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A family of rigid body models: connections between quasistatic and dynamic multibody systems

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Title: A family of rigid body models: connections between quasistatic and dynamic multibody systems


1
A family of rigid body models connections
between quasistatic and dynamic multibody systems
  • Jeff Trinkle
  • Computer Science Department
  • Rensselaer Polytechnic Institute
  • Troy, NY 12180
  • Jong-Shi Pang, Steve Berard, Guanfeng Liu

2
Motivation
Dexterous Manipulation Planning
Valid quasistatic plan exists
No quasistatic plan found, but dynamic plan exists
3
LIGA Tribology Test Vehicle
  • LIGA German acronym for process for making
    micro-scale parts from metals, ceramics, and
    plastics.
  • Typical dimensions are on the order of
  • Sandia wants to understand function, efficiency,
    robustness before building. Optimal design.

4
Micro-Machine Assembly
  • Pawl (2.3 mm) and washer (1.0 mm) subassembly.
  • Pins (0.169 mm) in holes (0.165 mm).
  • Need fixture to hold and align washer and pawl.
  • Fixture should guarantee unique positions and
    orientations of parts.

5
Pawl in Fixture
6
Simulation of Pawl Insertion
7
Past Work in Quasistatic Multibody Systems
  • Grasping and Walking Machines late 1970s.
  • Used quasistatic models with assumed contact
    states.
  • Whtney, Quasistatic Assembly of Compliantly
    Supported Rigid Parts, ASME DSMC, 1982
  • Caine, Quasistatic Assembly, 1982
  • Peshkin, Sanderson, Quasistatic Planar Sliding,
    1986
  • Cutkosky, Kao, Computing and Controlling
    Compliance in Robot Hands, IEEE TRA, 1989
  • Kao, Cutkosky, Quasistatic Manipulation with
    Compliance and Sliding, IJRR, 1992
  • Peshkin, Schimmels, Force-Guided Assembly, 1992

8
Past Work in Quasistatic Multibody Systems
  • Mason, Quasistatic Pushing, 1982 - 1996
  • Brost, Goldberg, Erdmann, Zumel, Lynch, Wang
  • Trinkle, Hunter, Ram , Farahat, Stiller, Ang,
    Pang, Lo, Yeap, Han, Berard, 1991 present
  • Trinkle Zeng, Prediction of Quasistatic Planar
    Motion of a Contacted Rigid Body, IEEE TRA, 1995
  • Pang, Trinkle, Lo, A Complementarity Approach to
    a Quasistatic Rigid Body Motion Problem, COAP
    1996

9
Hierarchical Family of Models
  • Models range from pure geometric to dynamic with
    contact compliance
  • Required model resolution is dependent on
    design or planning task
  • Approach
  • Plan with low resolution model first
  • Use low resolution results to speed planning with
    high resolution model
  • Repeat until plan/design with required accuracy
    is achieved

10
Components of a Dynamic Model
  • Newton-Euler Equation
  • Defines motion dynamics
  • Kinematic Constraints
  • Describe unilateral and bilateral constraints
  • Normal Complementarity
  • Prevents penetration and allows contact
    separation
  • Friction Law
  • Defines friction force behavior
  • Bounded magnitude
  • Maximum Dissipation
  • Leads to tangential complementarity
  • Maintains rolling or allows transition
  • from rolling to sliding

11
Complementarity Problems
  • Let be an element of and
  • let be a given function in .
    Find such that

12
Newton-Euler Equation
Non-contact forces
13
Kinematic Quantities at Contacts
Normal and tangential displacement functions
14
Normal Complementarity
Define the contact force
Normal Complementarity
where
15
Dry Friction
16
Instantaneous-Time Dynamic Model
17
Scale the Times of the Input Functions
Scale the driving inputs. Replace with
in the driving input functions.
18
Time-Scaled Dynamic Model
Change variables
Application of chain rule and algebra yields
19
Time Stepping Methods
  • Approximate derivatives by
  • where is the time step,
    , and is the
  • scaled time at which the state of the
    system was obtained.

20
LCP Time-Stepping Problem
21
Example Fence and Particle
  • Assume
  • Particle is constrained from below
  • Non-contact force
  • Fence is position-controlled
  • Wall is fixed in place
  • Expected motion
  • Quasistatic no motion when not in contact
    with fence.
  • Dynamic if deceleration of paddle is large,
    then particle can continue sliding without fence
    contact

22
Time-Scaled Fence and Particle System
Dynamic
Quasistatic
Boundary
23
Time-Scaled Fence and Particle System
Dynamic
Quasistatic
24
Cast Model as Convex Optimization Problem
  • Introduce the friction work rate value function

Linear in
Introduce the friction work rate minimum value
function
25
Equivalent Convex Optimization Problem
OPT
Hypograph of is convex.
Therefore is concave and
is convex. KKT conditions are
exactly the discrete-time model.
26
Theorem
If
solves the model with quadratic friction cone,
then is a globally optimal solutions of
OPT corresponding to . Conversely, if
is a globally optimal solution to OPT for a
given and if is equal to an
optimal KKT multiplier of the constraint in OPT,
then defining as below, the
tuple
solves the model with quadratic friction cone.
27
Proposition Solution Uniqueness
Corresponding to the solution
of the discrete-time
model with quadratic friction cone, is
the unique solution of OPT, if and only if the
following implication holds
Added motion does not decrease work
Added motion does not change friction work.
Added motion does not cause penetration
where is a small change in
28
Example
  • Solution is unique with non-zero quadratic
    friction on plane
  • Solution is not unique without friction
  • Solution is not unique with linearized friction
    on plane

29
Future Work
  • Convergence analysis
  • Experimental validation
  • Design applications

30
  • Fini

31
Maximum Work Inequalty Unilateral Constraints
Linearize the limit curve at contact
where the columns of are the vectors
transformed into C-space.
  • is the vector of the components of
    relative velocity at the contact in the
    directions.

32
Tangential Complementarity Example
33
Instantaneous Rigid Body Dynamics in the Plane
34
Example Sphere initially translating on
horizontal plane.
35
Simulation with Unilateral and Bilateral
Constraints
36
Time-Stepping with Unilateral Constraints
Without Constraint Stabilization
Solution always exists and Lemkes algorithm can
compute one (Anitescu and Potra).
37
Solution Non-uniqueness LCP Non-Convexity
Two Solutions
38
Solution Non-Uniqueness Contact Force Null Space
External Load
  • Both friction cones can see the other contact
    point.
  • Assume
  • Blue discs are fixed in space
  • Red disc is initially at rest
  • Solution 1 disc remains at rest
  • Solution 2 disc accelerates downward
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