surgical thread simulation j. lenoir, p. meseure, l. grisoni, c. chaillou alcove/lifl inria futurs,...
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Surgical Thread Simulation
J. Lenoir, P. Meseure, L. Grisoni, C. Chaillou
Alcove/LIFL INRIA Futurs, University of Lille 1
Outline
• Context• Geometric model• Mechanical model• Physical constraints management• Results• Conclusion and Perspectives
Context
• Surgical Simulators
• Need models of thread [Pai02]
• 3-sided model– Geometric model (rendering)– Mechanical model– Collision model
Mechanical model
Geometric model
Collision model
positions
forcespositions
Outline
ContextGeometric model
• Mechanical Model
• Physical constraints management
• Results
• Conclusion and Perspectives
Geometric model (a)
• Visual model = Axis with a volumetric skinning
• Axis = a spline curveDesired continuity with few control points
0
( , ) ( ) ( )n
i ii
P s t b s q t
s : parametric abscissas [0..1]
t : time
qi : control pointsbi : basis functions
Geometric model (b)
• Implemented splines:– Catmull-Rom (De Casteljau) (C1)– Cubic uniform B-Spline (C2)– NUBS (generic)
• Skinning by a generalized cylinder
Outline
ContextGeometric modelMechanical Model
• Physical constraints management
• Results
• Conclusion and Perspectives
Mechanical model (a)
• Mass-spring model [Provot95]– Discrete models are hard to identify
• Finite Element Model [Picinbono01]– No rest shape for a thread
• Lagrangian model [Rémion99]– Well adapted for curve– Various energies support (including
continuous)Identification is automatic
Mechanical model (b)
• Lagrangian equations :
With:K Kinetic energy,βi Degree of freedom, Qi Work of the external forces,E Deformation and gravitational energy,n Number of degrees of freedom.
( ) for i 0..nii i i
d K K EQ
dt
Mechanical model (c)
• Degrees of freedom = control points positions
• Lagrangian equations applied to splines:
z
y
x
z
y
x
B
B
B
A
A
A
M
M
M
00
00
00 With :
1
0
)()( dssbsbmM jiij
iqAi
B{x,y,z}, terms of potential energies.
z}y,{x, and0..n i with iq
Mechanical model (d)Deformation energies
• Discrete deformation energy:Stretch and bend springs [Provot95], no twisting yet
• Continuous deformation energy [Terzopoulos 87], [Nocent 01]
• Approximation of a continuous stretching energy:– Current length l and rest length l0 , computed by
sampling– Evaluation of by numerical variation of
2200 ))/(1(
2
1llkElE
iq
E
iq
Mechanical model (e)Resolution
• Properties of the matrix M:– symmetric– constant over time– band (thanks to the spline locality property)
• Real-time aspect:System resolved by pre-computing a LU decomposition
• A=M-1B => resolution in O(n)
• A is numerically integrated to get qi(alpha)
1
0
)()( dssbsbmM jiij
Outline
ContextGeometric modelMechanical ModelPhysical constraints management
• Results
• Conclusion and Perspectives
Physical constraints management (a)Unilateral constraints
Collisions and self-collisions
Collision sphere of another object
• The collision model is constrained by the simulation test-bed– Approximation by spheres– Penalty method
Physical constraints management (b)Bilateral constraints
• Constraints by Lagrangian multipliers Extension of the Lagrangian equations:
E
B
B
B
A
A
A
LLL
LM
LM
LM
z
y
x
z
y
x
zyx
Tz
Ty
x
T
0
00
00
00
n
kckck
c
kkki
ii
ii
qqEqL
Lq
EQ
q
K
q
K
dt
d
0
0
),(
.)(
=> extended matrix equation system:
for c=0..nb constraints-1
for i=0..n
Physical constraints management (c) Bilateral constraints
• Some constraints managed by Lagrangian multipliers on a thread :– Fixing 3 degrees of freedom
of a point = a fixed point
– Fixing 2 degrees of freedom of a point =the point can move in 1 direction
– Fixing 1 degree of freedom of a point =the point can move on a plane
Outline
ContextGeometric modelMechanical ModelPhysical constraints managementResults
• Conclusion and Perspectives
Results (a)
Computer: Pentium IV1.7 GhzNumerical integration: Implicit Euler [Hilde01]Energy: Springs
Number of controlpoints (n)
Number ofconstraints
Time computation(ms)
30 6 4.5
30 9 6.1
50 9 10.7
Cost analysis :
Resolution without constraints in O(n)
Resolution with c constraints in O(cn2+c2n+c3)
Results (b)
Some videos :
Collisions The 3 types of implemented constraints
Results (c)
Some videos :
Self-collisions
Conclusion and future works
• Conclusion:– Mechanical simulation of threads in interactive time
• Future works:– Use of a correct continuous deformation energy
including twisting– Manage self-collisions via the Lagrangian
multipliers and implement others constraints– Offer a mechanical multi-resolution for more precise
interaction (knot creation, sewing…)
Thank you !!
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