In robot navigation tasks, such as unmanned aerial vehicle (UAV) highway traffic monitoring, it is important for a mobile robot to follow a specified desired path. However, most of the existing path-following navigation algorithms cannot guarantee global convergence to desired paths or enable following self-intersected desired paths due to the existence of singular points where navigation algorithms return unreliable or even no solutions. One typical example arises in vector-field guided path-following (VF-PF) navigation algorithms. These algorithms are based on a vector field, and the singular points are exactly where the vector field diminishes. Conventional VF-PF algorithms generate a vector field of the same dimensions as those of the space where the desired path lives. In this article, we show that it is mathematically impossible for conventional VF-PF algorithms to achieve global convergence to desired paths that are self-intersected or even just simple closed (precisely, homeomorphic to the unit circle). Motivated by this new impossibility result, we propose a novel method to transform self-intersected or simple closed desired paths to nonself-intersected and unbounded (precisely, homeomorphic to the real line) counterparts in a higher dimensional space. Corresponding to this new desired path, we construct a singularity-free guiding vector field on a higher dimensional space. The integral curves of this new guiding vector field is thus exploited to enable global convergence to the higher dimensional desired path, and therefore the projection of the integral curves on a lower dimensional subspace converge to the physical (lower dimensional) desired path. Rigorous theoretical analysis is carried out for the theoretical results using dynamical systems theory. In addition, we show both by theoretical analysis and numerical simulations that our proposed method is an extension combining conventional VF-PF algorithms and trajectory tracking algorithms. Finally, to show the practical value of our proposed approach for complex engineering systems, we conduct outdoor experiments with a fixed-wing airplane in windy environment to follow both 2-D and 3-D desired paths.

中文翻译：

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机器人导航的无奇异性引导向量场


在机器人导航任务中，例如无人机（UAV）高速公路交通监控，移动机器人遵循指定的所需路径非常重要。然而，由于奇点的存在，大多数现有的路径跟踪导航算法不能保证全局收敛到期望路径，或者能够跟随自相交的期望路径，导致导航算法返回不可靠的解甚至没有解。一个典型的例子是矢量场引导路径跟踪 (VF-PF) 导航算法。这些算法基于矢量场，奇异点正是矢量场减小的地方。传统的 VF-PF 算法生成与所需路径所在空间维度相同的矢量场。在本文中，我们证明传统的 VF-PF 算法在数学上不可能实现全局收敛到自相交甚至简单闭合（准确地说，与单位圆同胚）的所需路径。受这种新的不可能性结果的启发，我们提出了一种新颖的方法，将自相交或简单闭合的期望路径转换为高维空间中的非自相交和无界（准确地说，与实线同胚）对应路径。对应于这个新的期望路径，我们在更高维空间上构造了一个无奇点的引导向量场。因此，利用这个新的引导向量场的积分曲线来使得能够全局收敛到更高维度的期望路径，并且因此积分曲线在较低维度子空间上的投影收敛到物理（较低维度）期望路径。利用动力系统理论对理论结果进行了严格的理论分析。 此外，我们通过理论分析和数值模拟表明，我们提出的方法是传统 VF-PF 算法和轨迹跟踪算法相结合的扩展。最后，为了展示我们提出的方法对复杂工程系统的实用价值，我们在有风的环境中使用固定翼飞机进行户外实验，以遵循 2D 和 3D 所需路径。