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Modeling interaction with deformable objects in real-time Diego dAulignac GRAVIR/INRIA Rhone-Alpes France

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Keyhole Surgery Surgery involves soft tissues Need to model deformation simulation

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Liver Model [Boux et al., ISER, 2000] HeterogenousNon-linear skinParenchyma

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Echography In collaboration with TIMC laboratory in Grenoble, France Interpolation (translation, rotation, deformation) Echographic images at sample points

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Thigh Model Identification (error minimization) In collaboration with UC Berkeley Presented at IROS 1999

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Integration 2nd order non-linear differential equation Convert to 1st order system

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Explicit Integration Runge-Kutta method with s stages Order of consistency vs. stages

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Linear Stability Im Re At least 2 solutions: Design better computer Design better algorithm

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Simulation Achitecture –SGI Onyx2 Compexity –370 facets –1151 tetrahedrons –3399 springs Frequency –150Hz

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Implicit Integation 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

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Simulation Haptic interaction with physical model Echographic image generation Timestep: 0.01s Octane 175Mhz

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Static Resolution Principle of virtual work: internal and external forces are balanced Linear case: Pre-inversion (if enough space) No large strain No rotation No material non-linearity Non-linear case: Stiffness matrix changes with displacement

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Newton Iteration Full Newton-Rapson method: Reevaluation of Jacobian Faster convergence Modified Newton-Rapson method: Constant Jacobian Slower Convergence

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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

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Result 1157 tetraheadrons Iterative non-linear resolution Rotational invarience (N.B. Real-time animation) 1157 tetraheadrons Iterative non-linear resolution Rotational invarience (N.B. Real-time animation) 60 iterations/sec on SGI Octane 175Mhz Pseudo-dynamic

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Static vs. Dynamic Static –Have clearly defined boundary conditions –No liver throwing contest Dynamic –Control of viscosity and inertia –Transient response

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Future Directions Multi-grid methods –More rapid propagation Parallelisation –Divide into sub-regions –e.g. Block Jacobi iteration

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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

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Ordinary Differential Equations

Ordinary Differential Equations

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