A.H. Gosline (andrewg [at] cim.mcgill.ca)

S.E. Salcudean (tims [at] ece.ubc.ca)

J. Yan (josephy [at] ece.ubc.ca)

Haptic Simulation of Linear Elastic Media with Fluid Pockets

Robotics and Control Laboratory 2

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.

Photos appear courtesy of Iman Brouwer and Simon DiMaio

Robotics and Control Laboratory 3

Fast Deformable Methods• Spring-Mass-Damper

Cotin et al. (2000)D’Aulignac et al. (2000)

-Pros:1. Simple to implement.

1. Easy to change mesh.

-Cons: 1. Sensitive to mesh topology

1. Coarse approximation to continuous material.

• BEM, FEMJames & Pai. (2001)DiMaio & Salcudean.

(2002)

-Pros:1. Accurate description of

elastic material.

-Cons:1. Large computational cost.

1. Difficult to change mesh.

1. Requires pre-computation.

Robotics and Control Laboratory 4

Fluid Modeling with FEMNavier-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.

Robotics and Control Laboratory 5

Hydrostatic Fluid Pressure

• Force boundary condition applied normal to fluid-elastic interface.

• Static force balance to distribute force over each element.• Pressure-Volume relationship.

Robotics and Control Laboratory 6

Pressure-Volume Relationship

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

Negative Pressure

Positive Pressure

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

V [%]

P

P vs. V DataLinear Polynomial Fit

Robotics and Control Laboratory 7

• Approximate nonlinear P-V relationship with line fit.

• Slope ~24kPa

• Use as optimal gain for control law.

Pressure-Volume Relationship

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

V [%]

P

P vs. V DataLinear Polynomial Fit

Robotics and Control Laboratory 8

Numerical Method • Proportional feedback update: Pi+1 = Pi + Kp Errori

• Errori = Vo - Vi

• Pressure to Volume transfer function:1. Distribute pressure over boundary

2. Solve FEM

3. Compute volume

• Iterate until Error < Tolerance. KpErrori

Kp

FEM

Vo

Vi

Errori

-

Disturbancefrom tool

Robotics and Control Laboratory 9

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%

CompressibleFluid

IncompressibleFluid

Robotics and Control Laboratory 10

Phantom Construction• 13% type B Gelatin.• 3% Cellulose for speckle.• Glove finger tip filled with fluid.

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Experimental Apparatus

• 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.

US Probe Phantom

Motion Stage

Force Sensor

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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

9mm Displacement

Largest deviation~ 11%

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Surface Tracked Results

0 0.01 0.02 0.03 0.04 0.05 0.060

0.01

0.02

0.03

0.04

0.05

0.06

x [m]

z [m

]FEM Node PositionsTracked Markers

Displaced Surface

Fixed Surface

Robotics and Control Laboratory 18

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

Robotics and Control Laboratory 21

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

Robotics and Control Laboratory 23

Questions ??

Acknowledgements•Rob Rohling for OptoTrak and Ultrasound.

•Simon DiMaio and RCL Labmates

•Simon Bachman and Technicians

Robotics and Control Laboratory 24

Pressure, Volume and Flow

• Bernoulli’s Equation:For incompressible, steady nonviscous flow,

P + ½ V2 + gh = constant along streamline

• Navier-Stokes Equations:

VgpVVt

V 2

Robotics and Control Laboratory 25

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

0 5 10 15 20 25 300

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Strain [%]

Str

ess

[Pa]

Compression ExperimentYoung's Modulus = 15.2kPa

• Linear elastic to ~ 15% strain.

• E ~ 15.2 kPa

Robotics and Control Laboratory 27

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

K u = f

Robotics and Control Laboratory 28

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

Pi+1Kp

FEM

Z-1

Vo

Vi Pi

Errori

-

Disturbancefrom tool

Robotics and Control Laboratory 29

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.

• Interactive Haptic Simulation in 3D.• Investigate validity of pressure prediction.• Validation for modeling of vascular anatomy.• Psychophysics experiments.

Future Work

Robotics and Control Laboratory 30

Acknowledgements

• Rob Rohling for OptoTrak and Ultrasound.

• Simon DiMaio and RCL Labmates

• Simon Bachman and Technicians