在李群上构建最优概率传输控制器，避免局部坐标缺陷。
Schrödinger Bridge Over A Compact Connected Lie Group
- 基于李群几何结构构造无坐标依赖的控制框架
- 证明了薛定谔桥系统解的存在唯一性，可直接导出最优控制律
- 适用于旋转群SO(2)/SO(3)的轨迹规划，适合机器人/视觉领域
本文研究在紧连通李群上的运动学方程的薛定谔桥问题，目标是在给定初始与终端概率密度分布的前提下，通过最小化控制代价实现受控扩散过程的转移。我们提出了一种不依赖坐标系的几何化公式，保持了李群的内在结构特性，克服了传统局部参数化或嵌入欧氏空间的局限性。建立了对应薛定谔系统的解的存在性与唯一性。结果具有构造性：可推导出在李群上最优插值概率密度的几何控制器。通过SO(2)和SO(3)上的数值实验验证方法有效性，代码与动画已公开于https://github.com/gradslab/SbpLieGroups.git。
原文摘要 · Abstract (English)
This work studies the Schrödinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while minimizing the control effort. We develop a coordinate-free formulation of this stochastic optimal control problem that respects the underlying geometric structure of the Lie group, thereby avoiding limitations associated with local parameterizations or embeddings in Euclidean spaces. We establish the existence and uniqueness of solution to the corresponding Schrödinger system. Our results are constructive in that they derive a geometric controller that optimally interpolates probability densities supported over the Lie group. To illustrate the results, we provide numerical examples on $\mathsf{SO}(2)$ and $\mathsf{SO}(3)$. The codes and animations are publicly available at https://github.com/gradslab/SbpLieGroups.git .
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