TY - JOUR
T1 - Topological Analysis of Vector-Field Guided Path Following on Manifolds
AU - Yao, Weijia
AU - Lin, Bohuan
AU - Anderson, Brian D.O.
AU - Cao, Ming
N1 - Publisher Copyright:
© 1963-2012 IEEE.
PY - 2023/3
Y1 - 2023/3
N2 - 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 $n \geq 3$. Examples are provided to illustrate the theoretical results.
AB - 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 $n \geq 3$. Examples are provided to illustrate the theoretical results.
KW - Convergence
KW - domain of attraction
KW - manifold
KW - path following
UR - http://www.scopus.com/inward/record.url?scp=85124840863&partnerID=8YFLogxK
U2 - 10.1109/TAC.2022.3151236
DO - 10.1109/TAC.2022.3151236
M3 - Article
SN - 0018-9286
VL - 68
SP - 1353
EP - 1368
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
IS - 3
ER -