Haptic Simulation of Linear Elastic
Media with Fluid Pockets
A.H. Gosline (andrewg [at] cim.mcgill.ca)
S.E. Salcudean (tims [at] ece.ubc.ca)
J. Yan (josephy [at] ece.ubc.ca)
Introduction
Haptic simulation becoming
increasingly popular for
medical training.
Issues addressed:
• Tissue models assume
continuous elastic
material.
• Fluid structures ignored.
• Haptics requires update
rates of order 500 Hz.
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Photos appear courtesy of
Iman Brouwer and Simon DiMaio
Fast Deformable Methods
• Spring-Mass-Damper
• BEM, FEM
Cotin et al. (2000)
D’Aulignac et al. (2000)
James & Pai. (2001)
DiMaio & Salcudean. (2002)
-Pros:
-Pros:
1. Simple to implement.
1. Accurate description of
elastic material.
1. Easy to change mesh.
-Cons:
-Cons:
1. Sensitive to mesh topology
1. Large computational cost.
1. Coarse approximation to
continuous material.
1. Difficult to change mesh.
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1. Requires pre-computation.
3
Fluid Modeling with FEM
Navier-Stokes Fluid. Basdogan et al. (2001), Agus et al. (2002).
– Dynamic analysis, large computational effort.
– In surgery simulators for graphics only (10-15Hz).
Irrotational Elastic Elements. Dogangun et al. (1993, 1996).
– Statics and Dynamics (not flow).
– Decoupling of fluid-elastic.
– Poor scaling.
Hydrostatic Fluid Pressure. De and Srinivasan (1999).
– Quasi-static.
– Arbitrary pressure/volume relationship.
– Force boundary condition.
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Hydrostatic Fluid Pressure
• Force boundary condition applied normal to fluid-elastic
interface.
• Static force balance to distribute force over each element.
• Pressure-Volume relationship.
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Pressure-Volume Relationship
0.01
0
10000
-0.01
-0.02
P vs. V Data
Linear Polynomial Fit
8000
-0.03
6000
-0.04
4000
-0.05
2000
-0.02
-0.01
0
0.01
0.02
0.03
Negative Pressure
0.04
0.01
0
P
-0.06
-0.03
0
-2000
-0.01
-4000
-0.02
-6000
-0.03
-8000
0.6
-0.04
-0.05
-0.06
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Positive Pressure
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0.7
0.8
0.9
1
V [%]
1.1
1.2
1.3
1.4
Pressure-Volume Relationship
10000
• Slope ~24kPa
• Use as optimal
gain for control
law.
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P vs. V Data
Linear Polynomial Fit
8000
6000
4000
2000
P
• Approximate
nonlinear P-V
relationship with
line fit.
0
-2000
-4000
-6000
-8000
0.6
7
0.7
0.8
0.9
1
V [%]
1.1
1.2
1.3
1.4
Numerical Method
•
•
•
Proportional feedback update: Pi+1 = Pi + Kp Errori
Errori = Vo - Vi
Pressure to Volume transfer function:
1.
2.
3.
•
Distribute pressure over boundary
Solve FEM
Vo
Compute volume
Iterate until Error < Tolerance.
Errori
Kp
-
Vi
KpErrori
FEM
Disturbance
from tool
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Performance
• With P-V slope as gain,
the performance is good.
• Convergence to 1%
tolerance in maximum 1
iteration for small strains.
• Robust to large
deformations of up to
30%
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Compressible
Fluid
Incompressible
Fluid
Phantom Construction
• 13% type B Gelatin.
• 3% Cellulose for speckle.
• Glove finger tip filled with fluid.
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Experimental Apparatus
Force Sensor
Motion Stage
US Probe
•
•
•
•
•
Phantom
Ultrasound probe to capture fluid pocket shape (left).
Top surface of phantom marked for surface tracking (center).
Force sensor (right).
3DOF Motion Stage for compression (far right).
All components rigidly mounted to aluminum base plate.
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Mesh Generation
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US Contour Results
No Displacement
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US Contour Results
3mm Displacement
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US Contour Results
6mm Displacement
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US Contour Results
Largest deviation
~ 11%
9mm Displacement
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Surface Tracked Results
0.06
FEM Node Positions
Tracked Markers
Displaced Surface
0.05
z [m]
0.04
0.03
0.02
0.01
Fixed Surface
0
0
0.01
0.02
0.03
x [m]
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0.04
0.05
0.06
Real-time Haptic Simulation
• Incompressible fluid added to the needle insertion simulator
by DiMaio and Salcudean (2002).
• Software runs at fixed update rate of 512 Hz.
• Haptic loop fixed at 2 iterations per update.
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Simulation: Volume Response
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Simulation: Pressure Response
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Conclusions
• Linear FEM with hydrostatic pressure predicts the
deformation of an incompressible fluid-filled
phantom in a realistic manner up to approximately
15% strain.
• Fast numerical method optimized with
understanding of P-V relation gives fast
convergence.
• Matrix condensation allows for real-time haptic
rendering of a fluid-filled deformable object at
512Hz.
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Future Work
• Interactive haptic simulation of fluid-filled structures
in 3D
• Investigate validity of pressure computation
• Validate for vascular anatomy
• Psychophysics experiments
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Acknowledgements
•Rob Rohling for OptoTrak and Ultrasound.
•Simon DiMaio and RCL Labmates
•Simon Bachman and Technicians
Questions ??
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Pressure, Volume and Flow
• Bernoulli’s Equation:
For incompressible, steady nonviscous flow,
P + ½ V2 + gh = constant along streamline
• Navier-Stokes Equations:
 V

2

 V  V    p  g   V
 t

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Approach
• Linear FEM with condensation
– Accurate elastic model.
– Condensation.
– Interior nodes.
• Hydrostatic Fluid Pressure
– Incompressible fluid enclosures.
– Flow relationships.
– Force boundary condition.
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Gelatin Properties
5000
4500
Compression Experiment
Young's Modulus = 15.2kPa
4000
Stress [Pa]
3500
3000
2500
2000
1500
•
•
1000
500
0
0
5
Linear elastic to ~ 15% strain.
E ~ 15.2 kPa
10
15
20
Strain [%]
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25
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Linear Elastic Finite Elements
Hooke’s Law, σ = D ε
E(u)strain
= ½∫Ω εTσ dx, ε = Bu
= ½∫Ω(Bu)T DBu dx
δE(u)strain = 0 = ∫ΩBeTD Beu dx – f
Ku=f
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Numerical Method
• Proportional feedback
control method.
• Pressure update law:
Pi+1 = Pi + K Errori
• FEM transfer function
computes V with P as
input.
• Iterate until Error < Tol.
• “Tune” the controller for
optimal performance
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Errori
Vo
Kp
-
Vi
Z-1
FEM
Disturbance
from tool
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Pi+1
Pi
Conclusions
• Linear FEM predicts 3D deformation of an incompressible
fluid-filled cavity in realistic manner.
• Optimized gain allows fast convergence.
• Linear FEM and matrix condensation allow for haptic display.
Future Work
•
•
•
•
Interactive Haptic Simulation in 3D.
Investigate validity of pressure prediction.
Validation for modeling of vascular anatomy.
Psychophysics experiments.
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Acknowledgements
• Rob Rohling for OptoTrak and Ultrasound.
• Simon DiMaio and RCL Labmates
• Simon Bachman and Technicians
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