Abstract
Using a designed vector field to control a mobile robot to follow a given desired path has found a range of practical applications, and it is in great need to further build a rigorous theory to guide its implementation.
This chapter is based on
\(\bullet \) W. Yao, Y. A. Kapitanyuk, and M. Cao, “Robotic path following in 3D using a guiding vector field,” in 2018 IEEE 57th Conference on Decision and Control (CDC), IEEE, 2018, pp. 4475–4480.
\(\bullet \) W. Yao and M. Cao, “Path following control in 3D using a vector field,” Automatica, vol. 117, p. 108–957, 2020.
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Notes
- 1.
A point where a vector field becomes zero is called a singular point of the vector field [p. 219] [77]. The set of singular points of a vector field is called the singular set of the vector field.
- 2.
Due to Lemma 4.6, desired paths can be conveniently classified into two categories: those homeomorphic to the unit circle \(\mathbb {S}^1\) if they are compact and those homeomorphic to the real line \(\mathbb {R}^{}\) otherwise [76, Theorem 5.27].
- 3.
This is justified as follows: one can choose a set \(\Omega _\beta \) as defined in (4.10), which is compact. Then there exists \(\gamma ' > 0\) such that \(\mathcal {E}_{\gamma '} \subseteq \Omega _\beta \) (this is true because by choosing \(\gamma ' \le \sqrt{2 \beta / k_{\textrm{max}}}\), \(\forall \xi \in \mathcal {E}_{\gamma '}, \Vert e(\xi ) \Vert \le \gamma ' \implies V(\xi ) \le k_{\textrm{max}} \Vert e(\xi ) \Vert ^2 / 2 \le k_{\textrm{max}} \gamma '^2 / 2 \le \beta \implies \xi \in \Omega _\beta \)). Therefore, \(\mathcal {E}_{\gamma '}\) is compact. Finally, by selecting \(0<\delta <\min \{\gamma , \gamma '\}\), it can be guaranteed that \(\mathcal {E}_\delta \) is compact as desired (since \(\mathcal {E}_\delta \subseteq \mathcal {E}_{\gamma '} \subseteq \Omega _\beta \)).
- 4.
Since \(\forall \xi \in \Omega _\iota , k_{\textrm{min}} \Vert e(\xi ) \Vert ^2 / 2 \le V(e(\xi )) \le \iota \implies \Vert e(\xi ) \Vert \le \delta \implies \xi \in \mathcal {E}_\delta \).
- 5.
The first inequality of (4.15) is justified since one can always choose the sequence \(\{ t_k \}\) such that \(t_{k+1}-t_k > \Delta \) for all \(k\ge 1\).
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Yao, W. (2023). Path Following Control in 3D Using a Vector Field. 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_4
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