Modeling interaction with deformable objects in real-time
Diego d’Aulignac
GRAVIR/INRIA Rhone-Alpes
France
Keyhole Surgery
Surgery involves soft tissues
Need to model deformation
simulation
Liver Model [Boux et al., ISER, 2000]
Heterogenous Non-linear
skin Parenchyma
EchographyIn collaboration with TIMC laboratory in Grenoble, France
Interpolation (translation, rotation, deformation)
Echographic images at sample points
Thigh Model
Identification(error minimization)
In collaboration with UC Berkeley
Presented at
IROS 1999
Integration
externalfKxxDxM
x
xY
x
xYf
)(
2nd order non-linear differential equation
Convertto
1st ordersystem
Explicit Integration
s
iijij kahYfk
10
Runge-Kutta method with s stages
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8
stage s
Order of consistency vs. stages
j
s
jjkbYY
1
01
Linear Stability
tM
K
M
D
M
Dz
etx z
2
22
)(Im
Re At least 2 solutions:
• Design better computer
• Design better algorithm
externalfKxxDxM
Simulation• Achitecture
– SGI Onyx2
• Compexity– 370 facets– 1151
tetrahedrons– 3399 springs
• Frequency– 150Hz
Implicit Integation
0)( 011 YYhfY
)()(1
00 YffIt
Y YY
linearisation
Semi-implicit euler
Implicit euler (non-linear system)
A-stable
… but not B-stable
If you know your history, then you would know where you are coming from.
Bob Marley
Over-damped case
Simulation
•Haptic interaction with physical model
•Echographic image generation
Timestep: 0.01s
Octane 175Mhz
Static ResolutionPrinciple of virtual work: internal and external forces are balanced
externalfKu
externalfuuK )(
Linear case:• Pre-inversion (if enough space)
• No large strain
• No rotation
• No material non-linearity
Non-linear case:•Stiffness matrix changes with displacement
Newton Iteration
Full Newton-Rapson method:
•Reevaluation of Jacobian
•Faster convergence
Modified Newton-Rapson method:
•Constant Jacobian
•Slower Convergence
Calculate forces on nodes
Evaluate stiffness matrix K?
Iteratively solve linear system for displacements u
Ku = f
by successive over-relaxation (SOR)
until residual forces < epsilon through Newton-Rapson iteration
Iterative Solution
Divergence•If objects are very soft
•Undercorrection
Result
1157 tetraheadronsIterative non-linear resolution
Rotational invarience
(N.B. Real-time animation)
1157 tetraheadronsIterative non-linear resolution
Rotational invarience
(N.B. Real-time animation)
60 iterations/sec on SGI Octane 175Mhz
Pseudo-dynamic
Static vs. Dynamic
• Static– Have clearly defined boundary conditions – No liver throwing contest
• Dynamic– Control of viscosity and inertia– Transient response
Future Directions
• Multi-grid methods– More rapid propagation
• Parallelisation – Divide into sub-regions– e.g. Block Jacobi iteration
Conclusions
• «Soft» soft-tissues may be simulated using explicit integration
• «Stiff» soft-tissues benefit from implicit methods
• Static analysis– well defined boundary conditions– transient response negligable