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An Implicit TimeStepping Method for QuasiRigid Multibody Systems with Intermittent Contact

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Title: An Implicit TimeStepping Method for QuasiRigid Multibody Systems with Intermittent Contact


1
An Implicit Time-Stepping Method for Quasi-Rigid
Multibody Systems with Intermittent Contact
  • Nilanjan Chakraborty, Stephen Berard, Srinivas
    Akella, and Jeff Trinkle
  • Department of Computer Science
  • Rensselaer Polytechnic Institute

2
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)
3
Related Work (Dynamics Modeling)
  • 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

4
Contact Modeling Assumptions
  • Point Contact
  • Quasi-Rigid or locally compliant objects each
    object has a rigid core surrounded by a thin
    compliant layer. (Song and Kumar, 2003, Song,
    Kraus, Kumar and Dupont, 2001)
  • Deflections are small
  • There is a maximum deflection beyond which the
    body acts as a rigid body
  • Lumped parameter model of compliance (Force based
    methods)
  • Kelvin-Voigt Model
  • Hertz Model
  • Hunt and Crossleys Model

5
Motivation Integrate Collision Detection
Collision Detection
Solve Dynamics
Update Position
6
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
7
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)
8
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
9
Complementarity Problem
10
Continuous Time Dynamics Model (Rigid Body)
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
11
Discrete Time Dynamics Model (Rigid Body)
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)
12
Compliant Contact Model (Quasi-Rigid Body)
(For tangential directions)
13
Contact Constraints (Rigid Body)
Contact Constraint in Configuration Space
14
Contact Constraints (Rigid Body)
From KKT Conditions
15
Contact Constraints (Quasi-Rigid Body)
16
Discrete Time Dynamics Model
The mathematical model is a Mixed Nonlinear
Complementarity Problem.
17
Results
Rigid Ellipsoid Falling on a deformable half-plane
The MCP solver PATH was used to solve this
problem.
18
Results
19
Results
20
Conclusion and Future Work
  • We presented a geometrically implicit
    time-stepper for quasi-rigid multi-body systems
    that combines the collision detection and dynamic
    time step to deal with a source of inaccuracy in
    dynamic simulation.
  • 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

21
THANK YOU!
22
Results
23
Results
24
Results
25
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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