An application-based characterization of dynamical distance geometry problems

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

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 2017

  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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  4. 4.

    de Leeuw, J.: Differetiability of Kruskal’s stress at a local minimum. Psychometrika 49, 111–113 (1984)

    MathSciNet  Article  Google 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)

    Article  Google 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)

  7. 7.

    Glunt, W., Hayden, T.L., Raydan, M.: Molecular conformations from distance matrices. J. Comput. Chem. 14(1), 114–120 (1993)

    Article  Google 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)

  9. 9.

    Helbing, D., Farkas, I., Vicsek, T.: Simulating dynamical features of escape panic. Nature 407, 487–490 (2000)

    Article  Google 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)

  11. 11.

    Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs. J. Combin. Theory Ser. B 94, 1–29 (2005)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Krislock, N., Wolkowicz, H.: Explicit sensor network localization using semidefinite representations and facial reductions. SIAM J. Optim. 20, 2679–2708 (2010)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Kruskal, J.B.: Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29, 1–27 (1964)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4(4), 331–340 (1970)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52, 115–146 (2012)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56(1), 3–69 (2014)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

  21. 21.

    Mucherino, A., de Freitas, R., Lavor, C.: Distance geometry and applications. Spec. Issue Discrete Appl. Math. 197, 1–144 (2015)

    MathSciNet  Article  Google 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)

  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)

  24. 24.

    Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.): Distance Geometry: Theory, Methods and Applications. Springer, Berlin (2013)

    Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google 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)

  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)

  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)

  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)

    Article  Google 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)

    MathSciNet  Article  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Antonio Mucherino.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mucherino, A., Omer, J., Hoyet, L. et al. An application-based characterization of dynamical distance geometry problems. Optim Lett 14, 493–507 (2020). https://doi.org/10.1007/s11590-018-1302-6

Download citation

Keywords

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