AI 中文总结
本文提出基于平坦性的几何控制（FGC），融合引导向量场与微分平坦性控制，消除时间参数化轨迹需求，实现路径跟踪的指数收敛，并扩展至无奇点情形。
AI 中文摘要
路径跟踪要求智能体在没有预设时间律的情况下收敛到并沿几何路径行进。引导向量场（GVFs）直接解决这一问题，但作为制导律，它们无法补偿智能体自身的动力学。另一方面，基于微分平坦性的控制（DFBC）补偿智能体的非线性动力学，但依赖于时间参数化的参考轨迹。本文提出一种统一算法，称为基于平坦性的几何控制（FGC），该算法直接从GVF及其高阶时间导数生成DFBC参考层级，消除了对预规划时间参数化轨迹的需求，同时保留了完全基于动力学的控制。级联李雅普诺夫论证建立了跟踪误差和物理轨迹到期望路径的指数收敛性。该框架进一步扩展到无奇点的GVFs，这消除了拓扑障碍，并保证了在闭合或自相交路径上全局指数收敛的条件。所提出的算法在仿真中得到了验证。
英文摘要
Path following requires an agent to converge to and traverse a geometric path without a prescribed timing law. Guiding vector fields (GVFs) address this directly, but as guidance laws they offer no mechanism to compensate for the agent's own dynamics. Differential flatness-based control (DFBC), on the other hand, compensates for the agent's nonlinear dynamics but relies on a time-parameterized reference trajectory. This letter presents a unified algorithm, which we term flatness-based geometric control (FGC), that generates the DFBC reference hierarchy directly from a GVF and its higher-order time derivatives, eliminating the need for a pre-planned time-parameterized trajectory while retaining full dynamics-informed control. A cascade Lyapunov argument establishes exponential convergence of the tracking error and the physical trajectory to the desired path. The framework is further extended to singularity-free GVFs, which removes the topological obstruction and guarantees the condition for global exponential convergence on closed or self-intersecting paths. The proposed algorithm is validated in simulation.
