发表机构
University of Pennsylvania（宾夕法尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对SE(3)轨迹生成的计算成本问题，提出基于学习的框架，用神经网络参数化多项式系数与时长，生成的轨迹近似数值优化解且推理达毫秒级，可用于机器人运动原语及四旋翼飞行优化。
AI 中文摘要
李群上刚体运动的最优轨迹生成可表述为变分问题，最小化由黎曼度量定义的能量泛函。尽管乘积度量、静止到静止边界条件等特殊情况存在闭式解，但求解具有任意边界状态、耦合旋转-平移度量的一般问题，通常需要计算成本高昂的数值边界值求解器，这些限制了几何一致的轨迹生成在实时机器人规划与控制中的应用。本文提出一种基于学习的框架，用于近似一般左不变黎曼度量下SE(3)上的高阶光滑轨迹。该方法用高阶多项式参数化体扭转轨迹，依赖神经网络学习部分多项式系数与轨迹时长，其余系数通过解析确定以满足边界条件。网络训练由欧拉-拉格朗日最优条件、度量加权平滑目标及可行性约束导出的损失引导。该度量条件框架可在不同度量结构与运动条件间泛化。大量数值实验表明，所提方法生成的光滑轨迹与数值优化解高度近似，且实现了毫秒级推理时间。我们展示了该框架的两个实际应用：带航路点遍历的多样运动原语实时生成，以及动态条件下四旋翼飞行的轨迹优化。这些结果表明，具有几何结构的学习型运动可成为SE(3)轨迹生成中传统基于优化方法的高效替代方案。
英文摘要
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).
