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Space-Aware Reconfiguration

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Abstract

We consider the problem of reconfiguring a set of physical objects into a desired target configuration, a typical (sub)task in robotics and automation, arising in product assembly, packaging, stocking store shelves, and more. In this paper we address a variant, which we call space-aware reconfiguration, where the goal is to minimize the physical space needed for the reconfiguration, while obeying constraints on the allowable collision-free motions of the objects. Since for given start and target configurations, reconfiguration may be impossible, we translate the entire target configuration rigidly into a location that admits a valid sequence of moves, where each object moves in turn just once, along a straight line, from its starting to its target location, so that the overall physical space required by the start, all intermediate, and target configurations for all the objects is minimized. We investigate two variants of space-aware reconfiguration for the often examined setting of n unit discs in the plane, depending on whether the discs are distinguishable (labeled) or indistinguishable (unlabeled). For the labeled case, we propose a representation of size \(O(n^4)\) of the space of all feasible initial rigid translations, and use it to find, in \(O(n^6)\) time, a shortest valid translation, or one that minimizes the enclosing disc or axis-aligned rectangle of both the start and target configurations. For the significantly harder unlabeled case, we show that for almost every direction, there exists a translation in this direction that makes the problem feasible. We use this to devise heuristic solutions, where we optimize the translation under stricter notions of feasibility. We present an implementation of such a heuristic, which solves unlabeled instances with hundreds of discs in seconds.

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Notes

  1. The translations from T to \(T+\vec {v}\) do not count as moves. We often refer to it as the initial translation.

  2. The only exceptional directions are those of common inner tangents of pairs of tangent discs. If no tangency between the discs is allowed, all directions admit a valid translation.

  3. Recall that \(A=(A^S,A^T)\) and \(B=(B^S,B^T)\); thus each of A and B is a pair of start and target placements of a disc.

  4. Here \(\lambda _s(m)\) denotes the maximum length of a Davenport–Schinzel sequence of order s on m symbols; see [28] for details.

  5. As already mentioned, although we assume general position, circular arcs of vippodromes may still overlap (see Fig. 5). Notice, however, that the overlapping arcs bound vippodromes that induce the same constraint on the itinerary and hence crossing the overlapping arcs still incurs insertion or deletion of a single edge to the graph.

  6. In general, \(G_f\) can have exponentially many topological orders, each of which yields a valid itinerary.

  7. Recall that we assume that at translation \(\vec {v} = \vec {0}\) the centers of mass of S and \(T +\vec {v} = T\), or the centers of their smallest enclosing discs, coincide. This makes the shortest valid translation a plausible criterion for space-aware optimization.

  8. This is indeed an equivalent formulation to the one given earlier: The smallest enclosing disc of \(D(S)\cup D(T+\vec {v})\) has the same center as the disc that we find, and its radius is larger by 1. In contrast to other sections in the paper, here we consider the SED to be a closed disc.

  9. Unless \(\gamma \) is \({\partial }\mathcal {V}^{(1)}_{AB}\) or \({\partial }\mathcal {V}^{(2)}_{AB}\) such that \(A^S = s_0\) and \(B^T = t_0\). In this exceptional case, the entire circular subarc of \(\gamma \) is a continuum of minima. This case is easily handled by a slight modification to the algorithm, which we omit here.

  10. Recall that the optimization procedures in Sect. 3 require a preprocessing stage that computes Q in \(O(n^6)\) time. This cost is only a consequence of the (expensive) need to test each face of \(\mathcal {A}(\mathcal {V}^{\partial })\) for acyclicity, which is no longer needed under our new notion of validity. This is why constructing \(\widetilde{Q}\) is faster. Note also that restricting the analysis to the bad vippodromes cleans the presentation, but is not the source of the above improvement, as the number of bad vippodromes is still half the number of all vippodromes.

  11. The reason why the index in the above bound is \(s+1\) is that each of the domains of the partial functions \(f_t(\vec {v})\), \(g_t(\vec {v})\) is an interval with at least one endpoint coinciding with an endpoint of e; see [28] for details.

  12. Similar techniques can be applied to the variant of optimizing the area of the axis-aligned bounding rectangle. To keep the presentation somewhat shorter, we do not include this case in this section.

  13. The random choice of each configuration is modified so as to ensure that they are valid—no two points are at distance smaller than 2. Random choices that violate this property are discarded and replaced by other random choices.

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Authors and Affiliations

  1. The Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel

    Dan Halperin, Golan Miglioli-Levy & Micha Sharir

  2. Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands

    Marc van Kreveld

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  1. Dan Halperin
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  2. Marc van Kreveld
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  3. Golan Miglioli-Levy
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  4. Micha Sharir
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Correspondence to Golan Miglioli-Levy.

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Work by D.H. and G.M.L. has been supported in part by the Israel Science Foundation Grants 825/15, 1736/19, by the Blavatnik Computer Science Research Fund, and by the Yandex Machine Learning Initiative for Machine Learning at Tel Aviv University. Work by D.H. has also been supported in part by NSF/US-Israel-BSF (grant no. 2019754) and by the Israel Ministry of Science and Technology (grant no. 103129). Work by M.v.K. has been supported by the the Netherlands Organisation for Scientific Research under Grant 612.001.651. Work by M.S. and G.M.L. has been supported by Grant 260/18 from the Israel Science Foundation. Work by M.S. has also been supported by Grant G-1367-407.6/2016 from the German-Israeli Foundation for Scientific Research and Development, and by the Blavatnik Computer Science Research Fund. An abridged version of this paper has appeared in the 14th International Workshop on the Algorithmic Foundations of Robotics, 2020 [19].

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Halperin, D., van Kreveld, M., Miglioli-Levy, G. et al. Space-Aware Reconfiguration. Discrete Comput Geom 69, 1157–1194 (2023). https://doi.org/10.1007/s00454-022-00407-7

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