On the Design of a Precision Machine
for Closed-Loop Performance
by
Kripa K. Varanasi
B.Tech., Indian Institute of Technology Madras, 1998
submitted to the Department of Mechanical Engineering and the
Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degrees of
Master of Science in Mechanical Engineering
and
Master of Science in Electrical Engineering and Computer Science
at the
Massachusetts Institute of Technology
February 2002
@
2002 Massachusetts Institute of Technology
all rights reserved
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
MAR 2 5 2002
LIBRARIES
Author
Department of Mechanical Engineering
Department of Electrical Engineering and Computer Science
18 January 2002
Certified by
Samir A. Nayfeh
Assistant Professor of Mechanical Engineering
Certified by
Munther A. Dahleh
Professor of Electrical Engineering and Computer Science
Accepted by
Am A. Sonin
Chairman, Department Committee on Graduate Students
Department-of Mechanical Engineering
Accepted by
_
Arthur C. Smith
Chairman, Department Committee on Graduate Students
Department of Electrical Engineering and Computer Science
On the Design of a Precision Machine
for Closed-Loop Performance
by
Kripa K. Varanasi
submitted to the Department of Mechanical Engineering and
the Department of Electrical Engineering and Computer Science
on January 18, 2002 in partial fulfillment of the requirements for the degrees
of Master of Science in Mechanical Engineering and Master of Science in
Electrical Engineering and Computer Science
ABSTRACT
In high performance servo-machines the resonances arising from the flexibility
of the machine structures become increasingly difficult to avoid, and can lead
to poor performance and possible instability due to the "spillover" of control
energy onto the vibratory modes. Hence, in order to achieve high performance
and robustness, the classical methodology of designing controllers for a given
machine must be replaced by designing machines and controllers simultaneously to achieve the required closed-loop specifications. This inverse problem
is the main focus of the thesis.
We start by formulating the inverse problem for ball-screw-type servomechanisms. We then present an overview of the ball-screw servomechanism and
qualitatively understand the limitations imposed by the axial resonance of the
screw on control performance. Next, we set out to derive a reasonably accurate model supported by experimental evidence. We then use the Nyquist
condition to establish metrics which quantify performance and robustness,
and relate these measures to the machine parameters using perturbation and
graphical techniques. Finally, we combine these results with the requirements
for acceleration and speed to yield a set of acceptable designs for the machine
which satisfy the given closed-loop specifications.
ACKNOWLEDGMENT
First of all, I want to express my sincerest gratitude to my advisor Prof. Samir
Nayfeh for his invaluable guidance, support, and encouragement. I consider it
a great privilege to work with him. Samir is a dedicated teacher and is like
an elder brother to me; motivating, instructing and inspiring me to do my
best. His creativity in practical design matters or theoretical formulations has
always left me in awe. Through his own example, he has instilled in me the
zeal for perfection and deep understanding. I have never met a person in my
life who is as profound and as humble as he is. I am indeed indebted to him
for all that he has done for me.
I am immensely grateful to Prof. Munther Dahleh for his time and inputs to
the thesis. It is an honor to interact and learn from him. Discussions with him
helped me understand the power of abstraction in mathematics, importance
of creating counter examples, necessity to search for an intuitive and elegant
solution for a rigorous proof, among others. His lectures and discussions have
always left me inspired. I am extremely grateful to Prof. Sanjay Sarma for
all his help, encouragement and advice. Sanjay's enthusiasm, and vibrant
personality always motivates me to be at my best. I have greatly benefited
from my discussions with Prof. Trumper. He has helped me refine my ideas on
mechatronics and feedback design. I am grateful to him for his instruction and
advice. I want to express my heart-felt thanks to my undergraduate advisor
Prof. Ramamurti for instructing me on the design, dynamics, and control of
machines.
Next, I would like to thank Mark Belanger, Gerry Wentworth, and Victor
Lerman for their support in the LMP shop. I have learnt a lot from Mark
and am grateful to him for his patient and thorough instruction. I want to
thank Maureen Lynch and Leslie Regan for all the help. Maureen's motherly
affection is always a great comfort when I am so far away from my family.
I thank Ashok Mantravadi for all his help and guidance. I was fortunate to
interact and learn from a number of my talented friends at MIT. I thank David
Chargin and Stephen Ludwick for patiently answering my questions and helping me in the lab during my initial period at MIT. I thoroughly enjoyed working with and learning from Perry Banks while building the "rack and track"
machine. Discussions with Pradeep Subrahmanyan were also very helpful. I
thank Eberhard Bamberg and Elmer Lee for all their help on computer-related
issues. I would like to thank Mahadevan Balasubramaniam for his help especially when I was preparing for my Ph.D qualifying exams.
I thank all my friends for their wonderful company and support. I would
also like to thank everyone who was directly or indirectly involved in the
completion of this work.
Most of all, I would like to thank my parents, Kanthi and Mohan Varanasi,
and my sister Chandana Varanasi, for their constant love, support, and encouragement. Their affection, instruction, and inspiration have always provided
me the means to excel and have given a direction to my life. It is to them
and to Samir that I respectfully dedicate this thesis. I also want to thank my
grandparents, aunts, uncles, and cousins for their constant support.
Finally, I thank the Lord God for all that he has given me. I could not
possibly be luckier.
5
6
CONTENTS
I
2
3
Introduction
16
1.1
1.2
1.3
17
20
23
The Inverse Problem in Motion Control .....
................
Overview ........
.................................
Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . .
Ball-Screw Servomechanism
2.1 Introduction . . . . . . . . . . . . . .
2.2 Basic Physics of the Drive Resonance
2.3 Options to Increase Performance . .
2.4 Chapter Summary . . . . . . . . . .
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32
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39
42
MIMO Systems with Mixed Damping . . . . . . . . . . . . .
Equivalent Model. . . . . . . . . . . . . . . . . . . . . . . . .
44
45
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A Qualitative
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Models of Damped Systems
3.1 Complex Modulus Approach . . . . . . .
3.2 Mixed Viscous and Hysteretic Damping .
3.2.1 M otivation . . . . . . . . . . . . .
3.2.2 SDOF system with mixed damping
3.2.3 Equivalent Model . . . . . . . . . .
3.2.4
3.2.5
3.3
4
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Picture
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25
25
27
29
31
Chapter Summary
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. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modeling
4.1 Introduction . . . . . . . . . . . . . . . . .
4.2 Distributed-Parameter Model . . . . . . . .
4.2.1 Ball-Screw Servo with Free End . . .
4.2.2 Ball-Screw Servo with Damped End
4.2.3 Boundary Conditions . . . . . . . .
4.3 Approximate Methods . . . . . . . . . . . .
4.3.1
Method of Assumed Modes . . . . .
4.3.2 Method of Weighted Residuals . . .
4.4 Lumped-Parameter Approximations . . . .
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46
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53
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59
7
5
4.4.1
Ball-Screw Servo with Free End
. . . . . . . . . . . . . . . .
60
4.4.2
Ball-Screw Servo with Damped End . . . . . . . . . . . . . .
65
Robust Controller Design and Bandwidth Formulae
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Expressions for Open-Loop Poles using Perturbation Method . . .
5.3 Collocated Transfer Function . . . . . . . . . . . . . . . . . . . . .
Control of um (t) . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1
5.3.2 Limitations of controlling uc(t) through Collocated Control
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5.4 Non-collocated Transfer Function
5.4.1
5.5
5.6
5.7
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75
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78
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81
83
Bandwidth and Phase Margin . . . . . . . . . . . . . . . . . .
83
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Margins.
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85
86
89
89
89
5.4.2 Sensitivity Properties . . . . . . . . . . . . . . . .
Significance of the Non-Minimum Phase Zero . . . . . . .
Maximum Achievable Bandwidth with Certain Robustness
5.6.1 Robust Stability and Performance . . . . . . . . .
5.6.2 Robustness Metrics and Maximum Bandwidth . .
Bandwidth Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.7.1
5.7.2
Closed-loop Bandwidth with a Lead Compensator . . . . . . 100
Closed-Loop Bandwidth with a Lead Compensator and an
Additional Pole
5.8
6
. . . . . . . . . . . . . . . . . . . . . . . . . 103
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
109
Model Parameter Reduction : Impact of Mechanical Design
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.1.1 Reduced-Parameter Model . . . . . . . . . . . . . . . . . . . 110
6.2
6.3
8
Chapter Summary
M echanical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
of Joints and Contact Interfaces . .
of the Machine Base and Linear Rail
Bearings and Assembling Techniques
of the Carriage . . . . . . . . . . . .
of the Bearing Blocks . . . . . . . .
of the Thrust Bearings . . . . . . .
6.2.1
6.2.2
6.2.3
6.2.4
6.2.5
6.2.6
Design
Design
Linear
Design
Design
Choice
6.2.7
Choice of the Coupling
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Assembly
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125
128
129
131
131
133
. . . . . . . . . . . . . . . . . . . . . 135
6.2.8 Design of the Damper . . . . . . . . . . . . . . . . . . . . . . 137
Choice of the Encoder and the Design of its Mounts . . . . . 137
6.2.9
Experimental Identification . . . . . . . . . . . . . . . . . . . . . . . 138
. . . . . . . . . . . . . . . . . . . . . . . . . 138
6.3.1
M odal Analysis
6.3.2
Sine-Sweep Experiments
. . . . . . . . . . . . . . . . . . . .
147
7
Solutions to the Inverse Problem
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3
7.2.1
Non-Minimum Phase Zero . . . . . . . . . . . . . . . . . . . . 154
7.2.2
7.2.3
7.2.4
7.2.5
Maximum Acceleration . . . . . .
Maximum Velocity . . . . . . . . .
Motor and Power Amplifier Limits
Machine Components . . . . . . .
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156
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159
160
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.3.1 Wb-PM-RGM Surface . . . . . . . . . . . . . . . . . . . . . . 160
7.3.2
7.3.3
8
153
153
154
Design Space and Constraint Surfaces . . . . . . . . . . . . . 162
Comment on Disturbance and Noise Rejection . . . . . . . . 162
Conclusions and Future Work
165
8.1 Sum mary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 165
A Proof of Theorem 1
167
B A Review of Classical Control
B.1 Typical Control Tradeoffs and Constraints .
B.1.1 Sensitivity to Parameter Changes . .
B.1.2 Algebraic Constraints . . . . . . . .
B.1.3 Properties of a First-Order System
B.1.4 Properties of a Second-Order System
B.1.5 Closed-Loop Behavior . . . . . . . .
B.2 Classical Compensators . . . . . . . . . . .
B.2.1 Design Notes . . . . . . . . . . . . .
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C Robust Stability and Performance
C.1
Robust Stability
C.2
Robust Performance
D Engineering Drawings
Bibliography
170
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172
172
175
176
182
184
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. . . . . . . . . . . . . . . . . . . . . . . . . . .
188
190
205
9
LIST OF FIGURES
1.1
1.2
Flowchart showing the forward and inverse problems . . . . .
Typical loop transmission . . . . . . . . . . . . . . . . . . . .
17
18
1.3
Influence of plant resonance on closed-loop performance . . . .
19
2.1
Ball-screw Designs: Return tube
ball-nuts (courtesy NSK Corp.) .
Schematic of a ball-screw drive .
Frequency response of undamped
and internal deflector type
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
screw; the carriage is posi-
26
26
tioned at (1) 10% (2) 42% (3) 90 % . . . . . . . . . . . . . . .
28
Frequency response with a damper in the preload path. Figure
shows a comparison of damped and undamped cases for three
carriage positions . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.2
2.3
2.4
3.1
3.2
3.3
3.4
3.5
Complex Modulus Representation . . . . . . . . .
Schematic of a SDOF system with mixed damping
Four solutions of the eigenvalue problem . . . . .
Correct Eigenvalues . . . . . . . . . . . . . . . . .
Geometric interpretation of frequency response . .
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35
36
37
38
41
4.1
4.2
4.3
4.4
Free-body diagram of the ball-screw servo with a free end . . .
Free-body diagram of the ball-screw servo with a damped end
Quasi-static displacements for the free-end boundary condition
Quasi-static displacements for the damped-end boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
51
61
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66
5.1
Feedback control of um(t)
. . . . . . . . . . . . . . . . . . . .
78
5.2
Bode plot of the collocated transfer function Gp1 (s) . . . . . .
Bode plots of Uc(s)/Um(s) and Uc(s)/De(s) . . . . . . . . . . .
Feedback control of uc(t) . . . . . . . . . . . . . . . . . . . . .
79
5.3
5.4
10
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82
83
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
Bode plot of the non-collocated transfer function GP2 (s) . . . .
Bode plot of Uc(s)/Fm(s) with M1 2 = 0
. . . . . . . . . . . .
Root locus comparisons of Uc(s)/Fm(s) with (thick line) and
without (thin line) the non-minimum phase zero; zero frequency
is much larger than the maximum crossover frequency . . . . .
Root locus comparisons of Uc(s)/Fm(s) with (thick line) and
without (thin line) the non-minimum phase zero; zero location
closer to the imaginary axis . . . . . . . . . . . . . . . . . . .
Bode plot of the non-collocated transfer function at different
crossovers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nyquist plot of the non-collocated transfer function for different
crossover frequencies . . . . . . . . . . . . . . . . . . . . . . .
Instability by adding phase at crossover . . . . . . . . . . . . .
Instability by adding phase at crossover: increasing crossover
frequency with constant phase addition leads to increased instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Type 2 crossing results in a stable plant with phase addition .
Geometrical interpretation of PM and RGM . . . . . . . . .
Radius of the uncertainty circles for w, < w < we . . . . . . . .
Bode plot with a lead compensator . . . . . . . . . . . . . . .
Lowering of the resonance peak by adding an extra pole before
resonance .......
.............................
Increase in bandwidth by adding an extra pole before resonance
Bode plot of the loop transfer function with a lead compensator
and an additional pole . . . . . . . . . . . . . . . . . . . . . .
84
87
88
88
90
91
92
93
95
97
99
101
104
105
106
6.1
Photo of the old test set-up
. . . . . . . . . . . . . . . . . . .
113
6.2
6.3
Measured collocated transfer function for the old set-up . . . .
Representative modal transfer function of the old set-up . . .
114
115
6.4
Measured bending mode of the bearing block in the old set-up
(191 Hz). Figure shows snapshots of the mode starting from
the undeformed position. . . . . . . . . . . . . . . . . . . . . .
Measured twisting mode of the bearing block in the old set-up
(214 Hz). Figure shows snapshots of the mode starting from
the undeformed position. . . . . . . . . . . . . . . . . . . . . .
Measured axial mode of the old set-up (260 Hz). . . . . . . . .
Measured yaw mode of the carriage in the old set-up (375 Hz).
6.5
6.6
6.7
116
117
118
119
11
Photo of the new test stand . . . . . . . . . . . . . . . . . . . 122
Exploded view of the machine assembly . . . . . . . . . . . . . 123
Representative modal transfer function of the new set-up . . . 124
Schematic of a bolted-joint configuration . . . . . . . . . . . . 126
Stress discontinuities due to the absence of the stress-cone interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.13 Photograph of the cross section of base showing viscoelastic inserts 128
6.14 Photo of new bearing block . . . . . . . . . . . . . . . . . . . 132
6.15 Photo of 600 angular-contact bearings preloaded in a back-toback fashion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.8
6.9
6.10
6.11
6.12
6.16 Photo of bearings preloaded using an external locknut
6.17
6.18
6.19
6.20
6.21
6.22
6.23
6.24
6.25
6.26
6.27
6.28
6.29
12
. . . .
134
Photo of bearings preloaded using a push-up plate . . . . . . 134
Photo of the bellow-type coupling . . . . . . . . . . . . . . . . 136
Photo of the disc-type coupling . . . . . . . . . . . . . . . . . 136
Photo of the external locknut (left) and the damper (right).
The damper is formed by gluing a viscoelastic washer to the
.137
. .......
.............
external locknut. .....
Experimental set-up for modal analysis . . . . . . . . . . . . . 139
Measurement positions for modal experiment . . . . . . . . . . 141
Representative modal transfer functions from accelerometer to
shaker at two different locations on the machine. . . . . . . . . 142
Measured axial mode shape of the new design (349 Hz). Figure
shows snapshots of the mode starting from undeformed position.
The small squares indicate measurement locations. . . . . . . 143
Measured twisting mode of base (415 Hz). Figure shows snapshots of the mode starting from undeformed position. The small
squares indicate measurement locations. . . . . . . . . . . . . 144
Measured yaw of the carriage and bending of base (485 Hz).
Figure shows snapshots of the mode starting from undeformed
position. The small squares indicate measurement locations. . 145
Measured bending mode of the base (635 Hz). Figure shows
snapshots of the mode starting from undeformed position. The
small squares indicate measurement locations. . . . . . . . . . 146
Schematic of sine-sweep experiments . . . . . . . . . . . . . . 148
Measured and predicted collocated transfer function for the case
of the free-end boundary condition . . . . . . . . . . . . . . . 150
6.30 Measured and predicted non-collocated transfer function for the
case of free-end boundary condition . . . . . . . . . . . . . . .
6.31 Measured and predicted collocated transfer function for the case
of damped-end boundary condition . . . . . . . . . . . . . . .
6.32 Measured and predicted non-collocated transfer function for the
case of damped-end boundary condition . . . . . . . . . . . .
6.33 Measured collocated transfer function with and without damper
6.34 Measured non-collocated transfer function with and without
dam per . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Non-minimum phase zero surface . .
Motor torque versus motor inertia for
Acceleration surfaces . . . . . . . . .
Velocity surfaces . . . . . . . . . . .
wb-PM-RGM Surfaces . . . . . . . .
Design Space and Constraint Surfaces
Typical Bode obstacle course . . . . .
B.1
B.2
B.3
B.4
B.5
Standard unity feedback system . . . . . . . . . . . .
Bode asymptotes for a first-order system . . . . . . .
Bode asymptotes for a second-order system . . . . .
Plot of the overshoot MP versus the damping ratio ( .
Frequency response characteristics of the closed-loop
function . . . . . . . . . . . . . . . . . . . . .
.
Bode plot for a lead compensator . . . . . . .
.
Variation of #ma with a . . . . . . . . . . . .
.
Bode plot for lag compensator . . . . . . . . .
.
Variation of 0mi with# . . . . . . . . . . . .
.
Bode plot of a PID compensator . . . . . . . .
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Bode plot of a lag-lead compensator . . . . . .
.
B.6
B.7
B.8
B.9
B.10
B.11
. . . .
various
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product
. . . . .
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. . . . .
C.1 Block diagram of the perturbed plant . . . . .
C.2 M - A structure and the small gain theorem.
C.3 Geometric interpretation of robust stability .
C.4 Geometrical interpretation of robust stability .
D.1 Drawing of the machine base weldment (page 1)
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transfer
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13
D.2
D.3
D.4
D.5
D.6
D.7
D.8
D.9
D.10
D.11
D.12
D.13
D.14
14
Drawing
Drawing
Drawing
Drawing
Drawing
Drawing
Drawing
Drawing
Drawing
Drawing
Drawing
Drawing
Drawing
of the
of the
of the
of the
of the
of the
of the
of the
of the
of the
of the
of the
of the
machine base weldment (page 2)
machined base (pagel) . . . . .
machined base (page2) . . . . .
carriage (pagel) . . . . . . . . .
carriage (page2) . . . . . . . . .
ball-nut holder (pagel) . . . . .
ball-nut holder (page2) . . . . .
motor-side bearing block (pagel)
motor-side bearing block (page2)
damped bearing block (pagel)
damped bearing block (page2)
encoder mount (pagel) . . . . .
encoder mount (page2) . . . . .
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LIST OF TABLES
2.1
Typical Ball-screw System . . . . . . . . . . . . . . . . . . . .
6.1
Predicted axial resonant frequency and damping ratio using the
reduced-parameter model. . . . . . . . . . . . . . . . . . . . .
Important parameters of the ball-screw drive shown in Fig. 6.8
Predicted and measured results for the axial resonance for the
case of free end from modal experiments . . . . . . . . . . . .
Predicted and measured results for axial resonance from sinesweep experiments . . . . . . . . . . . . . . . . . . . . . . . .
6.2
6.3
6.4
27
113
121
147
149
15
CHAPTER
1
Introd uction
Precision and rapid motion are the essential elements of today's motion industry. Precision machines play a vital role in the areas of semi-conductor
and optical component manufacturing, high-speed machining, and so on. In
order to meet the required performance criteria, most of the machines operate
in closed-loop. When machines operate in closed-loop, the most important
limiting factor on performance comes from the inherent dynamics of the machine. Hence, the classical methodology of designing controllers for a given
plant must be replaced by designing the machine for required closed-loop performance. Therefore, one should design the plant and the controller simultaneously; that is, one should design the loop transmission. This design strategy
is shown in the form of a flowchart in Fig. 1.1. The forward direction indicates
the typical forward problem, where, for a given machine and a controller, we
attain certain performance. But in order to meet the required performance
criteria, we have to take the inverse path, and hence solve the inverse problem
of designing the loop transmission for given performance specifications. The
flow chart in the Fig. 1.1 gives a road-map of the thesis. We will investigate
each block in the flow chart, close the forward-inverse loop, and show through
experimental evidence how this methodology can be implemented on a real
machine.
16
Forward Problem
Real Machine
Idealization
4
EO+otolrPerformance
(Loop transmission)
Inverse Problem
Figure 1.1: Flowchart showing the forward and inverse problems
1.1
The Inverse Problem in Motion Control
For a given mass of the payload and length of travel, the requirements of a
typical motion-control problem are listed below
" Group 1
1. Maximum Acceleration
2. Maximum Velocity
* Group 2
1. Stability
2. Trajectory Tracking
3. Disturbance Rejection
4. Noise Rejection
5. Stability and Performance Robustness
We notice that the Group 1 requirements can be easily parametrized in terms of
the machine parameters. The maximum acceleration is usually dependent on
the total inertia and the maximum force that can be delivered by the actuator.
There are a number of factors which can limit the traverse speed such as critical
speeds, speed limits on position sensors, and motor commutation limits.
We illustrate the importance of this approach by considering a simple SISO
example. In Fig. 1.2, we show a typical minimum-phase, proper, and rational
17
md
Low frequency
Bode obstacle
20 db/dec (for PM)
WnO
Frequency
Wd
High frequency
Bode obstacle
Figure 1.2: Typical loop transmission
18
type a
Low frequency
Bode obstacle
-o
-20 db/dec (for PM)
I
type b
clHigh
typn
type
c
Frequency
Bode
frequency
obstacle
Figure 1.3: Influence of plant resonance on closed-loop performance
19
loop transmission along with the disturbance and noise requirements. To be
able to meet these requirements the loop transmission cannot penetrate into
the so-called Bode obstacles as shown in Fig. 1.2. It is also necessary to have a
slope of -20 dB/dec (or utmost -40 dB/dec) at the crossover frequency in order
to ensure sufficient margin of stability (from the Bode gain-phase relation).
The typical problem in motion control would be to design a controller for a
given plant to attain the specified Groupi and Group 2 requirements. Consider
now a plant with possible resonances at three different positions as shown in
the Fig. 1.3. As we shall see in Chapter 5, if a plant possesses resonances of
either type a or b, it becomes practically impossible to design a controller to
meet the required closed-loop specifications due to factors such as actuator
saturation, noise amplification, and lack of robustness. When a plant has a
type c resonance the noise criterion cannot be met as the loop transmission
penetrates into the noise obstacle. Hence it becomes important to ask the
question: Why solve the forward problem rather than design the plant and
controller simultaneously? The Group 2 requirements require an integrated
dynamics and controls approach i.e., the focus should be on the design of the
loop transmission to achieve the Group 1 and 2 requirements. This inverse
problem will be the focus of the thesis.
We apply our approach to the ball-screw servomechanism. This is because
most servomechanisms can be treated as special cases of the ball-screw servomechanism (Chapter 2) in the sense that it offers most of the features such
as drive flexibility, stiffness nonlinearities, friction, and so on. Hence, if we
successfully at apply our approach to this mechanism, it can be extended to
other systems with relative ease. Moreover, manufacturers of machine tools
and precision manipulators have long relied on the ball-screw drives for accurate positioning of linear-motion systems. The ball-screw is found to be the
integral part of most linear-motion systems.
1.2
Overview
In Chapter 2, we provide an overview of the ball-screw servomechanism and
understand the limitations posed by the dynamics of the drive on closed-loop
performance. We then explore different possibilities of improving the performance and robustness of the drive. At the end of the chapter, we propose a
novel method to increase the closed-loop performance by introducing damping
into the machine.
20
Damping in machine structures plays a vital role in maintaining stability
and performance (see for e.g., Book [8]). Hence, a designer trying to meet a
dynamics-related specification needs to know how much damping to expect.
Damping in a machine can arise from a number of physical mechanisms. Unless one deliberately introduces dampers into a system, damping in machines
arises almost entirely from the rubbing at material interfaces. The damping
obtained by such a mechanism is completely unreliable and hence does not
guarantee any robustness. Therefore, the focus of the Chapter 3 is to understand models of damping. We give a brief description of the two classical
models of damping: viscous and hysteretic. Many machines consist of components some of which are viscous while others hysteretic. Upon review of
literature, we find that there is no documentation on mixed damping: combined viscous and hysteretic. Hence, we solve the mixed damping problem
and use the results to model damping in the ball-screw servomechanism (see
Chapter 4).
The demands on precision machines and their motion control is ever increasing due to the advances and stringent motion requirements in the semiconductor, optical and high-speed machining industries. In the past, researchers
such as Chen and Tlusty [11] and D. Smith [48] have noted that for high
speed and wide bandwidth applications, the dynamical behavior of lead-screw
mechanisms takes on increasing importance in determining the stability and
accuracy of machines. Others such as M. Smith [49] have tried to implement adaptive control techniques to enhance the performance of machine tool
axis. In all these cases even though the need to improve control strategies
for achieving high performance was established, very simple dynamic models
of the machines have not allowed one to exploit the control strategies to the
fullest extent. For example, in [11] and [48] the axial dynamics of the screw is
placed outside the control loop for collocated control which is not exactly true
(refer to Chapter 5). In [49] the effect of the screw flexibility was completely
ignored (which as we shall see in Chapters 4 and 5 is the most important limitation for motion control). Moreover, bad models lead to detrimental effects
such as controller and observer spillover as described by Balas [4].
To solve the inverse problem it is very important to have an accurate model
of the dynamics of the machine. An ideal machine-model match will allow us
to close the loop between the first two blocks in the flowchart of the Fig. 1.1.
Hence, in Chapter 4 we set out to derive an accurate dynamic model for the
ball-screw mechanism. We start with the wave equations to correctly account
21
for the distributed inertia of the screw. Next, we employ various approximation
methods such as the "method of assumed modes" and "method of weighted
residuals" to solve the distributed eigenvalue problem. The distributed inertia manifests as a non-minimum phase zero in the system. This effect can
be observed in any servomechanism with a distributed actuating mechanism.
A survey of literature shows that the presence of non-minimum phase zeros
has been documented for flexible robotic manipulators and flexible beam-like
structures (see for e.g., Schmitz [44]; Spector and Flashner [50]). However,
the presence of non-minimum phase dynamics and its influence on the system sensitivity properties has not been documented for lead-screw drives and
linear-motion systems. We therefore, devote some sections in Chapter 5 to
discuss the influence of the non-minimum phase dynamics caused by the distributed inertia of the screw on the closed-loop performance.
We start Chapter 5 by obtaining approximate expressions for the characteristic roots of a generic servomechanism using a perturbation approach.
This helps us to form a link between the mechanical and controller design and
ultimately provides us means to solve the inverse problem. We then discuss
the constraints imposed by the dynamics of the machine on the collocated
and non-collocated control. We also show how the non-minimum phase zero
arising from the distributed inertia places constraints on the inverse problem.
Finally, we establish metrics for stability and performance robustness using
"small gain" and Nyquist arguments and draw parallels with the weighteduncertainty approach of robust control. These metrics then become the design
parameters in the inverse problem.
In Chapter 6, we discuss the mechanical design details and experimental
identification methods. We layout a basic framework for a "well-designed" machine and provide extensive design details. We show the impact of the design
details on the control performance by considering two design examples. In the
first case, we consider a machine in which poorly designed components lead to
a modal spillover (see [4]) and ultimately to a significant drop in performance
(40% lower bandwidth than predicted as we shall see in Chapter 6). By modifying the design and paying special attention to the mechanical design details
(which we describe in Chapter 6), we are able to establish high performance
levels, increased robustness and a very close machine-model match. Next, we
discuss the experimental methods used to identify the plant and present the
results of these experiments. In Chapter 7, we describe the feasible solutions
of the inverse problem.
22
1.3
Summary of Contributions
1. Inverse problem in motion control: We formulate the inverse problem of
designing a precision machine for closed-loop performance. We derive
metrics and constraints which can capture Group 1 and 2 requirements.
An accurate dynamic model along with the closed-form expressions for
bandwidth allow us to link the closed-loop performance to mechanical
design parameters. By imposing the constraints, we obtain a family of
feasible solutions in the design-parameter space which can help a designer to rapidly layout a machine which meets the desired closed-loop
specifications.
2. Accurate dynamic model of the ball-screw servomechanism: We develop
an accurate model of the ball-screw drive dynamics which correctly
accounts for the distributed inertia of the screw and various sources
of damping. This results in a non-minimum phase zero in the noncollocated transfer function and a non-minimum sensitivity in the collocated transfer function. This then leads to constraints on the sensitivity
properties of the system.
3. Mixed damping model in vibration theory: We derive a model for the
combined viscous and hysteretic damping. This mixed model then allows
us to model damping more realistically in machines and provides a unified
way to leverage the advantages of both the standard models of damping.
4. High bandwidth control of distributed servomechanisms with guaranteed
robustness margins: We derive closed-form expressions for the maximum closed-loop bandwidth using a perturbation approach for a general
fourth-order servomechanism with cross terms in the mass matrix. The
bandwidth expressions are functions of the plant parameters and certain
robustness parameters. This helps to size the plant parameters for a
given bandwidth and robustness.
5. Documented the impact of mechanical design on control performance
via extensive experimentation: We have conducted several identification
experiments and documented the effect of contact stiffness, component
stiffness, modal behavior, preloading mechanisms on the control performance. After studying these effects, we layout certain guidelines on
23
the design of components, joints, and choice of standard parts such as
couplings, bearings, etc., in order to attain a well-designed machine.
6. Novel damper design: We propose a novel method for enhancing the axial
dynamics of the ball-screw. This involves preloading the usually floating
bearing end with a viscoelastic washer. The ball-screw damper has resulted in increasing the bandwidth by a factor of five. The damper also
enhances robustness as we can predict the magnitude of the resonance
peak with a higher degree of accuracy and certainty.
24
CHAPTER
2
Ball-Screw Servomechanism
2.1
Introduction
The principle of using a lead-screw and a nut to convert rotary motion to
linear motion has been used since many years. By turning the screw and
holding the nut, so that it does not rotate, linear motion can be imparted to
the nut. Alternatively, the shaft can be held and the nut turned. When the
sliding friction encountered in conventional lead-screws is replaced by rolling
friction in a manner analogous to replacing simple journal bearings with ball
bearings, we obtain what is commonly known as a ball-screw. Ball-screws are
perhaps the most common type of lead-screws used in industrial machinery
and precision machines. Ball-screws can easily be used to achieve repeatability
on the order of a micron; specially manufactured and tested ball-screws can
attain sub-micro-inch motion resolution (e.g., Slocum [47]). Typical ball-screw
designs are shown in Fig. 2.1.
A typical linear motion system incorporating a ball-screw is shown schematically in Fig. 2.2. It consists of a ball-screw which is mounted to the machine
base by means of rotary bearings and is driven by a motor via a flexible coupling. The ball-nut is mounted to the carriage which is constrained to move
axially on linear bearings. When such a system is operated under closed-loop
control, the position of the carriage, or the rotation of the motor, or both can
25
Figure 2.1: Ball-screw Designs: Return tube and internal deflector type ball-nuts (courtesy NSK Corp.)
screw
motor coupling
bearingnt
Figure 2.2: Schematic of a ball-screw drive
26
Ball-screw average diameter
Ball-screw lead
Ball-screw length between bearing supports
Ball-screw material
Ball-nut stiffness
Bearing stiffness
Torsional stiffness of coupling
Inertia of motor
Mass of carriage
16 mm
5.08 mm
406.4 mm
steel
260N/Mm
140N/pm
693 N-m
7.8 x 10-kg-m2
21 kg
Table 2.1: Typical Ball-screw System
be used for feedback. As we shall see in Chapter 5, if the feedback signal involves the linear position of the carriage, the maximum achievable bandwidth
is limited by the axial resonance frequency and the magnitude of the resonance peak. If it involves the rotation of the motor, precise positioning of the
carriage is not possible due to the axial resonance and the disturbance forces
on the carriage. In either case, a well-damped, high-frequency axial mode is
vital for rapid and accurate motion of the stage.
2.2
Basic Physics of the Drive Resonance - A Qualitative
Picture
During operation of the mechanism, friction between the screw and the nut
generates significant heat, leading to a temperature rise and thermal expansion
in the screw. To accommodate this expansion, the screw is usually allowed
to slide freely in the axial direction at the support bearing farthest from the
motor. Thus, the entire thrust load is transmitted to the base through the
serial combination of the elements between the motor-side thrust bearings and
the carriage. Because the compliance of the screw increases with increasing
length, the total axial compliance of the mechanism exhibits a similar behavior.
Hence, it is not unusual to expect the axial resonance to shift with the position
of the carriage. Using the results from Chapter 4, we plot in Fig. 2.3 the
response of the carriage position to motor torque as a function of frequency
for the representative system of Table. 2.1. As the carriage moves away from
27
I
Wl .
-120
-140
-160
-180-
-200-
-220-
102
10
1104
Frequency (rad/s)
Figure 2.3: Frequency response of undamped screw; the carriage is positioned at (1)
10% (2) 42% (3) 90 %
28
the motor, the resonant frequency drops and its peak amplitude rises.
In Chapter 5, we show that an estimate for the maximum achievable bandwidth of the mechanism with a given controller can be graphically determined
from the frequency response of the loop transfer function by drawing a line
horizontally through the resonant peak, finding its intersection with the portion of the curve to its left, and reading off the frequency. This is because a
further increase in gain results in multiple crossovers of the loop transfer function and instability of the closed-loop system. Also, the phase after resonance
drops to -360' and therefore, crossover after resonance requires the phase to
be raised by at least -180' for stability. Practical compensators can seldom
achieve 180' of lead (positive phase) due to factors such as noise and saturation
(e.g., Spector and Flashner [50]). This is because an increase in phase comes
only at the cost of an increase in the magnitude of the loop transmission due
to the Bode gain-phase condition (Bode [7]). From the above discussion, we
recognize that the axial resonance in the ball-screw mechanism places severe
limitations on its closed-loop performance. In the next section, we will explore
several means of increasing the performance of such a system and understand
their pros and cons.
2.3
Options to Increase Performance
One possibility of increasing the closed-loop bandwidth is to use a higher-order
controller to be able to have a crossover after the axial resonance. It is often
found that such a design is too sensitive to parameter changes to be useful for
any practical applications (e.g., Truckenbrodt [52]; Cannon and Schmitz [10]).
Apart from robustness issues, we have noted in the previous section that such
a design is impractical due to noise and saturation problems. Hence, a better
alternative would be to change the mechanical design to achieve the required
objective. Because the axial resonance is the key limiting factor, we consider
design possibilities which can result in increasing the resonant frequency and
damping the resonant peak.
To meet the requirement of a high-frequency and well-damped axial mode,
one usually seeks to raise the stiffness and damping while minimizing the
inertia of the system. For most configurations, the nut, support bearings, and
the coupling can be sized for high stiffness without significantly increasing
inertia. But the screw has longitudinal stiffness proportional to the square of
its radius and the rotary inertia proportional to the fourth power of its radius.
29
-1nn.
-120 --
--140 --
-160 --
-180-
-200-
-220-
-240
102
3
10
10"
Frequency (rad/s)
Figure 2.4: Frequency response with a damper in the preload path. Figure shows a
comparison of damped and undamped cases for three carriage positions
Thus, the option of using a stiff screw is limited by the tradeoff between
stiffness and inertia.
An approach sometimes employed to increase the stiffness is to use thrust
bearings at both the end of the screw, and pre-stretch the screw in order to
accommodate thermal expansion. Using this approach, the overall stiffness of
the system is increased due to the participation of both the portions of the
screw between the carriage and the bearing blocks in transmitting the load
from the carriage to the base. The disadvantage of this approach is that it
requires that a very high preload force be applied in stretching the screw. Such
a high preload force increases the starting torque, heat generation, and can
appreciably warp the bearing housing and the machine base.
30
A novel approach
In the present work, we propose (as originally suggested by Nayfeh and Slocum
[37]) to place a damping element in the preload path of the bearings which
are at the end farthest from the motor. For a relatively compliant but lossy
preload medium, significant attenuation of the resonant peak is observed as
shown in Fig. 2.4. Because the preload medium is complaint there is only
a marginal increase in the resonant frequency compared to the case of ballscrew with free end. But there is a significant increase in damping due to the
damped support and this helps achieve high performance and robustness; the
lowering of the resonance peak increases the maximum achievable bandwidth
and the presence of a predictable damping mechanism eliminates uncertainties
associated with the resonant peak as we will see in Chapters 5 and 6.
2.4
Chapter Summary
In this chapter, we have provided a brief overview of the ball-screw servomechanism and have qualitatively discussed the effect of the axial mode on the
closed-loop performance. We have also discussed several options to increase
the performance of the system and have proposed a novel approach to enhance
the axial dynamics of the mechanism.
31
CHAPTER
3
Models of Damped Systems
A system in which the damping force at any instant is proportional to the
velocity of motion is said to posses viscous damping. For a single-degree-offreedom oscillator the equation of motion takes the following familiar form
mn.z + C., + kx = F(t)
(3.1)
which is usually written as
+ ( oi
+ w2x
=
F(t)/m
(3.2)
where ( = C/2/km is the damping ratio and wn = Vk/m is the undamped
natural frequency of oscillation.
The energy dissipated in such a system during a period of harmonic motion
x(t) = B sin wt is given by
DV
=
c±dt
=
7rcwB 2
(3.3)
From Eq. (3.3), we find that the energy dissipated in a viscous damper varies
linearly with frequency. But, according to experimental evidence, the energy
dissipated in structures and many materials does not follow the above linear
dependence of the viscous model. Moreover, it is found that the damping
32
properties vary slowly with frequency under isothermal conditions (e.g., Nashif
et al. [34] and Lazan [27]). Hence, by using viscous damping to model energy
dissipation in structures, we will underestimate damping at low frequencies and
overestimate it at high frequencies. To circumvent this problem and to leverage
the extensive experimental data available for many damping materials, we use
the complex modulus approach, which we discuss in the following section.
3.1
Complex Modulus Approach
In order to understand the complex modulus approach, we start with a discussion of a first order model for stress and strain given by
-(t) +
du(t)
dt
Ectt) + /E
dt
(3.4)
We note that Hooke's law, or the simple dash-pot-spring combination are only
special cases of Eq. (3.4). When we introduce harmonic stress and strain of
the type -(t) = Re{-(w)ew t } and c(t) = Re{c(w)ewOi} in Eq. (3.4) we obtain
1 +jwa\
where -(w) and E(w) are complex variables representing the magnitude and
phase of the stress and strain, respectively. Werewrite Eq. (3.5) as
-(w)
=
(E'(w) + jE"(w)) (w)
(3.6)
where
(w 2a2 +
/l(/3
E"
=
-
2
ca)w
+ 1
(3.8)
)
The above exercise for a first-order model has provided us with a significant
result in the form of Eq. (3.6) even though the expressions for E' and E" have
much faster variation with w than what is observed in experiments. In order
to overcome this problem, one can introduce additional derivatives in Eq. (3.4)
as described by Bagley and Torvik [3] and Rogers [41]. But the essential point
33
is that in the frequency domain the relation between complex stress and strain
can always be reduced to the form of Eq. (3.6). One can then obtain real and
time-dependent stress and strain by taking the real part of the expressions
u(w)eiwt and E(w)eiwt.
We rewrite Eq. (3.6) as
a-(w) = E(w)(1 + jq(w)) E(w)
(3.9)
where E(w) = E'(w) and q(w) = E"(w)/E'(w) is the so-called loss factor.
The quantity E(w) (1 + jT(w)) is often referred to as the complex modulus in
vibration literature. From Eq. (3.9), we note that the inverse tangent of the
loss factor represents the phase difference between stress and strain. Another
possible interpretation of the loss factor can obtained by noting that it also
represents the ratio of the average energy dissipated per radian to the peak
strain energy in a cycle of harmonic vibration.
By introducing the complex notation for stress and strain, we have characterized damping in its most fundamental form: energy dissipation via the loss
factor. Further, this naturally leads to defining the equations-of-motion in the
frequency domain, where the properties of a structure are absorbed into E(w)
and q(w). One can then obtain the time-domain parameters by computing the
inverse Fourier transform of the corresponding frequency-domain parameters.
Once one has thought about the matter for a while, this approach is just as
satisfactory from a philosophical point of view as starting in the time domain
and transforming to the frequency domain. We recognize that by using the
complex modulus approach, we have provided means to leverage the vast experimental data and characterize damping in its most fundamental form. But
the cost we pay is that the time-domain representation has to be obtained
using transform theory, which places constraints on the functions E(w) and
,q(w) in order to have real and causal time-domain solutions.
We now solve the vibration problem for a single-degree-of-freedom system
using the complex modulus approach. Consider the system of Fig. 3.1 in which
a mass m is connected to the ground via a structure of complex modulus
E(w) (1 + jq(w)). The system is excited by a force f(t) = Y 1 F(w), which
results in a displacement x(t) = F-1 X(w) at steady state. We start by writing
the equation-of-motion for this system in the frequency domain as
-mw 2 X(w) + k(w)(1 + j'q(w))X(w) = F(w)
(3.10)
where k(w) (1 + jq(w)) represents the so-called complex stiffness of the structure. As energy dissipation in structures and most polymers is almost inde34
E(w) (1 + jr7(w))
Figure 3.1: Complex Modulus Representation
pendent of frequency, we let k and r to be constants in Eq. (3.10). This is
known as the frequency-independent hysteretic damping model (e.g., Bishop
and Johnson [6]; Crandall [14]). However, it is necessary to multiply the loss
factor r by sgn w = w/ wj to avoid fallacious results in the inverse Fourier
transform (Crandall [14]; Nashif et al. [34]). Hence for the case of frequencyindependent hysteretic damping, Eq. (3.10) takes the following form
-mw 2 X(w) + k(1 + jq sgn w)X(w) = F(w)
(3.11)
The steady state time-domain solution for Eq. (3.11) for F(w) = F is given by
x(t) = B 1 cos(wt - #)
(3.12)
where
B1
=
=
F
Vr{k - MW2)2 + k2rn2
arctan
km
2
k - mw
(3.13)
(3.14)
The energy dissipated per cycle for this system is given by
D =
Fdx = 7rkiB2
(3.15)
which is a constant as expected.
35
F(w)
k(1+r7sgnw)
X(w)
m
C
Figure 3.2: Schematic of a SDOF system with mixed damping
3.2
3.2.1
Mixed Viscous and Hysteretic Damping
Motivation
Consider the ball-screw servo discussed in Chapter 2. In order to increase
performance we proposed that the floating bearings be preloaded with a viscoelastic washer. When we want to obtain a dynamic model of such a machine,
a complication arises while modeling damping. This is because the viscoelastic washer behaves as a hysteretic spring whereas the damping in the bearing
races, motor, etc., are modeled as viscous dash-pots. Hence, we have a system
which exhibits mixed viscous and hysteretic damping. Many machines exhibit
this mixed damping because they comprise both viscous and hysteretic components. Another common example is an isolation system in which the isolation
table behaves as a hysteretic element whereas the air legs on which the table
stands are viscous. Hence, it is important to provide a framework to handle
this mixed damping problem. A search of literature reveals no studies on this
problem. Therefore, it is our goal in this section to develop a model for mixed
viscous and hysteretic damping.
36
jor
A2
A,
x.
(-
2)2 +72]1/4
L [l (2)2+772]1/4
(W.
A3
A4
Figure 3.3: Four solutions of the eigenvalue problem
37
jor
A2
A,
c~w 1
2
WnI
I
Figure 3.4: Correct Eigenvalues
38
w
3.2.2
SDOF system with mixed damping
Consider a single-degree-of-freedom system with mixed viscous and hysteretic
damping as shown in Fig. 3.2. Using the complex modulus approach of Sec. 3.1,
we write the equation-of-motion for the SDOF system of Fig. 3.2 as
-w 2 X(w) + 2j(owwX(w) + W2(1 + jr sgn Re(w))X(w) = F(w)/m
where 2
(3.16)
= C/rn and w2 = k/m. When there is no excitation (F = 0), we
obtain the following equation which constitutes an eigenvalue problem
[-A2 + 2j(nA + oW2(1 +jsgnw)] Q = 0
(3.17)
where A = w + jc- is the eigenvalue and Q is the eigenvector. For non-trivial
Q we obtain
-A 2 + 2j (nA + W (1 + jq sgn w) = 0
(3.18)
When we solve Eq. (3.18) we obtain the following four solutions for A:
For w > 0:
A,
=
wnA1/2 cosO + j(cWn + wnA 1/ 2 sin
A3
=
-WnA1/2 cos
A2
=
-Wn
A4
=
2
2
(3.19)
-)
2
+ J((Wn - WnA 1/ 2 sin
)
(3.20)
)
(3.21)
2
For w < 0:
1/2 cos
nA1/2 cos
2
2
+ j((Wn + WnA 1/ 2 sin
+ j((Wn - wnA 1/ 2 sin -)
2
2
(3.22)
(3.23)
where A = [(1 - (2)2 + q 2] 1/ 2 and # = arctan [j/(1 - (2)].
The solutions A1 4 are shown in Fig. 3.3 on the complex A plane. We
see that the solutions form a "quad" pattern unlike the standard complex
conjugate pair. By examining Eqs. (3.19)-(3.22), we immediately notice that
A3 and A4 do not satisfy the corresponding sign condition on w. Hence, they
fail to qualify as legitimate solutions to Eq. (3.18). Therefore, A1 and A2 are
the correct solutions of Eq. (3.18) and hence, are the eigenvalues or the poles
of the system. Thus, the correct eigenvalue/pole plot of this system is as
shown in Fig. 3.4 (not Fig. 3.3). This is indeed a very interesting result for
the eigenvalues: A1 and A2 do form a complex conjugate pair.
39
Frequency Response
The frequency response of a system is defined as the output of the system for
a harmonic input. This is characterized in terms of the magnitude and phase
of the response as the frequency is swept from 0 to oc. Consider a system with
a transfer relation of the form
rI A-
H= 0
=
G(A)
Aj)
(3.24)
rUn=O(A -kA)
The geometrical interpretation of the magnitude and phase of the transfer
relation at an arbitrary test point A, on the complex A plane is described
below. The magnitude IG(A)l at a test point A = A0 can be obtained as the
ratio of the lengths of vectors from the poles and zeros to the test point (see
Fig. 3.5) as given below
j= li
IG(Ao) =
=lm
0
(3.25)
where l, represent the length of the vector from ith zero to the AO and lpk
represents the length of the vector from the kth pole to AO as shown in Fig. 3.5.
Likewise, the phase ZG(Ao) is given by
m
LG(Ao)
=
n
#-
, pk
(3.26)
k=1
where O's are the angles between the corresponding vectors and vertical as
shown in Fig. 3.5. Hence, one can view frequency response as evaluating
Eqs. (3.25) and (3.26) when the test point Ao traverses the real axis from 0 to
oc. From this point of view, frequency response of the SDOF system given in
Eq. (3.16) is given by
X
m
F
F
Z
F
nA
(1co - A,1) (1w =
#1(W) + 02(W)
A21)
(3.27)
(3.28)
In the light of the above discussion, it is important to note that if we directly
use the straight-forward expressions for magnitude and phase to evaluate the
40
jcx
2
p
p3
*
3
*l
1
~p1
P1
lz1
-
Pz
w
z2
Oz2
qOzi
Figure 3.5: Geometric interpretation of frequency response
41
frequency response for the SDOF system of Eq. (3.16) given by
m
X
X- =
F
Z
F
x 2 )2 + (2(cnw + wn2)
-w
=
arctan
2
(3.29)
(3.30)
n2_ W
we will obtain erroneous results. This is because when we use Eqs. (3.29) and
(3.30) for characterizing the frequency response we are actually evaluating
m/(|w - Ai)(1w - A3 |) for magnitude, and #1 + #3 for phase, instead of using
the expressions given in Eqs. (3.27) and (3.28). This discrepancy is due to
the presence of the signum function in the definition of hysteretic damping.
Hence, one must be careful not to use the straight-forward expressions while
evaluating the frequency response of a mixed system; one should always obtain
such a response from the geometric pole-zero method discussed in this section.
Comment on pure hysteretic damping
The eigenvalue problem for the case of pure hysteretic damping is obtained by
letting (
0 in Eq. (3.16) and is given by
[-A 2 + W2(1 +jlsgnw)] Q
=
0
(3.31)
The solutions to the above eigenvalue problem can be obtained by letting ( = 0
in Eqs. (3.19)-(3.22). We note that the solutions are symmetric about the real
axis. Among the four solutions, w, and w 2 qualify to be the eigenvalues of
the system for reasons discussed before. We also note that the use of straightforward expression to evaluate the magnitude of the frequency response results
in an identical expression to the one obtained using the geometric pole-zero
method due to the symmetry of the solutions about the real axis. However,
erroneous results for phase follow at low frequencies.
3.2.3
Equivalent Model
In the previous section, we have solved the eigenvalue problem for a SDOF
system with mixed damping. We have found that the legitimate eigenvalues of
such a system ultimately appear in complex conjugate pairs. We note that it
is essential to use this correct configuration of poles of the system (Fig. 3.4) to
42
calculate important properties of the system such as the frequency response.
We also note that the pole configuration of Fig. 3.4 resembles the one attained by a SDOF system with pure viscous damping. This observation is
very valuable because one can imagine replacing the given mixed damping
model with an equivalent viscous model with identical eigenvalue configurations such that the frequency response is preserved. One of the main problems
with the frequency-independent hysteretic damping model is that the impulse
response of the system is non-causal and therefore, it cannot be used to describe the transient response of a causal LTI system (e.g., Crandall [13] and
Milne [33]). Hence, even though we are able to capture the damping of the
structure realistically over a wide frequency range we cannot use this model in
the framework of the general control theory because of its non-causal impulse
response.
One can now clearly envision the many advantages of the observation made
earlier regarding an equivalent viscous model. Firstly, as the frequency response is preserved the equivalent model fulfills the functionality of the mixed
model. Secondly, since the equivalent model consists of pure viscous damping
causality problems are eliminated. This helps us to solve transient problems.
Thirdly, this approach allows integrating structural dynamics with controller
design. In what follows, we derive an equivalent system with identical eigenvalues and frequency response.
To obtain this we preserve the mass and alter damping and stiffness. Let
the equivalent system be given by
-w2X (w) +
) + W21)X(w) = F(w)/m
(3.32)
where the new parameters are related to the old ones by
(
1/2 Sin
s
)2 +
A1/2 cos
)2
(3.33)
(3.34)
2
Therefore, the new damping coefficient is given by C1 = 2m(Wni and the new
stiffness is given by K 1 = mw 1 . From the expression for Wni given above we
notice that w2 = w2(1 + 0((2, r)). Hence, for small damping the equivalent
model preserves the static problem.
In this section, we have discussed the mixed damping problem for a singledegree-of-freedom system. We have solved the eigenvalue problem and obtained the correct eigenvalue structure. Noting that the eigenvalues occur in
(iwn
=
(Wn + WnA1/2 sin
43
complex conjugate pairs, we have introduced the idea of an equivalent model.
We see that the advantages of the equivalent model are many fold as it takes
into account the real damping behavior of structures and also yields a causal
impulse response. We have also seen that for small damping the static problem
is preserved. In the next section, we generalize this idea to a multi-degree-offreedom system.
3.2.4
MIMO Systems with Mixed Damping
In this section, we discuss mixed damping for the case of MIMO systems.
Consider an nth order system with mass matrix M, viscous damping matrix
C and the complex stiffness matrix K = Kr + jsgn w H. The equations-ofmotion for this system in the frequency domain are given below
-w 2 M{X(w)} + jwC{X(w)} + K{X(w)} = {F(w)}
(3.35)
When there is no excitation (F = 0), we obtain the following equation which
constitutes the eigenvalue problem for the system
[-A
2
M + jAC + K]{Q} = 0
(3.36)
where A = w + jc- is the eigenvalue and Q is the corresponding eigenvector.
We premultiply Eq. (3.36) with {Q'} to obtain
-A 2 + jaiA + a 2 (1 + ja3 sgn w) = 0
where a,
(3.37)
Q'CQ'/Q'MQ, a2
= Q'KrQ/Q'MQ and a3 = Q'HQ/Q'KrQ.
We note that a 1 , a 2 , and a3 are real and positive because M, C, Kr, and
H are symmetric and positive-definite matrices. Equation (3.37) resembles
Eq. (3.17), and therefore, the solutions for A have similar structure. Of the
four solutions, we again note that only two of them qualify to be legitimate
eigenvalues and the corresponding eigenvectors the legitimate eigenvectors.
Hence, the eigenvalue problem for a MIMO system with mixed damping
results in complex-conjugate eigenvalues and eigenvectors.
=
Frequency Response
In Sec. 3.2.2, we have found that straight-forward expressions for magnitude
and phase result in erroneous results for frequency response. We have also
44
found that the correct frequency response can be obtained from first principles viz., by using the poles of the system as explained in Sec. 3.2.2. The
same explanation holds for a MIMO case as well; if we use straight-forward
expressions from the receptance matrix, we will obtain erroneous results. The
receptance matrix has to be modified with the correct characteristic polynomial which is obtained using the right poles. We can achieve the same result by
constructing an equivalent model with pure viscous damping which preserves
the eigenvalues and frequency response. The advantages of such an approach
have been discussed before.
In summary, it is necessary to compute the equivalent model in order to
obtain the correct frequency response. In the next section, we construct an
equivalent model from the original mixed model.
3.2.5
Equivalent Model
Let the equivalent model be
-w 2 M{X(w)} + jWCe{X(W)} + Ke{X(w)} = {F(w)}
(3.38)
where Ce and Ke are the new damping and stiffness matrices. We have retained the original mass matrix in the equivalent model. The eigenvalue problem for the equivalent model consists of solving the following equation
[-A 2 M + jACe + Ke] {Q} = 0
(3.39)
For Eqs. (3.39) and (3.36) to have identical solutions, we require
[a(Ce- C) - H] U + [-3(Ce - C) + (Ke[-3(Ce - C) + (Ke-Kr)]U -[a(Ce
-C)
Kr)] V = 0
0
-H]V
(3.40)
(3.41)
where a and 3 are the respective real and imaginary parts of the diagonal
matrix containing the eigenvalues, and the matrices U and V are the respective
real and imaginary parts of the matrix containing the eigenvectors. Solving
Eqs. (3.40) and (3.41), we obtain
C + &-1 H
Ke = Kr + 13a 1 H
Ce
=
(3.42)
(3.43)
We again note that for small damping Ke approaches Kr.
45
3.3
Chapter Summary
In this chapter, we have introduced the most important models of damped
systems: viscous and hysteretic. As we have discussed before, most machines
exhibit mixed damping and therefore, we have investigated the mixed damping problem in great detail for SDOF and MIMO systems. Finally, we have
outlined an approach to handle the mixed damping problem.
46
CHAPTER
4
Modeling
4.1
Introduction
In this chapter, we set out to derive a dynamic model for the ball-screw drive
mechanism sketched in Fig. 2.2. This system has infinite degrees of freedom
whose time and space evolution is governed by wave equations. However, for
the frequency range of motion control (see Chapters 2 and 5), we can model
many components as lumped-parameter elements: For example, the carriage
and motor armature can be modeled as rigid bodies, bearings and coupling as
ideal springs, and so on. If the elastic modulus of the screw is E and its mass
density is p, longitudinal waves in the screw travel at a speed of (E/p)1/ 2 . If
the frequency w of motion is much lower than the time it takes for such a wave
to travel the length L of the screw (that is, if wL < (E/p)1/ 2 ), longitudinal
waves in the screw can be neglected and the deformation u(x, t) varies linearly
with x on each portion of the screw between the nut and the thrust bearings
as sketched in Figs. 4.3 and 4.4. Likewise, if the shear modulus of the screw is
G, torsional waves travel at a speed of (G/p) 1/ 2 and they can be neglected if
wL < (E/p)1 /2 . Under this condition, the torsional deformation O(x, t) varies
linearly with x on the portion of the screw between the nut and the motor and
is a constant on the remainder of the screw.
As we shall see in Chapter 5, the frequency range for motion control is
47
limited by the first axial mode of the system. Hence, our concern here is to
obtain an accurate dynamic model of the first longitudinal mode of the system.
Unless the carriage and the motor have small inertias, this mode will occur
at a frequency well below the frequencies at which longitudinal or torsional
waves become important. Therefore, we can safely use the piecewise-linear
deformation shown in Figs. 4.3 and 4.4 as the basis for a dynamic model.
We start this chapter by formulating a distributed-parameter model for the
screw in Sec. 4.2. Then, in Sec. 4.4 we perform a quasi-static analysis of the
system and obtain expressions for the longitudinal and torsional deformation
of the screw in terms of the carriage displacement uc(t) and the motor rotation
0
m (t). Next, we use appropriate approximation methods such as the "method
of assumed modes" and "method of weighted residuals" to obtain a dynamic
model for uc(t) and 0m(t) that correctly accounts for the distributed inertia of
the screw and various sources of damping.
4.2
Distributed-Parameter Model
As we have seen in Chapter 2, there can be various boundary conditions for
the end farthest from the motor: free, fixed, or partially constrained by a
damper. We will consider two cases here: free and damped ends, respectively,
to illustrate the time- and frequency-domain formulations of the motion problem. We note that any other case can be obtained by appropriately scaling
the stiffness of the damper; for example, the dynamic model for the case of the
fixed end can be obtained from the one for damped end by letting the stiffness
of the damper go to infinity.
4.2.1
Ball-Screw Servo with Free End
The free-body diagram for the ball-screw with a free end is sketched in Fig. 4.1.
The longitudinal deformation u(x, t) and torsional deformation 9(x, t) of the
screw are each governed by a second-order wave equation given by
(X, t)
E X2
G a____
2
ax
48
=
=
p
P
2(x, 2
t
t)
O(Xt)
(4.1)
(4.2)
kn
k
kb1
kbl
f (t)
LL
K
u(O1t)
screw between motor and carriage
Tm(t)
T(t)
Cm
(xeLt) 0C(t)
0(0,t)
Om(t)
f(t)
Mc
fc(t)
Cc
uc(t)
neU
)-
un(t) +
LOc(t)
Figure 4.1: Free-body diagram of the ball-screw servo with a free end
49
4.2.2
Ball-Screw Servo with Damped End
The free body diagram for the case of the ball-screw with damped end is
sketched in Fig. 4.2. As proposed in Chapter 2, the damped end is formed
by preloading the thrust bearings at the end farthest from the motor by using a strain-based lossy damping medium (e.g., a viscoelastic washer). We
model the viscoelastic washer as a frequency-independent hysteretic spring of
complex stiffness k,(1 + jrv sgn w), and write the equations of motion in the
frequency domain to comply with the requirements of hysteretic damping (see
Chapter 3).
Consider steady harmonic vibration of the ball-screw drive at a frequency
w, where the vibratory displacement of the carriage is uc(t) = F-'Uc(w) and
the angle of rotation of the motor is 0m(t) = F- 1Em(w). The complex variables
U, and Em, each represent the magnitude and the phase of the motion as
a function of frequency w. The longitudinal deformation and the angle of
twist of the screw vary along the length of the screw and hence are written
as u(x,t) = F- U(x,w) and O(x,t) = F-16(x,w). Substituting the above
expressions for u(x,t) and O(x,t) in Eqs. (4.1) and (4.2), we obtain wave
equations in the frequency domain as
Sd 2 U(X,
4.2.3
W) + pw 2 U(x, w)
=
0
(4.3)
2
Sd E(x,
w) + pw 2 E(x, w)
=
0
(4.4)
Boundary Conditions
At the end of the screw nearest the motor (at x
0), the longitudinal force
must match the longitudinal force in the thrust bearing. If kbi represents
the combined longitudinal stiffness of the thrust bearing and its housing, this
condition takes the form
EAu'(0, t)
= kbiu(0,
t)
(4.5)
where A is the average cross-sectional area of the screw and the prime denotes partial differentiation with respect to x. Similarly, denoting the effective
stiffness at the end farthest from the motor as kb2, we have
EAu'(L, t) = kb2 U(L,t)
50
(4.6)
U(O, w)
screw between motor and carriage
combined stiffness
of housing and
thrust bearings
kbI
k
F(w)
j?7v)
r(
kb2
screw between carriage
and damped end
Un~s
screw between motor and carriage
T(w)
CM
0,m
)8(0,
E8(x, w) On (W)
CeMC
_Fe(w)
Uc(w)
Uc(w) = Un(w) +
c
w)
Figure 4.2: Free-body diagram of the ball-screw servo with a damped end
51
At x = L, the twisting moment on the screw must vanish. This condition
takes the form 0'(L, t) = 0. At x = 0, the twisting moment in the screw must
match that in the coupling. Hence, the condition at the end of the screw is
related to the angular displacement 0m of the motor by
GJO'(0, t) = Kc [0(0, t) - Om(t)]
(4.7)
where G is the shear modulus of the screw, J is the (average) second polar
moment of its cross section, and , is the torsional stiffness of the coupling
between the motor and the screw.
If the carriage (more precisely, the nut) is located at a position x = x,
along the length of the screw and the interface between the nut and the screw
is perfectly rigid, the carriage displacement u, would be obtained from the
screw's longitudinal displacement plus its angle of twist times the drive ratio:
uc(t) = u(xc, t) + 0(xc, t)2wr
(4.8)
where f is the lead of the screw (i.e., the distance by which the thread advances
in one rotation). But in fact the interface between the screw and the nut has
some compliance. We define the nut's axial stiffness k, as the ratio of the axial
force f(t) developed in the screw and the resulting displacement u,(t) -u(xc, t)
of the carriage relative to the screw if neither the screw nor the nut are allowed
to rotate. Similarly, we define the nut's torsional stiffness r" as the ratio of
an applied torque T(t) and the relative angular displacement Oc(t) - 0(xc, t) if
neither the screw nor the carriage are allowed to move in the x direction. The
angle of twist Oc(t) of the carriage can safely be neglected if the carriage rides
on a pair of reasonably stiff bearing rails. Hence, we include the compliance
of the interface between the screw and the nut and rewrite the kinematic
relationship given in Eq. (4.8) as
uc(t) = un(t) + Oc(t)
(4.9)
2r
where un(t) -u(xc, t) represents the axial deformation of the nut. By matching
the axial force and the twisting moment in the nut to those of the screw at
x = xc (see Fig. 4.1), we obtain
EA [u'(x-, t) - u'(x+, t)]
= k, [un
GJ'(x-, t) = Ko [0c-
52
-
U(xc, t)]
(xc, t)]
(4.10)
(4.11)
The axial force and the torque developed in the screw are given by the
combination of the forces and torques on the portions of the screw to the left
and to the right of the nut according to
f(t) = EA [u'(x , t) - u'(x -, t)]
and
T(t) = -GJO'(x-,t)
(4.12)
The motor armature is subject to the twisting moment 'c [0(0, t) - Om(t)1
imposed by the screw on the coupling in addition to the actuation torque Tm(t)
and an effective viscous damping torque Cm~m(t). The equation of motion of
the motor armature therefore takes the form
JmOm(t) + CmOm(t) + icOm(t)
= Tm(t)
+ t'ic(O, t)
(4.13)
where Jm is the rotary inertia of the motor armature. We assume that the
motor is driven by a current amplifier so that its electrical dynamics can be
neglected.
The carriage is subject to the force exerted by the screw on the nut given
by k, [u(xc, t) - u,(t)] in addition to a disturbance force fc(t) and an effective
viscous damping force Cczic(t) (refer to Fig 4.1). We therefore write
mci(t) + Ce'de(t) + knuc(t)
=
fc(t) + kn u(xe~t) +0c(t)--
k U(XC
t) +OC~t)27r _
(4.14)
414
where me is the mass of the carriage.
The boundary conditions for the case of the ball-screw with free end can
be obtained by letting kb2 = 0 in Eq. (4.6) and u'(x4,t) = 0 in Eqs. 4.10
and 4.12. For the case of the damped end we rewrite Eqs. (4.5)-(4.14) in the
frequency domain and let kb2 represent the combined stiffness of the thrust
bearings, housing and damper at the end farthest from the motor.
4.3
Approximate Methods
The distributed-parameter formulation discussed in the previous section cannot yield closed-form solutions, and therefore, we seek approximate solutions
in the form of lumped-parameter models. As discussed in Sec. 4.1, our concern
here is with the first axial mode of the system. If the the axial mode occurs
at a frequency well below the frequencies at which wave propagation becomes
important, we can safely use the piecewise-linear deformation (Figs. 4.3 and
53
4.4) obtained from a quasi-static analysis as "trial functions" in the approximate methods. Lumped-parameter approximations for distributed-parameter
systems are usually obtained using energy methods. We will use one such
method-the method of assumed modes-to obtain dynamic equations for the
case of the ball-screw with a free end. In the case of the damped end, energy
methods cannot be directly applied due to the presence of a hysteretic element
in the model. Hence, we use the method of weighted residuals which works
directly in the frequency domain to solve the problem. In the following sections, we provide a brief review of the methods of assumed modes and weighted
residuals. For a detailed discussion on this subject the reader is referred to
Meirovitch [32].
Let us consider a distributed-parameter system with the following eigenvalue problem
Lu = AMu
(4.15)
where the displacement u is a function of the spatial variable x, A is the
eigenvalue, and L and M are linear homogeneous differential operators of
order 2p and 2s, respectively. Further, p and s are integers with s < p, such
that we can write L and M as
L
M
=
d
Ao(x) + A1(x)
Do(x) + D(x)
d 2p
+...
A 2 p(x)
d 2s
d
+... + D 2S(X)
dx
dxs
(4.16)
(4.17)
Equation (4.15) is defined over the open interval 0 < x < L. In addition to
the differential equation given by Eq. (4.15), the function u must satisfy the
boundary conditions
Biu - ACu=f,
i = 1, 2, ..., p,
x = 0,L
(4.18)
where Bi and Ci are linear homogeneous differential operators with maximum
orders 2p - 1 and 2q - 1, respectively, and fi represents the ith external force
acting at the boundary.
4.3.1
Method of Assumed Modes
The "method of assumed modes" is a procedure for discretizing a distributedparameter system given by Eq. (4.15) to obtain a lumped-parameter model. In
54
order to obtain an nth order lumped-parameter approximation for the given
distributed system, we assume the displacement function u(x, t) of the distributed system to be a linear combination of a finite series of n space dependent functions qi(x):
n
u(x, t)
#i(x) qj(t)
=
(4.19)
i=1
where qi(t) is the ith generalized coordinate of the system. The functions O(x)
belong to a space whose members have finite-energy derivatives of order p
and satisfy the geometric boundary conditions. More generally, one can write
the relation between the displacement at a point (x , y , z) and the generalized
coordinates as
u(x , y , z , t) = N(x, y, z) q(t)
(4.20)
where each element of the vector u corresponds to a degree-of-freedom at
the point (x , y , z) in the three dimensional space, the vector q consists of the
generalized coordinates, and the matrix N is the shape matrix such that N,, =
#ij (x,
y , z). This method is similar to the classical "Rayleigh-Ritz" method
(see for e.g., [32]) except that in the Rayleigh-Ritz method the coefficients
in the summation of Eq. (4.20) are constants. The displacement function in
Eq. (4.20) is substituted in the system kinetic co-energy and potential energy
functions to obtain the mass and stiffness matrices of the system. We now
illustrate this procedure for a general system.
Consider a distributed-parameter system with the following constitutive
relation
o(x , y, z , t) = D E(x , y, z, t)
where
(4.21)
r and c are the stress and strain vectors at a point (x , y , z) in the
structure and D is the elasticity matrix. The strain-displacement relation can
be written as
E(x, y , z , t) =
B(x, y, z) q(t)
(4.22)
where B is the strain-displacement matrix. Note that the elements of the
matrices N and B are functions of qij and its derivatives. Hence the choice
of #ij determines how well the lumped-parameter system approximates the
distributed-parameter system. The kinetic co-energy and the potential energy
55
of the system are given by
T* =
V
(4.23)
jfidm
ETa
=
dvol
(4.24)
Using Eqs. (4.20)-(4.22), we rewrite Eqs. (4.23) and (4.24) as
T*
=
V
=
-T
2
2
M e
(4.25)
qT K q
(4.26)
We can now obtain the Lagrangian L = T* - V, from which the equations
of motion of the discrete system can be obtained as
M4 + Kq =
(4.27)
where E is the vector of generalized forces. The matrices M and K are the
mass and stiffness matrices of the system and are given by
M
=
p NT N dvol
(4.28)
K
=
BT D B dvol
(4.29)
where p is the density of the material. Note that the method of assumed modes
is at its heart a variational method because we obtain the equations of motion
by rendering the action integral stationary.
4.3.2
Method of Weighted Residuals
The method of weighted residuals is a discretization method which works
directly with the differential equations and the boundary conditions. The
method is applicable to differential equations in general, although our interest
here lies in the eigenvalue problem given by Eq. (4.15).
We assume that Eq. (4.15) does not lend itself to closed-form solution and
that we are interested in an approximate solution. To this end, we consider
a "trial function" u from the space C2p whose members have finite-energy
56
derivatives of order 2p and satisfy the geometric boundary conditions. In
general, the trial function does not satisfy Eq. (4.15), so that a measure of
error introduced by substituting the trial function for the actual solution can
be written in the form
R(u, x) = Lu - AMu
(4.30)
where R(u, x) represents the error in satisfying Eq. (4.15) and is known as
the residual. If the trial function is the eigenfunction ui and A the eigenvalue
Ai, then the residual is zero. Next, we multiply the residual R with a finiteenergy function known as the test function or the weighting function to form
the weighted residual given by
vR = v(Lu - AMu)
(4.31)
Now, we assume an approximate solution for u as
n
u
(4.32)
#q$
3 (x) qj (t)
=
j=1
where qj's are the generalized coordinates and #j's are n functions chosen from
a complete n-dimensional subspace S" of IC2P. We also choose n functions ' 1 ,
2
2,.., Onfrom another complete n-dimensional subspace Vn of K p. The spaces
Sn and Vn are referred to as the trial space and test space, respectively. We
determine the unknowns qj's by imposing the condition that the functions bi
(i=1,2,..., n) be orthogonal to the residual, that is, we require
R)
(i,
=
j
(4.33)
(Lu - AMu) dx = 0
Substituting Eq. (4.32) into Eq. (4.33), we obtain the algebraic eigenvalue
problem
(O),R)
i
j
-
q( LO -A
7(kij - Amij)qj = 0,
)
5M
dx
i = 1, 2, ... , n
(4.34)
j=1
57
where kij and mij are given by
ki = (I,
j
f=
L01)
iL#j dx,
i, j =1, 2, ..., n
(4.35)
0L
mij= (oi, M#5)
=
o OiM~j dx,
ij =1,2, ...
,n
(4.36)
This method results in a solution converging to that of Eq. (4.15) because
the test functions Oi's are from a complete set and the residual R is required
to be orthogonal to every Oi. Hence, if the number of test functions is increased without bounds (i.e., when n -+ oc) the only possible way for R to be
orthogonal to the complete set V" is for R to be identically zero.
Galerkin's Method
In this method the test functions coincide with the trial functions, or
i ,i= i = 1, 2,,..., n
(4.37)
and the stiffness coefficients kij and mass coefficients mij can be obtained from
Eqs. (4.35) and (4.36), respectively, using the above equation.
We note that in the above approach we have not required the trial functions to satisfy the natural boundary conditions. Hence, in problems where the
natural boundary conditions are important, the lumped-parameter approximations will not be able to capture the dynamics of the system correctly. Also,
we may sometimes be unable to construct trial functions which satisfy all the
boundary conditions. Therefore, in order to minimize the error in satisfying
the boundary conditions (given by Eq. (4.18)) by the assumed expansion for
displacement given in Eq. (4.32), we can include the boundary conditions in
the weighted residual with appropriate weighting. Substituting Eq. (4.32) in
Eq. (4.18), we obtain the following equations
(bij - A cij)q=
f,
i = 1, 2, ..., p
(4.38)
j=1
We now obtain the lumped-parameter approximation for a given distributedparameter system by minimizing the error in a Galerkin-type weighted residual
58
R given by
u(Lu - AMu) dx +
q
(bij - Acij)q7 -
(4.39)
The first term in the above expression represents the error in satisfying Eq. (4.15)
weighted by the approximate solution u (Eq. (4.32)). Likewise, the second
term is the summation of the errors resulting from satisfying the boundary
conditions weighted by the corresponding generalized coordinate. Next, we
substitute the approximate solution for u given by Eq. (4.32) into the residual
given by Eq. (4.39) and set &R/&qi, i = 1, 2, ..., n, to zero, which yields a set
of n equations that minimize the residual of the dynamic equations over all
possible trial functions given by Eq. (4.32).
4.4
Lumped-Parameter Approximations
In this section, we obtain lumped-parameter approximations for the ball-screw
drive using the methods described in Sec. 4.3.1 and 4.3.2. As discussed in
Secs. 4.1 and 4.3, our concern is to obtain an accurate model for the first axial
mode of the system. Hence, we use the piecewise-linear shape for u(x, t) and
O(x, t) obtained from a quasi-static analysis as trial functions in the approximation methods. In quasi-static analysis, we neglect the inertia of the system
and examine the deformation one obtains if the carriage is subjected to a longitudinal displacement uc(t) and the motor is subjected to a rotation 0m(t).
That is, we consider what happens at very small w, where the longitudinal
and torsional deformation of the screw appear as shown in Figs. 4.3 and 4.4.
Let us now impose a quasi-static (slowly varying) carriage displacement
uc(t) and motor rotation 0m(t) by means of a longitudinal force f exerted
on the carriage and a torque -ff/27r exerted on the motor. The forces and
displacements are related by
f = k [c(t) - Om(t
(4.40)
where kt is the total stiffness of the system and we have neglected the inertia
effects. The expression for kt (which we obtain later) depends on the boundary
conditions at the end farthest from the motor. We wish to solve for the
quasi-static displacement of the system for the two boundary conditions under
consideration: free and damped.
59
4.4.1
Ball-Screw Servo with Free End
Quasi-Static Displacements
Under quasi-static conditions the longitudinal deformation u(x, t) and torsional deformation 6(x, t) in the screw vary linearly with x on each portion of
the screw as shown in Fig. 4.3 and are given by
u(x, t) = NuU(Oj t)
-
O(x, t) = No
(4.41)
U(0c, t)
0(0, t)
(4.42)
where Nu and No are shape matrices given by
NU=
[
N
[0 , 1] ,x> x
=
No =
No
=
-
)
x< x
(4.43)
(4.44)
-±)
,
ze
[0 , 1] , x > x.
(4.45)
(4.46)
Using the above relations, we rewrite the boundary conditions (see Sec. 4.2.3)
for a free end as
(EA/x)[u(0, t) - u(x,, t)] + kbl[U(0, t)] = 0
(G J/x)[0(0, t) - O(xc, t)] + K,[0(0, t) - Om(t)]
0
(EA/x)[u(x,, t) - u(0, t)] + k,[uc(x, t) - wun(t)] = 0
(G J/x)[6(x,, t) - 0(0, t)] + Kn [(x,, t) - Oc(t)] = 0
(4.47)
(4.48)
(4.49)
(4.50)
Making use of the above equations and Eq. (4.9) we solve for the angular
displacements at x = 0 and x = xc in terms of Om(t) and uc(t) as
0(0, t)/n
O(xce t)/n
_
I - (kt/n 2cK)
kt/n 2c
1-(kt /n 2KC)
kt /n 2 te
Om (t) /n
UC~t
M
(4.51)
T,
where n represents the drive ratio 27r/e of the screw, Kt, = (1/c + xc/GJ)-1 ,
the equivalent torsional stiffness of the serial combination of the coupling and
60
n M)
tkn
u(xc, t) ........................
EA/xc
u(0, t) ...............................
kbI
0
0
L
xc
x
0C (t)
n
O(xe, t)
......
0(0,t)
..
. c..
. . . .. . . . . . .
.. GJ/xc
.
.C
Om (t)
0
L
xC
x
Figure 4.3: Quasi-static displacements for the free-end boundary condition
61
screw, and kt is the total stiffness (refer to Eq. (4.40)). The expression for kt
is obtained as
1
kt
-
k1
1
+
kn
( f
-+
2
27r
I
1
-
Ke
I
c
+
Kn
(4.52)
G J)
where k, = (1/kbl + xc/EA)-- and represents the stiffness of the serial combination of the portion screw between the nut and the motor and the motor-side
thrust bearings. The first two terms in the above expression represent the
total longitudinal stiffness and the last three terms the total torsional stiffness
multiplied by the square of the drive ratio. Hence, the total stiffness kt is the
serial combination of the total longitudinal stiffness and the torsional stiffness
multiplied by the square of the drive ratio.
Similarly, we solve for the longitudinal displacements at x = 0 and x = xc
to obtain
u(0, t) \
-kt/kbl
kt/kbl
Om (t)/n
(4.53)
U(xe7 t)
-ktlk1
ktlk1
UCMt
T,,
Next, we solve for the longitudinal and torsional deformations of the nut
in terms of Om(t) and u,(t) as
un(t) - u(xe, t)
Oe(t) - O(xe, t)
)
(
-kt/kn
-kt/n
2
rn
kt/kn
kt/n
Om(t)/n
u(t)
)
(4.54)
(4
Finally, using Eqs. (4.41), (4.42), (4.51), and (4.53), we express the longitudinal deformation u(x, t) and torsional deformation O(x, t) of the screw in
terms of Om(t) and uc(t) as
u(x,t)
O(x, t)
=
=
Nu Tu
No To
Om(t)/n
(4.55)
c(t)
(456
(4.56)
(
\Om(t)/n
uc(t)
}
Dynamic Equations
In this section, we derive expressions for the kinetic co-energy T* and the
potential energy V of the system and construct the Lagrangian L = T* - V.
62
The dynamic equations for the system can then be obtained from Lagrange's
equations.
The kinetic co-energy T* of the system is given by
T*Jm
2
m
2
2
i+pJ
ct) 2 p 1
xt)d+
2(
pA
2
]uxf)o
t) dx (4.57)
The first term in the above expression represents the contribution from the
inertia of the motor armature Jm to the kinetic co-energy of the system. Similarly, the second, third, and fourth terms represent the respective contributions
from the mass of the carriage, the distributed rotary inertia of the screw and
the distributed linear inertia of the screw.
The potential energy V of the system is given by
V =_
(4.58)
1kblu 2 (0, t) +
+ GJ
2o2
e
[OrM(t)
-
[O'(xjt)] 2 dx +
(0,
t)]2 +
k,[u,(t)
-
EA
[U'(Xt)]2 dx
u(xe, t)] 2 + IKn[e(t) 2
(xe, t)]2
The first and the second terms in the above expression represent the elastic energy in the thrust bearings and the coupling. Similarly, the third and
the fourth terms represent the elastic energy arising from the stretching and
twisting of the screw. Finally, the fifth and the sixth terms indicate the elastic
energy arising from the deformation of the nut.
Because the measurable inputs and outputs of the system are at the motor
and the carriage, their displacements uc(t) and Om (t) are convenient generalized
coordinates for the model. We therefore substitute the quasi-static solution
in terms of uc(t) and Om(t) given by Eqs. (4.51)-(4.56) into the Lagrangian
,C = T*- V and obtain the following dynamic equations (Lagrange's equations)
for Om(t) and uc(t):
d ( 0,C
dt 00m
d (0=L
dt ONc
I\
=
00m
0
OUc
- Cm m(t) +
Tm(t)
-CCI(t) + fM(t)
(4.59)
(4.60)
where the disturbance torque Tm(t) and the viscous damping force Cmm(t)
at the motor (see Fig. 4.1) are the generalized forces corresponding to the
63
generalized coordinate 0m (t). Similarly, the disturbance force fc(t) and the
viscous damping force Ccitc(t) at the carriage (see Fig. 4.1) are the generalized
forces corresponding to the generalized coordinate uc(t). Finally, we combine
and rewrite Eqs (4.59) and (4.60) to yield the following familiar second-order
form
iim(t)
iic(t)
irnm(t)
I
UM(t)
uc(t)
I
itc(t)
nTm(t)
I
I
fc(t)
(4.61)
where we have introduced an equivalent motor displacement un = (f/27r)Orn.
The matrices M, C, and K are the mass, damping and stiffness matrices of
the system, respectively, and are given by
M
=
J
=
n2
n2J12-
n 2 J2 + m
J+
n2Kc
n2
3
kt
+ pJ(L - x,)
kt)
1
n2KC)
2
C
64
=
kt )2
k1
2C
(
kt
n2KC
+
k
0
+ kk
2
(4.63)
]1
-C
t
+2Ktc
k2
k
ctc
n2Kt c
kt
+ pJ(L -xc)
n 2 tC
+
kt kt
2
n2
3
kt
kt
- 2 t
r~t)
-
kt
1 -t 2 t
(4.64)
n2k)
n2KC 12 tc
(n2.)
pAxc
n2s,
) (1
kt
tc
pJxe
=+
(t
(
+ (1(
+ pJ(L -
n2KtC)
k
2
n2KC)
2
(4.62)
+Tmc
kt
kt
3pJxc
3
-J12
(
3
+
J2
m
PXC
(I
+
J1 2
m
n 2 J, + m
(4.65)
2
tc
kbik1
+
K
+ pA(L - xe)
k1
kt
kt
)
(k1
(4.66)
(4.67)
4.4.2
Ball-Screw Servo with Damped End
As we have discussed in Sec. 4.2.2, the damped-end problem is formulated in
the frequency domain due to the presence of a hysteretic element (the damper)
in the system. Therefore, we cannot directly apply energy methods as we did
in the previous section. Hence, we use the method of weighted residuals which
works directly with the differential equations and the boundary conditions to
obtain dynamic equations for the carriage and the motor in frequency domain.
Further, from among the several weighted residual methods we use the socalled Galerkin's method in which the test functions coincide with the trial
functions (see Sec. 4.3.2). We start this section by constructing a trial function
which in a quasi-static sense conforms with the first axial mode of the system.
Quasi-static Displacements
As we have seen in Sec. 4.4.1, under quasi-static conditions the longitudinal
and torsional deformations of the screw vary linearly with x on each portion of
the screw. This means that the complex phasors Uc(x, w) and Em(x, w) which
represent the magnitude and phase of motion (see Sec. 4.2.2) also vary linearly
with x on each portion of the screw as shown in Fig. 4.4 and are given by
U(x,w)
(x,w)
=
Nu (U(,
N
E
w)
U(L, w) )
(4.68)
(
(4.69)
)
where NU and No are shape matrices given by
Nu
=
Nu =
1 -
(]
0 ,
Ne
- Xc
No
=
e(4.70)
, 1=
[0 , 1] , x > xe
Xc.
,
,z
> Xe
(4.71)
oc(4.72)
(4.73)
Using the above relations, we rewrite the boundary conditions (refer to
65
Un(w)
................
. ... .. .... ... .... ... ..
kn
U (L , w )
....
U(x,w)
EAI.L -
SEA/x>
kb2
ko1
0
0
L
xe
housing+thrustbearings
at damped end
x
n
E ( ,w )
.....
*...
Kc
0
L
xP
x
Figure 4.4: Quasi-static displacements for the damped-end boundary condition
66
Sec. 4.2.3) for a damped end in the frequency domain as
EA[U(O, W) - U(xw
)] + kbl[U(0,W)]
xc
EA
xc
[U(xc, w) - U(O, w)] + L
EA
L-xe
= 0
(4.74)
=
0
(4.75)
=
0
(4.76)
0
(4.77)
0
(4.78)
[U(xc, w) - U(L, w)]
+k, [U(xc, w) - UN]
EA[U(c, w) - U(L, w)] + kb [U(Lw)]
2
L - xc
GJ
G[(,W)
xc
e(xc, )] + Kc[e(, W)
-
m]=
GJ[6(x w) - 9(0, w)] + K"[6(Xc,
W) - Ec] =
xc
Making use of the above equations and Eq. (4.9), we solve for the angular
displacements at x = 0 and x = xe in terms of 0m(w) and Uc(w) as
(
E(0, t)/n
E)(x, t)/n
_
1-
1-
2
(kt/n 2 K)
kt/nc
K
2
2
(kt/n Kt,) kt/n
tC
Em(w)/n
Uc(w)
)
(4.79)
Te
The above expressions for angular displacements are similar to the ones obtained for the case of the free-end boundary condition (see Eq. (4.51). This is
because in both cases we have the same form for the piecewise-linear functions
describing the torsional deformation in the screw (see Figs. 4.3 and 4.4). However, due to the difference in the axial boundary conditions, the expression
for the total stiffness kt is modified (see discussion following Eq. (4.40)) and
is given by
kt =
k, + k2
+
kn
2,7
+
Ke
-+
Kn
_+
GJ
(4.80)
where k 2 = [(L - xc)/EA + 1/kb2f 1 and represents the stiffness of a serial
combination of the screw (between the nut and the damped end) and the
damped end. We note from Eqs. (4.52) and (4.80) that the expression for
total stiffness kt takes on a similar form for both free and damped boundary conditions. It represents the serial combination of the total longitudinal
stiffness and the total torsional stiffness multiplied by the square of the drive
67
ratio. This effect can be clearly seen in the free-body diagrams (see Fig. 4.1
and 4.2). Next, we solve for the longitudinal displacements at x 0, x = xc,
and x = L to obtain
U(0, w)
U(xc, w)
U(L,w)
k
-kl/kbi
kl/kbl
-1
1
=k
-k 2 /k
+2
k2 /kb
2
/
e(w)
/n
(4.81)
Uc(w)
2
TU
Finally, using Eqs. (4.68), (4.69), (4.79), and (4.81), we express the complex
variables U(x, w) and e(x, w) of the screw in terms of Om(w) and U,(w) as
U(x, w)
=
Nu Tu O(w)/n
(4.82)
E(x, w)
=
Ne Te(
O(w)/n
(4.83)
Dynamic Equations
In this section, we will use the method of weighted residuals based on the
quasi-static deformation of the screw to obtain dynamic equations for @m(w)
and U,(w).
For steady harmonic vibration of the ball-screw drive at frequency w the
equations of motion for motor armature and carriage given by Eqs. (4.13) and
(4.14) become
(-W 2 jm + jwCm +
Gm(w)
=
(-W 2mc + wCc +kn) Uc(W)
=
Kc)
Tm(w)
/
+ K
(4.84)
(, w)
Fe(w) +kn(U(Xc,,W) +Ec(w) 2r)(4.85)
27
The longitudinal and torsional deformations of the screw are each governed
by a second-order wave equation, which in the frequency domain take the form
(as discussed in Sec. 4.2.2)
EU"(x, w) + pw2 U(x, w)
=
0
and
GY"(x, w) + pw2 e(x, w)
=
0
(4.86)
Based on the discussion given in Secs. 4.1 and 4.3, the solution of these equations coupled with Eqs. (4.13) and (4.14) can be approximated by the quasistatic shape shown in Fig. 4.4 provided that the frequency of motion W is well
below (E/pL2 )1 / 2 and (G/pL 2) 1/ 2 .
68
We perform this approximation by minimizing the error in a Galerkin-type
weighted residual R of the form
R =
0.(W) [(-W
+Uc(w)
-
jL
-
jL
2
Jm + jw C-
[(-w2mc +
+ K,) 0m(w) - T
Cc + k) Uc(w)
-Fc
-- Kc6(0, w)]
- kn(U(xc, w) +
c(w)
27
U(x, w) [EU"(x, w) + pw2 U(x, w)] dx
|L e (x, w) [GO5"(x, w)
+ pw 2
e0(x, w) ] dx
(4.87)
If the motor rotation Em(w) is an approximate solution to Eq. (4.13) governing
the dynamics of the motor, the first line of this expression represents the error
in satisfying Eq. (4.13) weighted by the approximate solution em(w). The
second, third, and fourth lines are similarly formed weighted residuals for the
carriage, longitudinal motion of the screw, and torsional motion of the screw.
Because the measurable inputs and outputs of the system are at the motor
and carriage, their displacements Uc(w) and em (w) are convenient generalized
coordinates for the model. We therefore substitute the quasi-static solution in
terms of Uc(w) and em(w) given by Eqs. (4.79), (4.81), (4.82) and (4.83) into
the residual given by Eq. (4.87) and set OR/OUc and R/Oem to zero, which
yields a set of equations that minimize the residual of the dynamic equations
over all possible quasi-static deformations. These equations can be written in
the familiar second-order form
(-W 2 M+jwC+K)
(
Um(w)
Uc (W)
_
(Tm(W)
Fe (w)
(4.88)
where we have introduced an equivalent motor displacement Um = (f/27r)Om.
The mass matrix M, damping matrix C, and the stiffness matrix K are given
69
by
M
2J, + m
n2 J12m-m[
=
J12
=
-
n
n
c
1 -
2
kt
+pJ(L - xc)
k
c n
k
+ (
kt
kt
n
(4.90)
k2
t
1
.2tc
_n
+
c
-n
kt
+ pJ(L - xc)
n2KtC
PJXC
n2 tC
±
(i
+[t
+ -(
2 n2c
n2 tC
c
kt
1t-
n2KtC
2
k
-
n2ec
(ilkg
k-
(i
3
3
, ) (I
+ (
)
nec)
+ pJ(L - xc)
[
pJxe
+
- t
1-
+
(4.89)
12 J 2 + m + mn
(
3
J, im +
m
2 J12 -
- t
n2Ktc
(4.91)
_
)2
n2()C
tC
2
(4.92)
(n2Ktc)
m
=
pAxe
3
kt )2 (1+k1
(k1 + k2
kb1
pA(L -xc)
3
( 2C
0
0
c
kt k2 1
k1 + k2
k2
k 2b
k2
kb2
+k
(4.93)
k2
K = k(1I+ j sgn w) (
1
-1
-1)(9
1 ,J(4.94)
where k(1 + j sgn w) represents the total dynamic stiffness of the system.
Under dynamic conditions, the stiffness elements in the structural loop exhibit
damping due to material hysteresis. In order to account for this hysteretic
damping in the dynamic model, we add to the stiffness value ki of every spring
the damping contribution jkjrjj sgn w, where qj represents the corresponding
loss factor. The total dynamic stiffness can then be computed from the quasistatic expression given in Eq. (4.80) by replacing the real spring constants
with their complex counterparts. Finally, the expression for the total dynamic
stiffness can be recast as k(1 + j7 sgn w), where k represents the total stiffness
and 77 the effective loss factor of the structural loop.
70
We notice from the expression for the stiffness matrix that the dynamic
model for the ball-screw drive with the viscoelastic damper exhibits mixed
damping (see Sec. 3.2). This is because the viscoelastic washer behaves as a
hysteretic spring whereas the damping in the motor and the bearings follows
a viscous law. As described in Sec. 3.2, it is important to obtain an equivalent viscous model for the given mixed model to avoid erroneous results in the
frequency and time domain calculations. Hence, we set out to obtain an equivalent model for the ball-screw drive mechanism using the method described in
Sec. 3.2.
Equivalent Model
As described in Sec. 3.2, the first step in obtaining an equivalent model for a
given mixed-damping system is to solve its characteristic equation and identify
the correct eigenvalues. The eigenvalues A = W + jo- of the system can be
obtained by solving the following characteristic equation
det [-A
2
M + jAC + K]
=
0
(4.95)
On substituting the expressions for mass, damping and stiffness matrices from
Eqs. (4.89) and (4.94) into the above equation, we obtain
A"
[mum 22
-A
[miiCc + rim22Cm]
[k(1 + ja sgn a) (mi + 2mn12 + M 2 2 ) + n 2 CmCc]
+jAk(1 + j sgn w) [n2 Cm + Cc]
- m12 ] - jA3
2
=
0 (4.96)
where mij is the ijth element of the mass matrix M. We require closedform solutions of Eq. (4.96) in order to obtain closed-form expressions for
the mass, damping and stiffness matrices of the equivalent model. As we
shall see in Chapters 5 and 7, closed-form expressions for open-loop poles are
important to obtain formulae relating performance and system parameters,
thereby providing us with the key to solve the inverse problem. We notice that
Eq. (4.96) is fourth-order in A and hence can be solved in closed-form. However,
when the order of the characteristic equation is greater than four, closed-form
solutions do not exist, and one has to resort to numerical techniques. We also
note that the expressions for solutions of polynomial equations of order greater
than two are too cumbersome to yield any useful formulae.
In order to avoid the above difficulties, we use a perturbation approach
(e.g., Nayfeh [35]) by recognizing that the damping in many machines and
71
structures is small. When damping is small, the imaginary part of the root is
much smaller than the real part. Hence, we can use the ratio of the imaginary
part to the real part of the root as a perturbation parameter in determining
an approximate solution to Eq. (4.96).
Perturbation Method
In this method, we assume the solution to Eq. (4.96) as an asymptotic expansion in terms of the above stated perturbation parameter. Therefore, we
write
Im(A)/Re(A)
A
k r-
0 (E)
Ar + jcAi , (Ar, Ai _ 0(1))
cAtr, (At ~ 0(1))
Cm = Am ,(Am
0(1))
0
Cc = AcI, (Aco0(1))
(4.97)
(4.98)
The second step involves substituting the above expressions for A, k r, Cm,
and Cc in Eq. (4.96). The result is
(Ar + jcAi) 3 [mum 22
m 2] - jE(Ar + jcA2 )2 [mu Ac + n 2 m 2 2 Am]
-(Ar + jcAj) [(k + jAt sgnw)(mil + 2m 1 2 + M 2 2 ) + 62 n 2 AmAc]
+jc [(k + jeAt sgn w) (n 2 Am + Ac)] = 0
-
(4.99)
The third step involves collecting coefficients of like powers of 6 in Eq. (4.99)
where only terms up to O(c) have to be retained, consistent with the assumed
expansion (Eq. (4.97)). In the next step, we equate the coefficients of 0l and
E0 to zero and solve the resulting equations. These equations are given as
A [mum22
-
m12] - Ark [i
+ 2m 12 +
(4.100)
M 2 2 ] =0
n 2m
3A Af [mum2 2 - mn2 ] - A2 [miiAc +
22 Am]
-(Aik + ArAtsgnw) [mul + 2n12 + M 2 2 ] + k[nr2 Am + Ac]
=
0 (4.101)
By solving Eqs. (4.100) and (4.101) simultaneously, we obtain six roots for
A, of which two are eliminated because they do not satisfy the conditions
imposed by sgn w (as discussed in Sec. 3.2). Therefore, the expressions for the
72
eigenvalues A are obtained as
A,
A2
A3
0
(4.102)
.l
=
in
=
n
1
± A3r
2Cm+Cc
(4.103)
+ 2m 1 2 + M 22
A3 i
+j
[k (m
+
(4.104)
2mn 2
+
iM2 2 )
(MIM22 - M1 2 )
1
(4.105)
1/2
[n2 Cm(M 2 2 + m12 ) 2 + Cc(mnII + Tn 12 ) 2
2 [(n
+ 2m 1 2 + m22)(mn M 22 - m1 2 )
1
k(mn + 2mn2 + M 2 2 )
(munm
22
1
- m1 2 )
As discussed in Sec. 3.2, we can now construct an equivalent viscous model
with pole locations identical to the ones given in Eqs. (4.102)-(4.106). Let the
equivalent viscous model be given by
Me
iimi(t)
1 c(t) 1
+ Ce
Em(t)
7c (t) I
+ Ke
Um
IUCWt
nrM(t)
fc(t)I
(4.107)
As the equivalent model preserves the undamped problem for small damping
(see Sec. 3.2) we obtain the following
Me
=
M
Ke
=
Re(K)
(4.108)
(4.109)
The next step is to choose the damping matrix in order to obtain identical
pole locations. We note that the choice of the damping matrix to attain this
objective is not unique. Hence, we propose the following structure for the
damping matrix based on physical reasoning
e
n 2C-C
=+ C
C -C
~410
(4.110)
where C can be determined by matching the poles of the equivalent system to
those given in Eqs. (4.102)-(4.106). To understand the meaning of C, we take
another look at the damping sources in the ball-screw system. Energy dissipation in the lumped-parameter model given by Eq. (4.88) is represented by
73
the dash-pot Cm between the motor and the ground, the dash-pot C, between
the carriage and the ground, and the imaginary part of the effective complex
stiffness k(1 + jiqsgn w). The damping produced by Cm or C, depends on the
corresponding generalized coordinates alone, whereas the damping from the
complex spring depends on the difference between the generalized coordinates;
hence, we preserve this effect by choosing the structure of the damping matrix
as given in Eq. (4.110).
We solve the characteristic equation of the equivalent model using a perturbation approach in a manner similar to the one used for the mixed system.
The results are given below
Aie
=
A 2e
j
A3 e
±
A 3 er
(4.111)
0
(4.112)
2Cm ± Cc
+ 2m1 2 + M 22
A 3 er + ' A3ei
inl 1
(4.113)
2mi 2 + 2m 22 )j 1/2
(mllm22 - M12)
.
2
2
[n Cm(m 2 2 + mi 2 ) + Cc(mu + m 1 2 )2
2 _[(mn + 2m 1 2 + m 22 )(mTnm 22 - m1 2 )
k(n +
1
+n 2m12 + M22)
(mlIM22 - M12).
C(m
(4.114)
(4.115)
(4.116)
(4.117)
By comparing Eqs. (4.102)-(4.106) and Eqs. (4.111)-(4.116), we obtain an
expression for C as
C
k(mllm 22
in
-
m1 2 )
+ 2m 12 +
M
(4.118)
22
Finally, we note that the damping matrix can also be obtained using
Eq. (3.42). In the limit of small damping the result reduces to what we have
obtained above.
74
CHAPTER
5
Robust Controller Design and
Bandwidth Formulae
5.1
Introduction
In Chapter 4, we obtained lumped-parameter models of the ball-screw servo
under various boundary conditions (see Sec. 4.4). The equations of motion
were cast in the following form:
M
[.m(t)
Mic(t)
+ C [M(t)
c(t)
+ K
um(t)
u(t)
_
fm(t) + dm(t)
dc(t)
-
(5.1)
(
where M, C, and K are the respective mass, damping and stiffness matrices
as obtained in Secs. 4.4.1 and 4.4.2. These matrices have the following form
M =[m
M12
m 12
M2
'
C = [ C1
. -C12
-012
C2
K=
k
-k
-k
k
(5.2)
The force vector in Eq. (5.1) comprises of the control and the disturbance
forces: fm(t) represents the control force at the motor end, dm(t) and dc(t)
represent the respective disturbance forces at the motor and the carriage ends.
Note that the stiffness matrix K is real since we are using the equivalent viscous
75
model derived in Sec. 4.4.2 for the damped screw. By taking the Laplace
transform of Eq. (5.1) and solving the resulting equations for Urn(s) and Uc(s)
we obtain
m 2 s2
UrnS)
UM (s)
~
A (s)
22+
12 s
A[()
2
+ C12 s +
SD(s)
D s
IS +Cjs+k D,(s)
[Fm(s) + Dm(s)] +
+ C 12
Uc(s) =
-m
[Fm(s) + Dm(s)] +
C2 s
A(s)
53
(5.3)
(5.4)
where the capital letters represent the Laplace transform of the corresponding
time variables, and A(s) is the characteristic polynomial given by
m1 2] s4 + [mIC 2 + m 2 C1 + 2m12 C12 ] S3
+[k(mi + 2m 1 2 + m 2 ) + C1C2] S2 + [k(C 1 + C2 - 2C12)] s
A(s)
[mim 2
-
(5.5)
For the general system represented by Eq. (5.1), two transfer functions
are of interest: one between fm(t) and um(t), and the other between fm(t)
and uc(t). The first represents the case where the sensor (here the motor
encoder) is placed on the same rigid body as the actuator (here the motor
rotor), but there exists a mechanical resonance elsewhere in the system that
is coupled to the mass to which the actuator and sensor are attached. The
second represents a case where there is a structural resonance between the
sensor (here the carriage encoder) and the actuator. These two situations
are referred to as collocated and non-collocated cases in the control literature
(e.g., Franklin et al. [19, 20]). Using Eq. (5.1), we obtain the collocated and
non-collocated transfer functions for the ball-screw drive as
5.2
Gi (s)
-UM(S)
Gp2 (S)
-
Fm(s)
Uc(s)
Fm(s)
m
2
-in
2 +C2
s+ k
(56)
+ C 12 s + k
A(s)
(5.7)
A(s)
12
S2
Expressions for Open-Loop Poles using Perturbation
Method
The open-loop poles of the system given by Eq. (5.1) are the roots of the
following characteristic equation
A(s) = 0
76
(5.8)
We solve the above equation using the perturbation approach described
in Sec. 4.4.2 to obtain closed-form expressions for the open-loop poles of the
system. As we shall see later in this chapter, closed-form expressions for the
open-loop poles provide the key link between the system parameters and the
performance measures, which then allows us to solve the inverse problem.
Because the characteristic equation is formulated in the Laplace domain the
perturbation parameter is different from the one used in Sec. 4.4.2; for the case
of small damping, the perturbation parameter is the ratio of the real part of
the root to the imaginary part. Therefore, we assume the solution to Eq. (5.8)
as an asymptotic expansion in terms of the perturbation parameter c, and we
write
O(c)
Csr + JSi , (Sr, Si ' 0(1))
0(1))
EA1 , (A1 0
Re(s)/Im(s)
s
C1
C 12
=
cA 12
C2
=
EA 2
,
,
(A 12 ~ 0(1))
(A 2 ~ 0(1))
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
Substituting the above expressions for s, C1, C12 and C2 into Eq. (5.8), we
obtain
(ESr
+ jsi)3 [mim 2 - m1 2] + C(csr + jsI) 2 [miA 2 + m 2 A 2 + 2m 1 2AI 2]
+(cs, + jsi) [[k(mi + 2m 12 + M 2 )] + E2 AIA 2]
+6 [k(A
1+
(5.14)
A 2 - 2A 12 )] = 0
The third step involves collecting coefficients of like powers of C in Eq. (5.14)
where only terms up to O(c) have to be retained, consistent with the assumed
expansion (Eq. (5.10)). In the next step we equate the coefficients of 01 and Ec
to zero and solve the resulting equations. This gives the approximate roots as
si
=
0
(5.15)
S2
=
srl
s 3 ,4
=
Sr2 ±
(5.16)
(5.17)
jSc
77
De(s)
D..Dm(s)
Um(s
+0
++
+Um(s)
0
F(GPI(s)
Gc(s)
-=
Fm (s
Figure 5.1: Feedback control of um(t)
where
Sri
=
C1 + C2 -
2C12
n + 2m
+ m
sc
=
12
2
2
2
2
12 ) + C 2 (mi + M 12 ) + 2C 1 2 [n 1 2 (mi + 2m 12 + m 2 ) + mim 2 - m 2 ]
S122
2(mi + 2nI 2 + mn2 ) (mlm 2 - m1 2 )
k(mi + 2mI 2 + M 2) 1/2
MIM2 - M12
CI(m
+ m
Hence the characteristic polynomial can be factored as
A(s) =
5.3
5.3.1
(mim 2
- m1 2 ) s (s - Sri) (s2 -
2s,2s
2
+
se)
(5.18)
Collocated Transfer Function
Control of um(t)
The block diagram representation for the feedback control of un(t) is shown
in Fig. 5.1, and the Bode plot of the collocated transfer function G, 1 (s) given
by Eq. (5.6) is sketched in Fig. 5.2.
78
-20db/dec
-40db/dec
-40db/dec
rl
0
k
2
2
-0---------------
-90 -- --- -- -- --
-180
Figure 5.2: Bode plot of the collocated transfer function Gpi(s)
79
Bandwidth and Phase Margin
From Fig. 5.2, we see that the phase of the collocated transfer function G, 1 does
not fall below -1800 due to the presence of an anti-resonance before the axial
resonance. Thus, the axial resonance associated with the drive compliance
does not pose any limitation on the crossover frequency. This allows us to
design a controller which can achieve high crossover frequencies (and hence,
high bandwidth) and high phase margin.
Disturbance and Noise Rejection
From Fig. 5.1, we obtain the transfer function Um(s)/Dc(s) for the closed-loop,
as the weighted sensitivity function given by
D(S)
De(s)
=
Wm(s)Sm(s)
(5.19)
where Sm(s) represents the sensitivity transfer function ((1 + Gpi(s)G,(s))
for collocated control, and Wm(s) is a stable non-minimum filter given by
Wmrns) =
-m
1
2S
2
1)
(5.20)
+ C 12 s + k
(.0
A (s)
We modify the generalized Bode's sensitivity theorem (see Freudenberg
and Looze [21]), to account for the non-minimum filter in Eq. (5.19). This
result is stated in the following theorem
Theorem 1 Let Sm(s) = W(s)S(s) represent a weighted sensitivity function,
where W(s) is a stable non-minimum filter and S(s) is the sensitivity function of a stable feedback system with a loop transmission L(s). Assume that
L(s) possesses finitely many right-half plane poles including multiplicites. In
addition, assume that
lim W logS.(jw) = 0
(5.21)
w_+o
then
J
log ISw(w)I dw = 7r
Re[p] + Z
Re[zm])
where pi represents the ith open right-half plane pole of L(s), and z, the mth
open right-halfplane zero of W(s)
80
The proof of the above theorem is given in Appendix A. Using Theorem 1, we
obtain
g UM(OW)
C 12 +
C122 + 4mi 2 k
(522)
log
=(Dr((5.22
2M12
Dc(3w)
0
From Eq. (5.22), we see that the presence of a non-minimum phase zero
in the transfer function from the disturbance dc(t) to the closed-loop output
UMr(t), poses an important limitation: an attempt to make IUm(s)/Dc(s)I small
(< 1) at low frequencies, will result in values greater than unity at higher frequencies. This condition along with the bandwidth constraint-as discussed
by Freudenberg and Looze [22]-result in a trade-off between the system sensitivity properties and closed-loop bandwidth. We note that the presence of
the non-minimum phase zero in Um(s)/Dc(s)| has made this trade-off more
severe than otherwise. Hence, we may have to relax either the bandwidth or
the sensitivity requirement. We will find in the next section (Sec. 5.4) that the
non-minimum phase zero places even greater limitations on the disturbance
rejection in the case of a non-collocated transfer function. This limitation
arises not only due to the presence of such a zero but also from its location.
The disturbance at the motor end can be rejected by having a high controller gain in the required frequency range. We also note that by having
a high phase margin for the nominal transfer function Gi(s) we can obtain
reasonable robustness margin over plant uncertainties. However, we are interested in the precise control of the carriage position (uc(t)). We will see in
the next sub-section that even though we can have precise control over um(t)
over a wide range of frequencies by closing a feedback loop over the collocated
transfer function, we cannot precisely control the carriage position due to the
drive resonance and the disturbance force at the carriage.
5.3.2
Limitations of controlling uc(t) through Collocated Control
We use Eqs. (5.3) and (5.4) to eliminate Fm(s) and obtain an expression for
Uc(s) as
-i
Uc(s) =(
2
32
2+
+ C 12 s ± k\
Um(s)+(
± k
C
2
1
±k) Dc(s) (5.23)
Equation (5.23) presents us with following limitations on the control of uc(t),
when we close a feedback loop on um(t).
81
'~20
1 )g2011k)
40db/dec
-40db/dec
bJ0
0
CC
Vk/rn_2
~m
w
vk/rn2
r /
WW
Figure 5.3: Bode plots of Uc(s)/Um(s) and Uc(s)/Dc(s)
1. Maximum Frequency Limit
(
k/rm2 ):
From the Bode plot of Uc(s)/Um(s) as shown in Fig. 5.3, we see that
in the absence of external disturbances, we have reasonable control on
uc(t) through um(t) in a frequency range approximately up to
k/m 2 .
However, for frequencies above
k/rm 2 , we cannot control uc(t) through
um(t), and in this sense the frequency
k/rm 2 is an upper limit on what
can be achieved through collocated control. Hence, for design purposes
we would want to make this frequency as high as possible.
2. Effect of Disturbances:
From the Bode plot of Uc(s)/Dc(s) shown in Fig. 5.3, we find that in
order to reject disturbances at the carriage the stiffness k must be made
large. Also, in order to avoid amplification of disturbances in the vicinity
of the axial resonance, we require that the resonant peak be well damped.
But if the damping is made too large the frequency range described
in the previous point reduces and therefore, this trade-off results in a
frequency range lower than the one described in the previous point for
precise control of carriage position using collocated control.
3. Thermal Expansion of the Screw:
One of the most important limitations on using collocated control to
precisely position the carriage arises due to the thermal expansion of the
screw. Unless the screw is pre-stretched against stiff thrust bearings at
both ends, thermal expansion of the screw makes it difficult to accurately
82
De(s)
Dm~s)Wc(s)
Uc~Wj(S)
Uc.,(s)
+Ge(s)
++
F (
+T
Uc-(S)
Gp2(S)
Figure 5.4: Feedback control of uc(t)
determine the position of the carriage from the rotation of the motor.
Therefore, it is most common to use the carriage position for feedback
and we focus on this non-collocated feedback in the next section.
5.4
Non-collocated Transfer Function
In the previous section, we have seen that collocated control cannot achieve
precise control of the carriage position due to the drive resonance and the
carriage disturbances. Hence, we would like to directly close the feedback loop
on uc(t). The block diagram representation of the feedback control of uc(t) is
shown in Fig. 5.4, and the Bode plot of the non-collocated transfer function
Gp 2 (s) given by Eq. (5.7) is given in Fig. 5.5.
5.4.1
Bandwidth and Phase Margin
From Fig. 5.5, we see that the phase of the nominal non-collocated transfer
function (Gp2 (s)) tends to -360' as w -+ oo. We also see that significant loss
of phase above the resonant frequency makes it practically impossible to have
a crossover after resonance. This is because of the restrictions imposed by
the Bode gain-phase relation (Bode [7]), which would result in a significant
83
-20db/dec
-40db/dec
0)
ISrIlI
Vsr2 + S2
zI
Z2
-80db/dec
-90
/_11
--------- -
-180
-40db/dec
-270
-360
-- -~~~~ ---- --~~
---
-wi
Figure 5.5: Bode plot of the non-collocated transfer function Gp2 (s)
84
increase in the compensator gain to raise the phase at crossover, resulting
in noise amplification and saturation of actuators. In Sec. 5.6.2, we will see
that the maximum achievable bandwidth with certain robustness margins is
lower than the resonance frequency. We also note that the location of the
non-minimum phase zero appearing in Gp2 has no effect on the phase of the
transfer function. But, we will see in the next section that the location of this
zero plays a very important role in determining the sensitivity properties of
the system.
5.4.2
Sensitivity Properties
Let Sc(s) = (1+Lc(s))- 1 represent the sensitivity function of the non-collocated
system, where Lc(s) represents the non-collocated loop-transmission. We note
that Lc(s) has a non-minimum phase zero given by
z
012
=
+
/C2 + 4m 1 2 k
2m 1 2
(5.24)
(.4
We use the "maximum modulus theorem" from complex analysis (see for e.g.,
Churchill [12]), and the fact that Sc(z) = 1 to obtain the following result
||SCJK >_ISc(z)l = 1
(5.25)
where ISc Io is the N, norm of the sensitivity function given by
| Sc||0 = sup Sc(jOW)1
w>0
(5.26)
The result in Eq. (5.25), shows that we cannot uniformly attenuate disturbances over the entire frequency range. However, Eq. (5.25) gives us a qualitative notion of the trade-off in the sensitivity properties of the system due
to the presence of a non-minimum phase zero. We use the results obtained by
Freudenberg and Looze [21] (Theorem 4) to quantify this trade-off. Using their
results, we obtain the following bound for the infinity norm of the sensitivity
function as
|IScloo > (1/a) emiw(2rn
(5.27)
where a represents the desired level of sensitivity reduction given by
Sc(j)
< a <1
VW E Q
(5.28)
85
and W(x, Q) represents the weighted length of the interval where sensitivity
reduction is desired.
In summary, the above bound tells us that requiring the sensitivity to
be small throughout a frequency range extending into the region where the
non-minimum phase zero contributes a significant amount of phase lag implies
that there will, of necessity, exist a large peak in the sensitivity at higher
frequencies. But if the zero is located so that it contributes only a negligible
amount of phase lag at frequencies for which sensitivity reduction is desired,
then it does not impose a serious limitation upon the sensitivity properties of
the system. Hence, we require
z =
C12
+
C2 + 4m
12
___i_(5.29)__
2m
k
(5.29)
>>
12
where [0, wi] is the required frequency range for sensitivity reduction. The
above condition becomes a constraint in the inverse problem.
5.5
Significance of the Non-Minimum Phase Zero
We have seen in Chapter 4 that accounting for the distributed inertia of the
screw has resulted in the off-diagonal terms in the mass matrix. These offdiagonal terms have then resulted in the non-minimum phase zero which has a
potential to pose severe limitations on the performance of the system. To the
best of the author's knowledge, the effect of the distributed inertia of the screw
on the system performance via the limitations imposed by the non-minimum
phase zero has not appeared in literature. In this section, we will summarize
the effect of this zero and compare it with a situation where no such zeros are
present.
Consider the same system where the off-diagonal terms in the mass matrix
are zero. In such a case Eqs. (5.3) and (5.4) reduce to
M2s2 + C2s +
U(S)
Uc(s)
-
C12s
A(s)
k (Fm(s)+Dm(s)) C12
k (Fm(s) + Dm(s)) +mT
2
k D(s) (5.3 0)
+ 018 + k D,(s) (5.31)
A(s)
The Bode plot of the transfer function IUc(s)/Fm(s)I with Mi1 2 = 0 is shown
in the Fig. 5.6. On comparison with Fig. 5.5, we find that the presence of the
86
-20db/dec
-40db/dec
-80db/dec
|sI-
Isril
il-s2
+ s2
k/1
k/2
W
-60db/dec
-90
-180
-270
-360
__ _
- - - - - - - - -
Figure 5.6: Bode plot of Uc(s)/Fm(s) with Mi1 2 = 0
87
x10
08
0--
0.2
042
0,8
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
Figure 5.7: Root locus comparisons of Uc(s)/Fm(s) with (thick line) and without (thin
line) the non-minimum phase zero; zero frequency is much larger than the maximum
crossover frequency
x 10
[
-08
0.60.4
~
0.2
0
--.-.-C
-02
-0.4 -
-4000
3000
2000
-1000
0
1000
2000
3000
4000
Figure 5.8: Root locus comparisons of Uc(s)/Fm(s) with (thick line) and without (thin
line) the non-minimum phase zero; zero location closer to the imaginary axis
88
non-minimum phase zero lowers the phase after the resonance. In Figs. 5.7
and 5.8, we show a comparison of the root locus with and without the nonminimum phase zero. The Fig. 5.7 shows that when the non-minimum phase
zero is much larger than the maximum crossover frequency, both the systems
have similar behavior in the vicinity of crossover. However, when the zero
moves closer to the imaginary axis we can clearly see from the root locus plot
in the Fig. 5.8 that the maximum crossover frequency is reduced. Hence, it is
clear that the presence of the non-minimum phase zero can have a significant
effect on the sensitivity properties and the maximum closed-loop bandwidth
of the system (see Secs. 5.3 and 5.4).
5.6
Maximum Achievable Bandwidth with Certain Robustness Margins
In this section, we obtain conditions for the maximum achievable bandwidth
with a certain degree of robustness. We quantify robustness in terms of certain
parameters which have physical and geometric interpretation.
5.6.1
Robust Stability and Performance
The conditions for stability and performance robustness follow from the "small
gain theorem" (e.g., Dahleh et.al. [15]; Zhou et al. [54]). The geometric interpretations of stability and performance robustness are discussed in Appendix C. We also note that there exist standard algorithms and procedures
such as W.. control, y synthesis (Zhou et al. [54]), etc., for the design of robust
controllers for a given plant and its uncertainties. However, we are interested
in the design of the loop transmission, which involves the design of the plant
and the controller simultaneously. Hence, we try to obtain metrics for stability
and performance robustness so that these can be used as design parameters
in the inverse problem. In the following section, we derive conditions for the
maximum achievable bandwidth with a certain degree of robustness.
5.6.2
Robustness Metrics and Maximum Bandwidth
In this section, we determine the maximum possible crossover frequency with
a certain degree of robustness. In Fig. 5.9, we indicate five different types of
89
Type 2
A
A/
B2 .
B,
B
.......................
C
C
.
. .
.
Type.
Type4..
Type 5
W
Wc2
3
W,
bWc5
-100
-180
-200
-220
-240
-260
-280
-300
-320
-340
w
Figure 5.9: Bode plot of the non-collocated transfer function at different crossovers
90
Im(L(jw))
Unit Circle for
Crossover (2)
.
Unit Circle fo
Crossover (3)
--l i
-----
Re(I (jw))
Unit Circle for
Crossover (4)
Figure 5.10: Nyquist plot of the non-collocated transfer function for different crossover
frequencies
91
Im(L(s))
loop
transmission with
lead compensator
--
- -
loop transmission
with
proportional controller
Re(L(s))
unit circle
Figure 5.11: Instability by adding phase at crossover
crossover positions (1-5) on the Bode plot and their corresponding unit circle
positions on the Nyquist plot given in Fig. 5.10 of the nominal non-collocated
transfer function. It is clear from these plots that the crossovers of type 1
and 5 cannot represent the maximum crossover positions; in the case of the
former, we can increase the gain to achieve a higher crossover with reasonable
PM, and in the later the loop is unstable. Further, from the Bode plot of
Fig. 5.9, we clearly note that the type 5 crossover is not practical as the phase
drops to -360'. As a result, we need a phase lead of more than 1800 for stability and robustness, which can seldom be achieved by practical compensators
due to factors such as saturation and noise amplification (e.g., Spector and
Flashner [50]). Moreover, if one is successful at attaining such a crossover it
is often found that the design lacks robustness and is too sensitive to parameter changes to be of any use in practical applications (e.g., Truckenbrodt [52];
Cannon and Schmitz [10]).
92
Im(L(jR))
creasing
Ga
Re(L(jw))
Figure 5.12: Instability by adding phase at crossover: increasing crossover frequency
with constant phase addition leads to increased instability
93
The crossovers 2, 3, and 4 result in multiple crossings of the loop transmission. According to the Nyquist plot of Fig 5.10, all these crossovers are
stable for the nominal plant, with 4 being on the verge of instability. We note
that all these crossovers lack any decent margin of stability as the crossover
points (A', B', and C') are very close to having a phase of -1801. In the case
of a "well-damped" system, we notice that these crossovers could lead to a
potential instability as the phase would have already dropped below -180'.
However, we will see later that damping the resonant peak can lead to an increased crossover and robustness in the system. One method which is usually
employed in classical control is to add phase at the crossover to provide the
so-called phase margin PM. But, this can have a very detrimental effect on
our system, as adding phase at the crossover frequency comes only at the cost
of an increase in the magnitude due to the fundamental constraint imposed by
the Bode gain-phase relation (see Bode [7]). This results in rotating the circular portion of the Nyquist plot (the resonance region) to encircle the -1 point,
thereby resulting in closed-loop instability. This effect is very sensitive to the
changes in the damping of the resonant peak. As an example, in Fig. 5.11,
we see that by adding phase at B' using a lead compensator, the closed-loop
system becomes unstable.
We also notice that this effect is very sensitive to the changes in the
crossover frequency at B'; with the same PM, as we increase the crossover
frequency the degree of instability is increased as shown in Fig. 5.12. Hence, it
is clear that crossover 3 is potentially unstable. One could always argue that
a higher-order controller could alleviate this effect and help attain a stable
crossover 3. But this would require the controller to perform either of the
following tasks: add phase at crossover and drop it very rapidly in the vicinity
of resonance to avoid the resonance-circle in the Nyquist plot to rotate in the
anti-clockwise direction or, increase the phase from the first crossover point B'
to the third crossover point B 2 such that the resonance-circle does not encircle
-1 point. We note that the first possibility cannot be attained by controllers
which obey the Bode gain-phase relation and the second possibility results
in a controller with a very high-gain which can easily saturate the actuators.
Hence, we note that the maximum possible crossover frequency that can be
attained with a sufficient PM without rendering the system unstable is obtained by circumscribing the resonance peak in the Nyquist plot by the unit
circle, which is represented by the crossover 2 in the Bode plot of Fig. 5.9. In
Fig. 5.14, we see that a crossover of type 2 has resulted in a stable system with
94
Im(L(jw))
Unit Circle
Re(L2(jw))
Loop transmissi n with
with phase addit on
(using lead com ensator)
Original loop tra smission
Figure 5.13: Type 2 crossing results in a stable plant with phase addition
95
decent PM at crossover.
One can always argue that by crossing slightly below the resonance, one can
sacrifice PM to attain a higher crossover frequency. We understand that this
is possible, but must realize that our aim here is to establish rational measures
of robustness and performance which can then become design parameters in
our original inverse problem. Hence, in the above paragraph we have identified
a procedure to obtain the maximum-bandwidth with a given PM: it is the
frequency at which the unit circle which circumscribes the resonant peak intersects with the Nyquist plot of the nominal loop transmission. Alternatively,
it can be determined from the Bode plot by drawing a horizontal line through
the resonant peak, finding its intersection with the portion of the curve to the
left, and reading off the frequency. As we shall see later, the PM we obtain
using this procedure becomes a robustness parameter in a "small gain sense."
Robustness Measures and their Geometric Interpretation
In the previous section, we have seen that by providing a decent PM we have
accounted for the uncertainties in the vicinity of the crossover frequency. However, uncertainties in the plant parameters arising due to the changes in the
inertia, damping, and stiffness of the machine can lead to uncertainties in the
magnitude and the frequency of the axial resonance. The uncertainty associated with the magnitude of the resonance peak is usually higher because of our
inability to accurately model damping in a machine. In such a scenario, the
methodology presented in the previous section for finding the maximum bandwidth can result in detrimental effects; for example, if we had over-estimated
damping, or the resonance frequency, then choosing a crossover of the type 2
for the nominal plant can lead to a crossover of the type 3 in the actual plant,
which is unfavorable for the reasons described in the previous section. Therefore, we introduce a new robustness parameter, the "resonance gain margin"
(RGM), which can provide a margin for the uncertainties in the vicinity of the
resonance peak. The resonance gain margin (RGM) is defined as the factor
by which the loop transmission can be multiplied without resulting in multiple crossovers at resonance. When we implement this margin we obtain the
Nyquist diagram of Fig. 5.14. From Fig. 5.14, we can clearly see how RGM
and PM quantify robustness margin: they provide a region on the Nyquist
plot where the actual plant can lie without becoming unstable. This region is
bounded by the curve "D", which is the unit circle for Im(Lc(jw)) > 0 and a
96
m L(jw))
GMIL
W=
Robustness
Margin
Wr
Nominal Plant
PM
(L(jw))
2 sin
M/2)
unit circfr
D curve
W
W1
Figure 5.14: Geometrical interpretation of PM and RGM
97
circular arc with center at Le(jwc) and radius 2 sin(PM/2) for Im(Lc(jw)) < 0
. We shall see in the next section that the results we have obtained in this
section represent a limiting case of the small-gain theorem and there exist parallels between our results and the usual weighted-frequency approach of robust
control. The reader is referred to Zhou, Doyle and Francis [54] for a detailed
account on robust control.
Parallels with the Weighted-U ncertainty Approach
From Sec. C.1, we find that for stability robustness of a nominal plant GN(s)
with weighted multiplicative uncertainty A(s)W 2 (s), the family of circles of
radius IW2 (jw)Lc(jw)I with centers on the nominal loop transmission Lc(jw)
should never encircle the -1 point on the Nyquist plot. This statement is geometrically illustrated in Fig. C.3. From Fig. C.3 we also note that a curve
which is a common tangent to the family of these circles will also not encircle
the -1 point. The extent to which the design is conservative depends on how
"far" this common tangent is from the -1 point. We note that in a limiting
case, this common tangent passes through the -1 point. We see from Fig. 5.14
that PM and RGM have resulted in providing a robustness region bounded
by the curve D. Hence D forms the limiting tangent curve for stability robustness. Under such a description, we can layout a possible frequency dependent
weighting filter W 2 (s) which satisfies the following magnitude conditions:
RGM|LcI , W = Wr
|W2Lc + JLcj = 1 , w 2 < W < 00
=W2Lc
= 2 sin(PM/2) , w = we
1W2Lc= Rc(w) , w, < w < w,
|W
2
Lcj
=
(5.32)
(5.33)
(5.34)
(5.35)
where Rc(w) is the radius of the uncertainty circle for the frequency range
[wi, WC]. Using simple geometry arguments, we can obtain an expression for
R(w) (see Fig. 5.15) as
R(w) = 2 sin(PM/2) - [1 + ILc(jw)
2
- 21Lc(jw)I cos(PM + ZLc(jw))]1/2
(5.36)
Performance Robustness
As stated in Appendix C, the condition for robust performance is that the
two families of circles resulting from robust stability and bounded sensitivity
98
Im(L(jw))
W= W
PM
Re(L(w)
p transmission
2 sin(PM/2)
W =W
ILc(iw)I
(W)
D curve
Figure 5.15: Radius of the uncertainty circles for wi < w < w,
be disjoint. The reader is referred to Doyle et al. [16] for a detailed proof on
robust performance. When we apply this condition to our problem we obtain
the following conditions:
|W(w)l < 1 + LN(Jw)I ,
Wi(jw)| + R(w) < 1 + LN (w)
1
(5.37)
, W1 <W < W,
(5.38)
<
where W 1 (yw) is the frequency weighting on the system sensitivity function.
In summary, the above conditions tell us that we attain robust performance if
we are able to meet the required sensitivity criteria posed by W (jw) and at
the same time meet the given PM and RGM requirements. These conditions
then become constraints in the inverse problem.
In this section, we have quantified robustness in terms of two important
parameters: PM and RGM. We have seen that our idea of robustness essentially stems from the small gain theorem. We want the reader to note that
there is an essential difference in our approach and the standard robust control
techniques such as 7t, and p synthesis: we are solving the inverse problem
of designing the loop transmission for high dynamic performance rather than
99
designing a controller for a given plant and uncertainties. To this effect, we
have used stability-robustness to establish robustness metrics such as PM and
RGM which could be used as design parameters in the inverse problem and
have used the inequalities (Eqs. (5.37) and (5.38)) resulting from performance
robustness to place constraints on the inverse problem.
5.7
Bandwidth Formulae
In this section, we will use the results of previous section to obtain closed-form
expressions for the maximum closed-loop bandwidth in terms of the plant and
robustness parameters. These formulae will give us the means to solve the
inverse problem as discussed in Chapter 7. We perform this exercise for two
compensators to illustrate the procedure, noting that this can be extended to
any other controller structure.
5.7.1
Closed-loop Bandwidth with a Lead Compensator
The loop transfer function for the non-collocated case with a lead compensator
is given by K,
T,+'Gp2
(refer to Eq. (5.7) for Gp 2 (s)).
We know that the
maximum phase for a lead compensator occurs at the geometric mean of the
zero and the pole frequency (refer Appendix B). To achieve the maximum PM
for a given a, we impose the constraint of making the crossover frequency equal
to the frequency where the maximum phase occurs for the lead compensator.
Hence, we set the crossover frequency w, as
C
I
(5.39)
The representative Bode plot for this case is given in the Fig. 5.16. We use
the robustness conditions derived in Sec. 5.6.2 to parameterize bandwidth in
terms of the system parameters and robustness margins. From the discussion
on second-order systems given in Sec. B.1.4, we obtain the following result for
100
dbC in Fig. 5.16
dbc
=
RGM
dbE +20 log
(5.40)
1GM
Vfr2 + 82
dbE
s 2 +s
+20logaT
dbc
Sr2 + s - 20 lg
dbD - 20log
20log T
-
sr 2 +s+
(5.41)
(5.42)
db,
From the above equations, we obtain the maximum crossover frequency with
the required PM and RGM as
Wc
2r
2
(5.43)
SCT
RGM
Using Eqs. (5.39) and (5.43), we solve for a as
(RGM
a=
_____
)2
2
(5.44)
In the above analysis, we have assumed that
T
<
s
2
+ sC
(5.45)
We note that we have ruled out the other possibility of placing the lead pole
after the resonance because it leads to a lower bandwidth. In order to eliminate
the parameter T and obtain an expression for the closed-loop bandwidth we
use the phase condition. Since we have chosen a lead compensator with the
maximum PM we obtain the following general expression for the phase at the
crossover frequency
Tr
c
--
- arctan
We
+ arctan
l2y'
a - 1 - r n-21 sr2|IWc
W(5.46)
2
- arctan
(s 2 + s) - w 2
We simplify the above expression, noting the following:
1. For a first-order system the phase is nearly -7r/2 a decade above the
break point (refer to Sec. B.1.3). Hence, the contribution of the phase
from the first-order pole (s = s,1) is approximately -7r/2 at crossover.
102
-20db/deC
dbD
-40db/dec
-20db/dec
dbc
20 log R GM
7:1
dbE
100
-90-
IsrI
-----
1
aT
1
T
2
Sr2
+ s2- C
W
------------
P
-180
-270
-
-
-360
W
Figure 5.16: Bode plot with a lead compensator
101
. ........ .. . .
. . . . . . . . . . . . .... ............ ....
Loweing of
resonance peak
Extra pole
before resonance
Figure 5.17: Lowering of the resonance peak by adding an extra pole before resonance
possible PM for a given a, we impose the constraint of making the crossover
frequency equal to 1/V'aT. Hence, we set the crossover frequency as
1
(5.51)
g=
A representative Bode plot for this system is shown in Fig. 5.19. We use
the robustness conditions derived in Sec. 5.6.2 to parameterize the closed-loop
bandwidth in terms of the system parameters and robustness margins. This
is obtained as follows
dbc
=
RGM
1GM
dbE+ 20 log
2
dbE
=
D-
20log
+2O1ogoT
dbc = dbW
104
82 +
r2
s
(5.52)
2C
-; - 20log
s 2 -40logT
s 2+
(5.53)
(5.54)
2. For a second-order system with light damping the phase drops very
sharply from 0" to -180" at resonance. Hence, for a lightly damped system, we can assume that the phase contribution from the second-order
pole is approximately zero at the crossover frequency.
Hence, we can simplify the expression for
#,
qc ~ -7 + arctan
as
S- 1
(5.47)
2v/a
which gives
I - sin
-
a
#c
1 + sin qc
for -7r<
c
< 0
(5.48)
and using Eq. (5.43), we get
RGM )2 (1+ Sin 0
T
21s,2 1sc
sin 0,
1 -
1/
11/
(5.49)
By substituting the above expressions for a and T in Eq. (5.39), and noting that PM = r - 10c|, we get the following expression for the closed-loop
bandwidth Wb in terms of PM and RGM as
eWWbW
5.7.2
- 1/4
1sn
|sc\2( 1 -sin PM 1/
1+sinPM)J
L_-RGM
-(2
[2|sr- 2
(5.50)
Closed-Loop Bandwidth with a Lead Compensator and an
Additional Pole
In the previous section we have seen that a higher bandwidth can be achieved
by placing a lead pole before the resonance. When a pole is added before
resonance, we can lower the resonant peak (as shown in Fig. 5.17 and Fig. 5.18),
and thereby obtain a higher crossover frequency. In this section, we derive the
bandwidth formula when the controller consists of a lead compensator with
an additional pole placed just before resonance.
The loop transfer function for the non-collocated case with a lead compensator and an extra pole is given by Kp ,+
1
Gp2 (refer Eq. (5.7) for Gp 2 (s)).
We note that the maximum phase for this compensator occurs at the same
location as that of the lead compensator. Hence, to achieve the maximum
103
-20db/dec
dbD
-40db/dec
lowering of
resonance peak
-o
increase ixt
bandwidth
100
Isri
T1C
aT
c
s22 + s2
LAc
-lO0db/de
-80db/dec
i
-90
-
-
-
-180
i
-270
..
.
.
.
-360
--
- - ----------
-450
Lead compensator with an extra pole
-------------
Typical Lead compensator
Figure 5.18: Increase in bandwidth by adding an extra pole before resonance
105
dbD
-20db/dec
-
-40db/dec
-20db/dec
..... . - - - - - - - - - - - - - - .. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
dbc
20 log RGM
-- - -- - -- - -- - -- - -- - -- - -- - -- --.-
dbE
10 0
-90-
|SrI!
--.
1
1
aT
T
2
/-S
+
sr2
Sw
- -- -- -- --- - -- -- - P
-180
-270
--
------ - - - - -
-360
-450
ON
Figure 5.19: Bode plot of the loop transfer function with a lead compensator and an
additional pole
106
from which we get the maximum crossover frequency as
2T 2
2T
|sr2 |sc js
2±+si
RGM
RGM
(5.55)
Using Eqs. (5.51) and (5.55), we write
2TJ|sr2|Isc
(5.56)
G=
\Qsr2
+ si2
We now use the phase condition to eliminate T. The phase at the crossover
frequency w, is given by
?r
c=
--
- arctan
We
2IsrI
+ arctan
a - 1121sr2|Wc
2V&
-
arctan
-
arctan
(s2 +
sc) - W2
(5.57)
We simplify the above expression, noting the following:
1. For a first order system the phase is nearly -7/2 a decade above the
break point (refer to Sec. B.1.3). Hence the contribution of the phase
from the first-order pole (s = Sri) is approximately -- r/2 at crossover.
2. For a second-order system with light damping the phase drops very
sharply from 0 to -180' at resonance. Hence for a lightly damped system
we can assume approximately zero phase contribution from the secondorder pole at crossover.
Based on the above points, we can approximate 0, to be
c
+ arctan
1
a -i
arctan
(5.58)
1 - 6ce~ + 9ce-2
tan2tan
qo C
c =9a- 1 - 6a- 2 + a-3
(-9
(5.99)
~-r
-
which gives
From Eq. 5.58, we find that in order to have a phase margin of at least 301,
we need a to be around 10 (see Fig. B.7). Hence, we neglect lower powers of
a in the above expression. This results in
a ~~6 + 9 tan2 #c
(5.60)
107
We use Eq. (5.55) and PM = 7r - 1#c to obtain the following expression for
closed-loop bandwidth Wb
T
=
1 ,(8,2
(21s,2|s
Wb
5.8
Wc
+ s2) (6
+ 9 tan2o,
2|s,2|18
s 2 +
(RGM(6 + 9 tan2 PM)
(5.61)
(5.62)
Chapter Summary
We started this chapter by obtaining closed-form expressions for the open-loop
poles of the system using a perturbation approach. We then examined the limitations on collocated and non-collocated control from the plant dynamics. It
is worthwhile to mention here that our approach of starting from the equations
of motion to establish the system's response to the various inputs has been able
to capture all the significant effects in contrast to the one adopted by Chen
and Tlusty [11] and Smith [48]. We can clearly notice from the Bode plot of
the collocated transfer function (Fig. 5.2) that the axial dynamics of the screw
forms an integral part of the control-loop and cannot be placed outside the
loop as mentioned in [11] and [48]. If we use the approach mentioned in [11]
and [48], we cannot capture the modal spillover effects and sensitivity properties of the system. Finally, we were able to derive closed-form expressions
for the closed-loop bandwidth in terms of the system parameters and certain
robustness measures, thus providing a link between mechanical and controller
design.
108
CHAPTER
6
Model Parameter Reduction
Mechanical Design
6.1
Impact of
Introduction
Our aim throughout the thesis has been to develop tools for solving the inverse problem. To this effect, we have derived a dynamic model for the ballscrew drive (see Chapter 4) and have expressed performance and robustness
in terms of the machine parameters. But, the model developed in Chapter 4
which we refer to as the "full" model contains too many free parameters to
be used directly at the initial design stages for sizing of the screw and the
motor. Hence, we begin this chapter by simplifying the full model to obtain a
reduced-parameter model. This can be achieved by noting that the ball-screw
can usually be made the dominant compliance in the stiffness loop by careful design; i.e., one can usually select thrust bearings and torsional couplings
that are several times stiffer than the screw itself without incurring inertia or
packaging problems. This condition is given by
n 2 ic, kn, n 2n,
k>
EA/xc, EA/(L - xc), GJ/xe, kv
(6.1)
Hence, a further simplification of the model (in terms of the number of
design parameters) is possible, and we can express the elements of the mass,
109
damping, and stiffness matrices in terms of the length of travel, lead and
diameter of the screw, inertia of the motor and so on. As we shall see in
Sec. 6.2, such careful design not only simplifies the model but also results in
an increased dynamic performance. Next, we discuss some of the important
mechanical design details which can enhance the dynamic performance and
help attain a robust design. Finally in Sec. 6.3, we describe the identification
experiments conducted on the test stand to validate the theoretical model.
From these experiments, we find a close agreement between the measured and
predicted results.
6.1.1
Reduced-Parameter Model
In this section, we obtain reduced-parameter models for the ball-screw drive
under the free-end and damped-end boundary conditions.
Ball-Screw Servo with a Free End
Using the Eq. (6.1), we can simplify the expression for total stiffness given by
Eq. (4.52) as
-C
kt =
EA
+2
+
X
27r
GJ
-1
(6.2)
Hence, when all the elements in the structural loop are stiffer than the screw,
the total stiffness is the serial combination of the longitudinal and torsional
stiffnesses of the screw. As a result, the expressions for the elements of the
110
mass and stiffness matrices of Eqs. (4.62) and (4.67) simplify to
n 2 J1 + m
M(
J
m
n2J12 -
3
=Jm +
n2J12
E
F(
3
In2
n2GJxe
pAxc[
=
(
2
21
E Alxe
21
(6.4)
(
C
EAxc)]
2+ pJ(L -
(6.5)
(6.6)
xc)
(n2G lxe
21
+ pA(L -x,)
EA/xc)
C
)
k2
I(GJlxe) (EAlxc)
PJXCL
J2=
r(Ec
+e(k
EAlxe
+ pJ(L - x,)
J
(6.3)
m)
E Akx,
+
En2GJxc(2
_pJxc
m
n 2j2 + mn +
+ pJ(L - xc)
J12
-
;K
(
EAlx)
kt
-kt
(6.7)
(6.8)
Ball-Screw Servo with a Damped End
Under the conditions of Eq. (6.1), the expressions for kj, k2 , and kt for the
damped end given in Sec. 4.4.2 simplify to
k EA
xc
kt=
k 2 =[L-
_E A
F (
1
k1 + k2 +
+
(6.9)
)2]
2x
27r
I
kv
GJ
(6.10)
111
Using the above expressions, we simplify Eqs. (4.89) and (4.94) to obtain the
following expressions for the mass, damping and stiffness matrices
(
M
=
2
- m
n2J12
n2 J2 + m +
J, + m
n 2 J12 - m
Jm + PJj1+
1
(
+ pJ(L - x,)
J12
pJxe
kt
3
_n2 GJ/xe
l
(6.12)
tG
/x2
-
p2/ xc
/x)e
(n2 Glx
pAxe
k
3
_k1 +k2)
)
(6.13)
[x
2
+ pJ(L - xc)
kt
7.TJ/X,
(n
2]
(6.14)
2
k
k1+ k2
+ k2
kv
2
-C
-C
/x
kt
2 -n2GJ/xe
3
(2C+C
2G
3
pA(L - x )
3
C C=
/xc +
2
+ pJ(L - x±)
J2
(6.11)
Jc
'
CC+C)
K=
k 2
k2
-k
(6.15)
-k
k )
(6.16)
where C is obtained from Eq. (4.118). Under the conditions of Eq. (6.1), the
expressions for k and r/ simplify to the following
k(1 + ji)
=
(EA
[L
+
- xe +
EA
I
kv(1 +jrjv)
1
+(
§2
2r
X]
-
.G1J
(6.17)
Using the expression for the non-collocated transfer function given in Eq. (5.7)
and the above simplifications for the mass, damping, and stiffness matrices,
we obtain the results given in Table. 6.1 for the axial resonance.
112
free end
damped end
frequency (Hz)
360
400
damping ratio
0.09
Table 6.1: Predicted axial resonant frequency and damping ratio using the reducedparameter model.
Figure 6.1: Photo of the old test set-up
6.2
Mechanical Design
In Chapters 4 and 5, we have found that the frequency range of motion control
for the ball-screw drive is limited by the first axial mode of the system. We have
also found in Sec. 4.4 that the total axial stiffness of the system is the serial
combination of various elements in the structural loop (refer to Eqs (4.52) and
(4.80)). Hence, it is clear that for a given ball-screw maximum stiffness of the
structural loop and therefore higher performance can be obtained by making
the screw the dominant compliance in the loop. This is usually possible by
careful mechanical design such that the condition of Eq. (6.1) is met. Many
machines are usually made up of complex sub-assemblies, and different kinds
113
-5
.
-10-
3
I
axial mode
bending mode
of bearing block
-20-25-
gi
twisting mode
of bearing block
-30
102
10
10 2
10
0
-50 -100 -
U)
.C -150 --
-2001'
freq(Hz)
Figure 6.2: Measured collocated transfer function for the old set-up
114
10-
I
I
I
I
I
I
I
300
350
axial mode
10_2
bending mode
of bearing block
C 0
0)
o
.j 10
twisting mode
of bearing block
-4
10~
10-6
0
50
100
150
200
250
400
Frequency (Hz)
Figure 6.3: Representative modal transfer function of the old set-up
115
Bearing Block
C ai i age
Base
/
z
Figure 6.4: Measured bending mode of the bearing block in the old set-up (191 Hz).
Figure shows snapshots of the mode starting from the undeformed position.
116
Bearing Block
Carnaie
Base
1z~7
Figure 6.5: Measured twisting mode of the bearing block in the old set-up (214 Hz).
Figure shows snapshots of the mode starting from the undeformed position.
117
Bearing Block
Base
Carriage
I Z7
Figure 6.6: Measured axial mode of the old set-up (260 Hz).
118
Bearing Block
Carriage
Base
Figure 6.7: Measured yaw mode of the carriage in the old set-up (375 Hz).
119
of joints are used to bring these sub-assemblies together. Hence, the design of
joints and contact interfaces is as important as the individual sub-assemblies
themselves for proper functioning of the whole machine. Therefore, by careful
mechanical design, we are not only emphasizing on the design of the individual
components such as the bearing housings, machine base etc., but also on the
design of joints, contact interfaces, sensor mounts and so on.
In this context, we present an example of a machine (see Fig. 6.1) in which
bad joints and compliant bearing blocks result in a significant reduction in
the closed-loop bandwidth. In Fig 6.2, we present the measured collocated
transfer function which is obtained from the sine-sweep experiments (refer to
Sec. 6.3.2 for the sine-sweep experimental procedure). We see from Fig. 6.2
that there are three resonant peaks in this transfer function in the place of a
single peak corresponding to the predicted axial mode of the system. We also
note from Table 6.1 that the predicted natural frequency of the axial resonance
is higher than the ones corresponding to the resonant peaks of Fig. 6.2 by at
least 100 Hz. Other results obtained from tuning the controller for maximum
performance show a 50% lower bandwidth than the predicted values.
In order to better understand the dynamics of the machine and this deviation from the theoretical model, we performed modal experiments (see
Sec. 6.3.1 for a review on experimental modal analysis) on the test stand.
A shaker was used to excite the machine and an accelerometer to measure
the response. In Fig. 6.3, we show a representative transfer function from the
accelerometer to the shaker. From Fig. 6.3, we find that the frequencies of the
resonant peaks in the modal experiment match with those from the sine-sweep
experiment of Fig. 6.2 as expected. After processing the modal data, we found
that these modes correspond to modes in which there is a significant motion
of the bearing block relative to the machine base. We sketch the snapshots
of these modes in Figs. 6.4 and 6.5 which show respectively, the bending and
twisting of the bearing blocks. A closer examination of the shapes also reveals
a small movement at the bearing block and machine base interface indicating
poor contact stiffness. The next mode at 260 Hz is found to be the axial mode
of the system as shown in Fig. 6.6. From these results, we conclude that this
difference of about 100 Hz between the predicted and the measured values for
axial resonance along with the appearance of the extra bearing block modes
is the cause for the poor performance. We found that the other higher modes
correspond to the carriage or the machine base which are not projected onto
the axial motion. An example of this is the yaw mode of the carriage which
120
Ball-screw average diameter
Ball-screw lead
Ball-screw length between bearing supports
Ball-screw material
Ball-nut stiffness
Bearing stiffness(2)
Torsional stiffness of coupling
25.4 mm
25.4 mm
1100 mm
steel
3000 N/pm
1500 N/pm
6800 Nm
Motor inertia
13.9
Carriage Mass
80 kg
x10 5
kgm 2
Table 6.2: Important parameters of the ball-screw drive shown in Fig. 6.8
occurs at 375 Hz and is shown in Fig. 6.7. The modes which occur before the
bending mode of the bearing block are all rigid-body modes of the whole machine on its mounts. These modes do not drop phase and hence do not affect
performance. We can now see the reason behind the appearance of three resonances in the sine-sweep measurements. From the results of this experiments,
we can see that several components need to be carefully designed: the bearing
blocks need to be made stiffer, the joints have to reinforced to improve contact
stiffness and so on. For the remaining of the chapter, we will refer to this bad
design of Fig. 6.1 as the "old" set-up.
We have redesigned several of the components in the old design of Fig. 6.1 to
improve the dynamics of the machine and achieve a reasonable correspondence
between the real machine and the reduced-parameter model of Sec. 6.1.1, and
hence, in this process we have strived to meet Eq. (6.1) in our new design.
A photo of this redesigned machine which we will refer to as "new" set-up
for the remaining of the chapter is shown in Fig. 6.8. An exploded view of
the assembly of the machine is shown in Fig. 6.9. Modal test results show
that all the undesired modes which appeared in the old design (Fig. 6.1) are
eliminated (see Fig. 6.10). The axial mode is found to occur at 349 Hz and is
in close agreement with the predicted value (see Table. 6.1). Hence through
careful design, we are able to eliminate all the destabilizing modes which lower
performance and obtain a very good machine-model match (see Sec. 6.3 for
other evidence). In the following sections, we describe some of the important
design details which helped us improve the dynamics of the stage significantly.
Some of the key parameters of the new design are listed in Table 6.2.
121
machine base
Figure 6.8: Photo of the new test stand
122
Ball-Screw
Encoder Shield
Encoder
Mount
"|
Carriage
Linear Bearing
'a
0 Is--
g
-oo
.
.Linear
-'Encoder
Trucks
.Damped
0.
0
0
Bearing Block
--
*
-
Linear Guide
Machine
Base
M
Bearing Block
Coupling
-
Motor
Figure 6.9: Exploded view of the machine assembly
123
100
10--
axial mode
0
23
-j 10-
10-4
-10-
0
50
100
150
200
250
300
350
400
Frequency (Hz)
Figure 6.10: Representative modal transfer function of the new set-up
124
6.2.1
Design of Joints and Contact Interfaces
A machine is formed by assembling several parts. In this process, we always
encounter joints and contact interfaces. If joints are not designed properly,
they become dominant compliances in the stiffness loop. Hence, bad joints
can easily lead to poor performance in spite of well-designed individual components. In this section, we will layout some guidelines and precautions one
must take while assembling parts and forming joints. We will be using these
guidelines in the assembly of many components like linear guides, carriage,
bearing blocks, and so on. For a more detailed discussion on contact mechanics and joint design, the reader is referred to Johnson [25] and Slocum [47].
When two surface are mated to form a joint there are many variables
that can affect the performance of a joint. Hence joint design is one of the
most difficult aspects of machine design. The stiffness of a joint depends to a
great extent on the micro asperities, surface roughness and the preload force
holding the joint. Hence, depending on the stiffness of the joint, the process
of manufacturing the contact surface can vary from lapping to milling. In
the new design, we have ground all the mating interfaces to improve their
contact stiffness. In order to improve the contact stiffness of a joint it is
important to perform the so-called "cleaning and stoning" operations on the
mating interfaces before forming a joint. We describe these operations below:
Cleaning and Stoning
To begin, we start by cleaning the mating surfaces with a degreaser. A small
amount of degreaser is sprayed onto a scratch-free tissue (not paper towels)
and the mating surfaces are wiped. It is useful to put on a new pair of gloves
to avoid skin-contact with the mating surfaces. The presence of burr or any
other micro projection on the mating surfaces results in a great reduction
in the contact stiffness of the interface. Hence, we use a precision stone for
removing these asperities on the mating surfaces. The process of stoning the
surfaces before mating them helps remove burr and micro projections from the
surfaces and thereby improves the contact stiffness of the interface.
Stoning Process
We start the stoning process by cleaning the precision stones. We then use
the coarse side of the stone during the initial phase, and move it in a circular
125
450 compression cone
MEX
11-
-
111111KI,
Ir1111
450 compression cone
Figure 6.11: Schematic of a bolted-joint configuration
fashion on the mating surface. During this process, care has to be taken not
to push the stone down (the normal force exerted on the surface should come
from the weight of the moving stone alone). We continue with this process
until we feel that the stone is gilding smoothly over the surface. We then
clean the surfaces with alcohol and scratch-free tissues. In the next step, we
repeat the above process with the finer side of the stone. Once we achieve the
required "smoothness", we stop and again clean the surfaces with alcohol.
Bolted Joints
We have used bolts and screws for forming almost all the joints in the machine
assembly. While designing bolted joints, one should always concentrate on the
required preload and compression-zone cones. In Fig. 6.11, we show a bolted
joint configuration.
The compression zone can be approximated as a cone
with cone angle of 450 for stiffness calculations (e.g., Shigley [46]). In order
126
rrrn
...................
IT
/X/
..
EM
. I
I
I
....
L
L
J6
........
NN ......
stress discontinuity
Figure 6.12: Stress discontinuities due to the absence of the stress-cone interference
to ensure that we achieve high joint stiffness, we should let the compression
cones interfere at the interface. In the absence of such interference there
will be stress discontinuities as shown in Fig. 6.12 along the interface and
reduction of contact stiffness. This interference of stress cones has many other
advantages: it can be very helpful in minimizing straightness errors caused by
bolt tightening, and is very useful to reduce bearing noise and enhance bearing
life when the stress cones of the bolts used for fastening the bearing push-up
plate for preloading purposes interfere. Therefore, in our old design where we
have used push-up plates for preloading the outer race of the bearings, we
have used twelve bolts spaced evenly along a circle to let the pressure cones
interfere (see Fig. 6.17). However, making the stress cones interfere may result
in the use of a plethora of bolts. But if the contact is in the load path of the
machine we are not left with much of a choice. Therefore, in all of the joints
of the new design (e.g. ball-nut holder to carriage, bearing block to base etc.)
we have strived to achieve the interference of the pressure cones (see drawings
of parts in Appendix D).
127
viscoelastic inserts
viscoelastic inserts
Figure 6.13: Photograph of the cross section of base showing viscoelastic inserts
6.2.2
Design of the Machine Base and Linear Rail Assembly
The engineering drawings of the machine base are given in Appendix D. In
the following, we list some of the important features of the machine base.
In order to achieve the required performance specifications, it is important
to have the modes of the machine base to be of a higher frequency and not
to project onto the axial mode resulting from the compliance of the screw.
Hence our aim for the machine base is to have favorable dynamics and be cost
effective. Therefore, we have used constrained-layer damping (see for e.g.,
Kerwin [26]; Ross et al. [42]; Mead and Markus [30]; Torvik [51]). In the next
paragraph, we give a brief account of the design and manufacturing of the
constrained-layer damped machine base. The reader is referred to Nayfeh [36]
for a more detailed description of the design of damped machine elements.
Other references of interest are Ruzicka [43] and Marsh and Slocum [29] which
deal with simplified analyses for the design along with fabrication methods for
internal shear dampers.
The machine base is manufactured by welding steel tubes and bars as
shown in the drawings of the base weldment given in Figs. D.1 and D.2, and
128
the assembly schematic of Fig. 6.9. The welded structure is stress relieved
before any other operation is performed . We then insert viscoelastic-covered
steel rods into the tubes and fill the gaps with a replicating epoxy as shown in
the photograph of Fig. 6.13. The surfaces required to mount the linear guides
and bearings are formed by machining the top bars (see Figs. D.3 and D.4) to a
flatness, co-location and parallelism tolerance of 0.0005". The presence of the
viscoelastic inserts is not only helpful for the machine performance but also to
meet the required tolerance levels during the initial machining process, as the
inserts help damp the resulting vibration. In fact without the shear dampers
in place, it would be difficult to meet the tight tolerance specifications given
in Figs. D.3 and D.4. The supports for the machine base are placed at 0.22Lb
from each end (see Figs. D.3 and D.4), where Lb is the length of the machine
base due to the following reasons:
1. The nodal points of the first mode of a beam of length Lb occur at
0.22Lb from each end under free-free boundary condition. Hence, placing
supports at those points will make the first bending mode of the base
to conform with a free-free boundary condition, thereby resulting in a
highest possible resonance frequency.
2. The sag of the machine bed under its own weight, the so-called gravity
sag is minimized when supported are placed at 0.22Lb from each end
(e.g., Loewen [28]). These points are also known as the airy points.
6.2.3
Linear Bearings and Assembling Techniques
The linear bearings used in the machine are manufactured by Schneeberger
Inc. [45]. In the following paragraphs, we outline a procedure for installing
these rails to achieve best performance.
1. Cleaning the rails and ground surfaces of the machine base
We begin by cleaning the rails and corresponding mounting surfaces on
the machine base with a degreaser such as isopropyl alcohol. During
this process it is recommended to put on a new pair of surgical gloves
to avoid skin contact with the mounting surfaces on the base (this helps
prevent corrosion of the interface). Next, we place the rail such that the
holes line up, and mark a nearby non-precision surface for later use. We
then take the rail off the base.
129
2. Preparing bolts and cleaning threaded holes :
In the second step, we clean by blowing and vacuuming all the degreasing
fluid and particles from the threaded holes in the base which are to
receive rail bolts. We then proceed to prepare the bolts. We smear
a small amount of grease on the first four threads of the bolt. It is
extremely important to grease the bolts as otherwise they can potentially
fail in torsion. It is also important not to put too much of grease on the
bolt because this can lead to it oozing out around the bolt when it is
screwed in. Any grease which oozes into the rail base interface during
assembly will defeat all of the cleaning and stoning steps that follow.
Hence, in order to get a feel for the amount of grease to be smeared on
the bolts, take one of the bolts needed for fastening the rail to the bed,
smear a small amount of grease on the bolt threads, screw the bolt into
a hole in the bed and check to make sure that no fluid or grease oozes
out around bolt when it is screwed in.
3. Cleaning and Stoning:
We perform the cleaning and stoning operations on the mating surfaces
as described in Sec. 6.2.1 before mounting the rails on the base. It is
recommended not to spray the degreaser (during the cleaning process)
directly on the mating surfaces as there is a danger of the degreaser to
get into the bearing trucks. By performing these operations we improve
the contact stiffness of the rail-base interface as described in Sec. 6.2.1.
4. Assembling the Rails:
When the cleaning steps mentioned above are complete, we proceed to
mount the rails on the bed. We first apply a very thin layer of machine
oil to prevent the surfaces from getting corroded. We identify the marked
vertical edge (reference edge) on the rail and seat the rail in the groove
with the marked edge against the vertical reference edge on the groove.
During the installation process, one should always be careful to avoid
skin contact with the mating surfaces to reduce the risk of corrosion.
When the rail is in place, we slide it forward and backward by about
2 mm to ensure that it feels smooth. We then center the rail over the
holes, insert bolts and screw them in to within 1mm of seating. Next,
we push the rails against the vertical reference edge by means of gibs or
C-clamps sequentially from one end of the rail. In the next step, we start
torquing down the rail bolts sequentially from one end to the other. The
130
bolts (here M6) are torqued to 60% of proof strength which corresponds
to a torque of 16 Nm in three passes.
It is very important to align (in terms of parallelism and co-location) the
second rail with respect to the first rail. This can be performed in several
different ways (e.g., Schneeberger Manual [45]). In our case, we use the
carriage to align the second rail with respect to the first. We perform
all the cleaning operations on the second rail and the corresponding
machine surface as described earlier. We center the second rail with
respect to the bolt holes on the base, insert and screw in the bolts to
within 1 mm of seating. We then assemble the carriage onto the trucks
of the linear bearings. Before this is done the contact surfaces (on the
carriage and the bearings) are cleaned and stoned in a fashion similar to
the one described above. We then bolt the carriage onto the bearings on
the reference rail at this stage. During this process the vertical reference
edge on the bearing trucks is pushed against the reference edge on the
carriage. The next step is to bolt the bearing trucks of the second rail
to the carriage. The bolts on the second linear guide are then bolted
sequentially as the carriage is moved from one end to the other.
6.2.4
Design of the Carriage
The engineering drawings of the carriage are shown in Figs. D.5 and D.6. The
key features in the carriage design are
1. Resonant modes of the carriage are much higher than the axial mode.
2. The face on which the ball-nut support and bearing surfaces mount are
ground to ensure high contact stiffness of the interface.
6.2.5
Design of the Bearing Blocks
The drawings of bearing blocks are shown in Figs. D.9- D.12 in Appendix D.
The stiffness of the bearing block and the contact stiffness of the joint between
the bearing block and the machine base should be made high as they are in the
stiffness loop. The key features in the bearing block design are listed below
1. Structural Design : The bearing blocks are designed to have high
stiffness in order to prevent the bending and twisting modes as seen in the
131
Figure 6.14: Photo of new bearing block
old design (see Figs. 6.4 and 6.5). In Figs. 6.14 and 6.17, we show photos
of the new and old bearing blocks. The old bearing block was made of
aluminum with three bolts on either side for mounting it onto the base.
The overhang of the bearing block along with the lack of required stiffness
(both structural and joint) has led to a very poor dynamic performance
as described in the very beginning of this section (see Figs. 6.1-6.6). In
order to eliminate the bending and twisting of the bearing blocks we
redesigned them to increase structural and contact stiffness. In the new
design the bearing block is made of steel with almost no overhang and
the maximum possible number of bolts are used for mounting purposes.
The contact surfaces are ground, and the cleaning and stoning processes
described in Sec. 6.2.1 are performed to ensure high contact stiffness.
2. Precision Details : In the drawings of the bearing blocks shown in
Figs. D.9-D.12 of Appendix D, we see that the dimensions related to
the location of the bearing bores are very precise. The reason for this
is to ensure very good alignment of the bores of the two bearing blocks
with the ball nut (note that the location of the bore on the ball-nut
132
Figure 6.15: Photo of 600 angular-contact bearings preloaded in a back-to-back fashion
holder is also very precise). This is extremely important to reduce the
misalignment forces on the nut and bearings.
6.2.6
Choice of the Thrust Bearings
The ball-screw is supported at either end by four NSK sixty-degree angular
contact bearings in a back-to-back fashion. The back-to-back configuration is
used to provide high moment stiffness and thermal stability. A photograph
of the bearings mounted on the screw and preloaded with an internal locknut
is shown in Fig. 6.15. In order to obtain a deterministic preload the external
rings are preloaded with an external locknut which can be torqued using a
torque wrench to a given preload (see Fig. 6.16). This proves to be a great
improvement over the older design where the bearings are preloaded using a
push-up plate (Fig. 6.17). In the old design the preload is very sensitive to the
length-wise tolerance of the bearing bore and hence is very uncertain.
In the following, we show by means of an order-of-magnitude analysis that
when the correct preload is applied, the stiffness of the angular contact thrust
bearings is indeed magnitudes higher than that of the screw. Hence by using
133
Figure 6.16: Photo of bearings preloaded using an external locknut
Figure 6.17: Photo of bearings preloaded using a push-up plate
134
the preloading technique suggested above we can safely neglect the compliance
of the bearings with respect to that of the screw in the stiffness calculations.
The axial stiffness of angular-contact bearings (based on Hertz theory) is given
by
kb = 1.5E/ 3 Z 2 / 3 D1 /3 (sin(a))5/ 3 F1/ 3
(6.18)
where Z is the number of balls, D is the diameter of the ball, a is the contact
angle, and Fp is the preload force. We need
stiffness of bearing > stiffness of screw
(6.19)
Using Eq. (6.18), we recast the above condition in terms of the basic bearing
and screw parameters as
1.5E2/ 3 Z 2 / 3D 1 /3 (sin(a)) 5 / 3 F / 3 > 7rEd2 /4L
(6.20)
We further simplify Eq. (6.20) by noting that ZD ~ ird to obtain the following
condition
2D
Fp >
((6.21)
>
EL2L
L
Hence, while choosing the thrust bearings a designer should always check
Eq. (6.21) in order to ensure that the bearing stiffness is magnitude higher
than the screw stiffness. We note that one can, in most cases, satisfy Eq. (6.21)
by the right choice of bearings and deterministic preload. For a detailed discussion and analysis on rolling element bearings, the reader is referred to
Harris [24]. Other references of interest on this topic are Brandlein et al. [9]
and Slocum [47].
6.2.7
Choice of the Coupling
We recall from Sec. 6.1.1 that an important requirement for the choice of
coupling is to have
n2Kcoupling > kscrew
In order to obtain the required high torsional stiffness we use a bellow-type
coupling as shown in Fig. 6.18. The stiffness of the coupling used in the experiments is approximately 6800 Nm and is procured from GAM Servo Couplings
Inc. [23]. There are other kinds of high stiffness couplings like the one shown
in Fig. 6.19, the so-called disc servo couplings (refer to manual of Renbrandt
135
Figure 6.18: Photo of the bellow-type coupling
Figure 6.19: Photo of the disc-type coupling
136
Figure 6.20: Photo of the external locknut (left) and the damper (right). The damper
is formed by gluing a viscoelastic washer to the external locknut.
Inc. [39]). In our experience these couplings always had backlash problems
after several cycles of testing and would lead to a very poor performance. In
this respect the bellow-type couplings are far superior to the disc-type ones.
However, the disc servo couplings can tolerate larger axial misalignments than
the bellow couplings
6.2.8
Design of the Damper
The damper which is used to partially constrain the second end is formed by
preloading the second set of bearings via a viscoelastic washer. The bearings
are preloaded using an external locknut. In Fig. 6.20, we show a photograph
of the viscoelastic washer which is glued on to an external locknut. This
viscoelastic-glued locknut is used to preload the bearings at the second end
to form the damped end. We use two such lock-nuts on both the sides of the
second bearing block to adjust the bearing preload .
6.2.9
Choice of the Encoder and the Design of its Mounts
In the experimental set-up shown in Fig. 6.8, we use the output from two
encoders: the rotary encoder and the linear encoder. The rotary encoder is
used to measure the motor displacement for collocated control (see Sec. 5.3).
The rotary encoder used in our experiments has a resolution of 2000 counts per
137
revolution and has been procured from Aerotech Inc. [1]. We use an optical
linear encoder manufactured by Renishaw Inc. [40]) to measure the linear
position of the carriage. The linear encoder is of the sine-type (unlike the
usual square pulse ones) with a 20 micron wavelength resolution at a carriage
speed of 2 m/s. We can get very high resolutions by further subdividing the
20 micron wavelength. In our experiment, we have used encoder interpolation
of the dSPACE Inc. [17] signal processing card to obtain a resolution of about
0.04 microns at 2 m/s.
The drawings of the mount for placing the read head of the linear encoder
are shown in Figs. D.13 and D.14. The design of the sensor mounts plays a
crucial role in the performance of the machine. If the sensors mounts are not
designed for favorable dynamics, bad performance ensues. This is because the
machine tracks the sensor's output and if the sensor is unstable it will lead to
the total instability of the machine.
6.3
Experimental Identification
The experimental identification consists of two sets of experiments:
" Modal Analysis
" Sine-sweeps
6.3.1
Modal Analysis
Modal analysis is performed by exciting a machine or structure with an impact
hammer or a shaker, and measuring the response using an accelerometer. In
order to obtain the complete modal picture of the machine, transfer functions
between the response points and excitation point have to measured at different
locations on the machine. Alternately, one can fix the accelerometer position
and move the excitation point around to give identical results (recall Maxwell's
reciprocity theorem). However, the former method where the excitation point
is fixed and the accelerometer is moved to different locations has practical
advantages: a three axis accelerometer can be employed to collect response
data in all the three directions, whereas it will be impossible, most of the
times to excite a machine in all the three directions simultaneously, at all the
points of interest. The reader is referred to Ewins [18] and McConnell [31] for
a detailed discussion on modal testing.
138
-- - -
t
-1
-
Figure 6.21: Experimental set-up for modal analysis
139
The experimental set-up used for conducting modal experiments is shown
in Fig. 6.21. In this experiment we excite the machine using an electromagnetic
shaker, and measure the response at various locations on the machine using a
three-axis accelerometer (PCB35608, PCB Piezotronics [38]).
Precautions while using a shaker
" In general when one uses a shaker to excite a structure, care has to be
taken to design the shaker mount such that the mode corresponding to
the shaker on its own mount does not interfere with the actual modes
of the machine. In order to achieve this, we can either make the mount
extremely stiff or compliant so that the mode arising from the shaker
mount is outside the range of interest. Usually it is very difficult or even
impractical in many situations to achieve a very high stiffness mount.
Hence, we suspend the shaker by means of elastic tubing as shown in
Fig. 6.21 to attain a very low natural frequency (~ 1Hz) for the shakermount mode.
* When we excite a machine using a shaker, we change the modal picture
of the machine because the mass of the shaker is connected to the machine. This can significantly effect the modal frequencies depending on
the inertia of the machine and the shaker. However, if we measure the
force that is exerted by the shaker on the machine at the excitation point,
we can obtain the exact modal picture of the machine irrespective of the
mass of the shaker. Hence, we use the output of a force sensor between
the tip of the shaker's stinger and the machine for force measurement as
shown in Fig. 6.21. We could have also measured the current in the coils
of the shaker to determine the force it exerts on the machine, but this
will provide us with a modal picture which includes both the machine
and the shaker.
In order to avoid any misalignment force from being exerted by the
shaker on the machine, we have used a slender stinger which acts as a
flexure to transmit the force from the shaker to the machine as shown in
Fig. 6.21.
In Fig. 6.22, we show the location of the excitation position and the various
measurement positions. The points 1-4 are located on the four corners of the
carriage, 5-8 on the motor-side bearing block, 9-12, and 13-16 on the linear
140
13
12
2
000
***800
000
**0
20
21
19
22
10
3
-
-
-' '
.;
0
15
0
9
23
8
~17)
-- 24,
,--
5
14
16
.' .6
;
#;
Fiue .2:M asrmetpoiiosfo
odlexeim n
141
100
10-1
.1
axial mode-
10-2
yaw of carriage
and bending of
base
o
bending modeof base
-3
10-5
0
100
200
300
400
500
Frequency (Hz)
600
700
8 )0
100
yaw of carriage
and bending of
base
axial mode
10-1
bending mode
of base
10-2
.E
0)
-J
twisting mode
of base
10-3
10
10-
0
-
100
200
300
400
500
600
700
800
Frequency (Hz)
Figure 6.23: Representative modal transfer functions from accelerometer to shaker at
two different locations on the machine.
142
Carriage
Base
Bearing Block
0
EE3
E3LI
E3
S
ED
El
E3
E3
Figure 6.24: Measured axial mode shape of the new design (349 Hz). Figure shows
snapshots of the mode starting from undeformed position. The small squares indicate
measurement locations.
143
Bearing Block
Base
Carriage
\
Figure 6.25: Measured twisting mode of base (415 Hz). Figure shows snapshots of
the mode starting from undeformed position. The small squares indicate measurement
locations.
144
Carriage
Base
E3
Bearing Block
E3
--- E
]1
E3
E3
Figure 6.26: Measured yaw of the carriage and bending of base (485 Hz). Figure shows
snapshots of the mode starting from undeformed position. The small squares indicate
measurement locations.
145
Bearing Block
Base
Carriage
Figure 6.27: Measured bending mode of the base (635 Hz). Figure shows snapshots of
the mode starting from undeformed position. The small squares indicate measurement
locations.
146
Predicted
frequency (Hz) damping ratio
360
Measured
frequency (Hz) loss factor
349
0.02
Table 6.3: Predicted and measured results for the axial resonance for the case of free
end from modal experiments
guide surfaces of the machine base. On the screw itself, the measurement
points (17-24) are chosen to be diametrically opposite in order to capture
the stretching and twisting motion of the screw. The force sensor and the
accelerometer are connected to a Hewlett-Packard HP35754 signal analyzer
for obtaining the modal transfer functions. Figure 6.23 shows representative
transfer functions from the accelerometer to the shaker. In Fig. 6.24, we
show a series of snapshots of the axial mode starting from the undeformed
position from which we can clearly see the axial motion of the carriage, and
the stretching and twisting of the screw. We note that this is exactly the shape
which we used as the trial function in obtaining a lumped-parameter model in
Chapter 4. Therefore, we have a close agreement between the measured and
the predicted (using reduced-parameter model) results for the axial resonance
as given in Table 6.3. The other higher modes which arise out of the twisting
and bending of the base are shown in Figs. 6.25-6.27.
6.3.2
Sine-Sweep Experiments
In this experiment, we measure the collocated and non-collocated transfer
functions by exciting the motor with a sinusoidal torque and measuring the
response from the respective encoders. A schematic of the experimental setup is shown in the Fig 6.28. We again make use of the HP analyzer for
processing the signals and obtaining the required transfer functions. Hence,
it becomes necessary to measure analog signals in the feedback system. The
only place where we have analog signals in the loop is at the DAC (digital
to analog converter) of the DSP board (dSPACE, DS1103 [17]) which is used
to control the machine. Hence, we form a summing junction at this location
as shown in Fig. 6.28, and inject a sinusoidal signal which is swept over a
range of frequencies to perform the sine-sweep measurements. By collecting
the voltage signals at both sides of the summing junction (see Fig. 6.28),
147
Signal Analyser
(HP35670)
isturbance
input
swept sine)
x (t)
y(t)
DAC Output
+
Power Amp Input
Machine
Digital Controller
(dSPACE DS 1103)
Required transfer function: Y(s)/X(s)
Figure 6.28: Schematic of sine-sweep experiments
148
Encoder Outputs
and further processing them in the analyzer, we obtain the required transfer
functions.
In Figs. 6.29 and 6.30, we show a comparison of the predicted and the measured transfer functions for the case of the ball-screw with a free end when the
carriage is located farthest from the motor (x, = 32 in). As mentioned before
in Sec. 6.2.9, the collocated transfer function is obtained by taking feedback
from the rotary encoder on the motor shaft and the non-collocated transfer
function by taking feedback from the linear encoder on the machine base. In
Figs. 6.31 and 6.32 we show a comparison of the predicted and measured transfer functions for the case of the ball-screw with a damped end. In Figs. 6.33
and 6.34, we show comparisons of the collocated and non-collocated transfer
function for free and damped ends. The sine-sweep results for axial resonance
are summarized in Table 6.4. From Fig. 6.34 and Table 6.4 it is clear that
Predicted Results
frequency (Hz) damping
free end
damped end
360
400
0.09
(()
Measured Results
frequency (Hz) damping
349
382
(()
0.02
0.07
Table 6.4: Predicted and measured results for axial resonance from sine-sweep experiments
adding the damper has resulted in a significant attenuation of the resonant
peak thereby leading to high performance and robustness as discussed before.
We also note from Tables 6.3 and 6.4 that the modal and sine-sweep results
are in close agreement.
149
-100
measured
-1----
-- predicted
-D
_160 -
LE
i
-140- -rdce
E
-180-200
-220
-10-
10
2
3
10
50 0
0-50 -
E
-150-
-200-
o
C.
-250103
102
freq(Hz)
case of the
Figure 6.29: Measured and predicted collocated transfer function for the
free-end boundary condition
10
1021
-100
-120120
M
E
measured
-predicted
-
-140 -160-
2
-180 --200 -220
10,
102
0
-100
-
E
_200 --
C.
.c
-300 -~
-400 ---
'
'
'
10
1e
freq(Hz)
of
Figure 6.30: Measured and predicted non-collocated transfer function for the case
free-end boundary condition
150
-120 --
measured-predicted
-140 -E
L-160 --180 --200
102
10
50-
0E
-50-
E
-100-
4)
-150-
a.
-200-250.-102
10
freq(Hz)
Figure 6.31: Measured and predicted collocated transfer function for the case of dampedend boundary condition
-120-measured
predicted-
-140 -L
160-7-
-180
-200
102
103
0
r100e-S-200-
C.
-300--
-400
10 2103
freq(Hz)
Figure 6.32: Measured and predicted non-collocated transfer function for the case of
damped-end boundary condition
151
--- undamped
- - - damped
-120-140-160
2
0
-180-200
ta
.
-220
102
E
E2
10
0E
- -
-50-100 -150 -
-200 -250 102
freq(Hz)
Figure 6.33: Measured collocated transfer function with and without damper
-120
-
undamped-
--_ _
E
0
damped
-140
-160
-180
-200
10
0
VI
E
-100
*0
-200
.
-300
-400
freq(Hz)
Figure 6.34: Measured non-collocated transfer function with and without damper
152
CHAPTER
7
Solutions to the Inverse Problem
7.1
Introduction
In this chapter, we discuss the several constraints and the feasible solutions to
the inverse problem. As we have seen in Chapter 1, the motion control problem
requires meeting the Group 1 and Group 2 requirements. In Chapters 4, 5, and
6, we were able to establish metrics for robustness and performance and were
able to quantify the Group 2 requirements in terms of the machine parameters
and control measures (bandwidth, PM and RGM). We have shown in Chapter 6 through extensive experiments that a well-designed machine leads to
higher performance and an ideal machine-model match. As a result, we could
simplify the full model and reduce the number of free parameters. Under such
conditions the key design parameters become the lead t of screw, diameter d
of screw, and the inertia Jm of the motor.
We start this chapter by specifying the constraints on the inverse problem
and express the acceleration and velocity requirements in terms of the design
parameters. We also present other sets of constraints such as the amplifier
saturation limits, machine limits, etc., which a designer should meet in order
to obtain a satisfactory design. Next, we sketch the constraint surfaces in
the design parameter space and identify feasible regions which satisfy all the
constraints. These regions then provide us with a family of designs which
153
constitute the solutions of the inverse problem.
7.2
7.2.1
Constraints
Non-Minimum Phase Zero
In Chapter 5, we have seen that the non-minimum phase zero places severe
limitations on the sensitivity properties of the system. In particular, we found
in Sec. 5.5 that if the non-minimum phase zero occurs at a frequency comparable to the first axial resonance, it becomes extremely difficult to achieve
closed-loop performance. Therefore, a designer should always ensure that the
non-minimum phase zero occurs well above the desired closed-loop bandwidth
by satisfying Eq. (5.29) which is given by
C 12 +
C12 + 4mi2 k
2m 1 2
where Mi1 2 , C12 are the absolute values of the off-diagonal terms in the mass
and damping matrices, and k is the total axial stiffness.
Using the expression for C1 2 given in Eq. (4.118), we rewrite the expression
for the non-minimum phase zero as
Zmm
- M2
2m 1 2
min
4m 12
M2
1/2 +i
2
+ 2m 1 2 + M 2
min
1m2
+ 2mi1 2 +
+
M2
1/2-
(7.1)
(
where the expressions for k, MI1 , Mi1 2 , and M2 are given in Eqs. (6.11)-(6.17).
We note from Eq. (7.1) that the minimum value of the zero occurs when the
carriage is located at the end farthest from the motor. Hence, we evaluate the
expressions for k, Mi1 , Mi1 2 , and m 2 (as given in Eqs. (6.11)-(6.17)) at x, = L.
Noting that typically EA/L is an order of magnitude higher than ko, we can
further simplify the expressions for k and q at x, = L as
L + (
~--- +
EA
2
L
27V GJ
e~ g
kok
(EA/L) 2
(7.2)
In order to understand the limitations posed by the non-minimum phase
zero (Eq. (5.29)), we plot the non-minimum phase zero surface given by
Eq. (7.1) at x, = L using the simplification given in Eq. (7.2). Then, we
154
x 10-5
0.60.2%0 0.020.02
0.018
0.018
..-.
0.016
0.016
.---.
0.014
.
L
0.01
0.L0
-'.0.014
0.01
-
0.006
0.008
0.012
t
Figure 7.1: Non-minimum phase zero surface
can use the non-minimum phase zero surfaces whose zero-frequency value is
much higher than the desired closed-loop bandwidth to identify regions in the
design-parameter space which satisfy Eq. (5.29). In Fig. 7.1, we plot one such
surface which is a function of the normalized design parameters for a zero frequency which is about three decades higher than a closed-loop bandwidth of
100 rad/s. From Eq. (7.1), we find that the region in the design space which
satisfies Eq. (5.29) lies above the zero surface. We find from the Fig. 7.1 that
an increase in the diameter d of the screw requires an increase in the inertia Jm
of the motor to lie above the surface. This is because the non-minimum zero
frequency increases with increasing motor inertia and decreases with increasing screw diameter. Similarly, we find that a decrease in the lead f requires
an increase in Jm to be above the surface, though this is not as pronounced as
the d-Jm effect.
155
20
E
18
* Aerotech brushless
+ Aerotech brushed
A Pacific Scientific
o Electrocraft
161
o Kollmorgen Goldline(A)..................................
.... ...............
14 ......
S12 .........
cu
....................
........
...... .......... ..........
....... ... ...... .........................
......
-.............
............-
10 .....
T~
--
6
4
-
-
....
...........
.............
......
;.......
-
**-
2010
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Motor rotor inertia (kgm)
Figure 7.2: Motor torque versus motor inertia for various product families
7.2.2
Maximum Acceleration
The maximum achievable acceleration is determined by the ratio of the motor
torque (minus frictional losses) to the total inertia of the system. Hence, we
obtain the following constraint
me + n2 (J
±.pJL)<a
x(73
where Tm is the motor torque (minus frictional losses) and amax is the required
maximum acceleration. A survey of commercially available servo-motors shows
that for a given family of motors the motor torque scales approximately linearly
with inertia. This is shown in Fig. 7.2. Hence, for a given type of motor, we
can characterize the motor by its inertia. As a result Eq. (7.3) can be recast
156
x10
3%
S2,
0.008
0.020001
0.015
a=.g,
0.01
-
0.005
--
0
0.012
0.014
-,-..
0.02
0.016
0.018
Figure 7.3: Acceleration surfaces
as
r3
me + n2 (Jm + pJL)
< amax
(7.4)
where 3 is the best-fit slope of the curves relating Tm and Jm (see Fig. 7.2).
In Fig. 7.3, we plot acceleration surfaces given by Eq. (7.4) in the space
formed by the normalized design parameters: e/L, d/L and Jm/mcL 2 . According to the inequality given by Eq. (7.4), a design which meets the required
maximum acceleration requirement lies above the respective acceleration surface. This is because decreasing the motor inertia Jm results in reducing the
torque rating (see Fig. 7.2), decreasing the lead f results in increasing the effective inertia of the system and increasing the diameter d results in increasing
the inertia of the screw.
7.2.3
Maximum Velocity
There are a number of factors which can limit the traverse speed, but the most
common among them is the critical speed of rotation at which the ball-screw
157
-
=1 Bn/s-
-V
X105
0.00
0~
0.02
-..
0.015
0.012
,
Veoct sufae
Figre-
0.014
.''-.'-,
0.016
UL0.01
'&.18dL
0.005
0.02
008:1
Figure 7.4 Velocity surfaces
undergoes transverse vibration. This leads to the following condition
e<VMax
(7.5)
n
where Vmax is the required maximum traverse speed and w, is the first critical
speed of the screw. We also note that the critical speed is a function of the
carriage position x, and its minimum occurs when the carriage is at the end
farthest from the motor. Recall from Sec. 6.2.6 that we use four sixty-degree
angular-contact thrust bearings for supporting the screw. Therefore, we can
safely treat this as a fixed-end (zero displacement and slope) in the critical
speed calculations. Similarly, the second end of the screw (for the portion
between the motor-side thrust bearings and the nut) can also be regarded as
a fixed-end due to the displacement and slope conditions imposed by the nut.
We recast Eq. (7.5) using the formula for the first critical speed of a shaft with
a fixed-fixed boundary condition and length L as
(d)
L
158
2.66Vmax
(1
L
NkLE/p
(7.6)
The above inequality represents a region bounded by a hyperbola in the f-d
plane. By extruding these hyperbolas in the third direction, we obtain the
velocity surfaces as shown in Fig. 7.4. In order to meet the required velocity
specifications of the stage, the design must lie to the left of the corresponding
velocity surface.
Other important factors which can constrain the traverse speed arise from
the speed limit on the encoders, motor commutation speed limit and so on.
7.2.4
Motor and Power Amplifier Limits
In this section, we provide order-of-magnitude estimates for the motor and
power amplifier limits.
The main limitation on the motor comes from the heat generated in its
coils. This leads to a limitation on the maximum achievable acceleration as
given by
Imax
I2axR
=
<
mc+ pAL+n
m +
2
(Jm + pJL)
hAs(AT)max
amax
(7.7)
(7.8)
where KT is the torque constant of the motor, Imax the maximum allowable
current, R the coil resistance, h the convection coefficient, A, the surface area
of the coils, and ATmax the maximum allowable temperature difference. Hence,
for a given ATmax, we have a constraint on the maximum acceleration, or for
a given amax the thermal design should be able to result in a safe ATmax.
The main constraints on the power amplifier arise from the power and bus
voltage requirements. These requirements are stated below
" Voltage requirement to avoid saturation
VbU8
Imax
>1
fR2 + (wLinad) 2
* Power requirement to avoid saturation
P
Imnax -R
2
+ (wLinad) 2
>1
where Vus is the bus voltage, Lind is the inductance of the motor coil, w the
frequency in rad/s and P is the required power.
159
7.2.5
Machine Components
In Chapter 6, we have seen the impact of mechanical design on control performance via two examples. We showed that in order to attain high performance
and robustness careful design of all the individual components and the contact interfaces is extremely important. In particular, a designer should always
make sure that the dominant compliance in the structural loop arises from the
ball-screw, and should ensure that the other components of the stiffness loop
such as the bearings, bearing housing, coupling, ball-nut etc., are an orderof-magnitude stiffer than the screw. If the condition given in Eq. (6.1) is not
satisfied then there will be reduced performance as we have seen in the case
of the old design described in Chapter 6.
7.3
Solutions
Solving the inverse problem requires finding the machine parameters such as
f, d, Jm, etc., from the expressions for the closed-loop bandwidth given in
Chapter 5 under the several constraints described in the previous section.
Hence, for a lead compensator, we have to solve Eq. (5.39) given by
Ljb(RGM)
1 + sin(PM)
1 - sin(PM)
=l 218s21sc
(7.9)
subject to the constraints described in the previous section. We can use standard optimization methods such as the Simplex or the Gradient based schemes
(e.g., Bertsekas [5]) to solve the above problem. The solution to the above
problem can then be used as a starting point for sizing the screw and the
motor.
7.3.1
Wb-PM-RGM Surface
Noting that the frequency of axial resonance is lowest when the carriage is at
the end farthest from the motor, we set out to solve Eq. (7.9) at x, = L for
a given bandwidth Wb, phase margin PM, and resonance gain margin RGM
requirements. By doing so, we ensure that the system satisfies the wb-PMRGM specifications at any other position of the carriage. Using the expression
for the damping matrix, we rewrite the expression for s2 given in Eq. (5.17)
160
x10
8.
5
86r/s PM
6%
RGM 15
4.
2.%
w
0.02
.-
41
sr/s
0.015
0.01
.
P=50* R'M140
0.012
..--...
0.01
0.014
'--,
,,-'.
0.016
-.
.0 5
0
0 .0 2 0 .0 18
Figure 7.5: Wb-PM-RGM Surfaces
as
Sr 2
2 Cm(m
tn
=
2
+ mi 2 ) 2 + Cc(mi + m12) 2 + C(mi + 2m 1 2 + M 2 ) 2
2(mi + 2m 12 + M 2 ) (mlm 2 - m12 )
(7.10)
Next, we note that for small damping Isr21 is much smaller than sc, and therefore, we can express the damping ratio (, of the axial mode as
(a
(7.11)
~ |sr2I/Sc
Using Eqs. (4.118), (5.17) and (7.10), we rewrite the above expression for
the damping ratio of the axial mode (a as
F{\
(a =m
2
(m,
in 2
+
+
+ m12
+
2m12
2
ml +
/\
2
+ M2)
+ a2
m,
+
in 1 2
+ m 12
2m12 + M2)
+i
+ 1
(7.12)
where we have introduced the ratios: a, = n2Cm/C and a 2 = Cc/C. The
above expression gives us some insight into the contributions of the various
161
damping elements to the damping of the axial mode. We note that among
these damping elements, we can to a great extent be certain about the damping
obtained from C, whereas it is difficult to predict the damping from Cm and
C, with the same level of certainty. This is because C represents the damping
r
that we introduce externally into the system via the dmper (and s
bearing
motor,
in
the
damping
the
model
C,
and
Cm
predictable) whereas
races, etc., and hence can change with temperature, machine conditions and
so on. As we have seen in Chapter 5, the damping of the axial mode plays
a very important role in determining the maximum achievable bandwidth for
a ball-screw drive. Therefore, for the reasons discussed above, we will set
a, and a2 to zero in the expression given in Eq. (7.9) to obtain conservative
Wb-PM-RGM surfaces.
We rewrite the expression given in Eq. (7.9) in terms of the design parameters and use it to plot Wb-PM-RGM surfaces as shown in Fig. 7.5. Hence, in
order to achieve a given Wb, PM, and RGM requirement the design has to lie
below the corresponding surface.
To present a visual picture of the solution, we plot the constraint and
the bandwidth surfaces to graphically determine the feasible solutions in the
following section.
7.3.2
Design Space and Constraint Surfaces
We normalize the key design parameters: f, d, and Jm by the length of travel L
and the mass of the carriage mc and plot each on an axis as shown in Fig. 7.6.
To achieve a given velocity, acceleration, and bandwidth, a design must lie
to the left of the corresponding velocity surface, above the corresponding acceleration surface, and below the corresponding Wb-PM-RGM surface. The
design must also lie above the non-minimum phase zero surface to meet the
constraints imposed by Eq. (5.29). Hence, the enclosed volume represents the
family of designs which can achieve the required closed-loop specifications.
7.3.3
Comment on Disturbance and Noise Rejection
After choosing a design from the feasible design space one must check if the
design meets the required disturbance and noise criteria; that is, one must
ensure that the loop transmission does not penetrate into the disturbance
and the noise obstacles (refer to Fig. 7.7). To achieve these requirements one
162
-
-
V=1U.
-I-
x 10-5
8%
=0.5m/g
-
86ri, PMQI
6%
50-
RGM0=1.5
b
O1r/s PM=50
non--mir,lmum
0 -kero =10s
---
a .5g
-
--
0.008
0.025
0.01
''
0.02
''
0.015
,
''..0.5g
0.01
' ' ' .
0.005
0.012
0.014
-.
,-'
,
. 1
0.018
0
0.02
Figure 7.6: Design Space and Constraint Surfaces
163
mnd
Low frequency
Bode obstacle
*
20 db/dec (for PM)
no,
Wd
Frequency
WC
High frequency
Bode obstacle
Figure 7.7: Typical Bode obstacle course
can use different strategies: for example, one can add a lag compensator in
the controller to increase the low-frequency gain (if the lag zero is placed a
decade below crossover, the effect of the lag part on the phase at crossover is
negligible), or choose a different design in the feasible design space and so on.
164
CHAPTER 8
Conclusions and Future Work
8.1
Summary
In this thesis, we have outlined the solution for the inverse problem of designing
a precision machine for closed-loop performance. In so doing, we have closed
the forward-inverse loop of the flowchart given in the Fig. 1.1. In order to
attain high performance and robustness, we showed that the focus should
be on the design of the loop transmission. To this end, we have obtained
a reasonably accurate model of the system which is supported by extensive
experimental evidence. Graphical design techniques then allow us to link the
closed-loop performance to mechanical design so that the designer can rapidly
layout a machine that meet the desired closed-loop specifications.
8.2
Recommendations for Future Work
The servomechanism described in this thesis forms a basic building block for
more complicated machines with multiple axes. Hence, a natural extension of
this work would to be apply this methodology in the design of built-up systems.
This could be done by considering a sub-structuring architecture where each
substructure is designed to given dynamic requirements. The entire machine
165
then will be formed by combining all these sub-systems. This problem is much
more complicated as it would generate several constraints from the interacting
sub-systems, and also may place severe restrictions on the performance metrics
of each subsystem to ensure overall performance.
We note that the methodology in the thesis has been very general in terms
of the final formulae; they are functions of the elements of mass, damping,
and stiffness matrices, and the robustness criteria. The ball-screw mechanism
possesses all possible complexities of a general servomechanism, hence it will
be relatively easier to derive the corresponding matrices for other kinds of
servomechanisms such as a linear motor system, belt drive system and so on.
We can then use these results as modules in the final built-up system, and
thereby layout any general precision machine that meets the desired closedloop specifications.
166
APPENDIX
A
Proof of Theorem 1
Assume that L(s) is free of unstable hidden modes. Then we can factor L(s)
as
(A.1)
L(s) = L(s)Bl(s)B,(s)e-r
The terms Ba(s) and Bp(s) are given by
B,(s) = i
_ ~
i+S
; BP,(s)
NIPi
-
i+
S
(A.2)
where zi's and pi's represent the right-half-plane zeros and poles of L(s) respectively. We now factor W(s) and Sm(s) as
W(s) =
Sw(s) =
W(s)B.(s)
5w(s)Bs(s)
(A.3)
(A.4)
where
Bw(s)
=
iNw
zw
s
z i +
±S
Bs(s) = Bw(s)Bp(s)
(A.5)
(A.6)
Because SW(s) is analytic and nonzero in the closed right-half plane except
for possible zeros on the imaginary axis, we can write the following well-known
167
Poisson integral formula from complex analysis (e.g., Churchill [12]) at every
point s = x +jy, x > 0
logjIS" (S) = If0 log I'w(jW)
T-J~i
92
+
-c~
Letting y = 0 in the above, and noting that
Sw(-3w), we obtain the following
IS (Ow) 1=
dw
(A.7)
Sw(JO) 1and Sw(jw) =
2 f 00log Sw~jc),_
I~Jf
+2
7o
x 2 +W
log jS-(X)f= -
W
dw
(A.8)
Further, we can write the following limit
2 f0
lim Xo
X-+00
x-+00r
-
X2
x2 +W
log |S( (j)l2
0
dw
(A.9)
since log S,(jW)X2 /(X 2 +w 2 ) converges to logI S(jw)I as x --+ oc. We can use
the above integral to evaluate the Bode sensitivity integral. From Eq. (5.21),
we note that IlogI S (jw) can be bounded by a positive constant for some
w > w,. Therefore, for any c > 0 there exists a CD > w0 such that
J00
log ISw(jw)I dw < c
(A.10)
We also note that x 2 /(X 2 + w 2 ) converges uniformly to one as x approaches
infinity. Hence, we can write
2 f W2
limX--OO o
2fw
log|S(jw)|
lo I WU )1X
2
W27r
dw = -
log ISw(jw)I dw
(A.11)
0
Hence from the above results, we obtain
lim-
2
-+Co -
f*0
0
log|Sw(ja)
___2
2
2
2+W
2
dw = -
fw*
log |S,(jw)I dw
(A.12)
From Eq. (A.9) and (A.11) it follows that the Bode sensitivity integral is given
by
I0logISw(jw) dw= 2 lim-* x logI5w(x)
00
168
(A.13)
We rewrite the limit on the left hand side of Eq. (A.9) using Eq. (A.4) as
lim x log |5(x)= lim x (logIS,(x)I+ log IBw-(x)I + log IB,-1 (x))
(A.14)
When we use Eq. (5.21), and expansions for Bw(s) and Bp(s) the above equation reduces to
N~~z
Zi + X
+
lim x log IS(x)j = Z lim x log
xi+0
x_00
z1 -
X
By using the power series expansion of log Izif
log
x-zi
-+
zi +
x
1
pA
limx log
k=1
xI
Z-2
2
oo
(A.15)
X
we obtain for x > IziI
-j
2
x
_+
Pk -
+ ...
x
(A.16)
Hence, we obtain
lim x log ' + X
2Re(zi)
(A.17)
lim x log Ak+X
2Re(pk)
(A.18)
X-+0o
Z - X
and similarly
Substituting the above results in Eq. (A.15), we obtain
j
log ISw(3w)l dw = ir
(ZRe[zi]
+
3 Re[Pk])
(A.19)
169
APPENDIX
B
A Review of Classical Control
In this Appendix, we provide a brief review on classical control techniques.
Consider a unity feedback system shown in Fig. B.1. We define some important
transfer functions from the block diagram of Fig. B.1 below
" Gp(s)
* Ge(s)
plant transfer function
=
controller transfer function
" K = proportional gain
" L(s) = G,(s)Gc(s)K = loop transfer function
* S(s) = [1 + L(s)]-
1
= sensitivity
* T(s) = [1 + L(s)]- 1 L(s) = complementary sensitivity
B.1
Typical Control Tradeoffs and Constraints
The output of the control system of Fig. B.1 is given by
Y(s) = S(s)D(s) + T(s)R(s) - T(s)N(s)
170
(B.1)
R~s)
RK
E~s)
D(s)
Gp (s)
Ge(s)
---
(S
N(s)
Figure B.1: Standard unity feedback system
From the above equation, it is clear that good tracking requires loop transmission to be large, disturbance rejection requires sensitivity be small and noise
rejection requires complementary sensitivity be small.
B.1.1
Sensitivity to Parameter Changes
One of the advantages of feedback is that it can reduce the sensitivity of the
closed-loop transfer function to changes in the parameters of certain elements
in the loop transmission. If the loop transmission is made very large (i.e.,
ILI > 1) then Y = R, and the closed-loop gain is determined by the feedback
function alone. For a SISO system the fractional change in the closed-loop
transfer function that results from a fractional change in the loop-transmission
can be obtained as
d(Y/R) dL
(B.2)
(Y/R)
L
Hence, for the closed-loop to be insensitive to changes in the forward path we
require ILl
1.
B.1.2
Algebraic Constraints
We now qualitatively state the performance requirements as
171
e |L|
> 1 for good tracking.
" ISI < 1 for good disturbance rejection and insensitive to parameter
changes (robustness).
" |TI < 1 for good noise rejection.
In the design of a control system it usually desirable to make both S and
T small. However, note that
S+T= 1
(B.3)
The above equation represents the fundamental algebraic constraint in control
system design: it is impossible to minimize the sensitivity and complimentary
sensitivity simultaneously at the same frequency. Hence a feedback design
problem is almost always a compromise between accuracy and stability.
B.1.3
Properties of a First-Order System
A typical first-order system with a time constant T is given by
G(s) =
TS +1
(B.4)
The Bode plot of this system is shown in Fig. B.2. The actual magnitude at
the break point lies below the asymptotes by -3dB, and the actual phase curve
deviates from phase asymptotes by 110 at the intersection of the asymptotes
i.e., at WT 0.2 and WT = 5 as shown in Fig. B.2.
B.1.4
Properties of a Second-Order System
A typical second-order system with a natural frequency of w, and damping
ratio ( is given by
2
G(s) =
s2+
2(Wn +nw
(B.5)
Frequency-Domain Properties
The Bode plot of this system is shown in Fig. B.3. The break-point frequency
for the asymptotes occurs at Wn whereas the resonance peak occurs at w,. =
172
0
-20db/dec
45
11
'W
-90
Figure B.2: Bode asymptotes for a first-order system
wnV(1 - 2(2). The ratio of the magnitude of the resonance peak to the DC
gain is given by
(B.6)
2 (/1
-(2
A useful point to note here is that for a second-order system the magnitude of
the resonance peak can be obtained by adding 20 log [1/(2(
1
-
(2)] to the
db level corresponding to any w < wo on the asymptotes. We make use of this
fact in Chapter 5 to obtain closed-form expressions for closed-loop bandwidth.
Time-Domain Properties
The important time-domain specifications are given in terms of rise time, settling time and percent overshoot for a step input. For a second-order system
without any zeros these quantities are given below. An approximate expression for the rise time t, (corresponds to the time required for the output to
change from 10% to 90% of the final value) is given by
tr ~
8(B.7)
Wn
173
20 log
0
WT
wi
0
-
-90
' - ----
- 180
--- -- -
Figure B.3: Bode asymptotes for a second-order system
The settling time t, is the time required for the response of the system to be
bounded within some specified percentage of the final value. For a 1% bound,
we obtain
t=
4.6
(B.8)
The maximum overshoot Mp is difference between the peak and final value of
the response for a step input. The expressions for Mp are given below
MP= e
M,
~
1-
ffor 0 <
0.6
<1
for 0 < ( < 1
(B.9)
(B.10)
In Fig. B.4, we plot the exact and first-order approximation for Mp against
the damping ratio. This plot is useful for obtaining estimates of the required
damping ratio to meet the overshoot specifications.
In analysis and design, the above parameters are used to characterize the
transient response of any system though these are exact for the second order
system. For higher order systems, we can replace w, with the closed-loop
174
1
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
I
I
0.8
0.9
0.9 0.80.70.60.50.40.30.20.1 -
0
0.6
0.7
1
Figure B.4: Plot of the overshoot MP versus the damping ratio (
bandwidth Wb to obtain estimates for the transient characteristics. The above
results can be used in conjunction with the approximations for the first-order
system to obtain approximations for magnitude and phase phase for a general
higher order system. These approximate results are very useful for back-ofthe-envelope calculations at the initial design stages.
B.1.5
Closed-Loop Behavior
For a system G(s) acting in a closed-loop with negative unity feedback, the
closed-loop transfer function is given by
G, 1(s)
_L(s)
1 L(s)
I + L(s)
(B.11)
For many systems IL(yw) has the following typical property
IL(jw)
1,>
« we
(B.12)
175
"G(s)I
bnwdPM
-d
w
0
-d
-------
--
-
-- -- -- --- -3db
PM =90'
WC
JGer(s)j
K
0-bandwidth
W
Figure B.5: Frequency response characteristics of the closed-loop transfer function
where we is the crossover frequency. The above expressions can be used to
approximate the closed-loop response as
IG,(jw)l = 1,w<we
|Gej(jw)I = IL(jw)I , w > we
(B.14)
(B.15)
In the vicinity of crossover, the magnitude depends heavily on the phase margin
(PM) as illustrated in Fig. B.5.
B.2
Classical Compensators
In this section, we discuss the properties of four important classical compensators: lead, lag, PID and lead-lag compensators.
Lead Compensator
A typical lead compensator is given by
Gc(s) = K
176
Ts+1
Ts + 1
(B.16)
with a > 1. The Bode plot for this compensator is sketched in Fig. B.6. The
frequency at which the phase attains a maximum and the corresponding value
of the maximum phase for the lead compensator are given by
1
S=
(B.17)
la-1
qmax
=
arcsin
(
(B.18)
A plot of om versus a is given in Fig. B.7 is very useful in the design of a
lead compensator because by picking a a, we can fix the maximum phase lead
Okmax ,
Lag Compensator
A typical lag compensator is given by
GT(s) = K
Ts+1
(B.19)
with 3 < 1. The Bode plot for the compensator is shown in Fig. B.8. For this
compensator the minimum phase occurs at a frequency given by
(B.20)
W
V #T
and this minimum value of phase
bmin
is given by
#min =arcsin
(B.21)
A plot of qmin versus 3 is shown in the Fig. (B.9). The lag compensator
is added to reject low frequency disturbances by increasing the low frequency
gain. The advantage of the lag compensator over a pure integrator is that it
can reject low-frequency disturbances with very little affect on the PM, if the
lag zero is placed a decade below crossover. Another advantage is the absence
of integral wind-up (see for e.g. Franklin et al. [19]), a common effect in an
unbounded integrator.
177
go
1
1
e
S90
W
7/-iT
a;T
-
--------
eO
W
Figure B.6: Bode plot for a lead compensator
80
- -.
.--. .-.-.
70
-
~
-.-
~~ --
--.-. -.-.
..... - ..-.- -
-.-. -.
-.--
60
-.-.-.-.-.
,0
E
---.
---.-.-.-.-
-.
CO
-
~ ~- - ~- -~ -~~~~~
-
-q
--
40
30
20
10
L-L
10
10'
10
10'
cc
Figure B.7: Variation of
178
bma
with a
-20db/dec
T
17r
1i
W
0
-
-90
W
Figure B.8: Bode plot for lag compensator
-.-.-.-. -.- .-...
-.- ....
.
-.
- - -. -.--... ...
-10
- -.
-20
-.
---.
..-
-. ..-..
. ... ..-
-30
-40
- ~~
--
E
-.---.--.-.--.--------.-.
- - --
-...-q-..
- -
-50
-60
......
-70
-80
1-3
10'
10
10-,
13
Figure B.9: Variation of qmjn with
#
179
-20db/dec
-o
-20db/de
20log K
1
i
T,
LO
TD
90
-----------------------
-e
0
-90
-------------- '
Lw
Figure B.10: Bode plot of a PID compensator
PID Compensator
A typical PID compensator is given by
Gc~s) =
(TDs8 + 1) (S +
)TB.2
The Bode plot of this compensator is given in Fig. B.10. A PID compensator
can be used to reject low-frequency disturbances because of its high gain at low
frequencies. A decent PM can be ensured by choosing the crossover to be in
the positive parts of the phase curve. However, a PID compensator amplifies
high frequency noise because of its increasing gain at such frequencies. As we
move the low frequency zero towards the right i.e., as we reduce TI, the system
settles faster to a command input because of the increased low-frequency gain.
This comes only at a cost of reducing the PM which indicates the typical
trade-offs in a PID design.
180
20 1(g Kp
-20db/dec
17
9)0
-90
20db/de
1T
I
- --- ---
- - - - - - --
-
17r21
-- ---
- -- -
--- --- --
---
Figure B.11: Bode plot of a lag-lead compensator
Lag-Lead Compensator
The advantages of the lead and lag compensators are combined in the lag-lead
compensator. A typical lag-lead compensator is given by
Gc (s) = KpO,+I
a2
1
( T + 1 )(T2 +I
(B.23)
From the Bode plot of this compensator given in Fig. B.11, we find that
the lead pole eliminates noise amplification at high frequencies, a disadvantage
often encountered in the PID compensator. The lag term can be designed
to reject disturbances in the low frequency region and the lead term can be
designed to provide the required PM. If we place the lag zero a decade below
the crossover frequency, the effect of the lag part on the phase at crossover is
insignificant. Hence, we can design the lead part of the compensator based on
the bandwidth and PM requirements and then design the lag part by placing
the lag zero a decade below the crossover.
However, we again have similar
settling time and phase margin trade-offs as we did for the PID controller:
181
that is, we can move the lag zero towards right to lower settling time and
suffer an increased overshoot (due to a reduction in PM). Hence, the design
at this stage solely depends on the required specifications.
B.2.1
Design Notes
The crossover frequency we, is nearly equal to the bandwidth of the closedloop (-3db point in Fig. B.5). Hence, when we have to design stable feedback
systems for a specific closed-loop bandwidth, we can quickly do so by designing
a compensator to allow the loop transfer function to have crossover at the
bandwidth frequency and add sufficient phase to have a decent phase margin.
This initial design can be further tuned to get the exact values. Even though
PM is a measure of stability it can be used to directly specify the control
system performance. Therefore, great insights can be derived by relating PM
to other measures of stability and performance: For example in the case of
a second-order system defined by Eq. (B.5), the relation between PM (in
degrees) and ( can be approximated as
PM
100
(B.24)
Although the above relation is true for a second-order system it can be used
as an important rule-of-thumb to assess the properties of other higher-order
systems. An approximate relation between the closed-loop bandwidth Wb and
the rise time t, for a higher order system can be obtained by replacing Wo with
Wb in Eq. (B.7). This gives
tr
1.8
1.
(B.25)
Wb
We can use Eqs. (B.24) and (B.25) to relate time and frequency-domain
requirements. As the frequency-domain approach via loop shaping is the most
convenient way to design controllers, we can use Eqs. (B.24) and (B.25) to
obtain constraints on the frequency-domain variables from time-domain specifications. For example, an overshoot specification can be recast as a PM
specification by using Eq. (B.24). Similarly, a rise time requirement can be
related to a bandwidth specification using Eq. (B.25). Once we know the required bandwidth and phase margin, we can design a lead compensator to
meet these specifications and obtain a first-cut design which can be modified
further to satisfy the exact requirements. Similarly, by knowing the required
182
bandwidth and phase margin we can obtain estimates for the corresponding
time-domain properties. Further, a lag term can be added to enhance disturbance rejection as stated in Sec. B.2. This approach of designing controllers is
usually very successful in the case of most minimum-phase systems (or when
the non-minimum phase effects are far above the frequency range for motion
control). When one encounters non-minimum phase systems, unstable plants
etc., one has to perform a Nyquist analysis to correctly analyze the closed-loop
behavior.
183
APPENDIX C
Robust Stability and Performance
We consider the following perturbation model of the plant
G(s)
=
[1 + A(s)W 2 (s)]GN(s)
(C.1)
where G(s) is the actual plant, GN(s) is the nominal plant, and A(s) represents
the plant uncertainty with a frequency weighting W2(s). The function A(s) is
chosen such that
A (jw) <; 1
(C.2)
There is no loss of generality in assuming the above because W 2 (jw) can
be specified such that it absorbs magnitudes greater than unity. The reason behind specifying the above bound on A(jw) is to conveniently represent
plant perturbations as being deviations from unity. This is clear by rewriting
Eq. (C.1) as
G(jw) - GN
= A(jw)W 2 (jw)
(C.3)
GN jw)
We also note that we have only constrained the magnitude of A(jw) and
not the phase. In what follows, we will see that the phase introduced by
the perturbations will strongly affect the encirclements of the -1 point. The
block diagram of the system given by Eq. (C.1) in a feedback loop is shown in
Fig. C.1.
184
S W2(s)
Ge(s)
---
-
(S)
GN(s)
Figure C.1: Block diagram of the perturbed plant
IN
A (s)
M(s)
M (s) = W 2 (s)TN(s)
Figure C.2: M - A structure and the small gain theorem
185
C.1
Robust Stability
In order to understand the stability of the perturbed system, we let the input
R(s) go to zero and examine the remaining loop. The stability of the system
can now be understood in terms of the stability of the so-called Ml-A structure
(e.g., Zhou, Doyle and Francis [54]) shown in Fig. C.2. For our problem M(s)
is given by
M(s) = -W 2 (s)TN(s)
(C.4)
where TN(s) is the nominal complimentary sensitivity function. The stability
of the system shown in Fig. C.2 is determined by the well-known small gain
theorem. According to this theorem, when A(s), W 2 (s) and TN(s) are stable,
the perturbed system is stable if and only if
(C.5)
||M11|0 < I
The above condition results in
IW2 (jw)LN(jW)I
(C.6)
< I1 -+LN(jW)I
where LN(s) is the nominal loop transmission. In Fig. C.3, we present the
geometrical interpretation of Eq. (C.6) from which we find that for robust
stability, the -1 point must lie outside of the circle of radius
IW2
(jw)LN(jw)
centered at the point LN(iw). Hence, for the system to be robust stable the
family of circles shown in Fig. C.3 must never encircle the -1 point. The
radius of these circles depend on the choice of the weighting filter and the
loop transmission. Therefore, the weighting filter should chosen such that it
correctly captures plant uncertainties.
Understanding Robust Stability
We recall that the stability of a control system depends on the number of the
encirclements of the origin by the function 1 + L(s). In terms of the nominal
loop transmission, we need to check the number of encirclements of the origin
by the function
1 + [1 + A(s)W 2 (s)] LN(s) = [1 + LN(s) [1 + A(s)W 2 (s)TN(s)]
The above expression shows the role played by the plant perturbation in the
encirclements of the origin . Using the robust stability criterion of Eq. (C.5),
186
-1
LN(JW)
w3
U) 2
Wij
Figure C.3: Geometric interpretation of robust stability
187
and noting that IA(jw)I < 1, we obtain
(C-7)
JAW 2TN <1
Therefore, the point (1+ A(s)W 2 (s)TN(s)) lies inside a circle centered at +1
with a radius less than unity. Hence, for the Nyquist criterion to hold 'r
the perturbed plant, we need to satisfy Eq. (C.7), which implies that the net
encirclements by the factor (1+ A(s)W 2 (s)TN(s)) will be zero if and only if the
angle change of the line joining the point (1 ± A(s)W 2 (s)TN(s)) and 1 never
makes a full rotation.
C.2
Robust Performance
The nominal performance of the closed-loop system can be characterized by
the following equation
(C.8)
< 1
ISN(jw)Wl(jw)W
where SN(s) is the nominal sensitivity transfer function and W (s) the required
frequency weighting of the sensitivity function. Now, to consider the effect of
the uncertainty, consider the perturbed sensitivity function given by
S
=
1
1+ L
SN
-
1 +AW 2TN
The perturbed system will exhibit robust performance if and only if it is robustly stable, and if the perturbed sensitivity function also satisfies Eq. (C.8).
These conditions are given below
IW2 (jw)TN(jw)
<
1
(C.10)
(C.11)
S(jW)W 1 (jW)1 < 1
It is shown in Doyle, Francis and Tannenbaum [16] that the above inequalities
are satisfied simultaneously if and only if
|W1(jW)SN(jW)J+
W 2 (jW)TN()I < 1
(C.12)
We recast the above equation in terms of the nominal loop transmission LN
as
(C.13)
WI(jw)l + JW2 (jw)LN(jW)J < 1 + LN(W)J
The inequality expressed in Eq. (C.13) is illustrated in Fig. C.4. We note
from Fig. C.4 that for robust performance, the two families of circles resulting
from robust stability and nominal performance have to be disjoint.
188
Co
2
V
r-|Wil
circles
W2
LN(JW)
W1
IW2LN
circles
Figure C.4: Geometrical interpretation of robust stability
189
APPENDIx
D
Engineering Drawings
190
i
9
96
i
0
1~~
V-
V,
I ~ ~
_______
CO
A____
~~7LJ
T-
i
Figure D.1: Drawing of the machine base weldment (page 1)
191
I
I
U
I
d
I1
N
I
I:5P .
id
. -9
A
U
Ll
J.1
Figure D.2: Drawing of the machine base weldment (page 2)
192
I
Y-Y-I-w
1~
co
~
LLU
UJM
I
i
I
Ii
pJ
0U
Ii
Figure D.3: Drawing of the machined base (pagel)
193
i
Cd
H
a-
1
15
Sil
d
'1,0IL
tt-Zrr
-
D
11111-----a
---- rQ
r----
~~~~~A-aa a-a a a A L~ .~..
-aL
_a....-A
c..- .A-
O
L
r
:0
r1
11
911
-
-
Figure D.4: Drawing of the machined base (page2)
194
000'-
N£SV'
I
i
0o
IDU
-I---
iii
'ii
ii
Irl
I
Il
Il
II
I ~
CD,
cjf
IU
W
.j
ii
I-
--
Ca
_______
ii
d
: g
I
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204
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