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Wavebased control of flexible mechanical systems

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Experimental Crane Model. Space structures simulated. Flexing rig, no control ... Typical of robots, some cranes. But sometimes the end point is not known initially. ... – PowerPoint PPT presentation

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Title: Wavebased control of flexible mechanical systems


1
Wave-based control of flexible mechanical systems
ICINCO 2006 Setúbal, Portugal.
  • William OConnor
  • University College Dublin
  • National University of Ireland

2
A powerful, new, robust, generic solution
Control of flexible mechanical systems
3
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4
Growing importance
  • Robotics
  • Cranes
  • Disk drives
  • Long reach manipulators
  • Cleaning, construction equipment
  • Medical human assistance devices
  • Space structures

5
Traditional robots
  • Heavy
  • Short arms
  • Dynamically sub-optimal
  • Expensive
  • But rigid

6
Lighter robots
  • Dynamically more responsive
  • Faster
  • Cheaper
  • Less energy/force
  • Sometimes necessary

7
Cranes
Cable is inherently flexible
The Gantry Crane problem
8
  • Reconciling
  • position control
  • and
  • active vibration
  • damping

9
Rigid System
x1
x0
x0 x1 Control Easy!
10
Flexible System
x0
x1
x2
x3
m
m
m
c
c
c
Tip mass
Actuator
What actuator input, x0(t), to control
output, x3(t) ?
11
Second order systems
actuator
load
actuator
load
actuator
load
Distributed and/or Lumped
12
Fourth order systems
y2
q3
yn
k3
y1
y3
kn
qn
q2
k2
Load qn , yn
k1
q1
Actuator q0 , y0
13
A difficult problem?
14
to date a general solution to the control
problem of flexible structures has yet to be
found. One important reason is that
computationally efficient (real-time)
mathematical methods do not exist for solving the
extremely complex sets of partial differential
equations and incorporating the associated
boundary conditions that most accurately model
flexible structures.
p.165 of Flexible robot dynamics and
controls Kluwer Academic/Plenum, New York, 2002
15
Approaches to date
  • Classical state-space control
  • Modal control
  • Input / command shaping
  • Bang-bang control
  • Sliding mode control
  • FOC
  • Wave Virtual system ideas

16
no general solution to date
  • Major focus on system identification and
    modelling (cf extremely complex PDEs BCs )
  • Generally, at best get asymptotic position
    vibration control.

17
Wave-based techniques
18
Experimental Crane Model
19
Space structures simulated
20
Flexing rig, no control
Lab rig in University of Castilla-La Mancha,
Ciudad Real, Spain
21
Flexing rig, w-b control
22
Wave-based techniques
Control of complex system in a simple, natural way
23
Simple idea
  • The interface is key
  • a) Understand,
  • b) Measure,
  • c) Manage the interface
  • All using Wave concepts

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25
Generic flexible system
Actuator
Arbitrary flexible system
Load
load / end point
26
Same control strategy
Actuator
Flexible system
Load
27
Also for laterally flexing systems
Arbitrary flexible system
Load qn , yn
Actuator q0 , y0
28
What must controller do?
?
?
Computer controller?
?
Assume actuator / trolley can move themselves
29
Wave view of flexible system
(Assuming displacement waves)
30
  • Wave-control ideas

When actuator moves, it launches wavex.
  • Wave pass though each of the masses.
  • Wave reaches load mass. It deflects 2x.
  • Returning wave moves each mass by x again.
  • If actuator absorbs this returned wave,
  • system will comes to rest at displacement 2x!

31
Actuator does 2 jobs
x0(t) launch absorb
x0(t)
x0(t) a0(t) b0 (t)
32
Key idea 1 PushPull
Set launch component, a0(t) to reach ½(Target_x)
while simultaneously measuring and adding b0(t)
to absorb returning motion. (Guarantees success)
Absorbing action causes system to move the other
half-target distance. (For how? and why? See
below.)
Newton III?
33
1.4
1.2
End mass posn
1
Actuator posn
Response
0.8
Launch_x ½ Trgt
0.6
0.4
absorb_x
0.2
0
?t
0
2
4
6
8
10
12
14
34
Works very well, but
  • Assymptotic approach to target

35
Key idea 2
Launch_x time profile is arbitrary provided the
final value is correct (½Target)
36
Best launch profile?
Best way for Launch-x to arrive at ½Target
position is a time-reversed and inverted
re-play of absorbed wave from start-up
  • Wave-Echo control

37
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38

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40
  • Wave echo control, single mass

1.4
1.2
1.0
1
Mirror of absorb
Response
0.8
0.6
0.5
Absorb
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
?t
41
Wave echo control, 3 mass-springs
1.4
1.2
1
0.8
Response
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
wt
42
Magic!
Works wonderfully! Load stops dead!
  • Many other outstanding features
  • (see later)

43
Waves??...
44
x0(t) launch absorb
a0(t)
x0(t)
b0(t)
45
Implications
\/
For rest to rest manoeuvres
\/
(for any value of Z)
46
  • Waves in lumped systems?

47
Ref x0
X0

Gn-1
Gn
G1
G2
-
F
Hn-1
Hn
H1
H2
Xi Ai Bi
Wave transfer functions
48
Wave transfer functions
  • Have neither poles nor zeros
  • Steady-state gain of unity
  • Close to second order
  • Zero instantaneous response
  • Dominated by local dynamics
  • Lagging phase

49
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50
a0(t) A0(s)
x0(t) launch absorb
x0(t) X0(s)
X1(s)
b0(t) B0(s)
51
X0
X1


A0a0
G1(s)


B0 b0
H1(s)
52
C c
Xtrgt

½

X0
X1


A0a0
Absorb wave b0 added to launch wave set to half
ref
G1(s)


B0 b0
H1(s)
53
Xtrgt
C c
X0
X1
a0


b0
G1(s)


H1(s)
54
G1(S), H1(s) approximated

x

x

0
1
k


m


c
Ö
(km)
G(s) X1(s)/X0(s)
wn2 s2swn wn2
55
Absorbing motion
  • Ensures stability
  • Dampens vibrations
  • Moves system second half of motion
  • Gives real-time system identification
  • It opens the loop, cancelling poles

56
Ref x0
X0

Gn-1
Gn
G1
G2
-
F
Hn-1
Hn
H1
H2
57
Features of control system
  • Works for n masses
  • Uniform or non-uniform
  • With or without internal damping
  • No system model needed
  • Or System itself is model
  • and computer

58
continued
  • Sensing is minimal,
  • and local
  • Real Actuator OK
  • Zero steady-state error
  • Minimal vibration in transit

59
Continued
  • Very rapid
  • Very energy efficient
  • Does deliberately naturally what other
    approaches do, perhaps unconsciously, with
    difficulty.
  • Generic

60
Take care of the interface
  • And the system will take care of itself

61
Other applications
  • Stabilized platforms
  • Power assisted motion
  • Multiple actuator systems
  • A complex problem may have a simple solution!

62
The End
  • Thank you

william.oconnor_at_ucd.ie
63
Space structures simulated
64
Actuator-System interface
  • Two-way motion flow in all flexible system types
  • distributed lumped 2nd order systems
  • distributed lumped 4th order systems
  • uniform or non-uniform
  • mixed lumped and distributed
  • Energy, momentum, dynamics, sensing, control
    all via the interface

65
Wave-based models
  • A new way to model lumped mechanical systems
  • Two-way motion revealed,
  • made legitimate (defined),
  • made measurable, and
  • thereby allowing control.

66
Problem definition
?
  • How should you move the actuator
  • (or trolley)
  • to move the end-point from A to B, and
  • to control the vibrations?

?
?
67
Magic!
  • Zero vibration. Zero steady state error
  • No system model needed, nor modal info
  • Arbitrary order
  • Adapts automatically to system changes
  • Works fine with real actuator
  • System can be non-linear, non-uniform
  • Very energy efficient, very rapid
  • Computationally very simple
  • No jerk, nor chatter, nor precise switching
  • etc

68
System modelling
  • New method of modelling lumped flexible dynamic
    systems
  • Already proving itself very powerful and
    adaptable to new demands

69
Point to point control
  • Typical of robots, some cranes
  • But sometimes the end point is not known
    initially.
  • Typical of manual operation of cranes.
  • Open ended control

70
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