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