simultaneous three-dimensional impacts with friction and compliance

Department of Computer Science, Iowa State University

Simultaneous Three-Dimensional Impacts with Friction and Compliance

Yan-Bin Jia

Department of Computer ScienceIowa State UniversityAmes, IA 50010, USA

Sep 27, 2012


Why Impact? Because impact is everywhere …

Efficiency over static and dynamic forces

Foundation of impact not fully laid out

Impulsive manipulation being underdeveloped in robotics Higuchi (1985); Izumi and Kitaka (1993); Hirai et al. (1999) Huang & Mason (2000); Han & Park (2001), Tagawa et al. (2010)

Accomplishing tasks otherwise very difficult.

Reduction of harmful impulsive forces

Because it is relevant to robotics…

Collision between robots and environments, walking robots …


Describing Impact by Impulse

)( 01 vvm

“Infinitesimal” duration

“Infinite” contact force

Finite change in momentum

I

tFdt

0

before

impulse

after

1v0v


Impact Phase 1 – Compression

Compression ends when the spring length stops decreasing:

The spring changes its length by , storing elastic energy .

vE

0vmaxEE

px

Contact point virtual spring.


Transition at End of Phase 1

v penergy coefficient of restitution:]1,0[

max2EeE

Loss of energy at transition to restitution:

200 / ekk xex 2

EekxE 22

21

FkxF

Increase in stiffness at transition to restitution:

change in spring length


Impact Phase 2 – Restitution

v Restitution ends when 0E

During restitution the normal spring (n-spring) releases the remaining .Ee2


Impulse-Energy Relationship

mIvvx

dIdE

0

Energy is a piecewise quadratic function of impulse.

One-to-one correspondence between impulse and time.

Describe the process of impact in terms of impulse.


Example of General 3D Impact – Billiard Shooting

1I2I

normal impulses

tangential impulses

Two contacts: - cue-ball - ball-table

At each contact: - normal impulse - tangential impulse

How are they relatedat the same contact?(compliance & friction)

At different contacts:

How are the normalimpulses related? (simultaneous impacts)


Talk Outline

II. Simultaneous Impacts

III. Model Integration (Billiard Shooting)

Relationships among normal impacts at different contacts(no friction or compliance)

Simultaneous impacts with friction and compliance

I. Impact with Compliance Relationship of tangential impulse to normal impact at single contact(also with friction)


Related Work on Impact

Newton’s law (kinematic coefficient of restitution)

Routh (1905); Han & Gilmore (1989); Ahmed et al. (1999); Lankarani (2000)

Maclaurin (1742); Bernoulli (1969); Ivanov (1995)

Energy increaseWang et al. (1992); Wang & Mason (1992)

No post-impact motion of a still object

Brogliato (1999); Liu, Zhao & Brogliato (2009)

Poisson’s hypothesis (kinetic coefficient of restitution)

Darboux (1880); Keller (1986); Bhatt & Koechling (1994);

Glocker & Pfeiffer (1995); Stewart & Trinkle (1996); Anitescu & Portta (1997)

Energy increase


Related Work (cont’d)

Energy-Based Restitution (energetic coefficient of restitution)

Smith (1991); Bilbao et al. (1989); Brach (1989)

Stronge (1990); Wang et al. (1992);

Liu, Zhao & Brogliato (2008, 2009); Jia, Mason & Erdmann (2008)

Tangential Impulse & Compliance Mindlin (1949); Maw et al. (1976)

Stronge (1994; 2000); Zhao et al. (2009); Hien (2010) ; Jia (2010)

Simultaneous Collisions Chatterjee & Ruina (1998); Ceanga & Hurmulu (2001); Seghete & Murphey (2010)

Energy conservation


Impact with Compliance

IIfdtI n

t

0

Normal impulse: 1. accumulates during impact (compression + restitution) 2. energy-based restitution 3. variable for impact analysis

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


Compliance Model

Gravity ignored in comparison with impulsive force.

Extension of Stronge’s contact structure to 3D.

opposing initialtangential contact velocity

),,( nwu IIII tangential impulse

massless particle

F

Analyze impulse in contact frame:


Normal vs Tangential Stiffnesses

:k:k

stiffness of normal spring (value varying with impact phase)

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

kk /020

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

kk /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.

nnnn vdIdEE /'

n

uu E

EI

'n

ww E

EI

'

11

(1 if extension of tangential u- and w-springs –1 if compression)


Tangential Springs

Elastic strain energies:

20

2

4u

uGE 2

0

2

4w

uGE

Can keep track of integrals vu GG ,

Cannot determine changes and in length without knowing stiffness.

nIover nn Ev

Eu

,of

u v


System Overview

Impact Dynamics & ContactKinematics

',' vu IIvu II ,

I nE

ContactMode

Analysis

vu EE ,

nI

v

integrate

integrate nv


Sliding Velocity

svv

:v tangential contact velocityfrom contact kinematics

:sv sliding velocity represented byvelocity of particle p

)0,,( wuvvs

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 22


Contact Mode Transitions

Stick to slip when

nwu EEE 22

Slip to stick when

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


Sticking Contact

Rates of change in spring length.

)0,,(0 wuvvs

)0,0,1(vu

)0,1,0(vw

Particle p in simple harmonic motion like a spring-mass system. (No energy dissipation tangentially)


Sliding Contact

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

Energy dissipation rate (tangentially):

sn vIE '


System of Impact with Compliance

Differential equations with five functions and one variable :

),(' 1 unu GEfI

),(' 2 unw GEfI

),,,(' 3 wunnu GGEIfG

),,,(' 4 wunnw GGEIfG

),,(' 5 wunn IIIfE

nI

Tangential impulses

Length (scaled) of tangential springs

Energy stored bythe normal spring


Bouncing Ball

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

00 v All bounces are in one vertical plane


Impulse Curve (1st Bounce)

contact mode switch


Non-collinear Bouncing Points

00 v

)5,0,1(0 V

)0,6,6(0


Trajectory Projection onto Table


Bounce of a Pencil

)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:

z

x


Impulse Curve

Slipping direction varies.

slipstick slip

end of compression

)5302.5,3908.0,962.3(

y

z


Where Are We?

I. Impact with Compliance


Relationship of tangential impulse to normal impact at single contact(also with friction)





Simultaneous Collisions in 3D

Collision as a state sequence.

Within each state, a subset of impacts are “active”.

Energy-based restitution law.

Our model

High-speed photographs shows >2 objects simultaneously in contact during collision.

Lack of a continuous impact law.


Two-Ball Collision

Problem: One rigid ball impacts another resting on the table.

Question: Ball velocities after the impact?

virtual springs

22111222 xkxkFFvm

(kinematics)

(dynamics)

211 vvx

22 vx

11111 xkFvm


Impulses, Velocities & Stain Energies

11

011 Im

vv

dtFI 11Ball-ball impulse:

dtFI 22Ball-table impulse:

)(112

22 IIm

v

Velocities:

Rates of change in contact strain energies:

2

21

210

1

1 111 Im

Imm

vdIdE 21

22

2 1 IImdI

dE


State Transition Diagram The two impacts almost never start or end restitution at the same time. An impact may be reactivated after restitution.

4s

spring 1 endsrestitution first

spring 2 endsrestitution first

both springs endrestitution together

otherwise, spring 2ends restitution

otherwise, spring 1ends restitution

and beforespring 2 endsrestitution

01 e21 vv

and beforespring 1 ends restitution

02 e02 v


Some Facts

Impulse & strain energy for one impact also depend on those for the other (correlation).

A change of state happens when a contact disappears or a disappeared contact reappears.

Compression may restart from restitution within an impact state due to the coupling of impulses at different contacts.


Assumptions on Simultaneous Impacts

when restitution switches back to compression, or

Every end of compression of single impact:

when a contact is reactivated.

No change in stiffness

increase in stiffness

invariance of contact force


Stiffness, Mass, and Velocity Ratios

Theorem 2

Collision outcome depends on the contact stiffness ratio but not on individual stiffness.

The outcome does not change if the ball masses scale by the same factor.

Output/input velocity ratio is constant (linearity).

Consider upper ball with unit mass and unit downward velocity.


Impulse Curve

Theorem 3 During the collision, the impulses accumulate along a curve that is first order continuous and bounded within an ellipse.

21,II

1S

4S

2S

1S

11 m3

22 m

11

2 kk

121 ee

2I

1I

0)(2

121

102

212

21

1

IvIIm

Im

94.01 v30.02 v

lines of compression

10 v


Example with Energy Loss

1I

2I

1E

2E

energy loss

energy loss

11 m3

22 m 1

1

2 kk 9.01 e 7.02 e


Convergence

impulses at the end of the ith state in the sequence.

monotone non-decreasing

bounded within an ellipse

)},{( )(2

)(1

ii II

:, )(2

)(1

ii II

:, )(2

)(1

ii IISequence

Theorem 4 : The state transition will either terminate with or the sequence will converge with either or .

021 vv021 vv 021 vv


Ping Pong Experiment


Experiment

0v

)/( sm

)/( sm

31

2 kk

950043.01 e

846529.02 e

(ball-ball)

(ball-table)

Measured values:

Guessed value:

00023.0m kg

1v

2vsame trial


Where Are We?

I. Impact with Compliance


Relationship of tangential impulse to normal impact at single contact(also with friction)





Billiard Shooting

1I2I

n

z

c

Simultaneous impacts: cue-ball and ball-table!


Contact Structures

Normal impulses at the two contacts are described by the simultaneous impact model.

At each contact, normal impulse drives tangential impulse as described by the compliance model.

Cue-ball contact Ball-table contact


Combing the Two Impact Models

normal CBimpulse

normal BTimpulse

The two normal impulses take turns to drive the system.

,cv


Mechanical Cue Stick


A Masse Shot


predicted trajectory by model

reconstructed trajectory from video

if not ended by cushion

increasing cue-ballcompliance

Shot video Trajectory fitting

Post-shot ball velocities

Impact model Predicted post-shot ball velocities


Conclusion

• 3D impact modeling with compliance and friction• elastic spring energies• impulse-based not time-based• contact mode analysis (stick / slip)• sliding velocity computation

• Multiple impacts • state transition diagram• impulse curve • stiffness ratio• scalability• convergence

• Physical experiment.• Integration of two impact models


Extensions of Collision Model

Rigid bodies with arbitrary geometry

≥3 contact points

General simultaneous multibody collision

State transition templates

Measurement of relative contact stiffness

Robot pool player!


Acknowledgement

Matt Mason, Michael Erdmann, Ben Brown (CMU)

Amir Degani (Israel Institute of Technology)

Rex Fernando, Feng Guo (ISU students)

HR0011-07-1-0002


Online Papers

International Journal of Robotics Research, 2012:

http://www.cs.iastate.edu/~jia/papers/IJRR11a-submit.pdf

http://www.cs.iastate.edu/~jia/papers/IJRR11b-submit.pdf

1. Yan-Bin Jia. Three-dimensional impact: energy-based modeling of tangential compliance. DOI: 10.1177/0278364912457832.

2. Yan-Bin Jia, Matthew T. Mason, and Michael A. Erdmann. Multiple impacts: a state transition diagram approach. DOI: 10.1177/0278364912461539.


Appendix 1: 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

vI0

020

1)0('

Under Coulomb’s law, we can show that


Bouncing Ball – Integration with Dynamics

Contact kinematics

Im

zmIvv z

27

0

)1,0,0(z

Theorem 1 During collision, is collinear with . I 0v

Velocity equations:

mIVV /0

Izmr

2

50

(Dynamics)

Impulse curve lies in a vertical plane.


Impulse Curve (1st Bounce)

Tangential contact velocity vs. spring velocity

contact mode switch

v


Bouncing Pencil

1m 2.120 8.0 5.0e1r 31 h 5.02 h

3

simultaneous three-dimensional impacts with friction and compliance

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