Abstract
Under some broad conditions, a dichotomy convergence property in the vector-field guided path-following problem has been proved in previous chapters: the integral curves of a guiding vector field converge to either the desired path or the singular set, where the vector field becomes zero.
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\(\bullet \) W. Yao, B. Lin, B. D. O. Anderson, and M. Cao, “Refining dichotomy convergence in vector-field guided path following control,” in European Control Conference (ECC), 2021.
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Notes
- 1.
The sign of the wedge product depends on the order of the gradient vectors. However, this does not affect the convergence result.
- 2.
Suppose not, then there exists the smallest time instant \(t_2 > t_1\) such that \(\Vert \xi (t_2) - \xi ^* \Vert = r\) and \(\xi (t) \in \mathcal {U}\) for any \(t \in (t_1, t_2)\). Therefore, we have \(\Vert \xi (t_2) - \xi ^* \Vert \le \Vert \xi (t_2) -\xi (t_1) \Vert + \Vert \xi (t_1) - \xi ^* \Vert< L_{12} + r/2 < r\), a contradiction, where \(L_{12}\) is shown in (7.12).
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Yao, W. (2023). Refined Dichotomy Convergence in Vector-Field Guided Path-Following on \(\mathbb {R}^{n}\). In: Guiding Vector Fields for Robot Motion Control. Springer Tracts in Advanced Robotics, vol 154. Springer, Cham. https://doi.org/10.1007/978-3-031-29152-4_7
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DOI: https://doi.org/10.1007/978-3-031-29152-4_7
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