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An application-based characterization of dynamical distance geometry problems

  • Antonio Mucherino
  • Jeremy Omer
  • Ludovic Hoyet
  • Paolo Robuffo Giordano
  • Franck Multon
Original Paper
  • 49 Downloads

Abstract

The dynamical distance geometry problem (dynDGP) is the problem of finding a realization in a Euclidean space of a weighted undirected graph G representing an animation by relative distances, so that the distances between realized vertices are as close as possible to the edge weights. In the dynDGP, the vertex set of the graph G is the set product of V, representing certain objects, and T, representing time as a sequence of discrete steps. We suppose moreover that distance information is given together with the priority of every distance value. The dynDGP is a special class of the DGP where the dynamics of the problem comes to play an important role. In this work, we propose an application-based characterization of dynDGP instances, where the main criteria are the presence or absence of a skeletal structure, and the rigidity of such a skeletal structure. Examples of considered applications include: multi-robot coordination, crowd simulations, and human motion retargeting.

Keywords

Distance geometry Dynamical problems Motion retargeting Crowd simulations Multi-agent formation 

Notes

Acknowledgements

We wish to thank Douglas S. Gonçalves for the fruitful discussions. This work was partially supported by an INS2I-CNRS 2016 “PEPS” Project, and by the ANR Project ANR-14-CE27-0007 SenseFly.

References

  1. 1.
    Bernardin, A., Hoyet, L., Mucherino, A., Gonçalves, D.S., Multon, F.: Normalized Euclidean distance matrices for human motion retargeting. In: ACM Conference Proceedings, Motion in Games 2017 (MIG17), Barcelona, Nov 2017Google Scholar
  2. 2.
    Birgin, E.G., Martínez, J.M.: Large-scale active-set box-constrained optimization method with spectral projected gradients. Comput. Optim. Appl. 23, 101–125 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    de Leeuw, J.: Differetiability of Kruskal’s stress at a local minimum. Psychometrika 49, 111–113 (1984)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fox, D., Ko, J., Konolige, K., Limketkai, B., Schulz, D., Stewart, B.: Distributed multirobot exploration and mapping. Proc. IEEE 94(7), 1325–1339 (2006)CrossRefzbMATHGoogle Scholar
  6. 6.
    Gleicher, M.: Retargetting motion to new characters. In: ACM proceedings of the 25th annual conference on computer graphics and interactive techniques, pp. 33–42 (1998)Google Scholar
  7. 7.
    Glunt, W., Hayden, T.L., Raydan, M.: Molecular conformations from distance matrices. J. Comput. Chem. 14(1), 114–120 (1993)CrossRefGoogle Scholar
  8. 8.
    Hecker, Ch., Raabe, B., Enslow, R.W., DeWeese, J., Maynard, J., van Prooijen, K.: Real-time motion retargeting to highly varied user-created morphologies. In: Proceedings of ACM SIGGRAPH 2008, ACM Transactions on Graphics 27(3) (2008)Google Scholar
  9. 9.
    Helbing, D., Farkas, I., Vicsek, T.: Simulating dynamical features of escape panic. Nature 407, 487–490 (2000)CrossRefGoogle Scholar
  10. 10.
    Hodgins, J.K., Wooten, W.L., Brogan, D.C., O’Brien, J.F.: Animating human athletics. In: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH95), pp. 71–78 (1995)Google Scholar
  11. 11.
    Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs. J. Combin. Theory Ser. B 94, 1–29 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Krislock, N., Wolkowicz, H.: Explicit sensor network localization using semidefinite representations and facial reductions. SIAM J. Optim. 20, 2679–2708 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kruskal, J.B.: Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29, 1–27 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4(4), 331–340 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52, 115–146 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lavor, C., Liberti, L., Mucherino, A.: The interval branch-and-prune algorithm for the discretizable molecular distance geometry problem with inexact distances. J. Global Optim. 56(3), 855–871 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Liberti, L., Lavor, C., Maculan, N.: A branch-and-prune algorithm for the molecular distance geometry problem. Int. Trans. Oper. Res. 15, 1–17 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56(1), 3–69 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Liberti, L., Lavor, C., Mucherino, A., Maculan, N.: Molecular distance geometry methods: from continuous to discrete. Int. Trans. Oper. Res. 18(1), 33–51 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Montijano, E., Cristofalo, E., Zhou, D., Schwager, M., Sagues, C.: Vision-based distributed formation control without an external positioning system. IEEE Trans. Robot. 32(2), 339–351 (2016)CrossRefGoogle Scholar
  21. 21.
    Mucherino, A., de Freitas, R., Lavor, C.: Distance geometry and applications. Spec. Issue Discrete Appl. Math. 197, 1–144 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mucherino, A., Gonçalves, D.S.: An approach to dynamical distance geometry. In: Nielsen, F., Barbaresco, F. (eds.) Proceedings of Geometric Science of Information (GSI17), Lecture Notes in Computer Science, vol. 10589, pp. 821–829. Paris (2017)Google Scholar
  23. 23.
    Mucherino, A., Gonçalves, D.S., Bernardin, A., Hoyet, L., Multon, F.: A distance-based approach for human posture simulations. In: IEEE Conference Proceedings, Federated Conference on Computer Science and Information Systems (FedCSIS17), Workshop on Computational Optimization (WCO17), Prague, pp. 441–444 (2017)Google Scholar
  24. 24.
    Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.): Distance Geometry: Theory, Methods and Applications. Springer, Berlin (2013)zbMATHGoogle Scholar
  25. 25.
    Multon, F., France, L., Cani-Gascuel, M.P., Debunne, G.: Computer animation of human walking: a survey. J. Vis. Comput. Anim. 10(1), 39–54 (1999)CrossRefGoogle Scholar
  26. 26.
    Multon, F., Kulpa, R., Bideau, B.: MKM: a global framework for animating humans in virtual reality applications. Presence Teleoper. Virtual Environ. 17(1), 17–28 (2008)CrossRefGoogle Scholar
  27. 27.
    Olivier, A.H., Marin, A., Crétual, A., Pettré, J.: Minimal predicted distance: a common metric for collision avoidance during pairwise interactions between walkers. Gait Posture 36(3), 399–404 (2012)CrossRefGoogle Scholar
  28. 28.
    Omer, J.: A space-discretized mixed-integer linear model for air-conflict resolution with speed and heading maneuvers. Comput. Oper. Res. 58, 75–86 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Richards, A., How, J.P.: Aircraft trajectory planning with collision avoidance using mixed integer linear programming. In: IEEE Conference Proceedings, American Control Conference 2002. Anchorage, AK (2002)Google Scholar
  30. 30.
    Saxe, J.: Embeddability of weighted graphs in \(k\)-space is strongly NP-hard. In: Proceedings of 17th Allerton Conference in Communications, Control and Computing, pp. 480–489 (1979)Google Scholar
  31. 31.
    Schiano, F., Franchi, A., Zelazo, D., Robuffo Giordano, P.: A rigidity-based decentralized bearing formation controller for groups of quadrotor UAVs. In: Proceedings of the 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS16), pp. 5099–5106 (2016)Google Scholar
  32. 32.
    Zelazo, D., Franchi, A., Bülthoff, H.-H., Robuffo Giordano, P.: Decentralized rigidity maintenance control with range measurements for multi-robot systems. Int. J. Robot. Res. 34(1), 105–128 (2015)CrossRefGoogle Scholar
  33. 33.
    Zhang, H., Hager, W.W.: A nonmonotone line search technique and its applications to unconstrained optimization. SIAM J. Optim. 14(4), 1043–1056 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Univ Rennes, Inria, CNRS, IRISARennesFrance
  2. 2.Univ Rennes, INSA Rennes, CNRS, IRMARRennesFrance
  3. 3.Univ Rennes, Inria, M2S, University of Rennes 2RennesFrance

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