1. Ray tracing and stability analysis of parametric systems Fabrice LE BARS

2. Ray tracing and stability analysis of parametric systems10/06/2009 - 2 > Plan 1.Introduction 2.Ray tracing 3.Stability analysis of a parametric system 4.Conclusion

3. Ray tracing and stability analysis of parametric systems10/06/2009 - 3 Introduction

4. Ray tracing and stability analysis of parametric systems10/06/2009 - 4  Goal : Show similarities between 2 problems apparently different : ray tracing and parametric stability analysis  Use of interval analysis Introduction

5. Ray tracing and stability analysis of parametric systems10/06/2009 - 5 Ray tracing

6. Ray tracing and stability analysis of parametric systems10/06/2009 - 6 Ray tracing  Description –Ray tracing, ray casting –3D scene display –Method : build the reverse light path starting from the screen to the object

7. Ray tracing and stability analysis of parametric systems10/06/2009 - 7 Ray tracing  Hypothesis –Objects are defined by implicit functions –The eye is at the origin of a coordinate space R(O,i,j,k) and the screen is at z=1 –The screen is not in the object Eye Screen Object Ray

8. Ray tracing and stability analysis of parametric systems10/06/2009 - 8 Ray tracing  Problem description –A ray assiociated with the pixel satisfies

9. Ray tracing and stability analysis of parametric systems10/06/2009 - 9 Ray tracing  Problem description –The point is in the object if –A pixel displays a point of the object if the associated ray intersects the object

10. Ray tracing and stability analysis of parametric systems10/06/2009 - 10 Ray tracing The ray associated with intersects the object if with

11. Ray tracing and stability analysis of parametric systems10/06/2009 - 11 Ray tracing  Light effects handling –Realism => illumination model –Phong : needs the distance from the eye to the object –We need to compute for each pixel :

12. Ray tracing and stability analysis of parametric systems10/06/2009 - 12 Ray tracing  Computation of –If Then Moreover, if We can use a dichotomy to get

13. Ray tracing and stability analysis of parametric systems10/06/2009 - 13 Ray tracing  Computation of –Interval computations are used to find –A dichotomy finds

14. Ray tracing and stability analysis of parametric systems10/06/2009 - 14 Ray tracing  Parametric version – now depends on –If Then Moreover if We can use a dichotomy to get for each

15. Ray tracing and stability analysis of parametric systems10/06/2009 - 15 Ray tracing  From to

16. Ray tracing and stability analysis of parametric systems10/06/2009 - 16 Ray tracing

17. Ray tracing and stability analysis of parametric systems10/06/2009 - 17 Stability analysis of a parametric system

18. Ray tracing and stability analysis of parametric systems10/06/2009 - 18  Stability where is retrieved from the Routh table Stability analysis of a parametric system (Routh)

19. Ray tracing and stability analysis of parametric systems10/06/2009 - 19  stability Stability analysis of a parametric system (Routh)

20. Ray tracing and stability analysis of parametric systems10/06/2009 - 20 Stability analysis of a parametric system  Example : Ackermann is stable if

21. Ray tracing and stability analysis of parametric systems10/06/2009 - 21  Stability degree Stability analysis of a parametric system

22. Ray tracing and stability analysis of parametric systems10/06/2009 - 22 Stability analysis of a parametric system  Similarities with ray tracing

23. Ray tracing and stability analysis of parametric systems10/06/2009 - 23 Stability analysis of a parametric system

24. Ray tracing and stability analysis of parametric systems10/06/2009 - 24 Conclusion

25. Ray tracing and stability analysis of parametric systems10/06/2009 - 25 Conclusion  Ray tracing and stability degree drawing of a linear system are similar problems  A common algorithm based on intervals and dichotomy has been proposed

26. Ray tracing and stability analysis of parametric systems10/06/2009 - 26 References  L. Jaulin. Solution globale et garantie de problèmes ensemblistes; Application à l'estimation non linéaire et à la commande robuste. PhD thesis, Université Paris XI Orsay, 1994.  S. Bazeille. Vision sous-marine monoculaire pour la reconnaissance d'objets. PhD thesis, Université de Bretagne Occidentale, 2008.  J. Flórez. Improvements in the ray tracing of implicit surfaces based on interval arithmetic. PhD thesis, Universitat de Girona, 2008.  L. Jaulin, M. Kieffer, O. Didrit et E. Walter, Applied interval analysis, Springer-Verlag, London, Great Britain, 2001.  L. Jaulin, E. Walter, O. Lévêque et D. Meizel, "Set inversion for chi- algorithms, with application to guaranteed robot localization", Math. Comput Simulation, 52, pp. 197-210, 2000.  J. Ackermann, "Does it suffice to check a subset of multilinear parameters in robustness analysis?", IEEE Transactions on Automatic Control, 37(4), pp. 487-488, 1992.  J. Ackermann, H. Hu et D. Kaesbauer, "Robustness analysis: a case study", IEEE Transactions on Automatic Control, 35(3), pp. 352-356, 1990.

27. Ray tracing and stability analysis of parametric systems10/06/2009 - 27 Ray tracing

28. Ray tracing and stability analysis of parametric systems10/06/2009 - 28 Ray tracing => a = d3

29. Ray tracing and stability analysis of parametric systems10/06/2009 - 29 Ray tracing => b = d2

30. Ray tracing and stability analysis of parametric systems10/06/2009 - 30 Ray tracing  Division in d

31. Ray tracing and stability analysis of parametric systems10/06/2009 - 31 Ray tracing  Division in d

32. Ray tracing and stability analysis of parametric systems10/06/2009 - 32 Ray tracing  Division in d

33. Ray tracing and stability analysis of parametric systems10/06/2009 - 33 Ray tracing  Division in p

34. Ray tracing and stability analysis of parametric systems10/06/2009 - 34 Ray tracing  Division in p

35. Ray tracing and stability analysis of parametric systems10/06/2009 - 35 Ray tracing  Division in p

36. Ray tracing and stability analysis of parametric systems10/06/2009 - 36 Ray tracing  Division in p

37. Ray tracing and stability analysis of parametric systems10/06/2009 - 37 Ray tracing  Division in p

38. Ray tracing and stability analysis of parametric systems10/06/2009 - 38 Ray tracing  Division in p and in d

39. Ray tracing and stability analysis of parametric systems10/06/2009 - 39 Ray tracing

40. Ray tracing and stability analysis of parametric systems10/06/2009 - 40 Ray tracing  Several objects display handling –We apply the previous algorithm for the function : –Indeed, we have to consider only the first object crossed by the ray

41. Ray tracing and stability analysis of parametric systems10/06/2009 - 41 Ray tracing

42. Ray tracing and stability analysis of parametric systems10/06/2009 - 42 Stability analysis of a parametric system –Stability degree of an invariant linear system of caracteristic polynomial P(s) : –We consider an invariant linear system parametized with a vector of parameter p :

43. Ray tracing and stability analysis of parametric systems10/06/2009 - 43 Stability analysis of a parametric system –The stability degree becomes : –With the polynomial is stable if (Routh) :

44. Ray tracing and stability analysis of parametric systems10/06/2009 - 44 Stability analysis of a parametric system –If we note We get –Therefore, the stability degree is :