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 selfintersected 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 Manuscript

A path-following control algorithm enables a system's trajectories under its guidance to converge to and evolve along a given geometric desired path. There exist various such algorithms, but many of them can only guarantee local convergence to the desired path in its neighborhood. In contrast, the control algorithms using a well-designed guiding vector field can ensure almost global convergence of trajectories to the desired path; here, "almost" means that in some cases, a measure-zero set of trajectories converge to the singular set where the vector field becomes zero (with all other trajectories converging to the desired path). In this article, we first generalize the guiding vector field from the Euclidean space to a general smooth Riemannian manifold. This generalization can deal with path-following in some abstract configuration space (such as robot arm joint space). Then, we show several theoretical results from a topological viewpoint. Specifically, we are motivated by the observation that singular points of the guiding vector field exist in many examples where the desired path is homeomorphic to the unit circle, but it is unknown whether the existence of singular points always holds in general (i.e., is inherent in the topology of the desired path). In the n-dimensional Euclidean space, we provide an affirmative answer, and conclude that it is not possible to guarantee global convergence to desired paths that are homeomorphic to the unit circle. Furthermore, we show that there always exist nonpath-converging trajectories (i.e., trajectories that do not converge to the desired path) starting from the boundary of a ball containing the desired path in an n-dimensional Euclidean space where n ≥ 3. Examples are provided to illustrate the theoretical results.

Accurately following a geometric desired path in a two-dimensional space is a fundamental task for many engineering systems, in particular mobile robots. When the desired path is occluded by obstacles, it is necessary and crucial to temporarily deviate from the path for obstacle/collision avoidance. In this paper, we develop a composite guiding vector field via the use of smooth bump functions, and provide theoretical guarantees that the integral curves of the vector field can follow an arbitrary sufficiently smooth desired path and avoid collision with obstacles of arbitrary shapes. These two behaviors are reactive since path (re)-planning and global map construction are not involved. To deal with the common deadlock problem, we introduce a switching vector field, and the Zeno behavior is excluded. Simulations are conducted to support the theoretical results.

Path following and collision avoidance are two important functionalities for mobile robots, but there are only a few approaches dealing with both. In this paper, we propose an integrated path following and collision avoidance method using a composite vector field. The vector field for path following is integrated with that for collision avoidance via bump functions, which reduce significantly the overlapping effect. Our method is general and flexible since the desired path and the contours of the obstacles, which are described by the zero-level sets of sufficiently smooth functions, are only required to be homeomorphic to a circle or the real line, and the derivation of the vector field does not involve specific geometric constraints. In addition, the collision avoidance behaviour is reactive; thus, real-time performance is possible. We show analytically the collision avoidance and path following capabilities, and use numerical simulations to illustrate the effectiveness of the theory.

In the vector-field guided path-following problem, the desired path is described by the zero-level set of a sufficiently smooth real-valued function and to follow this path, a (guiding) vector field is designed, which is not the gradient of any potential function. The value of the aforementioned real-valued function at any point in the ambient space is called the level value at this point. Under some broad conditions, a dichotomy convergence property has been proved in the literature: the integral curves of the vector field converge either to the desired path or the singular set, where the vector field attains a zero vector. In this paper, the property is further developed in two respects. We first show that the vanishing of the level value does not necessarily imply the convergence of a trajectory to the zero-level set, while additional conditions or assumptions identified in the paper are needed to make this implication hold. The second contribution is to show that under the condition of real-analyticity of the function whose zero-level set defines the desired path, the convergence to the singular set (assuming it is compact) implies the convergence to a single point of the set, dependent on the initial condition, i.e. limit cycles are precluded. These results, although obtained in the context of the vectorfield guided path-following problem, are widely applicable in many control problems, where the desired sets to converge to (particularly, a singleton constituting a desired equilibrium point) form a zero-level set of a Lyapunov(-like) function, and the system is not necessarily a gradient system.