Refining dichotomy convergence in vector-field guided path-following control

Abstract

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.

Topological Analysis of Vector-Field Guided Path Following on Manifolds

“…Assumption 2 implies that as the norm of the path-following error e(ξ) approaches zero, the trajectory ξ(t) approaches the desired path P [31]. These assumptions are vital in the sense that if either of these assumptions is not satisfied, then different choices of surface functions φ i for the same desired path may lead to opposite convergence results, as the following example.…”

A Singularity-Free Guiding Vector Field for Robot Navigation

“…The use of e(•) is more convenient than that of the distance function dist(•, P). However, there are subtle differences between the norm of the path-following error e(•) and the distance dist(•, P); e.g., when the norm of the path-following error converges to zero, the trajectory might not converge to the desired path [22, Example 3], [43]. Assumptions will be proposed to avoid this pathological situation.…”

“…Assumption 2 is satisfied in many practical cases such as many polynomial or trigonometric functions; examples are demonstrated in [3], [4], [16], and [22]. Since the same desired path can be characterized by various choices of φ i (•), the assumption are crucial to exclude some pathological cases [22,Example 3], [43]. The mathematical formulation of Assumption 2 is presented in Appendix A-C in the full version [34].…”

Guiding Vector Fields for Following Occluded Paths

“…This assumption restricts one to choose a "valid" function φ such that when |φ(ξ(t))| → 0 as t → ∞ along an infinitelyextendable trajectory ξ(t), then dist(ξ(t), P) → 0 as t → ∞ (guaranteed by the first part of the assumption), and when ∇φ(ξ(t)) → 0 as t → ∞ then dist(ξ(t), C P ) → 0 as t → ∞ (guaranteed by the second part of the assumption). Without this assumption, one can choose a function φ such that trajectories diverge to infinity even when the absolute path-following error |φ(•)| converges to zero (see [30, Section IV.B], [31,Example 1]). In addition, the assumption is not restrictive as it is satisfied for many polynomial or trigonometric functions φ [7], [30]; the assumption holds for all examples presented in this paper.…”