An Implicit TimeStepping Method for Multibody Systems with Intermittent Contact

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An Implicit Time-Stepping Method for Multibody
Systems with Intermittent Contact

• Nilanjan Chakraborty, Stephen Berard, Srinivas
  Akella, and Jeff Trinkle
• Department of Computer Science
• Rensselaer Polytechnic Institute
• (To appear in Robotics Science and Systems 2007)

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Introduction

• Dexterous manipulation/Grasping
• Mechanical design
• Computer Games

Example Grasping experiment where the Circular
lock piece must be grasped by the parallel jaw
grippers as they close. (Brost and Christensen,
1996)
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Related Work

• Differential Algebraic Equation (DAE)
• (Haug et al. 1986)
• Differential Equations (Motion model) Algebraic
  Constraints
• Requires knowledge of contact interactions
  (sliding, rolling or separating)
• Contact Interactions not known apriori!
• Differential Complementarity Problem (DCP)
• (Stewart and Trinkle 1996 Anitescu, Cremer and
  Potra 1996 Pfeiffer and Glocker 1996 Trinkle et
  al. 1997 Trinkle, Tzitizouris and Pang 2001)
• Differential Equations (Motion model)
  Complementarity constraints (Contact Model,
  Friction Model) Algebraic Constraints

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Motivation Integrate Collision Detection
Collision Detection
Solve Dynamics
Update Position
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Motivation Disc rolling on plane

• Eliminate sources of instability and inaccuracy
• Polyhedral approximations
• decoupling of collision detection from dynamics
• approximations of quadratic Coulomb friction model

100 vertices
10 vertices
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Motivation Disc rolling on plane
Discretization of geometry and linearization of
the distance function lead to a loss of energy in
current simulators
Results of simulating the rolling disc using the
Stewart-Trinkle algorithm for varying number of
edges and varying step-size. The top horizontal
line is the computed value obtained by our
geometrically implicit time-stepper using an
implicit surface representation of the
disc. (Error Tolerance 1e-06)
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Motivation Integrate Collision Detection
Collision Detection
Goal Integrate collision detection with
equations of motion
Solve Dynamics
Assumption Convex objects described as an
intersection of implicit surfaces.
Update Position
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Complementarity Problem
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Continuous Time Dynamics Model
Newton-Euler equations
Mass Matrix
Contact forces and moment
Applied force
Coriolis force
Kinematic map
Contact constraints
Friction Model Set of Complementarity
Constraints
Joint Constraints Set of Algebraic Constraints
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Discrete Time Dynamics Model
Discrete Time Model
(Stewart and Trinkle 1996, uses linearized
friction cone, subproblem at each time step is
LCP)
(Tzitzouris 2001 for closed form distance
functions, subproblem at each time Step is NCP)
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Contact Constraints
Contact Constraint in Configuration Space
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Contact Constraints
From KKT Conditions
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Discrete Time Dynamics Model
The mathematical model is a Mixed Nonlinear
Complementarity Problem.
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Results
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Results
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Conclusion

• We presented a geometrically implicit
  time-stepper for multi-body systems that combines
  the collision detection and dynamic time step to
  deal with a source of inaccuracy in dynamic
  simulation.

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Future Work

• Address the question of existence and uniqueness
  of solutions
• Implementation for intersection of surfaces.
• Extend to nonconvex implicit surface objects
  described as an union of convex objects as well
  as parametric surfaces.
• Precisely quantify the tradeoffs between the
  computation speed and physical accuracy

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THANK YOU!