Embedding Nonsmooth Systems in Digital...
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Ryo KikuuweRyo Kikuuwe
Embedding Embedding Nonsmooth Systems Nonsmooth Systems
inin Digital ControllersDigital Controllers
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Nonsmooth Systems?Nonsmooth Systems? Systems subject to discontinuous differential
equations. A good example is ...
v (velocity)
f (force)
Mathematically very cumbersome. but very familiar in our daily lives. and useful for some control applications
Coulomb friction
Mv
f
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Today's TalkToday's Talk
1. about natural nonsmooth systems
2. about artificial nonsmooth systems
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Simulation ofSimulation ofNatural Nonsmooth Systems:Natural Nonsmooth Systems:
Systems Subject toSystems Subject toCoulomb FrictionCoulomb Friction
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Motivation: Request from Automobile Motivation: Request from Automobile IndustryIndustry
Robotic assistance for manual fine positioning in assembling.
Problem: What kind of resistive forces are appropriate for enhancing the performance ?
robotic device(with actuators)
force sensor
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Haptic FeedbackHaptic Feedback
To realize a prescribed position-force relation at the interface between a device and an operator contacting therewith.
Applications: Robotic assistance of manual tasks Virtual reality (involving haptics)
surgery simulators, computer games, CAD interfaces, ...
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Idea: “Coulomb Friction”Idea: “Coulomb Friction” It cancels external forces below static friction
level. (Thus, effective for fine positioning.) Most familiar dynamics in our daily lives.
But producing such force-velocity relation by digital control is very challenging!!
f vvelocity
force
f
v
?
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sensor
actuator
device
velocity
force
velocity
force
computer
Why Is Coulomb Friction Problematic?Why Is Coulomb Friction Problematic? In discrete time, the force is determined according to
the relative velocity. Due to the latency, chattering (repeated zero-velocity
crossings) occurs.
static friction
kinetic friction
kinetic friction
f
v?
Idea: put a delayless feedback in the loop to remove chattering.
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Delayless Feedback Including Signum?Delayless Feedback Including Signum?+ ¡
y
x
where
y
x
where
equivalent
(unit saturation function)
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SolutionSolution
output f
input u
Rigid, discontinuous friction between objects: difficult to simulate.
output force f
input velocity v+
Put a delayless feedback as a virtual viscoelastic element.
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Derivation of Simulation AlgorithmDerivation of Simulation Algorithm
input: velocity u
output:force f
K
B
e
output f
input u+
[Kikuuwe et al.: IEEE Trans. on Robotics, 2006]
step 1: continuous-time representation differential algebraic equations
step 2: discrete-time representation backward Euler; algebraic equations
step 3: Algorithm (discrete-time) solution to the above
¡
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ApplicationsApplications
clampingfriction force onthe needle shaft
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Implementation ofImplementation ofArtificial Nonsmooth Systems:Artificial Nonsmooth Systems:
““Proxy-based Sliding Mode Control”Proxy-based Sliding Mode Control”
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MotivationMotivation What happens?
Physical characteristic that does not exist in nature but can be produced by digital control
(velocity)
(force)
v
f
p + H v (position + velocity
multiplied by something)
f (force)
Sliding Mode Control ?
Coulomb Friction
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Sliding Mode ControlSliding Mode Control A “nonsmooth” control scheme.
The actuator force is “discontinuously” switched on a “sliding surface” in the state space.
time t
position p
position p
velocity vsliding surface
Simplest Example:
exponentially converges to the target position with time constant H
F : upper bound of force H : time constant
In discrete time, however, there is chattering due to the discontinuity!!
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““Proxy-based Sliding Mode Control”Proxy-based Sliding Mode Control”[Kikuuwe & Fujimoto, ICRA2006]
step 1: continuous-time representation differential algebraic equations
step 2: discrete-time representation algebraic equations
step 3: Discrete-time representation solution to the above
targetposition
PIDcontroller
sliding modecontroller
p
q
pd
f
f
f
f
controlledobject
proxy
external forces
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Benefit of PSMCBenefit of PSMC PSMC is as accurate as but
much safer than PID!! Before saturation, equivalent to
PID control. fast response to small positional error
After saturation, SMC-like response is dominant moderate recovery from large
positional error without overshoots.
fast fast
v
p
p
time slowslow
t
targetposition
PIDcontroller
sliding modecontroller
p
q
pd
f
f
f
f
controlledobject
proxy
external forces
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Response to Step Input Response to Large DisturbancePID-C vs PSMCPID-C vs PSMC
PSMC is as accurate as but safer than PID
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Voice of UsersVoice of Users [Van Damme et al., 2007] @ Vrije Univ. Brussel
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ConclusionsConclusions Introduced an algorithm for producing
Coulomb friction characteristics in discrete-time controller.
Introduced a new position control scheme, "Proxy-based sliding mode control."
Future (Current Ongoing) ProjectsFuture (Current Ongoing) Projects Utilization of friction algorithms
Driving simulator with realistic steering resistance. Extension of PSMC for safer, human-friendly robots.
Low-cost robot equipped with standardized gear transmissions.