Overview Class

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OverviewClass 5 (Thurs, Jan 30)

• Rigid body contact
• Read BW course notes on
• Collision and Contact
• Constraint formulation
• Lagrangian
• Reduced coordinate
• Cool applications!
• Graphics literature

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Rigid Body Dynamics
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Rotations from last dayOne last thing
Quaternions

• Obey same SO(3) group structure as the rotation
  matrices
• Efficient 4-vector representation of rotations
  that do not suffer mathematical singularities
• Briefly on the whiteboard
• See also
• Murray, Li, Sastry, A Mathematical Intro. to
  Robotic Manipulation, (p. 33) (Handout)
• Andrew Hanson, Visualizing Quaternions,
  SIGGRAPH 2001 Course Notes, 2001. (Fun with
  Clifford Algebras!)
• And Baraff course notes.

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Contact Separation Distance

• Simple/common idea for rigid contact
• Signed distance function, ??(gA,gB)
• ?gt0 Separated
• ?0 In contact
• ?lt0 Rigid interpenetration

A
B
B
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Overview of Contact Models
Contact Models
Deformable Bodies
Impulse Response Rigid Models, e.g., Mirtich
Canny
Force Response Rigid Models
NOTE x-?
Adapted from D.Pai 99
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Overview of Contact Models
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Overview of Contact Models
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Overview of Contact Models
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A little history Hertz Contact

• Idealized quasistatic contact between spheres
• H. Hertz, Über die Berührung fester elastischer
  Körper, Gesammelte Werke, Bd. 1, Leipzig 1895.

f ? ?3/2
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Reality of spherical contact

• Real surface contact problems are complex!
• Models of surface contact (and friction) are
  highly approximate in practice
• Not a serious problem for graphics
• important for contact sounds (will discuss
  later)

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Penalty Methods

• Penalty spring force
• More sophisticated models use damping
• For example

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Overview of Contact Models
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Contact Response Maps

• Ullrich Pai, Contact Response Maps for Real
  Time Dynamic Simulation, ICRA, 1998.
• Convolve contact forcing with the precomputed
  impulse response function of a linear
  elastodynamic model.
• Generalized force response model
• Includes effect of vibration and internal shock
  waves produced by contact

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Overview of Contact Models
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Impulse Response Contact Models

• Brian Mirtich and John F. Canny, Impulse-Based
  Simulation of Rigid Bodies, Symposium on
  Interactive 3D Graphics, 181-188, 1995.

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Overview of Contact Models
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The Linear Complementarity Problem (LCP)

• Formulation of the contact force problem
• Famous mathematical problem
• Briefly (on whiteboard)
• See papers on webpage, e.g., by Baraff, Trinkle,
  Anitescu, etc.

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Constrained vs Reduced Coordinate Approaches

• Fast Lagrange multipliers
• D. Baraff. Linear-time dynamics using Lagrange
  multipliers, SIGGRAPH 96.
• Ascher, Pai, Cloutier, Forward Dynamics,
  Elimination Methods, and Formulation Stiffness in
  Robot Simulation, 1997
• Reduced coordinates
• Difficult to exploit in general contact
  animations
• Robotic assemblies important special case see
  Featherstone Murray, Li, Sastry

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Dynamics with Smooth Surface Contact

• P. G. Kry, and D. K. Pai, Continuous Contact
  Simulation for Smooth Surfaces, ACM Transactions
  on Graphics, Jan 2003.

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Other Issues

• Constraints with closed loops
• Mathematically tricky DAEs (Differential
  Algebraic Equations)
• For details see, e.g.,
• François Faure, Fast Iterative Refinement of
  Articulated Solid Dynamics, IEEE Transactions on
  Visualization and Computer Graphics, 5(3), pp.
  268-276, 1999.
• U. Ascher and P. Lin, SIAM J. Scient. Comput. 21
  (1999), 1244-1262 Sequential Regularization
  Methods for simulating mechanical systems with
  many closed loops)

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Parallel Computation

• Large scale rigid body simulation
• Time-warp approach from discrete simulation

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Cool Applications

• Graphics is interested in more than just rigid
  body mechanics fundamentals

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Random Sampling of Rigid Motions

• Stephen Chenney and D.A.Forsyth, "Sampling
  Plausible Solutions to Multi-Body Constraint
  Problems". SIGGRAPH 2000 Conference Proceedings,
  pages 219-228, July 2000.
• MOVIE 30 balls fall into 105 bins. These
  solutions were hand chosen from a chain that
  produced several hundred examples in 200000
  iterations, taking seven days to compute on a PC.
  The longer computation time is due to the greater
  number of balls, because each simulation takes
  longer and the constraints are much harder to
  satisfy.

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Placement of Rigid Objects in Scenes
John Snyder, An interactive tool for placing
curved surfaces without interpenetration,
SIGGRAPH 95.
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Interactive Manipulation of Rigid Body Simulations

• Jovan Popovic, Steven M. Seitz, Michael Erdmann,
  Zoran Popovic, and Andrew Witkin. Interactive
  Manipulation of Rigid Body Simulations. In
  Computer Graphics (Proceedings of SIGGRAPH 2000),
  ACM SIGGRAPH, Annual Conference Series, 209-217.

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Evolving Virtual Creatures

• Sims, K., "Evolving Virtual Creatures," Computer
  Graphics (Siggraph '94) Annual Conference
  Proceedings, July 1994, pp.43-50.

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Sims, K., "Evolving Virtual Creatures," Computer
Graphics (Siggraph '94) Annual Conference
Proceedings, July 1994, pp.43-50.