department of computer science, iowa state university energy-based modeling of tangential compliance...

Department of Computer Science, Iowa State University

Energy-based Modeling of Tangential Compliance in 3-Dimensional Impact

Yan-Bin Jia

Department of Computer Science

Iowa State University

Ames, IA 50010

Dec 14, 2010


Impact and Manipulation

Impulse-based Manipulation

Potential for task efficiency and minimalism

Foundation of impact not fully laid out

Underdeveloped research area in robotics

Huang & Mason (2000); Tagawa, Hirota & Hirose (2010)

Linear relationships during impact ( )

Vmfi VmI i 0t

QQfr ii QIr ii


Impact with Compliance

tn

tIIfdtI

0Normal impulse: 1. accumulates during impact (compression + restitution) 2. Poisson’s hypothesis. 3. variable for impact analysis.

Tangential impulse: 1. due to friction & compliance 2. dependent on contact modes 3. driven by normal impulse

2D Impact: Routh’s graphical method (1913)

Han & Gilmore (1989); Wang & Mason (1991); Ahmed, Lankarani & Pereira (1999)

3D Impact: Darboux (1880)

Keller (1986); Stewart & Trinkle (1996)

Tangential compliance and impulse:

Brach (1989); Smith (1991); Stronge’s 2D lumped parameter model (2000);Zhao, Liu & Brogliato (2009); Hien (2010)

tI


Compliance Model

Gravity ignored compared to impulsive force – horizontal contact plane.

Extension of Stronge’s contact structure to 3D.

opposing initialtangential contact velocity

),,( nwu IIII tangential impulse

massless particle

F

Analyze impulse in contact frame:


Two Phases of Impact

Ends when the spring length stops decreasing:

The normal spring (n-spring) stores energy .

Compression

nv

nv

Ends when

Restitution

nE

0nv

maxEEn

0nE

p

energy coefficient of restitution:10 emax

2Ee

2

00 / ekk

nen 2EeE 2FF


Normal vs Tangential Stiffnesses

:k:tk

stiffness of n-spring (value depending on impact phase)

stiffness of tangential u- and v-springs (value invariant)

tkk /020

Depends on Young’s moduli and Poisson’s ratios of materials.

tkk /2

Stiffness ratio:

0 (compression)

e/0 (restitution)


Normal Impulse as Sole Variable

nnnn kEFdtdII 2/ Idea: describe the impact system in terms of normal impulse.

Key fact:

Derivative well-defined at the impact phase transition.

nnnnnn vInknIEdIdEE ///'

n

uu E

EI

'n

ww E

EI

'

11 (signs of length changes of u- and w-springs)


System Overview

Impact Dynamics

',' vu IIvu II ,

I nE

ContactMode

Analysis

vu EE ,

nI

'nE

tv

integrate

integrate


Sliding Velocity

sv

tv

:tv tangential contact velocityfrom kinematics

:sv velocity of particle p representing sliding velocity.

)0,,( wuvv ts

Sticking contact if .0sv


Stick or Slip? Energy-based Criteria

By Coulomb’s law, the contact sticks , i.e., if 0sv

nwu FFF 22nwu III 22

nwu EEE 22

Slips if nwu EEE 22ratio of normal stiffnessto tangential stiffness


Sticking Contact

Change rates of the lengths of the tangential u- and w-springs.

)0,,(0 wuvv ts )0,0,1(vu

)0,1,0(vw

Impossible to keep track of u and w in time space.

Only signs of u and w are needed to compute tangential impulses.

infinitesimal duration of impactunknown stiffness

Particle p in simple harmonic motion like a spring-mass system.


Sticking Contact (cont’d)

Tangential elastic strain energies are determined as well.

20

2

4u

u

DE 2

0

2

4w

u

DE

evaluating an integral involving nt Ev ,),( wu DDD

Keep track of as functions of . wkuk 00 2,2 nI


Sliding Contact

can also be solved (via involved steps).wu ,

Keep track of in impulse space. wkuk 00 2,2

Evaluating two integrals that depend on . (to keep track of whether the springs are being compressed or stretched).

wu GG , nEwu ,,

Tangential elastic strain energies:

20

2

4u

u

GE

20

2

4w

u

GE


Contact Mode Transitions

Stick to slip when

nwu EEE 22

Initialize integrals for sliding mode based on energy. wu GG ,

Slip to stick when

0sv )0,,( wuvt i.e,

Initialize integral for sliding mode. D


Start of Impact

),0,( 000 nu vvv

Initial contact velocity

sticks if20

40

220

20 nwu vvv

slips if20

40

220

20 nwu vvv )0('uI …

…

nn vE 0)0('

0)0(' wI

n

uu v

vI

0

020

1)0('

Under Coulomb’s law, we can show that


Bouncing Ball – Integration with Dynamics

Contact kinematics

tz I

mz

m

Ivv

2

70

)1,0,0(z

Theorem During collision, is collinear with . tI tv0

Velocity equations:

mIVV /0

Izmr

2

50

(Dynamics)

Impulse curve lies in a vertical plane.


Instance

Physical parameters:

1m

2.1)3.022/()3.02(20

4.0 5.0e

)5,0,1(0 V

1r

)0,2,0(0

Before 1st impact:

After 1st impact:

)5.2,0,570982.0(V

)0,92746.1,0(

x

z


Impulse Curve (1st Bounce)

Tangential contact velocity vs. spring velocity

contact mode switch

tv


Non-collinear Bouncing Points

Projection of trajectory onto xy-plane


Bouncing Pencil

1m

2.120

8.0 5.0e

1r

31 h 5.02 h

3


Video

Slipping direction varies.

end of compression

slipstick slip

)5302.5,3908.0,962.3(

)6

sin,0,6

(cos50

V

)5.0,5.0,1(0

Pre-impact:

)0302.3,3908.0,3681.0(0 V

)5.0,8021.1,2362.0(0

Post-impact:


Simultaneous Collisions with Compliance

Combine with WAFR ‘08 paper (with M. Mason & M. Erdmann) to model a billiard masse shot.

Trajectory fit


Simultaneous Collisions with Compliance

Estimates of post-hit velocities:

)361.0,654.1( v

)537.80,278.24(

Predicted trajectory

Predicted post-hit velocities:

)733.0,244.0,658.1( v

)676.3,988.52,938.15(


Conclusion

• 3D impact modeling with compliance extending Stronge’s spring-based contact structure.

• Impulse-based not time-based (Stronge) and hence ready for impact analysis (quantitative) and computation.• elastic spring energies• contact mode analysis• sliding velocity computable• friction

• Physical experiment.• Further integration of two impact models (for compliance and simultaneous

impact).


Acknowledgement

Matt Mason (CMU)

Rex Fernando (ISU sophomore)

department of computer science, iowa state university energy-based modeling of tangential compliance...

Documents

impact slide

iowa state university

tangential stiffness

iowa state university

impact system

d impact

impact analysis

compliance normal impulse