1. 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

2. Robotics and Control Laboratory2 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

3. Robotics and Control Laboratory3 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, FEM James & 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.

4. Robotics and Control Laboratory4 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.

5. Robotics and Control Laboratory5 Hydrostatic Fluid Pressure Force boundary condition applied normal to fluid-elastic interface. Static force balance to distribute force over each element. Pressure-Volume relationship.

6. Robotics and Control Laboratory6 Pressure-Volume Relationship Negative Pressure Positive Pressure

7. Robotics and Control Laboratory7 Approximate nonlinear P-V relationship with line fit. Slope ~24kPa Use as optimal gain for control law. Pressure-Volume Relationship

8. Robotics and Control Laboratory8 Numerical Method Proportional feedback update: P i+1 = P i + K p Error i Error i = V o - V i Pressure to Volume transfer function: 1.Distribute pressure over boundary 2.Solve FEM 3.Compute volume Iterate until Error < Tolerance. K p Error i KpKp FEM VoVo ViVi Error i - Disturbance from tool

9. Robotics and Control Laboratory9 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% Compressible Fluid Incompressible Fluid

10. Robotics and Control Laboratory10 Phantom Construction 13% type B Gelatin. 3% Cellulose for speckle. Glove finger tip filled with fluid.

11. Robotics and Control Laboratory11 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 ProbePhantom Motion Stage Force Sensor

12. Robotics and Control Laboratory12 Mesh Generation

13. Robotics and Control Laboratory13 US Contour Results No Displacement

14. Robotics and Control Laboratory14 US Contour Results 3mm Displacement

15. Robotics and Control Laboratory15 US Contour Results 6mm Displacement

16. Robotics and Control Laboratory16 US Contour Results 9mm Displacement Largest deviation ~ 11%

17. Robotics and Control Laboratory17 Surface Tracked Results

18. Robotics and Control Laboratory18 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.

19. Robotics and Control Laboratory19 Simulation: Volume Response

20. Robotics and Control Laboratory20 Simulation: Pressure Response

21. Robotics and Control Laboratory21 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.

22. Robotics and Control Laboratory22 Future Work Interactive haptic simulation of fluid-filled structures in 3D Investigate validity of pressure computation Validate for vascular anatomy Psychophysics experiments

23. Robotics and Control Laboratory23 Questions ?? Acknowledgements Rob Rohling for OptoTrak and Ultrasound. Simon DiMaio and RCL Labmates Simon Bachman and Technicians

24. Robotics and Control Laboratory24 Pressure, Volume and Flow Bernoulli’s Equation: For incompressible, steady nonviscous flow, P + ½  V 2 +  gh = constant along streamline Navier-Stokes Equations:

25. Robotics and Control Laboratory25 Approach Linear FEM with condensation –Accurate elastic model. –Condensation. –Interior nodes. Hydrostatic Fluid Pressure –Incompressible fluid enclosures. –Flow relationships. –Force boundary condition.

26. Robotics and Control Laboratory26 Gelatin Properties Linear elastic to ~ 15% strain. E ~ 15.2 kPa

27. Robotics and Control Laboratory27 Linear Elastic Finite Elements Hooke’s Law, σ = D ε E(u) strain = ½ ∫ Ω ε T σ dx, ε = Bu = ½ ∫ Ω (Bu) T DBu dx δE(u) strain = 0 = ∫ Ω B e T D B e u dx – f K u = f

28. Robotics and Control Laboratory28 Numerical Method Proportional feedback control method. Pressure update law: P i+1 = P i + K Error i FEM transfer function computes V with P as input. Iterate until Error < Tol. “Tune” the controller for optimal performance P i+1 KpKp FEM Z -1 VoVo ViVi PiPi Error i - Disturbance from tool

29. Robotics and Control Laboratory29 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

30. Robotics and Control Laboratory30 Acknowledgements Rob Rohling for OptoTrak and Ultrasound. Simon DiMaio and RCL Labmates Simon Bachman and Technicians