BSplines Properties with Interval Analysis for Constraint Satisfaction Problem: Application in Robotics

  • Rawan KalawounEmail author
  • Sébastien Lengagne
  • François Bouchon
  • Youcef Mezouar
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 867)


Interval Analysis is a mathematical tool that could be used to solve Constraint Satisfaction Problem. It guarantees solutions, and deals with uncertainties. However, Interval Analysis suffers from an overestimation of the solutions, i.e. the pessimism. In this paper, we initiate a new method to reduce the pessimism based on the convex hull properties of BSplines and the Kronecker product. To assess our method, we compute the feasible workspace of a 2D manipulator taking into account joint limits, stability and reachability constraints: a classical Constraint Satisfaction Problem in robotics.


Interval analysis Pessimism BSplines Kronecker product Constraint Satisfaction Problem Robot Feasible workspace 


  1. 1.
    Tay, N.N.W., Saputra, A.A., Botzheim, J., Kubota, N.: Service robot planning via solving constraint satisfaction problem. ROBOMECH J. 3(1), 17 (2016)CrossRefGoogle Scholar
  2. 2.
    Lozano-Pérez, T., Kaelbling, L.P.: A constraint-based method for solving sequential manipulation planning problems. In: 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3684–3691, September 2014Google Scholar
  3. 3.
    Fromherz, M.P., Hogg, T., Shang, Y., Jackson, W.B.: Modular robot control and continuous constraint satisfaction. In: Proceedings of IJCAI-01 Workshop on Modelling and Solving Problems with Constraints, pp. 47–56 (2001)Google Scholar
  4. 4.
    Lengagne, S., Ramdani, N., Fraisse, P.: Planning and fast replanning safe motions for humanoid robots. IEEE Trans. Robot. 27(6), 1095–1106 (2011)CrossRefGoogle Scholar
  5. 5.
    Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, London (2001)zbMATHGoogle Scholar
  6. 6.
    Lengagne, S., Vaillant, J., Yoshida, E., Kheddar, A.: Generation of whole-body optimal dynamic multi-contact motions. Int. J. Robot. Res. 17 (2013)Google Scholar
  7. 7.
    Merlet, J.-P.: Interval analysis and reliability in robotics. Int. J. Reliab. Saf. 3(1–3), 104–130 (2009)CrossRefGoogle Scholar
  8. 8.
    Yokoo, M., Hirayama, K.: Algorithms for distributed constraint satisfaction: a review. Auton. Agents Multi-Agent Syst. 3(2), 185–207 (2000)CrossRefGoogle Scholar
  9. 9.
    Gelle, E., Faltings, B.: Solving mixed and conditional constraint satisfaction problems. Constraints 8(2), 107–141 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sunaga, T.: Theory of interval algebra and its application to numerical analysis. RAAG Mem. Ggujutsu Bunken Fukuy-kai 2, 547–564 (1958)zbMATHGoogle Scholar
  11. 11.
    Moore, R.E., Bierbaum, F.: Methods and Applications of Interval Analysis (SIAM Studies in Applied and Numerical Mathematics (Siam Studies in Applied Mathematics). Siam Studies in Applied Mathematics, vol. 2. Soc for Industrial & Applied Math, Philadelphia (1979)Google Scholar
  12. 12.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  13. 13.
    Pérez-Galván, C., Bogle, I.D.L.: Global optimisation for dynamic systems using interval analysis. Comput. Chem. Eng. 107, 343–356 (2017)CrossRefGoogle Scholar
  14. 14.
    Jiang, C., Han, X., Guan, F., Li, Y.: An uncertain structural optimization method based on nonlinear interval number programming and interval analysis method. Eng. Struct. 29(11), 3168–3177 (2007)CrossRefGoogle Scholar
  15. 15.
    Ma, H., Xu, S., Liang, Y.: Global optimization of fuel consumption in J2 rendezvous using interval analysis. Adv. Space Res. 59(6), 1577–1598 (2017)CrossRefGoogle Scholar
  16. 16.
    Merlet, J.P.: Interval Analysis and Robotics, pp. 147–156. Springer, Heidelberg (2011)Google Scholar
  17. 17.
    Jaulin, L.: Interval analysis and robotics. In: SCAN 2012, Russia, Novosibirsk (2012)Google Scholar
  18. 18.
    Rohou, S., Jaulin, L., Mihaylova, L., Le Bars, F., Veres, S.M.: Guaranteed computation of robot trajectories. Robot. Auton. Syst. 93, 76–84 (2017)CrossRefGoogle Scholar
  19. 19.
    Desrochers, B., Jaulin, L.: Minkowski operations of sets with application to robot localization. In: SNR 2017, Uppsala (2017)Google Scholar
  20. 20.
    Benoît, D., Luc, J.: Computing a guaranteed approximation of the zone explored by a robot. IEEE Trans. Autom. Control 62(1), 425–430 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Le Bars, F., Bertholom, A., Sliwka, J., Jaulin, L.: Interval SLAM for underwater robots; a new experiment. In: NOLCOS 2010, France, p. XX, September 2010CrossRefGoogle Scholar
  22. 22.
    Wu, J.: Uncertainty analysis and optimization by using the orthogonal polynomials, Ph.D. dissertation (2015)Google Scholar
  23. 23.
    Netz, L.: Using horner schemes to improve the efficiency and precision of interval constraint propagation (2015)Google Scholar
  24. 24.
    Schäcke, K.: On the kronecker product (2013)Google Scholar
  25. 25.
    Loan, C.F.: The ubiquitous kronecker product. J. Comput. Appl. Math. 123(1), 85–100 (2000). Numerical Analysis 2000. Vol. III: Linear AlgebraMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Rawan Kalawoun
    • 1
    • 2
    Email author
  • Sébastien Lengagne
    • 1
    • 3
  • François Bouchon
    • 4
  • Youcef Mezouar
    • 1
    • 2
  1. 1.CNRS, UMR 6602, Pascal InstituteAubiéreFrance
  2. 2.SIGMA Clermont, Pascal Institute, BP10448Clermont-FerrandFrance
  3. 3.Clermont-Auvergne University, Pascal Institute, BP10448Clermont-FerrandFrance
  4. 4.Clermont-Auvergne University, Laboratory of Mathematics Blaise Pascal, BP10448Clermont-FerrandFrance

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