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Mutational equations for shapes and vision-based control

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Abstract

Basic idea of vision-based control in robotics is to include the vision system directly in the control servo loop of the robot. When images are binary, this problem corresponds to the control of the evolution of a geometric domain. The present paper proposes mathematical tools derived from shape analysis and optimization to study this problem in a quite general way, i.e., without any regularity assumptions or modelsa priori on the domains that we deal with. Indeed, despite the lackness of a vectorial structure, one can develop a differential calculus in the metric space of all non-empty compact subsets of a given domain ofR n, and adapt ideas and results of classical differential systems to study and control the evolution of geometric domains. For instance, a shape Lyapunov characterization allows to investigate the asymptotic behavior of these geometric domains using the notion of directional shape derivative. We apply this inR 2 to the visual servoing problem using the optical flow equations and some experimental simulations illustrate this approach.

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References

  1. G. Agin,Real time control of a robot with a mobile camera, Technical Note 179, SRI International, 1979.

  2. G. Agin, “Servoing with visual feedback,”Symposium on Industrial robotics, Tokyo, pp. 551–560, 1977.

  3. P. Allen, B. Yoshimi, and A. Timcenko, “Real-time visual servoing,”Proceedings IEEE Int. Conf. on Robotics and Automation, Sacramento, pp. 851–856, 1991.

  4. J.Y. Aloimonos, I. Weiss, and A. Bandyopadhyay “Active Vision,”International Journal of Computer Vision, pp. 333–356, 1988.

  5. S.-I. Amari and M. Maryuama “A theory on the determination of 3D motion and 3D structure from features,”Spatial Vision, Vol. 2, No. 2, pp. 151–168, 1987.

    Google Scholar 

  6. S.-I. Amari, “Features spaces which admit and detect invariant signal transformations,”Proceedings of 4th Intern. Journal Conf. on Pattern Recognition, 1978.

  7. J.-P. Aubin and H. Frankowska,Set-valued analysis, Birkhäuser, 1990.

  8. J.-P. Aubin and Cellina,Differential inclusions, Springer-Verlag. Grundlehren der math, 1984.

  9. J.-P. Aubin,Viability theory, Birkhauser, 1991.

  10. J.-P. Aubin, “Mutational equations in metric space,”Journal of Set-valued Analysis, No. 1, pp. 3–46, 1993.

    Google Scholar 

  11. N. Ayache, “Vision stéréoscopique et perception multisensorielle,”Intereditions Science Inforrmatique, Paris, 1989.

  12. J. Céa, “Problems of Shape Optimal Design,”Optimization of Distributed Parameter Structures vol. I and II, E.J. Haug and Céa, eds., Sijhoff and Noorhoff, Alphen aan den Rijn, The Netherlands, pp. 1005–1087, 1981.

    Google Scholar 

  13. F. Chaumette, “La relation Vision-Commande,” Ph-D Thesis, Rennes I University, France, 1990.

    Google Scholar 

  14. F. Chaumette, P. Rives, and B. Espiau, “Positioning of a robot with respect to an object, tracking it and estimating its velocity by visual servoing,”Proceedings IEEE Int. Conf. on Robotics and Automation, Sacramento, pp. 2248–2253, April 91.

  15. M.C. Delfour and J.P. Zolesio,Structure of shape derivatives for nonsmooth domains, CRM-1669, Centre de Recherche Mathématiques de l'université de Montréal, Canada, 1990.

    Google Scholar 

  16. M.C. Delfour and J.P. Zolesio, “Shape sensitivity analysis via minmax differentiability,”SIAM Journal on Control and Optimization, Vol. 26, pp. 834–862, 1988.

    Google Scholar 

  17. M.C. Delfour and J.P. Zolesio, “Velocity method and lagrangian formulation for the computation of the shape Hessian,”SIAM Journal on Control and Optimization, (to appear), 1989.

  18. L. Doyen, “Inverse functions theorem and shape optimization,”SIAM Journal on Control and Optimization, Vol. 32, No. 6, pp. 1621–1642, Nov. 1994.

    Google Scholar 

  19. L. Doyen, “Filippov and invariance theorems for mutational inclusions of tubes,”Journal of Set-Valued Analysis, 1:289–303, 1993.

    Google Scholar 

  20. L. Doyen, “Shape Lyapunov functions,”Journal of Mathematical Analysis and Applications, Vol. 184, No. 2, June, 1994.

  21. L. Doyen, “Evolution, contrôle et optimisation de forme,” Thèse de Doctorat, Université Paris-Dauphine, France, 1993.

    Google Scholar 

  22. B. Espiau, F. Chaumette, and P. Rives, “A new approach to visual servoing in robotics,”IEEE Trans. on Robotics and Automation, Vol. 8, No. 3, 1992.

  23. O. Faugeras and Hebert, “The representation, recognition and locating of 3-D objects,”The Int. Journal of Robotics Research, Vol. 5, No. 3, pp. 27–52, 1986.

    Google Scholar 

  24. J. Feddema and O. Mitchell, “Vision-guided servoing with feature-based trajectory generation,”IEEE transactions of Robotics and Automation, Vol. 5, No. 5, pp. 691–700, 1989.

    Google Scholar 

  25. K. Hashimoto, T. Kimoto, T. Ebine and H. Kimura, “Imagebased dynamic visual servo for a hand-eye manipulator,”Proceedings IEEE Int. Conf. on Robotics and Automation, Sacramento, pp. 2267–2273, 1991.

  26. S. Hutchinson, “Exploiting visual constraints in robot planning,”Proceedings IEEE Int. Conf. on Robotics and Automation, Sacramento, pp. 1722–1727, 1991.

  27. W. Jang and Z. Bien, “Feature-based visual servoing of an eye-in-hand robot with improved tracking performance,”Proceedings IEEE Int. Conf. on Robotics and Automation, Sacramento, pp. 2254–2260, 1991.

  28. M. Kabuka and P. Shironoshita, “Adaptive approach to video tracking,”IEEE transactions of Robotics and Automation, Vol. 4, No. 2, pp. 228–236, 1988.

    Google Scholar 

  29. N. Papanikolopoulos, P. Khosla, and T. Kanabe, “Vision and control techniques for robotic visual tracking,”Proceedings IEEE Int. Conf. on Robotics and Automation, Sacramento, pp. 857–864, 1991.

  30. O. Pironneau, “Optimal shape design for elliptic systems,”Computational Physics, Springer-Verlag, 1983.

  31. J. Rehg and A. Witkin, “Visual tracking with deformation model,”Proceedings IEEE Int. Conf. on Robotics and Automation, Sacramento, pp. 844–850, 1991.

  32. P. Rives and Pissart-Gibollet,Asservissement visuel appliqué à la robotique mobile, Rapport de recherche INRIA No. 1577, Sophia-Antipolis, France, 1991.

  33. C. Samson, B. Espiau, and M. Le Borgne,Robot Control: the Task Function Approach, Oxford University Press, 1990.

  34. J. Sokolowski and J.P. Zolesio, “Introduction to shape optimization,”Computational Mathematics, Springer-Verlag, 1991.

  35. L.E. Weiss, A.C. Sanderson, and C.P. Neuman, “Dynamic sensor-based control of robots with visual feedback,”IEEE transactions of Robotics and Automation, Vol. RA3, No. 5, pp. 404–417, 1987.

    Google Scholar 

  36. D. Westmore and W. Wilson, “Direct dynamic control of a robot using end-point mounted camera and Kalman filter position estimation,”Proceedings IEEE Int. Conf. on Robotics and Automation, Sacramento, pp. 2376–2384, 1991.

  37. J.P. Zolesio, “Un résultat d'existence de vitesse convergente en optimisation de domaine,” C.R.A.S. t282, 1976.

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  1. Place du Maréchal de Lattre de Tassigny, CEREMADE, Université Paris-Dauphine, Paris cedex 16, 75775, France

    Luc Doyen

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Doyen, L. Mutational equations for shapes and vision-based control. J Math Imaging Vis 5, 99–109 (1995). https://doi.org/10.1007/BF01250522

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