Abstract
Parametrized motion planning algorithms have high degrees of universality and flexibility, as they are designed to work under a variety of external conditions, which are viewed as parameters and form part of the input of the underlying motion planning problem. In this paper, we analyze the parametrized motion planning problem for the motion of many distinct points in the plane, moving without collision and avoiding multiple distinct obstacles with a priori unknown positions. This complements our prior work Cohen et al. [3] (SIAM J. Appl. Algebra Geom. 5, 229–249), where parametrized motion planning algorithms were introduced, and the obstacle-avoiding collision-free motion planning problem in three-dimensional space was fully investigated. The planar case requires different algebraic and topological tools than its spatial analog.
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01 December 2022
A Correction to this paper has been published: https://doi.org/10.1007/s10472-022-09821-2
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Acknowledgements
The first author thanks Emanuele Delucci, Nick Proudfoot, and He Xiaoyi for productive conversations, and the organizers of the virtual workshop Arrangements at Home for facilitating several of these conversations. Portions of this work were undertaken when the first and second authors visited the University of Florida Department of Mathematics in November, 2019. We thank the department for its hospitality and for providing a productive mathematical environment. We also thank the anonymous referees for their helpful comments.
Funding
D. Cohen was partially supported by an LSU Faculty Travel Grant.
M. Farber was partially supported by EPSRC grant EP/V009877/1.
S. Weinberger was partially supported by National Science Foundation grant DMS 1811071.
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Appendix
Appendix
In this appendix, we state and prove a general result which includes as a special case the fact noted in Proposition 2.1 that, for a fibration p: E → B, the map \({\Pi } \colon {E^{I}_{B}} \to E\times _{B} E\) is also a fibration.
Let p : E → B be a Hurewicz fibration. For a topological space X, let \({E^{X}_{B}}\) denote the space of all continuous maps f : X → E lying in a single fiber of p, that is, such that the composition p ∘ f : X → B is a constant map. Equip the space \({E^{X}_{B}}\) with the compact-open topology. The following result may be compared with [, Thm. 2.8.2].22
Proposition
Let (K,L) be a pair consisting of a finite CW-complex and a subcomplex. Then the restriction map is a Hurewicz fibration.
Proof
Let \(\lambda : \overline B \to E^{I}\) be a lifting function for the fibration p: E → B (see [, Sec. 2.7]). Here 22\(\overline B=\{(e, \omega )\in E\times B^{I} \mid p(e)=\omega (0)\}\) and λ(e,ω) ∈ EI satisfies p ∘ λ(e,ω) = ω and λ(e,ω)(0) = e. Our goal is to construct a lifting function
for π. Here \(\overline {{E^{L}_{B}}}\) is the set of pairs
Clearly, \(G\in ({E^{L}_{B}})^{I}\) can be viewed as a map G: L × I → E such that for any t \(\in\) I the image G(L × t) ⊂ E lies in a single fiber. The map \(F\in {E^{K}_{B}}\) satisfies F(x) = G(x, 0) for x \(\in\) L. Denote by ω(t) = p(G(x,t)) (where x \(\in\) L) the path in B obtained by applying the projection p. For x \(\in\) K, the formula \(\tilde F(x, t) = \lambda (F(x), \omega )(t)\) defines a map \(\tilde F\in ({E^{K}_{B}})^{I}\) satisfying \(\tilde F(x, 0) =F(x)\) and \(p(\tilde F(x, t))=\omega (t)= p(G(x, t))\). However, we may not guarantee that the condition \(\tilde F(x, t)= G(x, t)\) for x \(\in\) L and t \(\in\) I holds.
Let ω[τ,1] denote the path s↦ω[τ,1](s) = ω(τ + (1 − τ)s), where s \(\in\) [0, 1]. Define αG: L × I × I → E by
For x \(\in\) L, we have p(αG(x,τ,t)) = ω(t), \(\alpha _{G}(x, 0, t) = \lambda (G(x, 0), \omega )(t)=\tilde F(x, t)\), and αG(x, 1,t) = G(x,t). Thus αG is a fibrewise homotopy between \({\Pi }(\tilde F)\) and G in \(({E^{L}_{B}})^{I}\).
Let ρ : K × I → K × 0 ∪ L × I be a retraction. Denote by H the composition
The map h(x,t) = H(x, 1,t) is an element of \(({E^{K}_{B}})^{I}\) satisfying h(x, 0) = F(x) and h(x,t) = G(x,t) for x ∈ L. Hence we may define the lifting function (†) by setting Λ(F,G) = h. □
Proposition 2.1, asserting that \({\Pi }\colon {E^{I}_{B}} \to E\times _{B}E\), γ↦(γ(0),γ(1)), is a fibration, may be obtained by taking K = [0, 1] and L = {0, 1} in the above result.
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Cohen, D.C., Farber, M. & Weinberger, S. Parametrized topological complexity of collision-free motion planning in the plane. Ann Math Artif Intell 90, 999–1015 (2022). https://doi.org/10.1007/s10472-022-09801-6
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DOI: https://doi.org/10.1007/s10472-022-09801-6