectories
• neural trajectory approximation