改进运动规划中的轨迹生成方法，提升稳定性与效率。
EL-AGHF: Extended Lagrangian Affine Geometric Heat Flow
- 引入对偶轨迹，用扩展拉格朗日法解决控制方向不可行问题。
- 通过求解状态与对偶轨迹的耦合偏微分方程，保证轨迹可执行性。
- 适用于各类运动规划问题，尤其适合高维非完整系统。
我们提出一种受约束的仿射几何热流（AGHF）方法，旨在抑制不可行控制方向带来的动态间隙。该方法为多种运动规划问题（包括完整与非完整系统）提供了统一框架，但需对不可行控制方向施加无穷惩罚以生成可行轨迹。这一设计虽理论成立，但在实际中常导致计算成本过高或数值不稳定。为此，我们基于增广拉格朗日法扩展了AGHF，引入与不可行控制方向动态间隙相关的对偶轨迹。该方法将约束变分问题转化为在状态与对偶轨迹上定义的扩展抛物型偏微分方程，从而确保生成轨迹的可执行性。仿真结果验证了算法的有效性。
原文摘要 · Abstract (English)
We propose a constrained Affine Geometric Heat Flow (AGHF) method that evolves so as to suppress the dynamics gaps associated with inadmissible control directions. AGHF provides a unified framework applicable to a wide range of motion planning problems, including both holonomic and non-holonomic systems. However, to generate admissible trajectories, it requires assigning infinite penalties to inadmissible control directions. This design choice, while theoretically valid, often leads to high computational cost or numerical instability when the penalty becomes excessively large. To overcome this limitation, we extend AGHF in an Augmented Lagrangian method approach by introducing a dual trajectory related to dynamics gaps in inadmissible control directions. This method solves the constrained variational problem as an extended parabolic partial differential equation defined over both the state and dual trajectorys, ensuring the admissibility of the resulting trajectory. We demonstrate the effectiveness of our algorithm through simulation examples.
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