Transcript Ray tracing and stability analysis of parametric systems Fabrice LE BARS

Ray tracing and stability
analysis of parametric
systems
Fabrice LE BARS
> Plan
1.
2.
3.
4.
Ray tracing and stability analysis of parametric systems
Introduction
Ray tracing
Stability analysis of a parametric system
Conclusion
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Introduction
Ray tracing and stability analysis of parametric systems
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Introduction
 Goal : Show similarities between 2 problems
apparently different : ray tracing and parametric
stability analysis
 Use of interval analysis
Ray tracing and stability analysis of parametric systems
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Ray tracing
Ray tracing and stability analysis of parametric systems
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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
Screen
Eye
Object
Ray
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Ray tracing
 Problem description
– A ray assiociated with the pixel
satisfies
Ray tracing and stability analysis of parametric systems
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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
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Ray tracing
The ray associated with p intersects the object if

d 0, g
p, d0
with
g
p, df
p 1 . d, p 2 . d, d
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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 p :
d 
p min d
d0
g
p,d
0
d 
p
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Ray tracing
 Computation of d 
– If


g
0, a
0, 
g
b0
Then
Moreover, if
d  
a, b

, 0
g 
a, b
We can use a dichotomy to get d 
Ray tracing and stability analysis of parametric systems
d
a
b
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Ray tracing
 Computation of d 
a, b
– Interval computations are used to find 
– A dichotomy finds d 
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Ray tracing
 Parametric version
p
p, dnow depends on p  
– g
– If

g

p
,
0, a
0, 
g

p
, b, 0
Then

d 
p

a, b
Moreover if

g


, 0
p
,
a, b

d
We can use a dichotomy to get d  for each p
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Ray tracing
p
 From d  to d 
d
d 
p

a
Ray tracing and stability analysis of parametric systems
b
a
b
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Ray tracing
Ray tracing and stability analysis of parametric systems
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Stability analysis of a parametric
system
Ray tracing and stability analysis of parametric systems
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Stability analysis of a parametric
system
 Stability
P
s, pstable  all its roots have a real part 0
(Routh)
 r
p0
where r is retrieved from the Routh table
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Stability analysis of a parametric
system

 stability
P
s, pis  stable  all its roots have a real part 
(Routh)
 r
p, 0
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Stability analysis of a parametric
system
 Example : Ackermann
P
s, p  s 3 
p 1 p 2 2
s 2 
p 1 p 2 2
s

2p 1 p 2 6p 1 6p 2 2. 25.
#
is  stable if
r
p, min
p 1 p 2 2 3

p 1 12 
p 2 12 0. 25 2

p 1 p 2 2

p 1 p 2 3 442 
0
2
p 1 3

p 2 315. 75 

p 1 p 2 2

1 2 
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Stability analysis of a parametric
system
 Stability degree
p min  #
0
r
p,
0

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Stability analysis of a parametric
system
 Similarities with ray tracing
Ray tracing
d


g

r
d 
pmin
d0
g
p,d
0
Ray tracing and stability analysis of parametric systems
Stability degree
d  
pmin
0
r
p,
0

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Stability analysis of a parametric
system
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Conclusion
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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
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References
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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 chialgorithms, 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.
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Ray tracing
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Ray tracing
=> a = d3
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Ray tracing
=> b = d2
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Ray tracing
 Division in d
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Ray tracing
 Division in d
Ray tracing and stability analysis of parametric systems
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Ray tracing
 Division in d
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Ray tracing
 Division in p
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Ray tracing
 Division in p
Ray tracing and stability analysis of parametric systems
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Ray tracing
 Division in p
Ray tracing and stability analysis of parametric systems
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Ray tracing
 Division in p
Ray tracing and stability analysis of parametric systems
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Ray tracing
 Division in p
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Ray tracing
 Division in p and in d
a
Ray tracing and stability analysis of parametric systems
b
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Ray tracing
Ray tracing and stability analysis of parametric systems
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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
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Ray tracing
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Stability analysis of a parametric
system
– Stability degree of an invariant linear system of
caracteristic polynomial P(s) :
 
min
P
sunstable
. #
– We consider an invariant linear system parametized
with a vector of parameter p :
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Stability analysis of a parametric
system
– The stability degree becomes :
p
min
P
s,punstable
. #
– With
the polynomial is stable if (Routh) :
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Stability analysis of a parametric
system
– If we note
We get
P
s , punstable  r min 
p, 0 #
– Therefore, the stability degree is :
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