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Quasi-Rigid Objects in Contact

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Introduction Existing Models Physical models used to address contact fall into two broad categories: - Rigid body models and deformable body models. – PowerPoint PPT presentation

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Title: Quasi-Rigid Objects in Contact


1
Quasi-Rigid Objects in Contact
2
Introduction
  • Recent paper of M. Pauly, D. Pai, L. Guibas,
    Quasi-Rigid Objects in Contact,Eurographics/ACM
    SIGGRAPH Symposium on Computer Animation (2004)
    presents techniques for modeling contact between
    quasi-rigid objects
  • solids that undergo modest deformation in the
    vicinity of a contact, while the overall object
    still preserves its basic shape.
  • Contact is ubiquitous in real world interactions
    and is one of the most difficult problems in 3D
    computer animation.
  • Applications
  • Bio-medical
  • surgery simulation
  • artificial joints, dental implants
  • Mechanical design
  • wear and tear of industrial parts
  • Physics-based animation
  • movies
  • games

3
Existing Models
  • Physical models used to address contact fall into
    two broad categories
  • - Rigid body models and deformable body models.
  • Rigid body dynamics
  • small number of state variables
  • efficient collision detection
  • contact sensitivity problem (a stool with
    hundreds of legs)
  • Fully deformable (e.g. FEM, mass-spring)
  • accurate modeling of complex materials
    (elasticity, plasticity)
  • too costly for models that hardly deform

4
Quasi-Rigid Objects
  • A new type of model that called quasi-rigid
    combines the benefits of rigid body models for
    dynamic simulation and the benefits of deformable
    models for resolving contacts and producing
    visible deformations.
  • Quasi-rigid means objects whose surfaces can
    undergo modest deformations in the vicinity of a
    contact, while the overall object still preserves
    its basic shape.

5
Surface model
  • Surfaces are represented by point clouds, i.e.,
    sets of point primitives that sample the position
    and normal of the underlying surface.
  • The moving least squares (MLS) surface model to
    define a smooth continuous surface from a set of
    point samples is used.
  • Given a point cloud P as input, the corresponding
    MLS surface S is defined as the stationary set of
    a projection operator ?P(x), i.e.,
  • S x ??3 ? P(x) x.
  • The projection ? P(x) is evaluated by first
    fitting a local least squares plane that serves
    as a local parametrization domain.
  • A second least squares optimization then yields a
    bivariate polynomial g(u v) that locally
    approximates the surface. The projection of x
    onto S is given as ? P(x) qg(00) ?n, where q
    is the origin and n the normal of the reference
    plane.

6
Quasi-Rigid Objects
  • Physical model
  • A point force acting on a quasi-rigid solids
    leads to a deformation that is restricted to a
    small, local area (active region) around the
    point of contact, while keeping the overall shape
    intact.
  • Linear elasticity Global system response by
    superposition
  • forces and displacements evaluated on surface
    only

7
Quasi-Rigid Objects
  • Surface model
  • point cloud representation for modeling
    consistent, highly dynamic contact surface

8
Physical Model
  • A widely used method in contact mechanics is the
    Boussinesq approximation, which models the
    surface around a point of contact as an elastic
    half-space
  • Boussinesq approximation
  • infinite elastic half-space

Poissons ratio
force at x
displacement at y due to force at x
shear modulus
9
Physical Model
  • Boussinesq approximation
  • system response function exhibits a rapid
    fall-off with increasing distance r

10
Physical Model
  • Linear elasticity
  • superposition

total displacement at y
11
Volume Preservation
  • Volume preservation can be formulated as the
    following constraint on the response function f
    (r)
  • A plausible analytical function can be obtaine by
    shifting Boussinesq function in the positive
    z-direction and modulating it with a Gaussian

12
Volume Preservation
  • Figure illustrates volume preservation for
    quasi-rigid objects on a simple example of a ball
    in contact with a plane, using the transfer
    function
  • (a) original configuration prior to contact
    resolution, (b) plane deformable, sphere rigid,
    (c) sphere deformable, plane rigid, (d) both
    models deformable.

13
Discretization
  • In a discrete setting, we need to find relations
    between the displacement ui that node qi
    experiences due to the tractions pj acting on all
    other nodes.

  • system response matrix

vector of tractions p1,...,pNT
vector of displacements u1,...,uNT
matrix coefficient
14
Discretization
  • Approximate system response at discrete nodes
    (point samples)

shape function
force at node j
displacement at node i
15
Contact
  • Given the model for quasi-rigid objects defined
    above, our goal is to determine the contact
    surface S of two solids A and B and compute the
    forces that act on this surface.
  • The bodies A and B are represented by two point
    clouds PA and PB that define two corresponding
    MLS surfaces SA and SB.
  • During a simulation, the two bodies might collide
    and interpenetrate.

16
Contact
  • Collision detection
  • Static bounding volume hierarchies (small
    deformations). If a collision is detected, we
    compute two sets of active nodes QA and QB
  • Contact resolution
  • Compute forces and displacements that resolve
    contact
  • Contact surface
  • Find contact surface that is consistent for both
    models

17
Contact Resolution
  • Collision detection determines points that
    potentially experience displacements (active
    nodes)
  • Find corresponding point for each active node

active nodes
corresponding nodes
18
Contact Resolution
  • A conservative estimate (that might include some
    points that are not penetrating, but close to the
    intersection region) is used
  • To further improve performance, a bounding sphere
    hierarchy for fast intersection culling is used.

19
Contact Resolution
  • Separation of active nodes
  • Given the sets of active nodes QA and QB, we need
    to find tractions on these points such that the
    corresponding displacements lead to deformed
    surfaces SA and S B that are in contact, but do
    not interpenetrate.
  • Let S SA ? S B be the (as yet unknown)
    contact surface between the deformed models.
  • If the two surfaces touch at a point x ? S, the
    corresponding normal tractions at this point on
    each of the two surfaces are equal, i.e., pA(x)
    pB(x)

20
Contact Resolution
  • Let qA ? QA be an active node of model A and
    assume for now that after deformation this point
    will end up on the final contact surface.
  • Then there exists a corresponding point xA ? SB
    such that its position in the final configuration
    coincides with the deformed position of qA and
    the normal tractions at both points are equal.

21
Contact Resolution
  • Separation of active nodes
  • Initial separation
  • Final separation

22
Contact Resolution
  • Condition for contact resolution
  • Non-negative separation si 0
  • Non-negative force pi 0

23
Contact Resolution
  • Finally, we observe that the separation si and
    the traction pi are complementary at each
    point, at least one of them is zero to be able
    to have non-zero traction, we must have zero
    separation, and to have non-zero separation, we
    must have zero traction. Therefore, we can
    combine these complementarity conditions to
    obtain
  • Linear Complementarity Problem (LCP)
  • Solved using Lemkes method

24
Linear Complementarity Problem
  • The linear complementarity problem in linear
    algebra consists of starting with a known
    n-dimensional column vector q and a known nn
    matrix M, and finding two n-dimensional vectors w
    and z such that
  • q w - Mz
  • wi 0 and zi 0 for each i
  • wi zi 0 (i.e. either wi 0 or zi 0) for
    each i
  • There are several algorithms (e.g. Lemke's
    algorithm) dealing with specific cases of the
    linear complementarity problem. A linear
    complementarity problem has a unique solution if
    and only if M is a P-matrix.
  • A P-matrix is a complex square matrix with every
    principal minor gt 0.

25
Linear Complementarity Problem
  • An LCP in standard form is specified by a pair b,
    M with a vector b in ?n and an n ? n matrix M
    (see COTTLE, R. W., PANG, J.-S., and STONE, R. E.
    (1992). The Linear Complementarity Problem. San
    Diego Academic Press., p. 1). The problem is to
    find z ? ?n so that
  • z ? 0

  • b Mz ? 0 ?
  • zT (b Mz) ? 0
  • For Lemkes method, the system ? is rewritten and
    generalized as follows.
  • Let I be the n?n identity matrix and d be an
    n-vector with positive components(for example, d
    (1,,1)T ).
  • Using an auxiliary variable z0, the term b Mz
    in (? ) is replaced by
  • b d z0 Mz, which is denoted by the
    n-vector w.

26
Linear Complementarity Problem
  • The problem generalizing (?) is that of finding w
    ? 0, z0 ? 0 and z ? 0 so that
  • Iw - dz0 Mz
    b ??
  • and zT w 0 hold.
  • A solution w , z0 , z to this problem defines a
    solution to (?) if and only if z0 0 .
  • In (??), the vector b is represented as a
    nonnegative linear combination of certain columns
    of the matrix I,-d,-M.
  • Like the simplex algorithm for linear
    programming, Lemkes algorithm traverses basic
    solutions of this system.

27
Contact Surface
  • Consistent conforming contact surface.
  • For instance, even though the distance between
    both meshes might be zero at all the mesh
    vertices, there could still be intersections or
    gaps between the two surfaces
  • Adaptive moving least squares approximation
    requires no re-meshing or zippering.
  • At any time instance, the surface of model A is
    represented by its original points PA plus all
    active nodes of B that lie on the contact surface
    (analogously for model B)

28
Simulation
  • Treat objects as rigid while in free motion
  • Whenever a collision between two objects is
    detected, we retrace the simulation to the time
    instance of first contact and proceed with a
    smaller time step.
  • Integrate contact forces to compute total
    wrenches, i.e., the forces and torques, acting on
    the bodies.

29
Friction
  • A major benefit of explicitly modeling the
    contact surface is that we can accurately
    simulate effects that are primarily dictated by
    contact, such as friction.
  • Friction could be added to the elastic LCP in
    essentially the same way as it is added to
    dynamic LCPs , using a polyhedral approximation
    to the friction cone.
  • But this increases the size of the LCP
    significantly by adding m1 pairs of
    complementary variables at each contact point,
    where m is the number of facets of the friction
    cone.
  • However, if we make the simplifying assumption
    that tangential tractions do not cause
    significant deformations, we can decouple
    tangential and normal quantities during contact
    resolution.
  • This greatly simplifies the computations.

30
Example
  • Model acquisition
  • laser-range scan consisting of 55,061 sample
    points of which 11,317 experience a displacement
    due to the contact resolution.
  • Hierarchy construction
  • recursive clustering
  • efficient multi-level computation

31
Example
  • Simulation

32
Example
  • Validation

Measurement
Simulation
X2 FootSensor (xSensor Corp.) 37 x 13 sensors,
1.94 sensors/cm2
33
Bio-medical Applications
  • Simulate friction effects to predict attrition
  • ? design of artificial joints

34
Computer Animation
  • Quasi-rigid body simulation

35
Conclusion
  • Quasi-rigid objects bridge the gap between rigid
    bodies and fully deformable models
  • Simple and efficient model for contact resolution
  • Limitations
  • small deformations
  • linear elasticity
  • sharp corners

36
Future Work
  • Coupling with low-resolution FEM model
  • Incorporate friction into LCP
  • Application Contact simulation in knee joint
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