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 !!