Surgical Thread Simulation J. Lenoir, P. Meseure, L. Grisoni, C. Chaillou Alcove/LIFL INRIA Futurs, University of Lille 1

Outline Context Geometric model Mechanical model Physical constraints management Results Conclusion and Perspectives

Context Surgical Simulators Need models of thread [Pai02] 3-sided model –Geometric model (rendering) –Mechanical model –Collision model Mechanical model Geometric model Collision model positions forcespositions

Outline Context  Geometric model Mechanical Model Physical constraints management Results Conclusion and Perspectives

Geometric model (a) Visual model = Axis with a volumetric skinning Axis = a spline curve Desired continuity with few control points s : parametric abscissa s  [0..1] t : time q i : control points b i : basis functions

Geometric model (b) Implemented splines: –Catmull-Rom (De Casteljau) (C 1 ) –Cubic uniform B-Spline (C 2 ) –NUBS (generic) Skinning by a generalized cylinder

Outline Context Geometric model  Mechanical Model Physical constraints management Results Conclusion and Perspectives

Mechanical model (a) Mass-spring model [Provot95] –Discrete models are hard to identify Finite Element Model [Picinbono01] –No rest shape for a thread Lagrangian model [Rémion99] –Well adapted for curve –Various energies support (including continuous) Identification is automatic

Mechanical model (b) Lagrangian equations : With: K Kinetic energy, β i Degree of freedom, Q i Work of the external forces, E Deformation and gravitational energy, n Number of degrees of freedom.

Mechanical model (c) Degrees of freedom = control points positions Lagrangian equations applied to splines: With : B {x,y,z}, terms of potential energies.

Mechanical model (d) Deformation energies Discrete deformation energy: Stretch and bend springs [Provot95], no twisting yet Continuous deformation energy [Terzopoulos 87], [Nocent 01] Approximation of a continuous stretching energy: –Current length l and rest length l 0, computed by sampling –Evaluation of by numerical variation of

Mechanical model (e) Resolution Properties of the matrix M: –symmetric –constant over time –band (thanks to the spline locality property) Real-time aspect: System resolved by pre-computing a LU decomposition A=M-1B => resolution in O(n) A is numerically integrated to get qi(alpha)

Outline Context Geometric model Mechanical Model  Physical constraints management Results Conclusion and Perspectives

Physical constraints management (a) Unilateral constraints Collisions and self-collisions Collision sphere of another object The collision model is constrained by the simulation test-bed –Approximation by spheres –Penalty method

Physical constraints management (b) Bilateral constraints Constraints by Lagrangian multipliers Extension of the Lagrangian equations: => extended matrix equation system: for c=0..nb constraints-1 for i=0..n

Physical constraints management (c) Bilateral constraints Some constraints managed by Lagrangian multipliers on a thread : –Fixing 3 degrees of freedom of a point = a fixed point –Fixing 2 degrees of freedom of a point = the point can move in 1 direction –Fixing 1 degree of freedom of a point = the point can move on a plane

Outline Context Geometric model Mechanical Model Physical constraints management  Results Conclusion and Perspectives

Results (a) Computer:Pentium IV1.7 Ghz Numerical integration:Implicit Euler [Hilde01] Energy:Springs Cost analysis : Resolution without constraints in O(n) Resolution with c constraints in O(cn 2 +c 2 n+c 3 )

Results (b) Some videos : CollisionsThe 3 types of implemented constraints

Results (c) Some videos : Self-collisions

Conclusion and future works Conclusion: –Mechanical simulation of threads in interactive time Future works: –Use of a correct continuous deformation energy including twisting –Manage self-collisions via the Lagrangian multipliers and implement others constraints –Offer a mechanical multi-resolution for more precise interaction (knot creation, sewing…)

Thank you !!