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A Hierarchical Design Methodology for Multibody Systems with Frictional Contacts Vijay Kumar GRASP Lab Mechanical Engineering and Applied Mechanics Computer and Information Science University of Pennsylvania Joint work with Bharath

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A Hierarchical Design Methodology for Multibody Systems with Frictional Contacts

Vijay Kumar

GRASP Lab Mechanical Engineering and Applied Mechanics Computer and Information Science University of Pennsylvania

Jong-Shi Pang

Mathematical Sciences Rennselaer Polytechnic Institute

Peng Song

Mechanical Engineering Rutgers University

Bharath Mukundakrishnan

GRASP Laboratory University of Pennsylvania

Jeffrey Trinkle

Computer Science Rennselaer Polytechnic Institute

Jonathan Fink

GRASP Lab, Penn ECS, RPI Joint work with

Steve Berard

Computer Science Rennselaer 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 A Configuration B Test Fixture

2 mm 1.5 mm

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Design Process or Plan Task

Dynamical 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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Outline

  • 1. 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

Undeformed Shell Viscoelastic Viscoelastic Layer Layer Rigid Rigid Core Core Rigid Rigid Core Core Deformed Shell N N T

T

δ

N

δ

N

ϕ

T

ϕ slip

S R i T i T i T i T i T i T i N i N i N i N i N i N

n n i g f g f + = + = + = , , 1 ) , ( ) ( ) , ( ) (

, , , , , , , , , , , ,

L & & δ δ δ λ δ δ δ λ

Gross motion Fine deformation

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More Generally… Elastic Bodies

Linear Elastic, Counterformal Contacts

n

Ø i

n i

δ

A B

deformed 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 unique

trajectory 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 trajectory

exists

 Under “certain conditions” the discrete time trajectory

converges to converges to that obtained by using the rigid body model 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 inputs

  • r disturbances

Difficulties: (1) high dimensionality; (2) non smoothness

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Abstractions and hierarchy

System 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*) is reachable 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 space

Original state space augmented by all parameters

 Inputs/disturbances

Geometric model - virtual input Dynamic model - gravitational force

 Design space

Initial conditions (original state space) + parameter choices

 Search space

x2

Focus on “search” and “satisfaction” rather than optimization

x1

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Explorating the Design Space: The RRT method

Explore motions from the chosen vertex by trying all possible inputs

initial state  random state  random state

Grow the tree until a solution is found or the no. of vertices reaches a certain value

Choose the state “closest" to the random state, Xnew. Find the state, Xnear, “closest” to the random state among all explored state s. Explore motions from the chosen vertex by trying all possible inputs. Choose the state closest to the random state, Xnew

Key: A vertex with a larger Voronoi region has higher probability of being 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 the growth rate slows below a specified threshold. Plots show 8 designs being explored.

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Red (thick) geometrically feasible successful

  • path. Green (thin) geometrically feasible

trajectories.

RRT Generated from the Geometric Model 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 at 1/10 normal speed

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Summary

  • Explore design space using a family of models
  • Simpler models are used as abstractions for

more complex models initially

  • Can incorporate uncertainty in parameters
  • Enhancement: Optimization [ICRA 04]
  • Alternative: Use “unified” (implicit, NCP) model

to solve boundary-value problem [RSS 05]

Related

(cf. Peng Song’s talk tomorrow)