m. zareinejad 1. fundamentally, instability has the potential to occur because real-world...
TRANSCRIPT
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Haptics and Virtual Reality
M. Zareinejad
Lecture 11:
Haptic Control
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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?
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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?
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sampled-data system example
stability of thevirtual wall
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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.
Passivity and stability
A passive system is stable.
Any interconnection of passive systems (feedforward or feedback) is stable.
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
Direct coupling – network model
• 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
Interaction behavior of ideal mass (inertia)
mssv
sFsZ
sFsmsv
Fym L
)(
)()(
)()(
..
Interaction behavior of ideal spring
s
K
sv
sFsZ
svs
KsF
KyF L
)(
)()(
)(1
)(
Interaction behavior of ideal dashpot
Bsv
sFsZ
sBvsF
yBF L
)(
)()(
)()(
Impedance behavior of typical mechanical system
s
KBsms
sv
sFsZ
svs
KBmssF
KyyBymF L
2
)(
)()(
)()(
19
20
21
22
23
24
25
26
27
28
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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
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1
12
01
)()(
e
h
e
h
f
vbms
v
f
01
1
Virtual coupling – mechanical model
• Rigid
• Compliant connection between haptic device and virtual tool
Virtual coupling – closed loop model
Virtual coupling – network model
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h
vc
d
e
h
vc
d
e
h
e
d
vce
d
f
v
zZ
zZOHzZ
f
v
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zZOHzZ
v
f
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v
zZv
f
***
*
*
*
)(
11
)()(
)(
11
10
01
)()(
)(
11
10
where:
Tzzs
vcvcvc s
KBzZ
1
)(
)()(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
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.
Teleoperation
Unilateral:◦ Transmit
operator command.
Bilateral:◦ Feel
environment response.
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
Bilateral teleoperation architectures• Position/Position (P/P).• Position/Force (P/F).• Force/Force (F/F).• 4 channels (PF/PF).• Local force feedback.
General bilateral teleoperator
General bilateral teleoperator (ctd.)
ess
emmssmmth ZCCCCCZ
ZCCCZCCCZCZZ
2343
2141
1
• Impedance transmitted to user:
• Perfect transparency:
• Trade-off between stability & transparency.
eth ZZ
Haptic teleoperation
Teleoperation:◦ Master robot.◦ Slave robot.◦ Communication
(force, velocity).
Haptics:◦ Haptic device.◦ Virtual tool.◦ Communication
(force, velocity).
4 channels teleoperation for haptics• Transparency requirement:
– Stiffness rendering (perfect transparency):
– Inertia rendering (intervening impedance):
• Control/VE design separated.
eth ZZ
esth ZZZ
Direct coupling as bilateral teleoperation• Perfect transparency.
• Potentially unstable.emth ZZZ
Virtual coupling as bilateral teleoperation
cm
esc
escmth
ZZ
ZZZ
ZZZZZ
• Not transparent.
• Unconditionally stable.
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.
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.
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
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
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.,
)(
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)()(
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)(
)(
)(
sv
sv
sF
sFsS
sv
sv
sF
sF
e
h
e
h
e
h
e
h
1)( sS
1)()(10
01)(
IsHIsHsS
Passivity – absolute stability – potential instability[Haykin ’70]
)Re()Re(2 2211
2112
hh
hh
)Re()Re(2
)Re()Re(
2211
2112
hh
hh