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

Keyhole Surgery Surgery involves soft tissues Need to model deformation simulation

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

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

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

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

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

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

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

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

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 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

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