a hierarchical design methodology for multibody systems...

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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.

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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

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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]

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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)