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University of Pennsylvania 1GRASP
A Hierarchical Design Methodology forMultibody Systems with Frictional Contacts
Vijay Kumar
GRASP LabMechanical Engineering and Applied Mechanics
Computer and Information ScienceUniversity of Pennsylvania
Jong-Shi Pang
Mathematical SciencesRennselaer Polytechnic Institute
Peng Song
Mechanical Engineering Rutgers University
Bharath Mukundakrishnan
GRASP LaboratoryUniversity of Pennsylvania
Jeffrey Trinkle
Computer ScienceRennselaer Polytechnic Institute
Jonathan Fink
GRASP Lab, PennECS, RPI
Joint work with
Steve Berard
Computer ScienceRennselaer Polytechnic Inst.
University of Pennsylvania 2GRASP
1. Decentralized Multirobot Manipulation
Motion plans derived from geometric models
Can we generalize to dynamic models?
Pereira, Campos and Kumar, WAFR 02
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2. Part Feeding, Assembly
Design with geometric and kinematic models is possible.
Dynamic models are necessary.
[Boothroyd, 1968]
[Kraus, 2001]
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3. Micro Manipulation
100 µ dia probe attached to10g load cell
0.4mm x 0.8mm part assembly
Configuration AConfiguration B
Test Fixture
2 mm
1.5 mm
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Design Process or Plan TaskDynamical System
Intermittent contacts Contact state transitions Multiple contacts
Fundamental difficulties Static indeterminacy with traditional models (jamming,
wedging) Whitney, Dupont
No consistent models for frictional impacts Goldsmith, Pfeiffer, Keller, Brach, Wang and Mason, Stronge, Chatterjee and Ruina
No unified treatment for design and planning
University of Pennsylvania 6GRASP
Outline1. Background
Contact models Normal and tangential compliance Frictional contacts Time-stepping methods
2. Hierarchical Approach Models at different levels of fidelity Abstraction and model reduction Example
3. Algorithms for design optimization Randomized algorithms Time-stepping algorithms
4. Case Study: Part Feeder Modeling Iterative design process
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Systems with Frictional Contacts
Friction T
O
c
T
O
cT
cO
λF
λF
µλN
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Compliant Contact Models
UndeformedShell
ViscoelasticLayer
RigidCore
RigidCore
DeformedShell
N N
T
TδNδNϕ
Tϕslip
SRiTiTiTiTiTiT
iNiNiNiNiNiN
nnigf
gf
+=+=+=
,,1),()(
),()(
,,,,,,
,,,,,,
L&
&
δδδλ
δδδλ
Gross motion
Fine deformation
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More Generally… Elastic Bodies
Linear Elastic,
Counterformal
Contacts
nØ i
niδA
Bdeformed
undeformed
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Advantages of Compliant Contact Model
Proof of uniqueness and existence Contact forces can always be determined More realistic friction model
Tangential compliance Gross slip is preceded by small local deformations Hysteresis
Disadvantages Identification of parameters Computational time
u
λΤ
Coulomb actual
u
λΤ
rigid linear elastic
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Time Stepping Model
Equations of Motion
[Anitescu, Pang, Potra, Stewart, and Trinkle]
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Extension 1: Compliant Models
deformations separation/slip relative gross motion
Constitutive law
Contact compliance
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Extension 2: Frictional Contact
(cf. Peng Song’s talk tomorrow)
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Compliant Frictional Contacts:Technical Results
Single frictional contact [Song and Kumar, 2003]
For a single contact with a lumped compliance model, a uniquetrajectory always exists
Multiple frictional contacts [Song, Pang and Kumar, 2003]
A discrete-time solution trajectory always exists
There exists a µ*>0, such that if µ*> µi>0, a unique trajectoryexists
Under “certain conditions” the discrete time trajectoryconverges to converges to that obtained by using the rigid bodymodel time-stepping algorithm
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Example
Linear, visco-elastic contacts
Initial value problem
Five springs at each contact
m = 0.05 kg. ε=10-10 N/m2
Δt~10-4 seconds
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Example (continued)
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Design Optimization
Design Optimization
External inputsor disturbances
Difficulties: (1) high dimensionality; (2) non smoothness
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Abstractions and hierarchySystem S1
System S2
Transformation
S2 is an abstraction of S1 if for any δ > 0 and all inputs u(t),there exists v(t) such that for all
x* is reachable for a given design implies z*=h(x*) isreachable for the same design
S1
S2
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Example of Abstraction
Kinematic (first order model)
Geometric (zeroth order model)
More generally… Dynamics with compliance
Rigid body dynamic
Kinematic (quasi-static)
Geometric
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Case Study: Design of a Part Feeder
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Definitions
State spaceOriginal state space augmented by all parameters
Inputs/disturbancesGeometric model - virtual input
Dynamic model - gravitational force
Design spaceInitial conditions (original state space) + parameter choices
Search space x2
Focus on “search” and“satisfaction” ratherthan optimization x1
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Explorating the Design Space: TheRRT method
Explore motions from thechosen vertex by trying allpossible inputs
initial state
random state
random state
Grow the tree until a solutionis found or the no. of verticesreaches a certain value
Choose the state “closest" to the random state,Xnew.
Find the state, Xnear, “closest” to therandom state among all explored states.
Explore motions from the chosenvertex by trying all possible inputs.Choose the state closest to therandom state, Xnew
Key: A vertex with a larger Voronoiregion has higher probability ofbeing chosen as Xnear
[Lavalle and co-workers, 1999-2003]
University of Pennsylvania 23GRASP
xinit , qinit
Rapidly Exploring Random Tree
Target set(e.g., successful assembly)
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Rapidly Exploring Random Tree
xinit , qinit
xrand , qrand
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Rapidly Exploring Random Tree
xinit , qinit
xrand , qrand
xnear , qnear
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Rapidly Exploring Random Tree
xinit , qinit
xrand , qrand
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xinit , qinit
xrand , qrand
xnew , qnew
Rapidly Exploring Random Tree
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Coverage and Growth
New trees are started when thegrowth rate slows below aspecified threshold. Plots show 8designs being explored.
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Red (thick) geometrically feasible successfulpath. Green (thin) geometrically feasibletrajectories.
RRT Generated from the GeometricModel with a Given Design
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Example: different chute angles
Sampling the 12-Dimensional Design Space:Geometric Model
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Exploring the Design Space:Geometric Model
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Pruning the Design Space:Kinematic Model
First order model further restricts the choice of design parameters!
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Initial Design for Dynamic Analysis
Geometric Kinematic
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Dynamic Analysis: Inelastic Impacts
Heavy end last Heavy end first
1. LCP solver, time-stepping algorithm [Stewart & Trinkle]2. No external input/disturbance
Song et al, ICRA 2004
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Dynamic Analyis: Visco-Elastic Contacts
Visco-elastic contacts
LCP solver, time-stepping [Song, Pang, & Kumar]
Exact detection of collisions [Esposito & Kumar]
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Experimental Prototype
Experimental data digitized at 500 Hz., played back at1/10 normal speed
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Summary• Explore design space using a family of models
• Simpler models are used as abstractions formore complex models initially
• Can incorporate uncertainty in parameters
•Enhancement: Optimization [ICRA 04]
•Alternative: Use “unified” (implicit, NCP) modelto solve boundary-value problem [RSS 05]
Related
(cf. Peng Song’s talk tomorrow)