Quasi-Rigid Objects in Contact

Mark Pauly Dinesh Pai Leo GuibasStanford University Rutgers University Stanford University

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Contacts in Simulation

• Bio-medical applications: • surgery simulation• artifical joints, dental implants

• Mechanical design:• wear and tear of industrial parts

• Physics-based animation: • movies • games

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Existing 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

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Quasi-Rigid Objects

• Physical model • point force applied to object only leads to small,

local deformation• analytical system response model to define

displacements due to point force• linear elasticity: Global system response by

superposition• forces and displacements evaluated on surface

only

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Quasi-Rigid Objects

• Surface model• point cloud representation for modeling

consistent, highly dynamic contact surface

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Physical Model

• Boussinesq approximation• infinite elastic half-space

yx

xyx

)(

2

1),(

p

Gu

Poisson’s ratio

shear modulus

force at x

displacement at ydue to force at x

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Physical Model

• Boussinesq approximation• system response function

rGrf

1

2

1)(

yx r

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Physical Model

• Linear elasticity• superposition

xyx

xy d

p

Gu

S

)(

2

1)(

total displacement at y

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Volume Preservation

• Condition:

0

0)( rdrrf

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Volume Preservation

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

• Approximate system response at discrete nodes (point samples)

Discretization

force at node j

shape function

N

jjjpp

1

)()( xx

xqx

xdp

Gu

S i

j

jji

)(

2

1

displacementat node i

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Discretization

system response matrix

Rpuvector of

displacements[u1,...,uN]T

vector of tractions[p1,...,pN]T

xxq

xd

GR

S i

jij

)(

2

1

matrix coefficient

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Contact

• Collision detection• static bounding volume hierarchies (small

deformations)

• Contact resolution• compute forces and displacements that resolve

contact

• Contact surface• find contact surface that is consistent for both

models

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Contact Resolution

• Collision detection determines points that potentially experience displacements (active nodes)

• find corresponding point for each active node

active nodes

corresponding nodes

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Contact Resolution

• Separation of active nodes• initial separation

• final separationB

iA

iB

iA

ii uus qq

Bi

Aiis qq ~

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Contact Resolution

• Condition for contact resolution:

• non-negative separation: si ≥ 0

• non-negative force: pi ≥ 0

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

• Linear Complementarity Problem (LCP)

• solved using Lemke’s method

Contact Resolution

qRps

0s0p

0psT

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Contact Surface

• Consistent conforming contact surface

• Adaptive moving least squares (MLS) approximation requires no re-meshing or zippering

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Simulation

• Treat objects as rigid while in free motion

• Integrate contact forces to compute total wrench

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Example

• Model acquisition• laser-range scan

• Hierarchy construction• recursive clustering• efficient multi-level computation

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Example

• Simulation

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Example

Measurement Simulation

X2 FootSensor (xSensor Corp.) 37 x 13 sensors, 1.94 sensors/cm2

• Validation

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Bio-medical Applications

• Simulate friction effects to predict attrition

design of artificial joints

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Computer Animation

• Quasi-rigid body simulation

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Computer Animation

• Explicit representation of contact surface allows accurate simulation of friction effects

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Computer Animation

• Explicit representation of contact surface allows accurate simulation of friction effects

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

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

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Future Work

• Coupling with low-resolution FEM model

• Acquired system response functions

• Incorporate friction into LCP

• Application: Contact simulation in knee joint

Quasi-Rigid Objects in Contact SCA 04Mark Pauly

Acknowledgements

• NSF grants CARGO-0138456, ITR-0205671, IIS-0308157, EIA-0215887, ARO grant DAAD19-03-1-0331

• Anonymous reviewers