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Haptics and Virtual Reality M. Zareinejad Lecture 11: Haptic Control 1

M. Zareinejad 1. fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 1: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Haptics and Virtual Reality

M. Zareinejad

Lecture 11:

Haptic Control

Page 2: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

although these approximation errors are small, their potentially non-passive nature can have profound effects, notably:

instability limit cycle oscillations (which can be just as

bad as instability)

why do instabilities occur?

Page 3: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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a useful tool for studying the stability and performance of haptic systems a one-port is passive if the integral of power extracted over time does not exceed the initial energy stored in the system.

Passivity

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Passive system properties

A passive system is stable

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“Z-width” is the dynamic range of impedances that can be rendered with a haptic display while maintaining passivity

we want a large z-width, in particular: zero impedance in free space large impedance during interactions with

highly massive/viscous/stiff objects

Z-width

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lower bound depends primarily on mechanical design (can be modified through control)

upper bound depends on sensor quantization, sampled data effects, time delay (in

teleoperators), and noise (can be modified through control)

in a different category are methods that seek to create a perceptual effect (e.g., event-based rendering)

How do you improve Z-width?

Page 7: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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sampled-data system example

stability of thevirtual wall

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Page 10: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Direct coupling – limitations in 1DOF

No distinction between simulation and control:◦ VE used to close the force feedback loop.

In 1DOF:◦ Energy leaks. ZOH. Asynchronous switching.

Page 11: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Passivity and stability

A passive system is stable.

Any interconnection of passive systems (feedforward or feedback) is stable.

Page 12: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Real contact

◦ No contact oscillations.◦ Wall does not generate

energy (passive).

Haptic contact

◦ Contact oscillations possible.◦ Wall may generate energy (may

be active).

Haptic vs. physical interaction

Page 13: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Direct coupling – network model

Page 14: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

• Impedance– Measures how much a

system impedes motion.– Input: velocity.– Output: force.

• Varies with frequency.

Terminology – interaction behavior

Interaction behavior = dynamic relation among port variables (effort & flow).

• Admittance– Measures how much a

system admits motion.– Input: force.– Output: velocity.

)(

)()(

sv

sFsZ )(

)(

)()( 1 sZ

sF

svsY

Page 15: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Interaction behavior of ideal mass (inertia)

mssv

sFsZ

sFsmsv

Fym L

)(

)()(

)()(

..

Page 16: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Interaction behavior of ideal spring

s

K

sv

sFsZ

svs

KsF

KyF L

)(

)()(

)(1

)(

Page 17: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Interaction behavior of ideal dashpot

Bsv

sFsZ

sBvsF

yBF L

)(

)()(

)()(

Page 18: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Impedance behavior of typical mechanical system

s

KBsms

sv

sFsZ

svs

KBmssF

KyyBymF L

2

)(

)()(

)()(

Page 19: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 20: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 21: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 22: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 23: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 24: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 25: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 26: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 27: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 28: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 29: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

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Page 30: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Direct coupling – network model (ctd. II)

Continuous time system:

Sampled data system – never unconditionally stable!

e

h

e

hd

e

h

f

vz

zb

z

z

Tm

f

vzZOHzZ

v

f***

01

1

2

1

1

12

01

)()(

e

h

e

h

f

vbms

v

f

01

1

Page 31: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Virtual coupling – mechanical model

• Rigid

• Compliant connection between haptic device and virtual tool

Page 32: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Virtual coupling – closed loop model

Page 33: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Virtual coupling – network model

e

h

vc

d

e

h

vc

d

e

h

e

d

vce

d

f

v

zZ

zZOHzZ

f

v

zZ

zZOHzZ

v

f

f

v

zZv

f

***

*

*

*

)(

11

)()(

)(

11

10

01

)()(

)(

11

10

where:

Tzzs

vcvcvc s

KBzZ

1

)(

Page 34: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

)()(Re2

)(cos1

)(

1Re

0)(

1Re

0))(Re(

zZOHzZ

zZOH

zZ

zZ

zZ

dvc

vc

d

Virtual coupling – absolute stability

• Llewellyn’s criterion:

• Need:• Physical damping.• “Give” during rigid contact.• Larger coupling damping for larger contact stiffness.• Worst-case for stability: loose grasp during contact.• Not transparent.

evc

eZOHvcdh ZZ

ZZZZZ

Page 35: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Virtual coupling – admittance device

• Impedance / admittance duality:

• Spring inertia.• Damper damper.

• Need:• “Give” during contact.• Physical damping.• Larger coupling damping for larger contact stiffness.• Worst case for stability: rigid grasp during free motion.

Page 36: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Teleoperation

Unilateral:◦ Transmit

operator command.

Bilateral:◦ Feel

environment response.

Page 37: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Ideal bilateral teleoperator• Position & force matching:

• Impedance matching:

• Intervening impedance:

• Position & force matching impedance matching.

sm

sm

ff

xx

teth ZZ

s

m

s

m

f

v

v

f

01

10

temth ZZZ

s

mm

s

m

f

vZ

v

f

01

1

Page 38: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Bilateral teleoperation architectures• Position/Position (P/P).• Position/Force (P/F).• Force/Force (F/F).• 4 channels (PF/PF).• Local force feedback.

Page 39: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

General bilateral teleoperator

Page 40: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

General bilateral teleoperator (ctd.)

ess

emmssmmth ZCCCCCZ

ZCCCZCCCZCZZ

2343

2141

1

• Impedance transmitted to user:

• Perfect transparency:

• Trade-off between stability & transparency.

eth ZZ

Page 41: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Haptic teleoperation

Teleoperation:◦ Master robot.◦ Slave robot.◦ Communication

(force, velocity).

Haptics:◦ Haptic device.◦ Virtual tool.◦ Communication

(force, velocity).

Page 42: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

4 channels teleoperation for haptics• Transparency requirement:

– Stiffness rendering (perfect transparency):

– Inertia rendering (intervening impedance):

• Control/VE design separated.

eth ZZ

esth ZZZ

Page 43: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Direct coupling as bilateral teleoperation• Perfect transparency.

• Potentially unstable.emth ZZZ

Page 44: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Virtual coupling as bilateral teleoperation

cm

esc

escmth

ZZ

ZZZ

ZZZZZ

• Not transparent.

• Unconditionally stable.

Page 45: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Stability analysis

• Questions:

• Is the system stable?• How stable is the system?

• Analysis methods:

• Linear systems:• Analysis including Zh and Ze:

• Non-conservative.• Need user & virtual environment models.

• Analysis without Zh and Ze:• Zh and Ze restricted to passive operators.• Conservative.• No need for user & virtual environment models.

• Nonlinear systems – based on energy concepts (next lecture):• Lyapunov stability.• Passivity.

Page 46: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Stability analysis with Zh and Ze

• Can incorporate time delays.

• Methods:• Routh-Hurwitz:

• Stability given as a function of multiple variables.

• Root locus:• Can analyze performance.• Stability given as function of single parameter.

• Nyquist stability.

• Lyapunov stability:• Stability margins not available.• Cannot incorporate time delay.

• Small gain theorem:• Conservative: considers only magnitude of OL.

• m-analysis:• Accounts for model uncertainties.

Page 47: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Stability analysis without Zh and Ze

• Absolute stability: network is stable for all possible passive

terminations.

• Llewellyn’s criterion:• h11(s) and h22(s) have no poles in RHP.• Poles of h11(s) and h22(s) on imaginary axis are simple

with real & positive residues.• For all frequencies:

• Network stability parameter (equivalent to last 3 conditions):

• Perfect transparent system is marginally absolutely stable, i.e.,

1

ReRe2cos

2112

22112112

hh

hhhh

0ReReRe2

0Re

0Re

211221122211

22

11

hhhhhh

h

h

1

Page 48: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Stability analysis without Zh and Ze

(ctd. I)

• Passivity:• Raisbeck’s passivity criterion:

• h-parameters have no poles in RHP.• Poles of h-parameters on imaginary axis are simple with

residues satisfying:

• For all frequencies:

0ImImReReReRe4

0Re

0Re

22112

222112211

22

11

hhhhhh

h

h

12*

2121122211

22

11

0

0

0

kkwithkkkk

k

k

Page 49: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Stability analysis without Zh and Ze

(ctd. II)

• Passivity:• Scattering parameter S:

• As function of h-parameters:

• Perfect transparent system is marginally passive, i.e.,

)(

)(

)(

)()(

)(

)(

)(

)(

sv

sv

sF

sFsS

sv

sv

sF

sF

e

h

e

h

e

h

e

h

1)( sS

1)()(10

01)(

IsHIsHsS

Page 50: M. Zareinejad 1.  fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

Passivity – absolute stability – potential instability[Haykin ’70]

)Re()Re(2 2211

2112

hh

hh

)Re()Re(2

)Re()Re(

2211

2112

hh

hh