ray tracing and stability analysis of parametric systems fabrice le bars
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10/06/2009 - 2Ray tracing and stability analysis of parametric systems
> Plan
1. Introduction
2. Ray tracing
3. Stability analysis of a parametric system
4. Conclusion
10/06/2009 - 4Ray tracing and stability analysis of parametric systems
Goal : Show similarities between 2 problems apparently different : ray tracing and parametric stability analysis
Use of interval analysis
Introduction
10/06/2009 - 6Ray tracing and stability analysis of parametric systems
Ray tracing
Description– Ray tracing, ray casting
– 3D scene display
– Method : build the reverse light path starting from the screen to the object
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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
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Ray tracing
Problem description– A ray assiociated with the pixel
satisfies
10/06/2009 - 9Ray tracing and stability analysis of parametric systems
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
10/06/2009 - 10Ray tracing and stability analysis of parametric systems
Ray tracing
The ray associated with intersects the object if
with
d 0,gp,d 0
gp,d fp1 .d,p2 .d,d
p
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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 :
d p mind 0
gp,d 0
d
p
d p
10/06/2009 - 12Ray tracing and stability analysis of parametric systems
Ray tracing
Computation of– If
Then
Moreover, if
We can use a dichotomy to get
g0,a 0,
ga,b , 0
d a,b
gb 0
d
a b
d
d
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Ray tracing
Computation of
– Interval computations are used to find – A dichotomy finds
a,bd
d
10/06/2009 - 14Ray tracing and stability analysis of parametric systems
Ray tracing
Parametric version– now depends on– If
Then
Moreover if
We can use a dichotomy to get for each
gp, 0,a 0, gp,b , 0
gd
p, a,b , 0
d p a,b
gp,d p p
d p
10/06/2009 - 15Ray tracing and stability analysis of parametric systems
Ray tracing
From to d pd
a ba b
d d p
10/06/2009 - 17Ray tracing and stability analysis of parametric systems
Stability analysis of a parametric system
10/06/2009 - 18Ray tracing and stability analysis of parametric systems
Stability
where is retrieved from the Routh table
Stability analysis of a parametric system
rp 0
(Routh)
r
Ps,p stable all its roots have a real part 0
10/06/2009 - 19Ray tracing and stability analysis of parametric systems
stability
Stability analysis of a parametric system
(Routh)
rp, 0
Ps,p is stable all its roots have a real part
10/06/2009 - 20Ray tracing and stability analysis of parametric systems
Stability analysis of a parametric system
Example : Ackermann
is stable if
Ps,p s3 p1 p2 2s2 p1 p2 2s
2p1p2 6p1 6p2 2.25. #
rp, min
p1 p2 2 3
p1 12 p2 12 0.25 2 p1 p2 2p1 p2 3 4 4 2
2p1 3p2 3 15.75 p1 p2 21 2 0
10/06/2009 - 21Ray tracing and stability analysis of parametric systems
Stability degree
Stability analysis of a parametric system
p min 0
rp, 0
#
10/06/2009 - 22Ray tracing and stability analysis of parametric systems
Stability analysis of a parametric system
Similarities with ray tracing
Ray tracing Stability degree
d
g r
d p min d 0
gp,d 0
d p min 0
rp, 0
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Stability analysis of a parametric system
10/06/2009 - 25Ray tracing and stability analysis of parametric systems
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
10/06/2009 - 26Ray tracing and stability analysis of parametric systems
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.
10/06/2009 - 38Ray tracing and stability analysis of parametric systems
Ray tracing
Division in p and in d
a b
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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
10/06/2009 - 42Ray tracing and stability analysis of parametric systems
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 :
minPs unstable
. #
10/06/2009 - 43Ray tracing and stability analysis of parametric systems
Stability analysis of a parametric system
– The stability degree becomes :
– With
the polynomial is stable if (Routh) :
p minPs ,p unstable
. #