A N INVESTIGATION INTO THE REDUCTION OF STICK-SLIP FRICTION
IN HYDRAULIC
ACTUATORS
W i l l i a m Scott
Owen.
B . A . S c , The University of British Columbia, 1990
A THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R
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T H E F A C U L T Y OF G R A D U A T E S T U D I E S
D E P A R T M E N T OF M E C H A N I C A L
ENGINEERING
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Abstract
The stick-slip friction phenomenon occurs during the switch from static to dynamic friction.
Static friction is the force that opposes the sliding motion of an object at rest. Dynamic friction
is the force that opposes the sliding motion of a moving object. Thus, near zero velocity, there is
a switch from static to dynamic friction.
Generally, static friction is greater than dynamic
friction. In order to move an object the applied force must exceed the static friction. Once
movement starts the friction force typically decreases as it switches to dynamic friction.
However, if the applied force is still at the original magnitude, then the sudden increase in the
resultant forces results in an increase in the object's acceleration; namely a jerky motion.
In a similar manner, when an object is brought to rest the sudden increase in friction, as the
switch from dynamic to static friction occurs, results in an abrupt and premature stopping of the
object. Because of the rapidly changing and inconsistent nature of the friction force at low
velocities, accurate and repeatable position control is difficult to achieve. In some cases the
actuator position controller can reach a limit cycle (hunting effect).
Friction compensation at low speeds has traditionally been approached through various control
techniques. This work presents an alternative solution, namely, friction avoidance. By rotating
the piston and rod, the Stribeck region of the friction - velocity curve is avoided and the axial
friction opposing the piston movement is approximately linearized. As a result, simpler, linear
control techniques at low speeds may then be utilized. Simulation and experimental results are
presented to validate this approach and identify the operating limits for the rotational velocity.
The experimental results validate the model.
The results show that by rotating the piston, the friction is reduced and the Stribeck curve is
eliminated. As the rotational velocity is increased the static friction from the axial motion
approaches the static friction of the rotational motion. In order to eliminate the Stribeck curve,
the rotating velocity must be located outside the range of the Stribeck area of the rotating friction
- rotating velocity curve and into the full fluid lubrication regime.
ii
Table of Contents
Abstract
ii
Table of Contents
Hi
List of Tables
vii
List of Figures
viii
Acknowledgement
x
Dedication
Chapter 1
xi
Introduction
1
1.1
Preliminary Remarks
1
1.2
Motivation and Objective
2
1.3
Thesis Overview
4
Chapter 2
2.1
6
Friction in Lubricated Machines
6
2.1.1 Basic and Classical Models of Friction
6
2.1.2
8
2.1.3
2.2
Friction in Hydraulic Actuators
Stribeck Curve
2.1.2.1
Regime I: Static Friction
10
2.1.2.2
Regime II: Boundary Lubrication
11
2.1.2.3
Regime III: Partial Fluid Lubrication
12
2.1.2.4
Regime IV: Full Fluid Lubrication
12
Stick-Slip Friction
12
2.1.3.1
Static Friction and Rising Static Friction
13
2.1.3.2
Frictional Memory
14
Friction in Hydraulic Actuators
15
2.2.1 Control of Hydraulic Actuators in the Presence of Friction
16
2.2.1.1
Model Based Friction Compensation
17
2.2.1.2
Observer Based Friction Compensation
18
2.2.1.3
Observer Based Adaptive Friction Compensation
18
iii
2.2.2
2.3
Friction Avoidance
Summary
Chapter 3
19
20
The Hydraulic Actuator Model
21
3.1
Modeling
21
3.2
Non-Rotating Model
21
3.2.1
Servo Valve
21
3.2.2
Fluid Flow
22
3.2.3
Pressure Changes
23
3.2.4
Dynamics
24
3.2.5
Friction
25
3.2.6
State Space Model
26
3.3
Modeling of the System with Piston Rotation
27
3.3.1
DC Motor
28
3.3.1.1
Current
28
3.3.1.2
Dynamics
28
3.3.2
Helical Motion
29
3.3.3
State Space Model
31
3.4
Simulation
32
3.5
Summary
37
Chapter 4
Friction Identification in Hydraulic Actuators: Experimental Method
38
4.1
Introduction
38
4.2
Hydraulic Actuator Setup
39
4.3
Data Acquisition
40
4.3.1
Position Signal
41
4.3.2
Velocity Determination
42
4.3.3
Pressures
42
4.4
Friction Model
42
4.5
Determining The Friction Parameters
43
4.5.1
Equation of Motion
43
4.5.2
Friction Parameter Determination
44
4.5.2.1
Static Parameter Determination
45
4.5.2.2
Dynamic Parameter Determination
46
iv
4.6
Pure Rotational and Linear-Rotating Friction Parameters
47
4.7
Power Requirements
48
4.8
Summary
48
Chapter 5
Reducing Stick-Slip Friction in Hydraulic Actuators: Experimental Results.49
5.1
Pure Rotation
49
5.2
Hydraulic Actuator - 0 Rpm
50
5.3
Hydraulic Actuator Rotating
52
5.4
Friction Parameters
59
5.5
Model Validation
64
5.6
Power Requirements
67
5.7
Position Tracking
69
5.8
Summary
70
Chapter 6
Conclusions and Suggestions for Future Work
72
6.1
Conclusions
72
6.2
Recommendations for Future Work
72
Nomenclature
74
Bibliography
80
Appendix A Hydraulic Actuator Specifications
84
A . 1 Process and Instrumentation Diagram
84
A.2
Physical Parameters
85
A.3
Instrumentation and Hardware
85
A.3.1 Hydraulic Pump
85
A.3.2 Motor
86
A.3.3 Servo Valve
87
A.3.4 Pressure Transducers
88
A.3.5 Potentiometer
88
A.3.6 Data Acquisition
89
Appendix B Simulation and Experimental Values
90
Appendix C Jacobian Linearization
91
v
C. 1 Jacobian Linearization and Discretization
C.2
Non-Rotating Model State Equations
C.2.1
C.3
91
:
92
Position
92
C.2.2 Axial Velocity
92
C.2.3 Pressure
93
C.2.4 Friction
93
C.2.5
94
Servo Valve Input
C.2.6 Jacobians
94
Rotating Model State Equations
95
C.3.1
Position
95
C.3.2 Axial Velocity
95
C.3.3 Pressure
97
C.3.4 Friction
97
C.3.5
98
Rotating Velocity
C.3.6 Motor Current
99
C.3.7 Motor Voltage Input
100
C.3.8
100
Servo Valve Input
C.3.9 Jacobians
101
vi
List of Tables
Table 3.1 Viscous Friction Parameter vs. Rotation Speed
34
Table 5.1 LuGre Rotating Friction Parameters
50
Table 5.2 Hydraulic Actuator LuGre Axial Friction Parameters (Positive Velocity, 0 rpm)
51
Table 5.3 Hydraulic Actuator LuGre Axial Friction Parameters (Negative Velocity, 0 rpm)
52
Table 5.4 LuGre Axial Friction Parameters (Positive Axial Velocities, with Rotation)
58
Table 5.5 LuGre Axial Friction Parameters (Negative Axial Velocities, with Rotation)
58
Table A . l Physical parameters
85
Table A.2 Hydraulic Pump
85
Table A.3 Motor and Amplifier
86
Table A.4 Servo Valve
87
Table A.5 Pressure Transducers
88
Table A.6 Potentiometer
88
Table A.7 Data Aquisition
89
Table B . l Simulation and Experimental Values
90
List of Figures
Figure 1.1 Hydraulic Actuator
1
Figure 1.2 ISE's Hysub Remote Operated Vehicle
3
Figure 1.3 Hydraulic Actuated Manipulator
3
Figure 2.1 Da Vinci's Model of Friction
6
Figure 2.2 Basic Model of Friction
7
Figure 2.3 Classical Model of Friction
8
Figure 2.4 Stribeck Curve for Lubricated Surfaces
9
Figure 2.5 Stribeck Curve Regimes
10
Figure 2.6 Material Contact at Asperities
10
Figure 2.7 Dahl's Spring Model
11
Figure 2.8 Rising Static Friction
13
Figure 2.9 Frictional Memory
14
Figure 2.10 Hysteresis Effect
14
Figure 2.11 Double Acting Hydraulic Actuator
15
Figure 2.12 Sealless Tapered Pistons
19
Figure 3.1 Block Diagram for Hydraulic Actuator System
21
Figure 3.2 Hydraulic Actuator with DC Motor
27
Figure 3.3 Block Diagram for Hydraulic Actuator System with a Rotating Piston
27
Figure 3.4 Helical Motion of an Element on a Rotating Piston
29
Figure 3.5 Vector Components on an Element on the Piston
30
Figure 3.6 Piston Position for the Standard and Rotating Model
33
Figure 3.7 Friction Curves for the Standard and Rotating Model (460 rpm)
34
Figure 3.8 Viscous Friction Parameter vs. Rotation Speed
35
Figure 3.9 Friction versus Velocity for a Slow Angular Velocity (50 rpm avg)
36
Figure 3.10 Step Input - Piston Position
37
Figure 4.1 Experimental Setup
39
Figure 4.2 Hydraulic Actuator, Motor, and Servo-Valve
40
Figure 4.3 Quasi-Static Experiments: Applied Force and Acceleration Force
44
Figure 4.4 Quasi-Static Velocity Curve
45
Figure 5.1 Rotating Friction - Rotating Velocity Curve
49
Figure 5.2 Hydraulic Actuator Axial Friction - Axial Velocity Curve at 0 rpm.
51
viii
Figure 5.3 Axial Friction - Axial Velocity Curve at 10 rpm (0.033 m/s)
52
Figure 5.4 Axial Friction - Axial Velocity Curve at 25 rpm (0.083 m/s)
53
Figure 5.5 Axial Friction - Axial Velocity Curve at 50 rpm (0.17 m/s)
54
Figure 5.6 Axial Friction - Axial Velocity Curve at 75 rpm (0.25 m/s)
55
Figure 5.7 Axial Friction - Axial Velocity Curve at 100 rpm (0.33 m/s)
55
Figure 5.8 Axial Friction - Axial Velocity Curve at 125 rpm (0.42 m/s)
56
Figure 5.9 Axial Friction - Axial Velocity, Actuator Slipping at 100 rpm
57
Figure 5.10 Average Percent Axial Static Friction Reduction
59
Figure 5.11 Axial Coulomb Friction Parameter vs. Rotation Speed
60
Figure 5.12 Axial Stribeck Friction Parameter vs. Rotation Speed
60
Figure 5.13 Axial Static Friction Parameter vs. Rotation Speed
61
Figure 5.14 Axial Viscous Friction Parameter vs. Rotation Speed
62
Figure 5.15 Axial Stribeck Velocity Friction Parameter vs. Rotation Speed
63
Figure 5.16 Axial Bristle Spring Constant vs. Rotation Speed
63
Figure 5.17 Axial Bristle Damping Coefficient vs. Rotation Speed
64
Figure 5.18 Hydraulic Actuator Model Comparison at 0 rpm
66
Figure 5.19 Hydraulic Actuator Model Comparison at 10, 25, and 50 rpm
66
Figure 5.20 Hydraulic Power Requirements
67
Figure 5.21 Percent Reduction in Hydraulic Power (at 0.008 m/s axially)
68
Figure 5.22 Total Power Requirements
69
Figure A . l Hydraulic Actuator PID
84
Figure A.2 Motor amplifier signal conversion
86
Figure A.3 Servo valve signal conversion
87
Figure A.4 Anti-Aliasing Filter
89
ix
Acknowledgement
There are three people who deserve a very special thank you. I would like to start by thanking
my wife, Mary Wells, for her support throughout this degree. Her intellectual stimulation helped
to drive my thirst for knowledge. Andrea Zaradic deserves to be thanked as well, not only did
she introduce me to my wife, she also gave me the extra push I needed when I was
contemplating returning to school for graduate studies. She also told me about a professor at the
University of British Columbia who was looking for students. The third person at the top of my
list to be thanked is Elizabeth Croft. She took a chance by accepting me as a student and I have
not looked back since. Her enthusiasm is never ending and contagious.
James McFarlane of International Submarine Engineering must also be thanked.
Our first
meeting started with a 5:30 am phone call in October 1998 and led to this project. His financial
and intellectual contribution is appreciated.
The Science Council of British Columbia also
contributed to this project through a GREAT Scholarship and their support was invaluable.
My family deserves to be thanked. I will start with my brother Bob Owen, those years on 16
th
Avenue were full of good times; my sister Marne Owen, who still continues to pursue a higher
education while working; and my parents, Scott and June Owen, who always taught us to work
hard and go after what we want.
Doug Yuen from the machine shop and Glen Jolly from instrumentation must also be
recognized. Their technical contribution helped the project along.
Fellow students, Damien Clapa, Jason Elliott, David Langlois, and Sonja Macfarlane, with
whom I started my degree, should also be recognized. Daniela Constantinescu should also be
thanked for her insightfulness that she brought to the lab meetings.
x
Dedication
This thesis is dedicated to my wife, Mary Wells, and our daughter, Patricia Ann Mary Owen,
born on July 21, 2001. The joy that these two people bring to my life continues to grow.
Chapter 1
Introduction
1.1
Preliminary Remarks
Hydraulic actuators provide high force, stiffness, and durability suitable for applications in
mining, machining equipment, and remote manipulator operations in unstructured environments
such as ground, sea, and space applications. Although in some cases, these applications can also
utilize pneumatic actuators or electric motors, hydraulic actuators provide a strength and
durability that is unparalleled [1]. There is a growing need for such actuators to perform with
improved precision and repeatability for manipulation tasks such as remote assembly, repair, and
nuclear remediation. A recurring issue with hydraulic actuators is the level of friction present in
the system. This friction affects the controllability, accuracy, and repeatability of the actuator.
To achieve improved precision and repeatability, especially at low speeds, the problems related
to friction in hydraulic actuators must be overcome.
In a typical hydraulic actuator, as shown in Figure 1.1, movement of the rod, piston, and
hydraulic fluid are subject to friction. The contacts between the rods and the seals, and between
the piston o-rings and seals and the cylinder, and the viscous effects of the hydraulic fluid all
generate friction. This friction must be overcome before movement can occur. In hydraulic
manipulators, friction can reach 30% of the nominal actuator torque [2].
Figure 1.1 Hydraulic Actuator.
1
Friction is a complicated phenomenon that is still not fully understood. Friction models that
have been used in the literature range from being a simple constant force that opposes motion to
a seven parameter model including various behavioral characteristics such as stiction, a negative
viscous slope, frictional memory, and hysteresis. Armstrong-Helouvry et al. discuss several of
these models in detail [3]. Canudas de Wit et al [4] developed a new model in 1995, the LuGre
model, which incorporates many of these effects.
Before motion starts, the surfaces are in the static friction regime. A force greater than the static
friction is required for movement to commence. As the component starts to move the friction
suddenly decreases as it switches to the dynamic friction regime. This sudden change in friction
results in a jerky actuator motion, making positional control and repeatability difficult [2, 3, 5, 6,
7, 8]. This effect is commonly referred to as stick-slip friction. Stick-slip friction is a non-linear
friction phenomenon and can be found in hydraulic actuators around the zero velocity range.
Modeling of this sudden switching is difficult, and precise control of the system usually involves
complex system identification and prediction.
1.2
Motivation and Objective
The motivation behind this work is best provided through an example. Figure 1.2 is a remote
operated vehicle (ROV) from International Submarine Engineering Ltd. (ISE), a robotics and
submersible vehicle company located in Port Coquitlam, British Columbia. The ROV can be
used for subsurface applications such as maintenance, search and rescue, and underwater
research.
2
The objective of this research is to investigate a novel approach of avoiding and reducing friction
in hydraulic actuators. Stick-slip friction is encountered when the actuator switches directions
and must pass through zero velocity. Stick-slip friction is also encountered when the actuator
functions at speeds close to zero velocity where the switch from static to dynamic friction
occurs. In this case a cycle of sticking and slipping can occur [3]. If near zero velocity
operation can be avoided then it is expected that stick-slip friction will not be a problem. By
rotating the rod and piston [10] the actuator will be kept in motion, well away from zero
velocity, without moving axially. Since there is continuous motion, when the actuator is moved
axially, stick-slip friction will be avoided. The end result of reducing stick-slip friction is an
expected increase in controllability, accuracy, repeatability, and a reduction of jerk in hydraulic
actuators.
1.3
Thesis Overview
The research studied the effects of rotating the piston on the friction developed in a hydraulic
actuator. A non-rotating hydraulic actuator was modeled using a state space approach in order to
obtain a base line of performance. Next, a rotating actuator model was developed to provide
some guidance as to what effects the rotation would have on the friction. Finally, experimental
measurements validated the non-rotating model, confirmed that rotating the piston would reduce
the amount of friction present in a hydraulic actuator, and that the Stribeck curve would be
eliminated. The outline of the thesis is as follows:
Chapter 2: Friction in Hydraulic Actuators: Friction in lubricated machines is introduced.
Particular attention is paid to friction in hydraulic actuators with a discussion on the current
methodologies in the industry. The difference between controlling the actuator in the presence
of friction, through modeling and observers, and trying to reduce the problem of friction will be
highlighted.
Chapter 3: The Hydraulic Actuator Model: A state space model for a non-rotating hydraulic
actuator will be presented. This model will utilize a friction model that has been adopted from
the literature. Then a new model for a hydraulic actuator that incorporates a rotating piston will
be presented.
The same friction model will be used but it will be a function of multiple
velocities instead of just one velocity. The rotating model will provide guidance as to what to
expect from the experimental tests.
4
Chapter 4: Friction Identification in Hydraulic Actuators: Experimental Method:
The
method to determine the friction parameters will be presented. Quasi-static experiments were
conducted to obtain friction values over a range of velocities that includes the Stribeck curve and
extends into the viscous region of the friction - velocity curve. These experiments were used to
obtain the axial friction parameters for the hydraulic actuator with no rotation and while rotating
at various speeds. These techniques were also applied to identify the rotating friction parameters
of the motor, piston, and rod with no axial motion.
Chapter 5: Reducing Stick-Slip Friction in Hydraulic Actuators: Experimental Results:
The experimental results will be presented. This includes the rotating friction - rotating velocity
curve and rotating friction parameters of the system for rotating with no axial motion, and the
axial friction - axial velocity curve and the axial friction parameters of the actuator at speeds
ranging from 0 rpm to 125 rpm.
The axial friction parameters determined through
experimentation will be used in the non-rotating model of the hydraulic actuator to validate the
model. The power requirements to move axially and to rotate the system will be compared.
Chapter 6: Conclusions and Suggestions for Future Work: Conclusions stemming from this
research are summarized. Suggestions for further work are provided.
5
Chapter 2
Friction in Hydraulic Actuators
2.1
Friction in Lubricated Machines
Armstrong-Helouvry, Dupont, and Canudas de Wit [3] provided an extensive survey of friction
research, some of which is summarized in the following section. A clear picture of the friction
1
phenomena and the problems found with friction in lubricated machines is important to the
development of the friction avoidance technique presented in this thesis.
2.1.1
Basic and Classical Models of Friction
Da Vinci (1519) first postulated that the friction force is proportional to the normal load, as
shown in Figure 2.1. Da Vinci believed the coefficient of friction to be dependent on the
characteristics of the contact areas and remained constant. In reality, coefficients of friction are
dependent on surface characteristics that are dependant on time, temperature, lubrication, and
other variables.
A friction force opposes the direction of motion and is independent of the contact area between
the two surfaces. The net driving force is the difference between the applied force and the
frictional force.
F,Normal
F Net Drive = FApplied - F,Friction
F Applied
77777/
F,Friction
•
//////
Figure 2.1 Da Vinci's Model of Friction.
The bulk of the information in this section comes from reference [3]. Figures 2.6, 2.7, 2.8, and 2.9 are modeled
after those in reference [3].
1
6
Coulomb (1785) introduced the concept of a dry or Coulomb friction.
The friction force
opposing motion was constant and independent of the velocity,
M'Dynamic^^Normal>
^Dynamic
where F
\s
Dynamic
the dynamic friction force, F
(2*1)
is the normal force, and Moynamic is the
Nomal
dynamic coefficient of friction. Morin (1833) stated that there was a threshold friction force that
had to be overcome before movement occurred:
^Static ~ MSialic ^Normal '
where
fJ-
Dynamic
< fi
Slalic
(2-2)
. F tic is the static friction force and u,tatic is the static coefficient of
Sta
S
friction. This is the "Basic Model of Friction" and is shown in Figure 2.2.
Friction Force
Sialic
Coulomb "
>Velocity
Figure 2.2 Basic Model of Friction.
Reynolds (1866) made a significant contribution to understanding friction with his work in
viscous fluid flow. Viscosity is the ability of a fluid to resist shear. More formally, viscosity
relates momentum flux to the velocity gradient. It is the property of a fluid that relates applied
stress to the resulting strain rate [11]. Applying this property yields:
F.
Bx,
w
where B is the viscous damping coefficient, F
Viscous
(2.3)
is the viscous friction, and x is the velocity.
The friction model then becomes:
F
=F
Sialic
Applied
F
=F
+F
Dynamic
Coulomb
1
1
\x\ = 0,
T
1
x> 0
Viscous
(2.4)
A
^
7
This is the "Classical Model of Friction", Figure 2.3.
Friction Force
Figure 2.3 Classical Model of Friction.
Here the model shown is symmetrical. This is not always the case. Friction can be direction
dependent as discussed in [5, 12, 13].
2.1.2
Stribeck Curve
One of the main problems with the Classical model is the discontinuity between static friction
and dynamic friction. The Classical model does not provide a sufficient representation of
friction, especially for a lubricated application. Under lubrication the discontinuity is softened.
However, the nature of the discontinuity is such that it is still difficult to compensate for.
Stribeck (1902) developed the Stribeck Curve. The change in static friction to dynamic friction
was recognized as being continuous, as shown in Figure 2.4 [3, 14]. The steep negative slope or
negative viscous slope shows that there is a continuous change in the friction.
8
Friction Force
>•
Velocity
Figure 2.4 Stribeck Curve for Lubricated Surfaces.
As shown in Figure 2.4, the Coulomb friction parameter is measured at the intersection of the
viscous friction curve and the friction axis. The Stribeck friction is the difference between the
static friction and the Coulomb friction [3]. The Stribeck curve applies to lubricated surfaces. If
the surfaces are dry and unlubricated, then the change from static to dynamic friction can be
considered essentially discontinuous, as in the Classical model.
The Stribeck Curve can be separated into four different regions, each displaying the different
characteristics of friction. These different regimes are:
• Regime I:
Static Friction (also known as stiction)
• Regime II:
Boundary Layer Lubrication
• Regime III:
Partial Fluid Lubrication
• Regime IV:
Full Fluid Lubrication
These different regimes are shown in Figure 2.5.
9
Friction Force
A
Regime I
^1
Regime II
Figure 2.5 Stribeck Curve Regimes.
2.1.2.1 Regime I: Static Friction
Regime I is the static Friction regime where there is no apparent relative velocity between the
contact surfaces and no appreciable sliding or movement occurs. The contact between the two
surfaces occurs at asperities (microscopic roughness) as shown in Figure 2.6. The contact area
between the two surfaces is relatively small compared to the total area of each surface. This is
due to the surface imperfections on each surface and the difficulty in obtaining a purely flat and
smooth surface at the microstructure level.
Boundary Lubricant
Asperities
Figure 2.6 Material Contact at Asperities.
Figure 2.6 shows a boundary lubricant on the surfaces of the materials. Many lubricants have
additives that leave deposits on the surfaces. These deposits help to reduce the coefficient of
friction between the surfaces and therefore reduce friction. The choice of lubricant affects both
the surface friction and the surface wear. Some lubricants, called way oils, can actually lower
10
the level of static friction below the level of Coulomb friction. However, the characteristics of
these lubricants change over time and cannot always be relied upon. Through continuous use the
lubricant can break down and as wear increases the lubricant becomes dirty, losing the properties
that it was chosen for in the first place. Furthermore, in applications such as hydraulics the
lubricant is the hydraulic fluid, which will not necessarily have such desirable properties.
Before sliding occurs there can be elastic deformation between the asperities. This is known as
the Dahl Effect where there is a pre-sliding dislocation. Dahl (1968, 1976, 1977) modeled the
asperities as springs, Figure 2.7, where the friction force depends on the displacement and not
the velocity:
Ffriction
=
~k,x.
(2.5)
When noticeable movement occurs the friction force has reached the breakaway force, that is the
static friction level, and the 'springs' have then been broken.
Figure 2.7 Dahl's Spring Model.
The term static friction is often considered to be somewhat misleading. Friction is considered to
be a function of velocity and since there is no velocity or sliding in the static friction regime then
friction cannot exist. Polycarpou and Soom (1992) refer to the friction force at zero velocity as a
tangential force or a force of constraint.
2.1.2.2 Regime II: Boundary Lubrication
As previously mentioned, many lubricants have additives that leave a deposit on the surfaces of
the materials. This leads to Regime II: Boundary Lubrication. With hydrodynamic lubrication a
minimum velocity is required to draw the lubricant in between the surfaces. In Regime II the
relative velocity is below that minimum and there is no lubrication except that provided by the
boundary lubricant. Movement occurs when the applied force is greater than the static friction
11
and the asperities are sheared. This regime of the Stribeck curve experiences solid to solid
contact.
2.1.2.3 Regime III: Partial Fluid Lubrication
In Regime III the velocity has increased to a point where the lubricant begins to be drawn into
the area between the surfaces. The fluid lubrication increases and the solid to solid contact
decreases. There is a partial support of the surfaces by the fluid and a partial support by the
asperities. As the velocity increases the surfaces are supported more and more by the fluid and
less by the asperities. As a result the resistance to movement decreases (the friction decreases),
and under constant applied force, acceleration of the moving body increases. With increasing
acceleration, the velocity increases and more lubricant is drawn in. This positive feedback cycle,
results in an unstable system response.
The negative slope of the friction curve, referred to as a negative viscous slope, is responsible for
this unstable condition, and leads to most of the problems related to friction compensation.
When motion is initiated, the applied force increases until it is large enough to overcome the
static friction. Then, as motion begins the friction decreases rapidly. The sudden decrease in
friction results in the net force being higher than desired and a jerky motion results. A similar
phenomenon occurs as the body is brought to rest. As the body decelerates the friction force
suddenly increases. The sudden increase in friction results in the net force being lower than
desired and the body comes to a sudden halt prior to reaching the desired set point. These
starting and stopping effects result in overshooting or undershooting the desired trajectory.
Regime III also experiences a frictional memory phenomenon where there is a time lag between
a change in velocity or load conditions and the resulting change in friction. This results in a
hysteresis effect. Frictional memory will be discussed in Section 2.1.3.2.
2.1.2.4 Regime IV: Full Fluid Lubrication
When full fluid lubrication occurs all solid to solid contact has been eliminated and the surfaces
are supported entirely by the lubricant. In this area the friction is near linear.
2.1.3
Stick-Slip Friction
Stick-slip friction behaviour , which occurs near zero velocity, covers Regimes I, II, and III. The
various factors that contribute to stick-slip friction are:
12
• Static Friction
• Rising Static Friction
• Frictional Memory
• Negative Viscous Slope
Static friction, rising static friction, and frictional memory are considered in the following
sections.
2.1.3.1 Static Friction and Rising Static Friction
Static friction was discussed under Regime I. A sub-topic of static friction is rising static
friction. Several researchers considered rising static friction, such as Rabinowicz (1958) and
Kato et al. (1972). Static friction is a function of time as shown in Figure 2.8.
A
F
Sialic
^
|
Time at Zero Velocity
Figure 2.8 Rising Static Friction.
As shown in Figure 2.8 a short period of rest results in a breakaway force with a value between
the Coulomb friction and static friction. A long period of rest results in the breakaway force
approaching static friction. It is hypothesized that rising static friction is related to the time it
takes for the asperities to effectively weld together, while the surfaces are at rest.
Counter to the dwell time theory, several researchers have considered varying the rate of force
application. Rabinowicz, Kato, and other's work considered a constant rate of force application,
thus dwell time was a function of the force application rate.
Johannes et al. (1973) and
Richardson and Nolle (1976) investigated independent variation of the force rate and dwell time.
They demonstrated that static friction is not a function of dwell time but is a function of the force
13
application rate. Recent work by Canudas de Wit et al. [4] showed that static friction is
independent of the time at rest but is dependent on the rate of force application.
2.1.3.2 Frictional Memory
There is a time lag in the change of friction following a change in velocity. It is hypothesized
that the physical process is related to the time required to modify the lubricating film after the
determining parameters are changed. In other words, it takes time for a system to come to a new
steady state when the determining parameters are changed.
Empirical models represent
frictional memory as a delay in the friction response to velocity:
Fpricuon
=
?velocity (*('
~ ))
At
,
(2.6)
where xis the velocity and At is the time lag, Figure 2.9. Frictional memory results in a
hysteresis effect as shown in Figure 2.10.
Friction
Friction
o
o
"3
>
JO
"5
>
s=
.o
%H
_o
"C
fa
Time Lag
co
Velocity
Velocity
Time
No Frictional Memory
Time
Frictional Memory
Figure 2.9 Frictional Memory.
CD
CD
«H
O
fa
Acceleration
co
fa
Deceleration
Velocity
Figure 2.10 Hysteresis Effect.
14
The same environment can result in two different frictional forces leading to problems in
modeling and control. A large hysteresis effect can result in unmodeled disturbances, which,
when combined with the general instability of the Stribeck curve, can lead to undesirable limit
cycling (hunting) by the controller.
2.2
Friction in Hydraulic Actuators
Until recently, not much work had been done on friction in hydraulic actuators. Tafazoli states
in his thesis from 1997 "To our best knowledge, there is no published work on friction modeling,
estimation, and compensation for hydraulically actuated manipulators" [14].
The literature
review showed that there had been very little work on hydraulic actuators and friction prior to
1996.
Tafazoli determined that friction in hydraulic actuators consumes a large part of the applied
actuating force. Tafazoli worked on a work cell for the decapitation of salmon at the Industrial
Automation Laboratory at the University of British Columbia. He showed that there is a
considerable amount of static and Coulomb friction in the actuator and the guide ways that
position the fish [14]. With his work on a mini-excavator Tafazoli showed that the hydraulic
actuators again contain a large amount of static friction [14].
Lischinsky et al. [2] showed that in a Schilling Titan II manipulator the joint friction can reach
30% of the nominal actuator torque.
They attributed this friction to the tight seals that are
required to prevent leaks. Figure 2.11 shows a simplified diagram of a double acting hydraulic
actuator. The main contributors to friction will be the lip seals, the piston seals, and the piston oring.
Hydraulic Ports
Cylinder
Piston
Lip Seals
3
3
Rod
Piston Seal
Piston O-Ring
Figure 2.11 Double Acting Hydraulic Actuator.
15
Lischinsky et al. have also determined that the Schilling Titan II manipulator can have a 25%
drop between static and Coulomb friction [2]. The impact of the negative viscous slope in their
application is quite significant.
Hsu et al. [21] and Kwak et al. [22, 23] considered system identification of friction in hydraulic
actuators. Kwak et al. [22] states "The requirements imposed by today's high precision machines
motivates the precise simulation of friction between these seals and sliding components..."
Recent work by Bonchis et al. [13] showed that friction in asymmetric hydraulic actuators is
direction and location dependent. The direction dependency was attributed to the seal exhibiting
different characteristics depending on whether the rod was being extended or retracted. The
pressure differential between the atmospheric pressure and the chamber also influenced the
friction. The chamber will develop a different pressure depending on whether the rod is being
extended or retracted. The pressure differential affects the seal and its pressure on the rod.
The location dependency was attributed to wear in the actuator. The friction parameters for a
hydraulic actuator can change depending upon where in the actuator the piston is located.
Typically, for a well placed actuator, the center of the actuator will have more wear, and the ends
will not have as much wear.
There are two approaches to dealing with friction. The first is to design the control system to
compensate for the friction. The alternative is to design the hydraulic actuator such that it avoids
friction. Armstrong-Helouvry et al. [3] discuss friction avoidance as a possibility in the design
of equipment. This latter method is not used very often as control techniques are the standard
approach.
2.2.1
Control of Hydraulic Actuators in the Presence of Friction
Recent approaches to the stick-slip friction problem in hydraulic actuators have been through
various control techniques such as: model based friction compensation [14, 15], observer-based
friction compensation [14, 18, 19], model/observer-based adaptive friction compensation [2, 4],
nonlinear PI control [8], and generalized predictive control [1] where all the nonlinearities in an
electrohydraulic system are captured.
The above topics that explicitly include friction
compensation will be discussed further.
16
2.2.1.1 Model Based Friction Compensation
In [14] Tafazoli used both model based and observer based friction compensation in a hydraulic
system. The former will be discussed here and the latter will be discussed in the next section. In
both applications Tafazoli used the estimated friction to estimate the acceleration:
F
>v
Applied
- F
Friction
,^ _*
M
where F
Applied
is the applied force, F
is the estimated friction force, M is the system mass,
Friction
and a. is the estimated acceleration. The estimated acceleration was used in a control law that
included position feedback, velocity feedback, and acceleration feedback.
In the model based, approach Tafazoli [14, 15] used the modified Tustin model [16] to estimate
friction:
Dynamic
F
where F
Dynamjc
~ «lH" + OC \x\) Sign( ) ,
(2.8)
2
= ( « 0
2
X
is the dynamic friction, x is the velocity, a is the Coulomb friction, a is the
0
Stribeck friction parameter, and a
2
x
is the viscous friction parameter. Tafazoli then included a
term to represent the static friction near zero velocity:
F
Friction
=F
Dynamic
+(F
\
Applied
-F
\|exp
Dynamic I
f
V
(
• \ 2\
X
(2.9)
)
where D is a threshold velocity that represents where the switch from static friction to dynamic
v
friction occurs [17]. When the velocity was less than D equation (2.9) was used; when the
v
velocity was greater than D , the friction was dynamic and equation (2.8) was used. As the
v
control system did not measure velocity, Tafazoli used a velocity observer. The velocity was
estimated in real time using a low pass-filtered differentiator:
7> + l
where T is the filter time constant and s is the Laplace operator.
v
The model based approach proved to be better than a conventional proportional-derivative (PD)
controller. However, as Tafazoli realized, his model based approach was not adaptive and
17
considered friction to be time invariant. The friction parameters were estimated off-line and
were not updated on-line.
2.2.1.2
Observer Based Friction Compensation
To consider the time variance of friction Tafazoli used a modified Friedland-Park Coulomb
friction observer to estimate and compensate for friction in a hydraulic actuator [14, 18, 19].
The Friedland-Park friction observer [24] depends on a measured velocity. As before, Tafazoli
incorporated a velocity observer using an on line low pass-filtered differentiator, equation (2.10).
This estimated velocity was used in the Friedland-Park observer.
The observer provided a more adaptive approach, as the friction estimation relied more on the
current estimated velocity and acceleration and not a set friction parameter. The observer also
caught the hysteresis found in friction at low velocities whereas the modified Tustin model did
not [14]. Tafazoli's observer significantly outperformed a conventional proportional-derivative
(PD) controller and was comparable to the model based approach [14].
2.2.1.3
Observer Based Adaptive Friction Compensation
Lischinsky et al. [2] applied an adaptive control scheme to a hydraulic system using the LuGre
model of friction (reference Section 3.2.5). As the parameter z , representing the average bristle
deflection, is not measurable, it has to be estimated using an observer. The friction estimation is
then used in a control system with an outer position control loop and an inner torque control
loop. The position control is a conventional PD controller. The torque loop considers the
dynamics of the system including the estimated friction. Using friction compensation showed a
significant improvement over the same system without friction compensation.
To consider the time variance of friction Lischinsky et al. [2] included an adaptive control
scheme. Equation (3.31) is rewritten as:
dz .
o* \X\
— =x - e - ^ z ,
dt
g(x)
(2.11)
where 6 is an adaptive parameter that is estimated during controller operation. The friction
parameters, particularly Coulomb friction, may be updated on occasion using the current value
for 6 as a multiplier. Lischinsky et al. found the adaptive friction compensation scheme
provided the best results.
18
2.2.2
Friction Avoidance
Friction avoidance is the design of machines such that friction is reduced or avoided. Meikandan
et al. [7, 31] looked at sealless, tapered pistons where they found that the friction would be
reduced.
Meikandan et al. considered three types of pistons: diverging, converging, and a
converging-diverging piston, Figure 2.12. The arrows indicate the direction of fluid flow.
//////
//////
Diverging
//////
—
//////
Converging
//////
//////
Converging-Diverging
Figure 2.12 Sealless Tapered Pistons.
Their initial studies were theoretical [31]. The friction in the converging-diverging piston was
calculated to be one-fifth of a similar conventional piston with seals.
Considering the taper
angle, eccentricity of the piston, and the velocity, the converging-diverging piston is preferred.
Though it has more leakage than the other two pistons, it always has a positive centering force
that keeps the piston in the center of the cylinder. The other two pistons can develop a negative
centering force pushing them into the cylinder wall. The diverging piston develops a negative
centering force below a critical velocity. The converging piston develops a negative centering
force above a critical velocity.
Meikandan et al. [7] confirmed their theoretical work with experiments. The diverging piston
had high friction values at low velocities, corresponding to the negative centering force causing
piston contact with the cylinder wall. At high velocities the friction force was low and viscous in
nature.
The converging piston had high friction values at high velocities when the negative centering
force caused the piston to contact the cylinder wall. At low velocities the friction values were
low and viscous in nature.
The converging-diverging piston exhibited low friction values that were viscous in nature at all
velocities.
19
2.3
Summary
Friction is problematic in the accurate positioning and repeatability of hydraulic actuators. This
is a result of the Stribeck effect; namely the negative viscous slope portion of the friction velocity curve.
The usual approach to compensate for friction is through various control
techniques. For example, acceleration feedback using either model based friction estimation or
observers to estimate friction has been shown to be successful [14].
Friction avoidance using tapered and sealless pistons has also been employed [7, 31]. This
approach reduced the friction significantly but required a high hydraulic fluid flow rate.
The approach presented here considers the rotation of the piston and rod to reduce friction. It is
expected that once the piston and rod are moving the Stribeck effects will be avoided.
20
Chapter 3
The Hydraulic Actuator Model
3.1
Modeling
The primary purpose of friction models has been in the use of observers in control systems [2, 3,
4, 14, 15, 24]. The friction model prediction is used by the controller to compensate for
frictional disturbances in the system.
On the other hand, the purpose of modeling hydraulic actuators in this work is to predict the
effects that modifying the design of hydraulic actuators will have on the friction in the actuator.
The models will be validated through comparison with empirical results.
3.2
Non-Rotating Model
The model for the hydraulic actuator was developed with reference to a number of sources [2,
14, 25, 26, 27, 28]. The friction parameters for the hydraulic actuator are derived from data
made available by Tafazoli [14]. Since hydraulic actuators are highly non-linear, Jacobian
linearization, was used in the development of the model. The block diagram of the system is
shown in Figure 3.1.
*P
Controller
Gain = K
p
Servo
Valve
»
PL '
Z
Hydraulic Actuator
4 x 4 Model
X
P
Figure 3.1 Block Diagram for Hydraulic Actuator System.
3.2.1
Servo Valve
The flow of the hydraulic fluid to the actuator is controlled with a servo valve. This has been
modeled as a first order system [26]. The equation relating the spool position, x , to the control
iV
voltage, V , is:
m
21
where K is the valve position gain and T is the time constant of the valve. The control
SV
sv
voltage of the valve is limited to ±10 volts. A high gain in a position feedback loop may produce
a voltage greater than the upper and lower limits. To effectively model this limitation, the valve
control voltage in the model had upper and lower bounds of ±10 Volts.
3.2.2
Fluid Flow
The position of the servo valve spool controls the flow rate in, Q , and out, Q , of the hydraulic
x
2
actuator. However, the flow rate also depends on the supply pressure, P , the tank pressure, P ,
S
T
and the pressure in the actuator chambers, P and P . The following equations determine the
T
2
flow rate through the servo valve [14, 25, 26, 27]:
a = KX (S(XJ4P~^
SV
+ Si-x^Z-P,),
(3.13)
+ S(rx„)JP -P ),
(3.14)
Q = Kx„(s(x„)JP -P
2
2
T
S
2
where K is the servo valve flow gain and S(x ) is a switching function that indicates whether
sv
the actuator is extending or retracting:
S{x„)
=\
x„>0
= 0
*„<0
(3.15)
Under ideal conditions there is no leakage and Q = Q = Q,. Equating Q and Q (equations
x
2
2
(3.13) and (3.14)) produces the following relation:
P =P +P ,
S
L
(3.16)
2
where the tank pressure is equal to zero. In other words, the sum of the pressures in each of the
actuator chambers is equal to the supply pressure. Defining the pressure across the load as [25]
P =P,-P
L
(3.17)
2
the following relations can also be established:
P +P
P=
5
L
{
P
and,
(3.18)
-P
P =^y±.
(3.19)
Differentiating the above two equations yields the pressure rate of change in each chamber:
22
2
PL
and,
(3.20)
(3.21)
Substituting equations (3.18) and (3.19) into either equation (3.13) or (3.14) yields the servo
valve flow equation:
Q =Kx
L
(3.22)
s
Taking fluid compressibility into account, and using the continuity principle, the following
equations can be derived [14, 25, 26]:
Q =A x +
l
l
Qi - Ai
p
A,x + V .
'
and
n
1
h
h
(3.23)
A (L-x ) + V
7,
P-2 '
2
x
P
p
h
(3.24)
1
where A and A are the piston areas, L is the actuator stroke length, V is the hose volume
x
2
h
between the servo valve and the actuator, x is the piston axial position, x is the piston axial
p
p
velocity, and B is the effective bulk modulus of the oil.
If the two flows are added, Q + Q = 2Q , the result can be used to eliminate the dependency
t
2
L
on position:
QL
3.2.3
A +A
x
=
2
+
P
X
2V +A L
h
2
(3.25)
PL-
Pressure Changes
Equating the servo valve flow equations (3.13) or (3.14) with the continuity equations (3.23) or
(3.24), respectively, the pressure change in each of the hydraulic chambers can be determined:
Pi =
v
f
v
h
P =
2
V
V
M
h
+
(S(*„)4 s ~ .
p
A
A
+ Ax
+
(3.26)
<r*»)V^)-4* l
s
P
p
/! _
2\
A
p
L
, [KX {S(X )4F +
SV
SV
2
(-xjJP -P )-A x
S
s
2
2
p
(3.27)
p)
X
23
There are two reasons that the above two equations cannot be used. First, these equations show
the pressure change as being dependent on the piston position, where in fact it only depends on
the rate of change in position, ie. velocity. Second, under ideal conditions P = — P ; therefore
X
2
using equations (3.26) and (3.27) for P and P produces a dependency in the state space
x
2
matrices, and the state transition matrix is singular. The model becomes unstable. Under the
original assumption of ideal conditions with no leakage, P = P + P must be satisfied.
s
l
2
Both problems can be solved by considering the pressure to be one state represented by the load
pressure, P , instead of two different states [29].
L
Instead of working with the individual
chamber pressures, the load pressure and load flow can be used to eliminate the dependency on
position and to eliminate the singularity. By equating equations (3.22) and (3.25) for Q and
L
solving for P , , the rate of change of the load pressure is:
4/3
P,
=•
2V +A L
h
A +A
Kx.
x
2
(3.28)
2
A saturation experienced by the system is pressure [28]. During simulations the pressures in the
hydraulic chambers can exceed the supply pressure. As this is not realistic upper and lower
bounds, equal to the supply pressure, were placed on the chamber pressures.
3.2.4
Dynamics
Applying Newton's second law to a hydraulic piston yields the equation of motion for an
actuator:
FActuator
where
F
A c t u a t o r
^1 A
is the applied force,
F
^2
A
L o a d
—
MX
p
+ F
p F r t c U o
„
is the external load,
Flood
F
p F r t c t j o n
'
(3.29)
is the friction opposing
axial motion, M is the system mass, and x is the system acceleration. Setting the applied load
p
to zero and substituting in equations (3.18) and (3.19) the equation of motion may be rewritten
as:
A, A-.
M
A + A,
P
L
- F
pFriction
(3.30)
24
During the simulations the external load was set to zero. A n external load does not affect the
state space equations during linearization. The external load does determine the initial pressures
in each chamber during static equilibrium.
3.2.5
Friction
Contact between two surfaces occurs at surface asperities. Relative motion between the two
surfaces results in the asperities behaving like a spring-damper system. Movement is resisted
until the bond between the asperities breaks, or the asperities are sheared. The force required to
break the bond between the two surfaces for movement to start is the static friction [3, 4,6].
Haessig and Friedland [6] developed a model where the asperities were modeled as bristles.
Canudas de Wit et al [4] and Lischinsky et al. [2] represented the average deflection of the
bristles by a state variable z . This model is known as the LuGre (Lund-Grenoble) model [4]:
dz
— =i
dt
°~ \x\
-^-z,
g(x)
0
(
g(x) = a + a, exp
0
V
FH«io»
F
where F
(3.31)
• \ 2\
X
U J
(3.32)
J
dz
=°~oz + o- — + a x,
dt
x
(3.33)
2
is a generic friction force, dz/dt is the rate of bristle deflection, g(x) is a function
Frictlm
describing the steady state friction characteristics at a constant velocity [4], v
sk
is the Stribeck
velocity defined as being the most unstable velocity on the Stribeck curve [30], x is a generic
velocity, a
is the Coulomb friction, a is the Stribeck friction, a
0
x
is the viscous friction
2
parameter, cr is the bristle spring constant, and o is the bristle damping coefficient. The static
x
0
friction is equal to Cf + or,.
0
The friction parameters are divided into four static parameters, a , a , a , and v , and two
0
x
2
sk
dynamic parameters, o~ and CT, . These friction parameters are difficult to estimate since the
0
model is nonlinear in parameters and the average deflection of the bristles cannot be measured.
A method of obtaining the friction parameters will be discussed in Chapter 4.
25
The LuGre model is suitable for a state space approach and, unlike other models, does not
require a separate component for static friction [3, .14]. Instead, the deflections of the bristles are
used to calculate the pre-displacements and thus determine the pre-displacement forces (static
friction).
The above equations (3.31), (3.32), and (3.33) can be combined to give one equation for friction
that can be used in the actuator equations of motion:
O~ O~AX\Z
FFriction — O~ Z + <7 X •
0
0
X
a + a exp
0
(
• \2 \
X
x
(3.34)
+ a x.
2
)
V
The velocity along a specific direction of motion replaces the generic velocity (i) to obtain a
specific directional friction. For example, for FpFrjctjon „ x = xp.
3.2.6
State Space Model
Including equations (3.28), (3.30), and (3.31), a 4 x 4 state space model of the actuator can be
formed. The state variables are:
[x, x2 x3 x ] = \xp xp PL z\ ,
4
and the state space equations are:
A
x
A
(3.35)
—x,
Xj
2
A +A
2
x
CT CT,X
2
0
(
M
' a +a
0
4B
2V +A L
h
Kx. S(xv^^^
x
S(-x )J^l
+
v
4
X
2
,
,2\
22
a
X
,(3.36)
exp
\
A
V
skJ
A +A
r
l
2
and (3.37)
2
a + a exp
0
x
V
'X
2\
(3.38)
N
J
26
It can be seen from the selected equations that the state space model is highly non-linear. The
equations are linearized and discretized for a computer simulation. Jacobian linearization for
non-linear systems is discussed in Appendix C.
3.3
Modeling of the System with Piston Rotation
Rotating the piston and rod generates a relative velocity between the rod and seals and between
the piston seals and the cylinder wall. Once the velocity is sufficiently high, and the rotation is
maintained, the switch from static friction to dynamic friction will no longer occur, as the
actuator only operates in the dynamic regime. When the piston moves axially, the axial friction axial velocity curve is linear and the Stribeck region is avoided. However, without a sufficient
rotating velocity the axial friction versus axial velocity relation will remain nonlinear.
The hydraulic actuator model was modified to include rotating the piston, resulting in a 6 x 6
state space model. Rotation of the piston was modeled with a permanent magnet brushless DC
motor coupled to the rod, Figure 3.2.
I— Hydraulic Ports
DC Motor
/
EE
<5
Lip Seals
Coupling
Cylinder
Ll Piston
Piston Seal
Piston O-Ring
Figure 3.2 Hydraulic Actuator with DC Motor.
The block diagram for the system is shown in Figure 3.3.
v.
Controller
Gain = K
p
p'
x
Servo
Valve
Motor and
Hydraulic Actuator
6 x 6 Model
PL ' > e»I
z
x
>•
Figure 3.3 Block Diagram for Hydraulic Actuator System with a Rotating Piston.
27
3.3.1
DC Motor
Torques and angular velocities are usually used to model motors. To remain consistent with the
hydraulic forces and force units, the motor torques and angular velocities were changed into
forces and 'linear' velocities acting on the piston surface.
3.3.1.1
Current
Applying Kirchoffs voltage law and taking into account the back electromotive force the
following equation represents the DC motor [26]:
L
where K
bemf
L
m
(3.39)
L R,
m
m
cyl
is the back electromotive force constant, L is the motor inductance, R
m
cyl
is the
radius of the piston, R is the motor resistance, V is the voltage applied to the motor, i is the
m
m
m
motor current, i is the rate of change of the motor current, and x is the rotating velocity of the
m
e
system. The angular velocity, co , of the system is:
g
cyl
K
3.3.1.2 Dynamics
The equation of motion for rotation is:
J_
jy
M
where F
eFriclion
(3.41)
OFriction
cyl
K
is the friction opposing the rotating motion, K is the torque constant, and x is
t
e
the rotating acceleration. The torque of the motor, r ,\s:
m
r =K,i .
m
(3.42)
m
Dividing by the radius of the piston produces the equivalent force,
lied
, acting on the piston
surface:
Ki
^Supplied
'
(3.43)
cyl
K
F
gFricljon
is equation (3.34) where the velocity is the angular velocity, x :
e
28
p0Friction — 0~ Z
Q
Xg
+ (TjXg
f
2
x
U*
V
3.3.2
+ ax
g
(3.44)
( Xg
• }
a + a exp
0
Z
J
J
Helical Motion
Combining the rotating and axial motion yields a helical motion for each element on the piston,
as shown in Figure 3.4. The direction along the helical motion, as shown, is considered to be
positive. By maintaining the same direction of rotation the helical motion will always have the
same sign regardless of the axial motion.
helical, x
h
rotating, x
e
Figure 3.4 Helical Motion of an Element on a Rotating Piston.
During operation, the helical friction force opposing the helical motion is modeled as:
CT CT, \X \Z
h
0
FhFriction ~ °~0
Z
G\ h
X
• + oi x^,
2
(3-45)
g( h)
X
where F
hFriclion
is the friction opposing the helical motion and x is the velocity in the helical
h
direction.
The friction forces opposing the axial and rotating motions are approximated by the vector
components of the helical friction force, Figure 3.5.
29
pFrictior,
F
v
N
hFriction
^
Figure 3.5 Vector Components on an Element on the Piston.
The rotating friction and axial friction components may be rewritten as:
F,
^hFriction
a
F
3
n
<
(3.46)
^
=— F
pFriction
.
(3.47)
hFriction '
x
h
x
x
since -^- = sin(0) and -7^-= cos(0).
x
An assumption was made that the axial and rotating
h
components of F
hFriclion
are approximately equal to F
pFriclion
and F
, respectively.
eFriclion
To eliminate the dependency between x , x , and x in the state space equations, setting
p
h
x
+ e'
=
h
x
x
l s
eliminated
2
p
h
in equations
= sign(x ) - sign{^x + x )-1
h
g
e
(3.46)
and (3.47).
Noting
that,
and ^jx + x\ = ^jx + x , the two system equations
2
2
2
p
9
(3.30) and (3.41) are then rewritten as:
x
pr
1
M
A, A~,
A. + A,
Q~o Z
X
P-
(3.48)
P
•\j
x
+
P
e
g(^ l + l )
x
x
x
and
1 m
M
GQXQZ
- <j x +
+ xl
g
x
cyl
R
e
0*0\ 8
zx
CCjXg
(3.49)
The motor torque and the hydraulic pressure combine into a resultant force that produces the
helical motion. If there is no rotary motion then the helical velocity defaults to equal the axial
30
velocity and, likewise, if there is no axial motion then the helical velocity defaults to equal the
rotating motion.
During operation it is the friction force opposing the helical force that is relevant. It is expected
that the friction opposing the motion can have a range of parameters depending upon the
influence of both the rotating and axial motion.
3.3.3
State Space Model
Including equations (3.28), (3.31), (3.39), (3.48), and (3.49) a 6 x 6 state space model can be
formed that represents a hydraulic actuator complete with a rotating piston. The state variables
are:
[x, x x x x x ] = \x x P z x i \ ,
2
3
4
5
6
p
p
L
g
m
and the resulting state space equations are:
X^ "™ X 2
Ml
I
i
2
i
J
s +
p
(A + AA
2
J
(TQX2 X^
I
—" "~" ^TtX-y
XI
2
f
0
AB
2V +A L
H
I
X2
...+.
a
(3.50)
5
+ a, exp
2
' x l
V V
v
+
*
(3.51)
x ^ ~ ^
J)
Kx,
(3.52)
2
r~
2
X ^ —— ^ X 2
X
a + ax exp
0
'
^x +x
v v
2
2
k
y2
2
5
^
(3.53)
J)
31
J_
M
0~QO~\ 4 5
X
-<7 X
l
5
+•
X
, and
2
a + a expj
0
x
V
V
*k
v
K.
x.
^bemf^S
f
R 6
X
M
Rcy!
(3.54)
J J
(3.55)
The model of the motor and thus the current, equation (3.55) was included in the state space
model of the actuator. The time constant of the motor is much great than the bandwidth of the
hydraulic system. The model of the motor could have been eliminated from the state space
model but was left in for completeness.
The motor could have been modeled separately but that would have introduced a time lag
between the back electromotive force and the resulting current.
Such a model would have
required the speed of rotation from the actuator model to be fed back to the motor model in a
negative feedback loop. Simulation results showed the difference between the two models was
less than one percent so the combined model was chosen.
3.4
Simulation
The baseline axial (ie. non-rotating) simulation results are consistent with the published
behaviour of hydraulic actuators in the literature [1, 14, 23, 32]. The parameter values used in
the simulation are found in Appendix B. The position control loop for the piston was closed
using a simple proportional controller to facilitate evaluation of the model behaviour without the
controller influencing the outcome.
Simulations were then run with the piston rotating over a range of speeds from 0 to 800 rpm. A
sinusoidal position reference input was used to evaluate the tracking ability of both the rotating
and non-rotating models while the piston velocity passed back and forth through the Stribeck
region. Figure 3.6 shows the piston position for the standard model and the rotating model for a
simulation run time of 1.25 seconds. The lag between the models and the desired piston position
is apparent near time zero and can be attributed to stiction. When the piston is rotated the phase
32
shift decreases. As the angular velocity of the piston is increased there is a corresponding
increase in the improvement of the phase shift.
0.061
1
1
„
™
1
T
,
™
„
Piston Rotation = 460 RPM
0.05
K
P
,
= 5 0 0
standard
Rotating
1
—
1
Phase Shift
8.55 degrees
r
7.65 degrees
/
/
0.04
£ 0.03
a
o
0.01
X
-0.01
d
Input
0.6
0.8
1
Time (s)
Figure 3.6 Piston Position for the Standard and Rotating Model.
0
0.2
0.4
1.2
The friction curves for the two systems are shown in Figure 3.7 for a simulation run time of four
seconds. The top curve is the total (helical) friction versus the helical velocity. As the piston
starts to rotate the system friction passes through the Stribeck curve and then proceeds along the
viscous portion of the curve to an operating region where it remains.
The bottom curve shows the axial friction versus the axial velocity with the coefficient of
determination (R ) given for each model. The typical friction - velocity curve is apparent for the
2
non-rotating model where the velocity accounts for only a portion of the changes in friction.
When the piston is rotated the axial friction - axial velocity curve is a linearized, predictable, and
dependent function of velocity. The magnitude of the axial friction has decreased as well. This
translates into a smaller hydraulic power requirement.
33
Helical Friction vs Helical Velocity
3000
|
1
1
2000
o
i
^ ^ ^ ^ ^
Start Up
PH
i
Operating Region
| 10i
13
X
i
/
i
i
0.2
0
i
0.4
0.6
Helical Velocity (m/s)
Axial Friction vs Axial Velocity
1000
i
0.8
Standard Model R^ = 0.92612
500
e
0
o
Rotating Model R = 1
F = 2700 Vel + 0.004
2
P
P
CCJ
•S -500
0.1
0.15
-0.05
0
0.05
0.2
Axial Velocity (m/s)
Figure 3.7 Friction Curves for the Standard and Rotating Model (460 rpm).
-0.2
-1000
-0.15
-0.1
The value of the viscous friction coefficient that was used during the simulation was 2318 N/m/s.
The slope of the linearized friction curve is given as 2700 N/m/s. In the simulations the slopes
of the linearized friction curves are near the value of the viscous friction coefficient. This
suggests that all the factors that influence friction, except the viscous friction, are reduced to
some degree. The variation of the slope or viscous friction parameter is evident in Table 3.1 and
Figure 3.8.
Table 3.1 Viscous Friction Parameter vs. Rotation Speed.
RPM
Viscous Friction Parameter (N/m/s)
0
2320
120
3600
290
2940
460
2700
630
2580
800
2520
34
Viscous Friction Parameter Variation with Speed
4000
1
1500
s-
PH
§
o
o
1000
co
200
400
600
800
1000
Rotation Speed (rpm)
Figure 3.8 Viscous Friction Parameter vs. Rotation Speed.
As the angular velocity is decreased the operating region lengthens and approaches the Stribeck
region. As this happens, the relation between the axial friction and axial velocity becomes
nonlinear, as shown in Figure 3.9. In this case the friction dependence on velocity is no longer
first order. However, the non-linear phenomena such as hysteresis are still eliminated. If the
angular velocity slows down below the Stribeck limit, non-linear phenomenon begin to reappear.
During simulation, it was found that a minimum rotational velocity of more than twice the
Stribeck velocity (0.07 m/s, R = 0.99) was required for hysteresis to be avoided. This speed
2
indicates the outer limit of the Stribeck region. For the axial friction to be completely linearized
the required velocity was 0.5 m/s (R = 1.0).
2
35
Helical Friction vs Helical Velocity
1000
a
o
IC 500
13
X
1000
a
*HJ
o
F
0.1
0.15
Helical Velocity (m/s)
Axial Friction vs Axial Velocity
= -39358 V e l - 47 V e l + 5318 Vel +1.25
3
0.25
2
500
0
Rotating Model R = 0.99
Standard Model R = 0.93
-500
-1000
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Axial Velocity (m/s)
Figure 3.9 Friction versus Velocity for a Slow Angular Velocity (50 rpm avg).
To simulate a pick and place motion the simulation was run with several consecutive step inputs.
Figure 3.10 provides the results. When the piston is rotated the rise times are faster and the
steady state errors are improved. Simulations showed that rotating the piston is equivalent to
removing damping from the system.
36
Piston Position
0.06
0.05
SS Error
Standard 0.2378 mm
Rotating 0.0450 mm
SS Error Improvement = 50%
Piston Rotation = 460 RPM
K =400
P
SS Error
\
Standard
0.0096 mm
V Rotating
0.0014 mm
i\SS Error Improvement = 86%
0
Standard
Rotating
• Input
(
-0.01
0
0.2\
0.4
0.6
0.8
Time (s)
Figure 3.10 Step Input - Piston Position.
1.2
The steady state error improvement is small in magnitude. However, as discussed in Section 1.2
a small error in the joint actuators can result in a large error in the end effector's position. The
percent improvement does show significant improvement.
3.5
Summary
The simulations show that the axial friction is linearized when the piston and rod are rotated.
Linearizing axial friction and eliminating the Stribeck effect will improve the controllability of
hydraulic actuators.
The simulations showed that the steady state error and phase lag are
reduced. The results motivate the further investigation of rotating the piston and rod to reduce
friction in hydraulic actuators.
37
Chapter 4
Friction Identification in Hydraulic Actuators: Experimental Method
4.1 Introduction
The purposes of these experiments are two fold. The first purpose is to evaluate the benefit, and
characterize the effect of rotating the piston in a hydraulic actuator on the reduction and
linearization of friction. Linearizing the friction eliminates the non-linear effects found in the
Stribeck area of the friction - velocity curve. When linearized, the friction then becomes more
predictable.
The second purpose is to validate the hydraulic actuator model, which can be used to predict
friction for both design (friction elimination) and control (friction compensation) purposes. A
fourth order state space model was developed to simulate a hydraulic actuator under normal
operating conditions. A sixth order model was also developed to simulate the effect of rotating
the piston. The model showed that by rotating the piston at a sufficient velocity, hysteresis is
avoided, and the friction can be linearized.
One major difference expected between the model results and the experimental results is the
required rotational velocity. The friction parameters used in the model for rotating the motor and
piston are the same as the friction parameters used for axial motion of the piston, 588 N for static
friction. This results in a high static friction that the motor must overcome in order to rotate. It
is expected that the rotating friction parameters are considerably less. For example, Lischinsky
et al. [2] determined that the first joint, a hydraulic actuator, of their Titan Schilling II
manipulator had a static friction level of about 200 Nm. Canudas de Wit et al. [33] showed that
their DC motor had a static friction level of 0.29 Nm. These two friction levels are quite
different. It is expected that there will be a significant difference in the rotating friction and axial
friction levels when the correct values are determined.
This will result in a lower rotating
velocity.
38
Hz. This was deemed acceptable [34] as the fastest component in the system was the valve with
a maximum frequency response of 13 Hz.
The motor speed was set to a constant level and was not changed during each test run so its
response time, given its much faster time constant, did not affect the results. The control of the
motor speed was accomplished through a dedicated amplifier for the motor.
The signal to the servo valve was run in open loop for friction parameter identification
experiments.
A position feedback controller was used for tracking experiments.
A simple
proportional controller was used so that the controller did not influence the response of the
system. A low gain of 100 V/m was used so that the friction phenomena near zero velocity were
more clearly observed.
To eliminate noise, the position signal was averaged off line with a window size of 10 before
differentiating. This averaging behaved as a 100 Hz filter. The resulting velocity was again
averaged producing a signal that was effectively filtered at 10 Hz. As the open loop bandwidth
of the system was measured at 1 Hz, this was considered acceptable.
The data filtered to 10 Hz was used to determine the static parameters.
This produced, on
average, over 250 data points per test run.
In order to acquire a sufficient number of data points for the dynamic parameters the data filtered
to 100 Hz was used. This again provided over 250 data points per test run. Data points where
the applied force was greater than the static friction (a + Cf,) were removed in order to focus on
0
the static friction regime.
4.3.1
Position Signal
The position was obtained by using a linear potentiometer. The potentiometer was calibrated by
measuring the displacement of the motor and actuator in 1 mm increments. One hundred data
points were taken. The resulting graph was linear. The curve fit of the data produced an average
error of zero with a standard deviation of 0.07 mm. The 95% confidence interval for 95% of the
error population [35] was ±0.16 mm.
41
Given the above position signal accuracy, at a sampling frequency of 1000 Hz, the linear speed
of the actuator was limited to 160 mm/s in order to obtain accurate measurements within the
95% confidence interval. Velocity curves were compared at the sampling rate of 1000 Hz and
the filtered rates of 100 Hz and 10 Hz. There was no loss of data. In other words, the velocity
did not have high frequency components, and the slower effective sampling rates were sufficient
to accurately reflect the velocity data.
4.3.2
Velocity Determination
Using the same filter as Tafazoli [14] the velocity was obtained from the position signal off-line
using a low pass-filtered differentiator.
The filter was an order 16 digital Finite-Impulse
Response filter. It is a non-causal, high order, delay free differentiator.
4.3.3
Pressures
Pressure transducers were used to measure the supply pressure, return pressure, and the pressure
in each cylinder chamber. The supply pressure was held at 712 psi while the return pressure was
zero.
4.4
Friction Model
The model used for the friction term,
F
is the LuGre model as discussed earlier, repeated
,
pFrictjon
here for convenience:
dz .
— -x
dt
cr
0
x
g(x)
g(x) = a + a exp
0
(4.1)
, and
x
_,
z,
(4.2)
dz
Fpncior, =0-02 + 0-, — + a x.
dt
The model has static and dynamic parameters.
(4.3)
2
The static parameters are a , a cr ,and
0
v
2
while the dynamic parameters are o~ and cr,. The static parameters characterize the steady
0
state static map between velocity and friction [2, 4]. This includes static, Coulomb, Stribeck,
and viscous friction.
The dynamic parameters characterize the dynamic response of friction
42
including stiction. The parameters are difficult to determine as they appear nonlinearly in the
equations [36].
4.5
Determining The Friction Parameters
In order to confirm the accuracy of the friction model and the hydraulic actuator model, the
friction parameters were determined.
Model simulations were then run with the correct
parameters, duplicating the experimental test runs for comparison.
In [37] Wassink discusses the repeatability of friction.
Friction is history dependant and
experimental tests may often exhibit different friction values. Tests must be performed close
together to have any confidence in the comparison of values.
With this in mind, up to ten runs were conducted consecutively over a short period of time for
each rotation speed. The friction parameters were calculated for each run and the average of
these values are used in the following discussion and in the model simulations.
4.5.1
Equation of Motion
With the applied load set to zero, the equation of motion for an actuator (equation (3.29)) is rewritten to solve for friction:
F
„=P,A-P A -Mx
pFriclio
2
2
p
This equation requires knowledge of the acceleration.
(4.4)
During quasi-static experiments the
acceleration is low enough to consider the velocity to be constant at any instant in time. Figure
4.3 shows a typical curve for the applied force and acceleration force plotted against time. It can
be seen that omitting the acceleration is justified. This approach was also used by Tafazoli [14].
43
Applied Force
600
g400
o
£200
Acceleration Force
10
15
Time (s)
20
25
30
Figure 4.3 Quasi-Static Experiments: Applied Force and Acceleration Force.
The applied force can then be considered to be the friction force:
P A P A.
pFriction
x
4.5.2
x
2
2
(4.5)
Friction Parameter Determination
Quasi-static experiments were conducted by using a sinusoidal signal with a period of 200
seconds to control the servo-valve in open loop. The control voltage input to the servo valve
was:
V„=\.0-l
.0 sin(400;r t-nlT).
(4.6)
This input allowed the control voltage to increase from zero in a very slow manner, maximizing
the amount of data collected near zero velocity while minimizing the amount of data in the
viscous region. The experiments were run with a duration of 35 seconds.
This produced a slowly varying sine wave resulting in velocities varying from zero in the static
regime to velocities in the viscous regime. Figure 4.4 shows a typical velocity-time curve.
44
15
20
Time (s)
Figure 4.4 Quasi-Static Velocity Curve.
35
The resulting friction - velocity curve is split into a static regime and a dynamic regime. The
split is determined by observing how the LuGre friction model behaves. The switch from static
to dynamic was chosen as 0.6 mm/s. The dynamic regime and static regime are used to
determine the static parameters and the dynamic parameters, respectively. Using a set velocity
to represent the transition from static to dynamic motion was considered by Tafazoli [14] and
Karnopp [17].
4.5.2.1 Static Parameter Determination
With quasi-static tests with near zero acceleration the velocity can be considered to be constant.
dz
At constant velocity the friction enters steady state and — = 0 . Solving equation (4.1) for z
dt
and substituting into equation (4.5) produces:
(
P\ A
^2 A
-a0
+a
x
x
^
f
p
sign(x ) + a x
x
exp
p
2
p
(4.7)
J)
A nonlinear least squares optimization function from Matlab {Isqnonlin) [38] was utilized to
determine the four static parameters. With F - P A - P A
ss
X
X
2
2
taken as the measured steady state
45
friction and F the estimated friction (from the right-hand side of equation (4.7)), the cost
ss
function to be minimized can be written as [33]:
# data point s
.min.
[*".(**)-tH*)]
Z
(4-8)
2
The original parameter estimates for the non-linear least squares were obtained through trial and
error.
4.5.2.2 Dynamic Parameter Determination
The dynamic parameters are more difficult to determine because the internal state z is not
dz
measurable. At velocities between zero and 0.6 mm/s — is assumed to be constant. Then
dt
(
equation (4.2) is approximated by g(x ) = a + a since exp
0
{
V
• \2 \
X
\y )
sk
1 since x —> 0.
J
Assuming the rate of deflection of the bristles to be negligible at near zero velocities, equation
(4.3) can be written as [2]:
FFriction = ^
(4.9)
This provides a simpler equation for z :
z=^ ^ ,
(4.10)
dz
dz
which can be used in equation (4.1) to eliminate z and <r . A n estimate, — , for — for each
dt
dt
data point is obtained:
0
- = x-^" ''°"W.
c
a +a
dt
0
(4.11)
{
Then z is estimated using Euler integration [36] to obtain z . This allows the estimation, <J , of
0
<7 from equation (4.9), rewritten as:
0
a =
f
Fpri
0
m
.
(4.12)
An estimate, d , for the other dynamic parameter, cr,, is found through a linearized description
x
of the LuGre friction model and the equation of motion [2]:
46
\ A -P A =Mx +
p
2
2
p
(cr, + a )x
2
p
+
(4.13)
ax,
0
p
where the bristle deflection is estimated with the micro-displacements of the actuator and the rate
of bristle deflection is estimated with the near zero velocities. The parameter <7, is chosen such
that equation (4.13) has near critical damping,
(4.14)
In this case the damping coefficient was chosen to be 0.9. It should also be noted that the
viscous friction was neglected from equation (4.13).
At near zero velocities the friction -
velocity curve is in the static regime and approaching the boundary lubrication regime. Viscous
friction is not yet developed and is therefore neglected.
Curve fitting is applied to equation (4.3) to solve for a and CJ, using & and a, as the original
0
0
estimates. Here, again, viscous friction is neglected.
The friction parameters for the hydraulic actuator are presented in Section 5.2.
4.6
Pure Rotational and Linear-Rotating Friction Parameters
The approach to determine the rotating friction parameters for rotation with no axial motion was
the same as that used for the axial motion with no rotation. The torque and rotating velocity of
the motor and piston was obtained from the motor amplifier. The torque was converted into a
force acting on the piston surface. This force was used as the applied force in the equations from
Section 4.5 in place of the hydraulic pressures. The control voltage input to the motor was
V =1.0-1.0sin(400^r^-^/2)
m
(4.15)
The rotating friction parameters for rotation with no axial motion are presented in Section 5.1.
The axial friction parameters were determined for the hydraulic actuator while under a constant
speed of rotation. The approach is identical to that outlined in the preceding sections. Each rpm
has a corresponding set of axial friction parameters. These parameters are presented in Section
5.3.
47
4.7
Power Requirements
The mechanical power was calculated to determine the cost of rotating the piston. The power for
the non-rotating piston can be calculated from [39]:
(P,A.
-P A )Ax
2
2
n
Power =
y
- = {P A ~ A )x
The power for rotating the motor, piston, and rod is calculated from:
p
p
hyd
}
t
2
2
p
Power =r co .
e
m
(4.16)
(4.17)
e
Substituting in equations (3.40) and (3.43) the rotating power becomes:
e= *p n«i e-
Power
F
(- )
i
4
P
18
The total power is then equal to:
Power , ={P A -P A )x
Tola
X
X
2
2
p
+F ^ x , .
(4.19)
The hydraulic and total power requirements are presented in Section 5.6.
4.8
Summary
The determination of the friction parameters was split into two regimes, static parameters and
dynamic parameters. The static parameters were determined from the dynamic friction region of
the friction - velocity curve. The dynamic parameters were determined from the static friction
region of the friction - velocity curve.
Rotating friction parameters were determined for rotation of the hydraulic system with no axial
motion. Axial friction parameters were also determined for axial motion of the hydraulic system
with no rotation. Axial friction parameters were also determined for the hydraulic actuator for a
range of piston rotation speeds. The results are presented in Sections 5.1, 5.2, and 5.3.
The hydraulic power and total power (rotating and hydraulic) was calculated to determine if
rotating the piston was economically beneficial. The results are presented in Section 5.6.
48
Chapter 5
Reducing Stick-Slip Friction in Hydraulic Actuators: Experimental Results
5.1 Pure Rotation
The rotating friction - rotating velocity curve for rotation with no axial motion (i.e., no
translation of the hydraulic actuator) produced a typical Stribeck curve, Figure 5.1. The LuGre
rotating friction parameters are provided in Table 5.1. The rotating friction - rotating velocity
curve from each test run was consistent, as suggested by the standard deviations. The negative
viscous slope has a drop of 30%.
60
r
50
40
| 30
o
20
10
0.1
0.2
0.3
0.4
0.5
0.6
Velocity (m/s)
Figure 5.1 Rotating Friction - Rotating Velocity Curve
0.7
49
Table 5.1 LuGre Rotating Friction Parameters.
LuGre Friction Parameter
Average Value
Standard Deviation
Coulomb Friction
34.7 N
0.5
Stribeck Friction
14.6 N
3.9
Static Friction
49.3 N
4.4
8.3 N/m/s
1.4
0.042 m/s
0.012
Viscous Friction
Stribeck Velocity
OC
W
ske
v
Bristle Spring Constant
°oe
300 N/m
70
Bristle Damping Coefficient
°~\e
120 N/m/s
13
5.2
Hydraulic Actuator - 0 Rpm
The axial friction - axial velocity curve for the hydraulic actuator with zero rotation produced
the curve shown in Figure 5.2. The LuGre axial friction parameters for the hydraulic actuator for
positive velocities are provided in Table 5.2.
The axial friction parameters for negative
velocities are provided in Table 5.3. The subscripts p and n refer to positive and negative
velocities, respectively.
The results show that friction is direction dependent in hydraulic actuators, which is in
agreement with work done by Bonchis et al. [13]. The standard deviations indicate a wide range
of possible friction values, which is typical for friction and indicative of the problems found with
controlling hydraulic actuators. The standard deviations for the hydraulic actuator are much
higher than those for the motor.
The influence of the lubricant and its viscosity can be seen. During pure rotation, lubricant
shears against the moving surfaces, but is not under bulk displacement. While moving axially,
lubricant is being displaced, and the viscosity of the lubricant will have a greater influence on the
friction curve.
The negative viscous slope has a drop of 30 %. The drop occurs over a much shorter range of
velocities when compared to pure rotation. Thus, the negative viscous slope is steeper for the
50
hydraulic actuator, which suggests greater control problems would be encountered in the control
of the hydraulic actuator.
600 h
1
-0.01
1
I
1
L _
-0.005
0
0.005
0.01
Axial Velocity (m/s)
Figure 5.2 Hydraulic Actuator Axial Friction - Axial Velocity Curve at 0 rpm.
Table 5.2 Hydraulic Actuator LuGre Axial Friction Parameters (Positive Velocity, 0 rpm).
LuGre Friction Parameter
Average Value
Standard Deviation
Coulomb Friction
330 N
30
Stribeck Friction
150N
15
Static Friction
480 N
45
22 800 N/m/s
2 500
0.0024 m/s
0.0003
2.6 x 10 N/m
0.6 x 10
b
1.1 x 10 N/m/s
0.1 x 10
4
Viscous Friction
Stribeck Velocity
Bristle Spring Constant
«2„
skp
V
b
Co
P
Bristle Damping Coefficient
4
51
Table 5.3 Hydraulic Actuator LuGre Axial Friction Parameters (Negative Velocity, 0 rpm)
LuGre Friction Parameter
Average Value
Standard Deviation
Coulomb Friction
«o„
290 N
25.0
Stribeck Friction
«.„
HON
15
400 N
40
Viscous Friction
28 200 N/m/s
1 800
Stribeck Velocity
0.0031 m/s
0.0003
Bristle Spring Constant
2.6 x 10 N/m
0.8 x 10
Bristle Damping Coefficient
33.9 x 10 N/m/s
14.2 x 10
Static Friction
5.3
<*0«
+ « . „
b
4
b
4
Hydraulic Actuator Rotating
The axial friction - axial velocity curves for the hydraulic actuator with a piston rotation of 10
rpm (0.033 m/s) produced the curve shown in Figure 5.3.
6001-
c
o
Figure 5.3 Axial Friction - Axial Velocity Curve at 10 rpm (0.033 m/s)
52
The Stribeck curve has been reduced in size.
However, it still exists and can affect the
performance of the actuator. Comparing with the rotating friction - rotating velocity curve,
Figure 5.1, 10 rpm (0.033 m/s) is still within the negative viscous slope of the Stribeck curve for
the rotating piston. In this case only partial lubrication has occurred and as the actuator moves
axially it must finish lubricating the surface to surface contacts.
As the speed of rotation is increased the system moves towards full fluid lubrication and the
Stribeck curve is eliminated during axial movement. This can be seen in Figures 5.4 and 5.5
where the rotational speed has been increased to 25 and 50 rpm. Referring again to Figure 5.1,
at rotational speeds of 25 rpm (0.083 m/s) and 50 rpm (0.17 m/s) the rotating friction response
has moved away from the Stribeck curve into the full fluid regime. As a result, the effective
axial friction - axial velocity curve becomes near linear.
6001-
400 h
200
co
•S o
o
• l-H
Is
'3
M00
-400
-600
-0.01
-0.005
. . , , 0 .„ , . , 0.005
0.01
Axial Velocity (m/s)
Figure 5.4 Axial Friction - Axial Velocity Curve at 25 rpm (0.083 m/s).
53
600
400 h
Figure 5.5 Axial Friction - Axial Velocity Curve at 50 rpm (0.17 m/s)
Rotating the piston and rod breaks the bonds between the asperities. Then, when axial motion
commences the axial friction opposing axial motion is reduced. The negative viscous slope is
eliminated if the rotating velocity is in the full fluid lubrication area of the rotating friction rotating velocity curve. However, even at higher rotational speeds there appears to be a 'static'
friction component in the axial direction. But, the switch from the static regime to the dynamic
regime is continuous.
As the speed of rotation is increased further there is no significant improvement in the axial
friction reduction as shown in Figure 5.6 through to Figure 5.8. It is apparent in these results
that, when the piston and rod are rotated, the classical friction - velocity model (Figure 2.3), with
the static friction equal to the Coulomb friction, more closely represents the axial friction - axial
velocity curve. A more simplified model can then be used to estimate axial friction.
54
600
400 h
Figure 5.7 Axial Friction - Axial Velocity Curve at 100 rpm (0.33 m/s)
600
400
,200
co
•g o
UH
<200
-400
-600
-0.01
-0.005
0
0.005
0.01
Axial Velocity (m/s)
Figure 5.8 Axial Friction - Axial Velocity Curve at 125 rpm (0.42 m/s).
At 100 rpm, Figure 5.7, some system related problems were noticed. The actuator was observed
to slip during movement in some test runs and this is reflected in Figure 5.9. A discontinuity can
be found in several places. In this particular system, 100 rpm is close to a resonant mode and a
disturbance in the performance can be seen. The instability originates near 75 rpm and peaks at
100 rpm.
56
600 h
Figure 5.9 Axial Friction - Axial Velocity, Actuator Slipping at 100 rpm.
Table 5.4 shows the LuGre axial friction parameters corresponding to positive axial velocities
for a range of rotating speeds.
Table 5.5 shows the LuGre axial friction parameters
corresponding to negative axial velocities for the same range of rotating speeds. As the rotation
speed is increased the axial friction values become more stable and thus more predictable. For
example, the standard deviation for axial static friction at 0 rpm is 44.3 N . It decreases to 13.1 N
at 50 rpm and to 6.9 N at 125 rpm. The instability of the parameters at 100 rpm is evident with a
corresponding increase in the standard deviation.
57
Table 5.4 LuGre Axial Friction Parameters (Positive Axial Velocities, with Rotation).
Friction
Parameters
a
(N)
0p
0
(0)
330
Std Dev
a (N)
125
(0.42)
50
0
0
lp
2p
skp
6
0p
4
Xp
0
30
40
Std Dev
15
« o + oc (N) 480
Std Dev
45
a (N/m/s)
22800
Std Dev
2500
v (m/s)
0.0024
Std Dev
0.0003
er (10 N/m)
2.6
Std Dev
0.6
cr (10 N/m/s) 1.1
Std Dev
0.1
P
Speed of dotation (rpm (m/s))
25
50
75
100
(0.083)
(0.17)
(0.25)
(0.33)
30
150
Xp
10
(0.033)
0
50
25
50
15
0
40
15
50
10
1
50
15
40
70
0
50
10
50
0
0
40
40
90
0
10
40
15
25
10
37200
34800
31500
33400
31400
1100
1700
14
1300
4500
0.0017
0.0030
0.0029
0.0030
0.0035
0.0004
0.0004
0.0007
0.0005
0.0015
0.4
0.4
0.3
1.3
0.7
0.06
1.1
0.09
0.1
0.6
8.4
2.6
4.1
2.1
12.3
8.5
3.2
3.5
2.0
14.1
30300
1200
0.0032
0.0006
0.2
0.08
4.7
5.7
Table 5.5 LuGre Axial Friction Parameters (Negative Axial Velocities, with Rotation).
Friction
Parameters
0
(0)
10
(0.033)
Speed of Rotation (rpm (m/s))
25
50
75
100
(0.083)
(0.25)
(0.33)
(0-17)
a
290
10
0
0p
(N)
Std Dev
25
110
«,„ (N)
«o + a (N)
P
lp
15
400
(N/m/s)
2p
Std Dev
V
S
kp
( )
m/s
Std Dev
cr (10 N/m)
6
Qp
0.0031
0.0003
2.6
lp
0.0024
0.0004
14.2
1600
0.0026
0.0005
0.3
0.8
1.4
0.0025
0.0008
0.08
0.0023
0.0007
0.1
31300
0.0020
0.0009
0.03
30600
1700
0.0026
0.0007
0.2
0.2
0.4
0.02
15
5100
0.3
0.3
0.06
20
2900
0.2
0.4
0.09
33200
15
30
10
1400
0.3
0.4
1.8
32500
20
40
10
1
30
10
30
0
1
40
10
30
35600
0
30
10.0
1600
0.8
(10 N/m/s) 33.9
Std Dev
44500
0.9
4
30
30
0
1
10.0
70
1800
Std Dev
a
60
40
28200
0
0
30
135
Std Dev
a
10
125
Std Dev
0
125
(0.42)
0.05
0.3
0.1
0.04
58
O
$
600
Positive Velocity (solid)
Negative Velocity (dash)
Rotating Only
500
400 h 1
300
200
100
0
(D
®
(D
100
40 .
60
80
120
Rotation Speed (rpm)
Figure 5.11 Axial Coulomb Friction Parameter vs. Rotation Speed
o
20
O
^
600
Positive Velocity (solid)
Negative Velocity (dash)
Rotating Only
500
400
300
200
100
0
0
20
40
60
80
100
120
Rotation Speed (rpm)
Figure 5.12 Axial Stribeck Friction Parameter vs. Rotation Speed.
O
•
600
Positive Velocity (solid)
Negative Velocity (dash)
Rotating Only
500
C)
g,
c
o
400
300
r
o
*+-»
CO
t/3
200
100
f
0
0
20
§
?
?
100
60
80
Rotation Speed (rpm)
Figure 5.13 Axial Static Friction Parameter vs. Rotation Speed.
40
#
120
The effective axial Coulomb friction parameter converges to zero when the piston and rod are
rotated at speeds above 20 rpm. The Stribeck friction parameter is reduced and the static friction
approaches that of the rotating static friction with no axial motion. The axial friction - axial
velocity curves for axial motion have an inflection point, which, with the curve fitting of the
LuGre model, requires a non-zero value for the Stribeck velocity and thus, the Stribeck friction.
If the Classical model of friction were to be utilized instead of the LuGre model, the effective
axial Coulomb friction becomes equal to the axial static friction with the Stribeck friction equal
to zero.
61
5.5
5
•
4.5
4
o
3.5
3
CO
&
PH
C
o
1/3
3
O
o
2.5
V
2
O
^
1.5
1
Positive Velocity (solid)
Negative Velocity (dash)
Rotating Only
0.5
0
0
20
40
60
80
100
120
Rotation Speed (rpm)
Figure 5.14 Axial Viscous Friction Parameter vs. Rotation Speed.
The viscous friction parameter can be seen to increase and then decrease, much like the Stribeck
curve. The viscous friction parameter can be seen to change in a similar manner to that predicted
by the simulations, reference Figure 3.8.
62
0.07
0.06 -
2
Positive Velocity (solid)
Negative Velocity (dash)
Rotating Only
O
•
0.05 0.04
0.03
13
0.02
h
0.01
$
oh
0
20
i
40
9
i
i
*
60
80
100
120
Rotation Speed (rpm)
Figure 5.15 Axial Stribeck Velocity Friction Parameter vs. Rotation Speed.
O
$
i
0
20
40
i
Positive Velocity (solid)
Negative Velocity (dash)
Rotating Only
I
60
80
100
Rotation Speed (rpm)
Figure 5.16 Axial Bristle Spring Constant vs. Rotation Speed.
A
120
10
9
O
$
i
Positive Velocity (solid)
Negative Velocity (dash)
Rotating Only
8
O
7
6
CD
5h
i
O
O
bO
c
"a
E
4
CO
Q
PQ
4
o
20
100
120
60
80
Rotation Speed (rpm)
Figure 5.17 Axial Bristle Damping Coefficient vs. Rotation Speed.
40
The bristle spring and damping constants appear to approach the values of the motor. The
magnitudes have been reduced and are more consistent, but they are still higher than those of the
motor.
5.5
Model Validation
The rotating model developed in Chapter 3 provided valuable insight as to what the result would
be when the piston and rod are rotated. However, where the model showed that friction would
be linearized, the experiments showed that the friction is not completely linearized. The most
notable difference in the axial friction, as predicted by the model, is the elimination of the
Stribeck curve.
A significant difference between the rotating model and the experiments is the friction values for
pure rotation and pure axial motion. Using static friction as a base of comparison, the rotating
model used the same values for both forms of motion with the static friction estimated at 588 N .
When rotation occurred and axial motion commenced the transfer between the same friction
values was seamless and the model behaved well. However, the experiments showed that the
64
actual values for rotating and axial static friction are quite different. Static friction for rotating
with no axial motion was 50 N while static friction for axial motion with no rotation was 480 N .
When these two widely different values are used in the rotating model the discontinuity resulted
in the model becoming unstable.
Based on the above observation, the non-rotating model, developed in Chapter 3 was
implemented with the experimentally computed axial friction values. This model was compared
to the experimental curves. The results are plotted in Figure 5.18 for zero rotation and Figure
5.19 for rotation speeds of 10, 25 and 50 rpm. The results show that the LuGre friction model
and the non-rotating hydraulic actuator model agree well with the experiments. The non-rotating
model closely predicts the axial friction - axial velocity behaviour. The simulations for 75, 100,
and 125 rpm also agreed well with the experimental results.
A few modifications were required. One can consider the axial friction - axial velocity curve for
zero rotation, Figure 5.18. The value for <7,„, the bristle damping coefficient for negative
velocities, resulted in an unstable model. The value for <r,„ was reduced by two thirds to 11.3 x
10 N/m/s. This new value is within the 95% confidence interval for 95% of the population but
4
outside the 95% confidence interval for the mean. This new value produced stable results that
agreed with the friction values that were measured experimentally.
One can also consider the axial friction - axial velocity curves for various rotation speeds, Figure
5.19. The values of o , the bristle damping coefficient for positive velocities, are considerably
Xp
higher than o~ and again produced unstable results. Thus, the values for cr,„ were substituted
ln
for a .
lp
The simulations were stable and corresponded with the experimental results. The
experimental values for the bristle spring constant for positive velocities and negative
velocities,( <j
0p
and
<7 „),
0
are similar to each other suggesting that the bristle damping
coefficient for positive and negative velocities, (ox
and <7,„), should also be similar.
65
were over whelmed by the sluggish response of the pressure change. It is expected that for a
smaller actuator and a higher supply pressure, both outside the limits of this experimental
system, that improvements would be seen.
Canudas de Wit et al. [5] state that typical errors caused by friction are steady state errors and
tracking lags. Steady state error is related to static friction and the tracking lag is related to the
viscous friction. If static friction can be eliminated or reduced then the steady state error will be
reduced. If the friction in a system can be reduced or eliminated then the tracking lag will be
improved. However, since the viscous friction cannot be removed there will be an upper limit on
the possible improvement.
Since rotating the rod and piston reduces the static friction and system friction it is expected that
the tracking performance of a hydraulic system with rod and piston rotation would be improved.
However, as friction is reduced damping is removed from the system. This results in an increase
in transient oscillations. A slight increase in oscillation was observed in response to an axial step
input when the piston was rotated.
5.8
Summary
Rotating the piston and rod eliminates the Stribeck effect and reduces axial friction in a
hydraulic actuator. A minimum critical rotating velocity is required for the near linearization of
the axial friction. This critical velocity is located in the full fluid regime of the rotating friction rotating velocity curve.
Comparison with experimental results showed that the non-rotating hydraulic actuator model
accurately predicted the axial friction behaviour of the actuator with no rotation. The nonrotating model, using the appropriate friction parameters can also be used to predict the axial
friction behaviour at various speeds of rotation.
If a simple friction model is desired, with the piston rotating at sufficient speeds, the axial
friction - axial velocity curve approaches the Classical friction - velocity model with static
friction equal to Coulomb friction.
70
Rotating the piston reduces the hydraulic power requirements. However, the power required to
rotate the piston exceeds the power saved by reducing axial friction.
71
Chapter 6
Conclusions and Suggestions for Future Work
6.1 Conclusions
Hydraulic actuators are highly non-linear. One of the main contributors to the non-linearities is
friction, particularly near zero velocity. Simulations and experimental results of this work have
shown that rotating the piston at a sufficient velocity linearizes the axial friction opposing the
axial motion of the piston. Both the Stribeck effect and the hysteresis effect were shown to be
eliminated above a critical rotating piston speed. Rotating the piston also resulted in a reduction
of the axial friction and thus a reduction in the damping of the system.
The non-rotating hydraulic actuator model, incorporating the LuGre friction model, accurately
predicted the friction behaviour of the hydraulic actuator. This model effectively predicts the
axial friction behaviour when the piston and rod are rotated when the appropriate axial friction
parameters are utilized.
Reducing the friction in the hydraulic system reduced the hydraulic power requirements. This
equates to smaller hydraulic power packs and therefore reduced costs.
However, in this
particular application, rotating the piston required more overall power as the power required for
rotating was greater than the power saved from reducing the friction.
Thus, the benefits of
reduced friction and increased accuracy must be evaluated against an increase in power costs.
Another expected benefit of reducing or eliminating stick-slip friction is a reduction in fatigue of
the system. With a decrease in jerk, the amount of stress and fatigue will be reduced.
6.2 Recommendations for Future Work
Future work includes evaluating the tracking ability of a rotating hydraulic actuator. The effect
that rotating the piston has on hunting (limit cycling) in a system requires evaluation.
72
A further step will be to consider various control systems such as a feed forward control loop to
provide a canceling signal for friction, adaptive control, and acceleration feed back to capture the
dynamics of the system.
Control strategies for turning the rotation on and off to maintain a minimum helical velocity can
also be considered.
The combined axial and rotating velocities can be minimized during
operation.
The practicality of implementing a rotating piston and rod will also require evaluation. Instead
of rotating the rod and piston, one may consider rotating the cylinder and holding the piston and
rod secure.
It is expected that rotating will introduce extra wear in the system. The wear of the lip seals, orings, piston seals, and the cylinder wall will need evaluation.
73
Nomenclature
Parameter
Physical Property
Units
A
Piston surface areas
m
A
Piston surface areas
m
B
Viscous damping coefficient
C
Capacitance
juF
D
Velocity level for switch from static to dynamic friction
m/s
2
v
F
2
2
Ns/m
Actuation force applied to piston
N
F
Force applied to an object
N
F
Equivalent force applied by the motor
N
Coulomb friction force
N
Resultant driving force
N
Dynamic friction force
N
F
Generic friction force
N
F
Estimate for friction
N
F
Helical friction force
N
F
Axial friction force
N
F
Rotary friction force
N
External load on the actuator
N
Normal force
N
Steady state friction
N
Estimate of steady state friction
N
F
Static friction force
N
F
Friction as a function of velocity
N
F
Viscous friction force
N
H{s)
Velocity filter, low pass-filtered differentiator
Actuator
Applied
±
dApplied
F
Coulomb
F
Drive
1
F
Dynamic
[•riction
Friction
hb riction
pFriction
OFriction
F
Load
1
F
Normal
t
Static
velocity
Viscous
74
Parameter
Physical Property
/
Identity matrix
J
Manipulator Jacobian
J,
Jacobian state transition matrix
J
Jacobian input matrix
K
Servo valve flow gain
U
Back electromotive force
K f
oem
P
K
Units
m /s/JPa
2
V/rad/s
Proportional Gain
V/m
Servo valve position gain
m/V
K,
Motor torque constant
Nm/A
L
Actuator stroke length
mm
L
Motor inductance
M
System mass
kg
Px
Pressure in chamber one
Pa
Pi
Pressure in chamber two
Pa
PL
Load pressure
Pa
p
Supply pressure, constant
m
s
P
T
P°™ hyd
er
Power
Tolal
Power
g
Henries
MPa
Tank pressure
Pa
Hydraulic power
W
Total power
W
Rotating power
W
Qx
Flow rate into chamber one
m /s
Qi
Flow rate into chamber two
m /s
QL
Load flow
m /s
R
Resistance
ohms
R
Coefficient of determination
cyl
Hydraulic piston radius
K
Motor resistance
2
R
3
3
3
mm
ohms
75
Parameter
S(x„)
Physical Property
Units
Switching function
Velocity filter time constant
s
Daqbook analog output to motor
V
Daqbook analog output to servo valve
V
v
Hose volume between the servo valve and actuator
(Typical x2)
m
v
Unfiltered signal
V
v
Applied voltage to the motor
V
V
Filtered signal
V
v
Servo valve control voltage
V
X
Vector of end-effector coordinates in Cartesian space
m
AX
End effector's position error in Cartesian space
m
v
daqm
v
daqsv
h
in
m
out
w
a
dz
dt
dz
dt
3
Estimate for acceleration
m/s
Rate of change of bristle deflection
m/s
Estimate for the rate of change of bristle deflection
m/s
Describes part of the steady state friction characteristics at
a constant velocity [4]
N
h
Discrete time step
s
L
Motor current
A
L
Rate of change of motor current
A
k
Discrete time
s
K
Dahl's spring constant
q
Vector of joint positions in joint space
m
Aq
Joint position errors in joint space
m
q>
Joint position of the i joint in joint space
m
s
Laplace operator
t
Time variable
s
Time lag
s
At
th
2
N/m
76
Parameter
Physical Property
«/
Input variables
u,
Small deviations about the input variable's operating point
u
Input variables operating point
"k
Discrete input variable vector
i0
Units
Stribeck velocity
m/s
Estimate for Stribeck velocity
m/s
Axial Stribeck velocity for positive velocities
m/s
*kn
Axial Stribeck velocity for negative velocities
m/s
,k6
Rotating Stribeck velocity
m/s
v
skp
V
V
X
Generic position
m
X
Generic velocity
m/s
X
Generic acceleration
m/s
2
x,,
Reference position
m
x„
End effector's x-coordinate in Cartesian space
m
x
Piston helical position
m
x
h
Piston helical velocity
m/s
x
h
Piston helical acceleration
m/s
Xi
State variables
Xi
State variable derivatives
h
*
i
Small deviations about the operating point
i
Small deviations about the operating points derivatives
iO
Operating points of the state variables
iO
Operating points of the state variable derivatives
xi
Current state variable deviation vector
X
X
X
X
*
k+l
X
2
Future state variable deviation vector
P
Piston axial position
m
P
Piston axial velocity
m/s
X
X
77
Parameter
'P
X
Physical Property
Units
Piston axial acceleration
m/s
2
Servo valve spool displacement
m
Xg
Piston angular position
m
Xg
Piston angular velocity
m/s
Xg
Piston angular acceleration
m/s
y.
End effector's y-coordinate in Cartesian space
m
z
Average bristle deflection
m
z
Estimate for the average bristle deflection
m
End effector's z-coordinate in Cartesian space
m
*
z
A(h)
Forced response of a system
Torque produced by the motor
0(h)
2
Nm
Free response of a system
Damping coefficient
Coulomb friction
N
Estimate for Coulomb friction
N
Axial Coulomb friction for positive velocities
N
Axial Coulomb friction for negative velocities
N
09
Rotating Coulomb friction
N
«.
Stribeck friction
N
Estimate for Stribeck friction
N
<*lp
Axial Stribeck friction for positive velocities
N
«.„
Axial Stribeck friction for negative velocities
N
Rotating Stribeck friction
N
« 0
<x
0p
A
« 1
a
Viscous friction coefficient
N/m/s
a
Estimate for viscous friction coefficient
N/m/s
Axial viscous friction coefficient for positive velocities
N/m/s
Axial viscous friction coefficient for negative velocities
N/m/s
2
2
« 2 „
78
Parameter
oc
2e
Physical Property
Units
Rotating viscous friction coefficient
N/m/s
Static friction
N
«Op+«l„
Axial static friction for positive velocities
N
« 0 „
Axial static friction for negative velocities
N
Rotating static friction
N
+
« ! „
p
Effective bulk modulus (oil)
MPa
Bristles stiffness coefficient
N/m
Estimate for bristles stiffness coefficient
N/m
Axial bristles stiffness coefficient for positive velocities
N/m
Axial bristles stiffness coefficient for negative velocities
N/m
Rotating bristles stiffness coefficient
N/m
Bristles damping coefficient
N/m/s
Axial bristles damping coefficient for positive velocities
N/m/s
Axial bristles damping coefficient for negative velocities
N/m/s
°\8
Rotating bristles damping coefficient
N/m/s
O".
Estimate for bristles damping coefficient
N/m/s
7„
Servo valve time constant
<*x
°~ir
e
M Dynamic
M Static
ms
Adaptive parameter for friction observer
Dynamic coefficient of friction
Static coefficient of friction
Angular velocity of the system
rad/s
79
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nd
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S. Tafazoli, C. W. de Silva, P. D. Lawrence, "Friction Modeling and Compensation in
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19.
S. Tafazoli, C. W. de Silva, P. D. Lawrence, "Position and Force Control of an
Electrohydraulic Manipulator in the Presence of Friction", Proceedings of the IEEE
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1933-1936, 1998.
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29.
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an Industrial Hydraulic Rotary Vane Actuator", Proc. 32 Conf. Dcsn & Cntrl, pp. 19131918, Dec, 1993.
30.
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Control of Machines with Friction, Kluwer Academic
Publishers, Boston, M A , 1991.
31.
N . Meikandan, R. Raman, M . Singaperumal, and K . N . Seetharamu, "Theoretical
Analysis of Tapered Pistons in High Speed Hydraulic Actuators", Wear, Vol. 137, No. 2,
pp. 299-321, May 1990.
32.
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Validation of a Highly Nonlinear Hydraulic Servosystem Model", Proc. Amer. Cntrl.
Conf, Pittsburgh, PA, pp. 385-391, 1989.
33.
C. Canudas de Wit and P. Lischinsky, "Adaptive Friction Compensation with Partially
Known Dynamic Friction Model", International Journal of Adaptive Control and Signal
Processing, Vol. 11, pp. 65-80, 1997.
34.
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36.
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New York, N Y , 1988.
nd
Ed., McGraw Hill,
37.
D. A . Wassink, Friction Dynamics in Low Speed, Lubricated Sliding of Rubber: A Case
Study of Lip Seals, PhD Thesis, Department of Mechanical Engineering, University of
Michigan, 1996.
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User's Guide, Version 2, The Mathworks Inc., 1999.
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th
NJ, 1997.
83
Appendix A
Hydraulic Actuator Specifications
A.l
Process and Instrumentation Diagram
The process and instrumentation diagram (PID) is provided in Figure A . l .
1£0 V A C L P h
208 V A C 3 P h
Figure A . l Hydraulic Actuator PID.
84
A.2 Physical Parameters
The physical parameters of the system are given in Table A. 1:
Table A.1 Physical parameters.
Parameter
Measured / Calculated Value
Unit
System Mass
15.0
Stroke Length
0.172
kg
m
Hose Volume (x 2)
1.961 x 10"
m
Piston Diameter
0.0635
m
Rod Diameter
0.0254
m
Piston Areas (x2)
0.00266
m
5
3
2
A.3 Instrumentation and Hardware
A.3.1 Hydraulic Pump
A gear type hydraulic pump was used to supply a constant pressure to the hydraulic system.
Table A.2 Hydraulic Pump.
Manufacturer
Hyseco
Motor
3 hp
Voltage
208 V 3 Ph 60 Hz
Maximum Pressure
9.0 Mpa (1300 psi)
Operating Pressure
4.9 (712 psi)
85
A.3.2
Motor
A permanent magnet DC motor was coupled to the piston to provide rotation.
Table A.3 Motor and Amplifier.
Manufacturer
Indramat
Motor Model
M K D 071B-061-GP1
Amplifier Model
D K C 1.1-030-3
Motor Voltage
208 V 3 Ph 60 Hz
Amplifier Controller Voltage
24 VDC
Command Voltage
[-10 ... +10] V D C
Speed Output
0-10 V D C
Torque Output
0-10 VDC
Speed Range
1 - 30 000 rpm
Controller Speed Limit for Experiments
1 - 500 rpm
Torque Constant
0.77 Nm/A
Windings Inductance
0.0104 H
Windings Resistance
1.65 ohms
Back Electromotive Force
0.6685 V/rad/s
Mass
8.8 kg
Operating Ambient Temperature
0° to 45° C
Nominal Torque
8Nm
The following circuit was used to convert the 0 to 5 V D C analog signal from the controller into a
0 to +10 V D C signal for the motor amplifier, Figure A.2:
R = \0k& {Typxl)
I—V\A/
1
Figure A.2 Motor amplifier signal conversion.
86
A.3.3 Servo Valve
A direct drive servo proportional control valve regulated the flow to the hydraulic actuator.
Table A.4 Servo Valve.
Manufacturer
Moog
Model
D633 (R-16-K-01-M-0-N-S-M-2)
Time Constant
12 ms
Supply Voltage
24 V D C
Command Voltage
[-10 ... +10] V D C
Output
4 - 20 raA
Rated Flow
40 1/min at 1000 psi pressure drop
Max Flow
75 1/min
Fluid Temperature Range
-20° to 80° C
The following circuit was used to convert the 0 to 5 VDC analog signal from the controller into a
-10 to +10 V D C signal for the servo valve, Figure A.3:
i—wv
R = \QkQ(TypxA)
i
AA/V
'I
WV
V„, = -10 —»+\ov
5V
V
daqsv
= 0^5V
Figure A.3 Servo valve signal conversion.
87
A.3.4 Pressure Transducers
Pressure transducers were used to measure the supply pressure, tank pressure, and the pressure in
each hydraulic chamber.
Table A.5 Pressure Transducers.
Manufacturer
M S L Measurement Specialties Ltd.
Model
MSP-300-2500-P-Z
Pressure Rating
2500 psi
Accuracy
± 1% full scale output (includes nonlinearity,
hysteresis, and repeatability)
Excitation Input
12 VDC
Output
1-5 VDC
Operating Temperature Range
-40° to 85° C
Compensated Temperature Range
0° to 55° C
A.3.5 Potentiometer
The displacement of the hydraulic actuator and the motor was measured with a linear
potentiometer.
Table A.6 Potentiometer.
Supplier
Duncan Electronics, a BEI Electronics Company
Model
6300-100
Electrical Travel
0.200 m
Resistance
16 kilo-ohms
Manufacturer's Accuracy
Standard
±0.38 mm
Best
±0.075 mm
Calculated Accuracy
±0.16 mm
Repeatability
Within 0.013 mm
Actuation Force
0.56 N
Input
5 VDC
Output
1-5VDC
88
A.3.6
Data Acquisition
An IOTech Daqbook 120 was used for data acquisition. Labview software was used to control
the data acquisition.
Table A.7 Data Acquisition.
Supplier
IOTech
Model
Daqbook 120
Resolution
12 bit
Supply Voltage
15 V D C
Analog Output (x 2)
0-5 VDC
Analog Input
8 Differential
6 : 0 - 5 VDC
2: 0 - 10 V D C
Before entering the daqbook the signals were passed through an anti-aliasing filter, Figure A.4,
with a cut off frequency of 194 Hz. This is appropriate with a sampling frequency of 1000 Hz
and a resulting Nyquist frequency of 500 Hz.
R = S20n
Figure A.4 Anti-Aliasing Filter.
89
Appendix B
Simulation and Experimental Values
The values for the physical parameters used in the model simulations and experiments are found
in Table B . l . The simulation values are relevant for Chapter 3. The values for experimental and
model validation pertain to Chapter 4 and Chapter 5.
Table B.l Simulation and Experimental Values.
Parameter
Physical Property
Simulation Values
Experimental &
Model Validation
A
Piston surface areas
6.33 x l O m
2.66xlO m
A
Piston surface areas
6.33xlCT m
F
External load on the actuator
2
Load
1
Servo valve flow gain
K
-4
2
4
2
ON
5.9xlO
-3
2
2.66xlO m
-3
2
ON
m /s/^
-4
2
5.9X10" m lsl-JPa
4
2
Back electromotive force
0.6685 V/rad/s
0.6685 V/rad/s
K„
Servo valve position gain
0.005 m/V
0.005 m/V
K,
Motor torque constant
0.77 Nm/A
0.77 Nm/A
L
Actuator stroke length
60 mm
172 mm
0.0104 Henries
34 kg
0.0104 Henries
15 kg
2.94 MPa
4.91 MPa
OPa
~0Pa
19 mm
31.75 mm
1.65 ohms
1.65 ohms
L
Motor inductance
M
System mass
P
Supply pressure, constant
P
Tank pressure
m
s
T
cyl
R
Hydraulic piston radius
Motor resistance
K
v
Hose volume between the servo valve
and actuator (Typical x2)
Servo valve control voltage
8.9x 10~ m
1.961X10 m
±10 V
± 10 V
Stribeck velocity
0.032 m/s
m/s
Servo valve spool displacement
±0.6 mm
± 0.6 mm
Servo valve time constant
12 ms
12 ms
Coulomb Friction
370 N
N
«.
Stribeck friction
217N
N
a
Viscous friction coefficient
2318 N/m/s
N/m/s
588 N
N
Effective bulk modulus (oil)
1724 MPa
1724 MPa
Bristles stiffness coefficient
5.77xlO N/m
N/m
Bristles damping coefficient
2.28x10" N/m/s
N/m/s
h
y„
sv
X
t »
2
Static Friction
P
5
3
6
-5
3
90
Appendix C
Jacobian Linearization
C.l
Jacobian Linearization and Discretization
The state space equations were linearized as follows [34]:
x* = J x'+ J u*,
(C.l)
' = *, ~ x ,
(C.2)
x'=x,-x ,
(C.3)
u*=u,-u ,
(C.4)
J =^-,md
dx,
(C.5)
x
u
x
i0
l0
l0
x
J
u = { ^ >
(C6)
aw,
where x, are the state variables, x, are the state variable derivatives, u are the input variables,
j
x , x, , and u, are operating points, x*, x*, and u* are small deviations about the operating
i0
0
0
points, J is the Jacobian state transition matrix calculated about the operating points, and J is
x
u
the Jacobian input matrix calculated about the input operating points, x is a vector
representation of x,.
The state equations were then discretized for computer simulation [34].
x =0{h)xl+A{h)ul
{C.l)
M
where h is the time step, k is the current time, and k +1 is one time step into the future. 0(h)
is the free response of the system:
0{h) = exp{J h),
(C.8)
x
and A(h) is the forced response of the system:
A{h) = j; [0{h)-I\j .
x
u
(C.9)
Then the state variables of interest are:
=**+*<>•
(CIO)
91
Details of the individual linearizations of the non-rotating state space equations ((3.35), (3.36)
(3.37), and (3.38)) and the rotating state space equations ((3.50), (3.51), (3.52), (3.53), (3.54),
and (3.55))are given in the following sections.
C.l
Non-Rotating Model State Equations
C.2.1 Position
Equation (3.35) describes the position of the actuator:
(C.ll)
The state equations are:
dx,
= 0,
3x,
3x
(C.12)
(C.13)
2
(C.14)
dx.
dx,
= 0.
dx
(C.l 5)
4
C.2.2 Axial Velocity
Equation (3.36) describes the axial velocity of the actuator:
( A - A )
p
,
(A+A)
^ 0 ^ 1 "^4
f
M
a
0
+ «, exp
V
P^2
2\
.
(C.16)
fx ^
)
The state equations are:
3x
2
dx,
= 0,
(C.17)
92
f (
\2\
2a o~ o~ x x x exp_ 2
X
x
dx _ 1
2
dx
2
M
- cTj
O~ O~ X sign(x )
x
0
-a +
4
2
0
x
0
3x
J ,(C18)
x
exp
v *
v
j
J)
(C.19)
2M
3
(7 CT \X \
0
•<j +•
M
4
4
+4)
2
2
2
a +a
dx =
3x _ 1
2
V
exp
3x
x
+•
2
a +a
0
X
2
(C.20)
0
'x
1
a +a
0
x
exp
V
A
C.2.3 Pressure
Equation (3.37) describes the load pressure of the actuator:
f
46
f
m—:
A
+ S{-x P)j+x
s
3
sv
2V +A L V
h
n;
iA+A).
(C.21)
v
2
The state equations are:
dx
3
dx,
2B{A +A )
3x,
2V + A L '
X
h
dx _42BKxJ
2
(C.23)
2
S{x ) , S{-x )^
3
sv
2V +A L
h
(C.22)
dx
3
dx,
= 0,
2
sv
+•
V
3
3*3
dx.
(C.24)
J
= 0.
(C.25)
C.2.4 Friction
Equation (3.38) describes the load average bristle deflection (friction) of the actuator:
<7 |x |x
0
a
0
2
4
'x ^
+ ax exp _
V
2\
(C.26)
J
93
The state equations are:
3*4
= 0,
(C.27)
dx,
f
di
4
dx.
2a o~ x x x exp
x
<J sign(x )x
0
= 1-
2
2
2
4
V
4
a + a exp
0
0
v"sk a + a exp
2
x
0
J
V
(C.28)
<xx
2
^
x
\ skJ
V
V
v
dx
= 0,
dx.
4
dx
dx
J)
(C.29)
<7 x
4
0
2
4
a + a exp
V
fx ^
2\
(C.30)
2
x
0
C.2.5
'x ^
1
y
Servo Valve Input
The state equations with respect to the input (servo valve opening) are:
dx,
' =0,
dx
(C.31)
dx2
- = o,
dx
(C.32)
V
dx,
4/3K
dx
2V +A L
sv
H
S{xJ
S
P
~
3
X
+ S(-x„l
(C.33)
2
2
dx
A
ox. = 0.
(C.34)
C.2.6 Jacobians
The Jacobian state transition matrix is:
94
dx,
3x,
J
R
dx,
3x
dx, 3x,
dx, dx,
dx,
3x
3x
3x
dx
3x
3x,
3i
3x,
3x,
3x
3x
dx.
4
3x,
3x
2
3x,
3x.
3
4
3x,
3x
2
3
2
3
3
3
4
dx,
3x
4
3x
3x
4
(C.35)
3
3x
4
and the Jacobian input matrix is:
3x,
3x
2
Bx„
3x
J.. =
(C.36)
3
3x
4
3x„,
C.3 Rotating Model State Equations
C.3.1 Position
Equation (3.50) describes the axial position of the actuator:
(C.37)
The state equations are:
3x,
dx,
3x,
3x
= 0,
(C.38)
= 1,
(C.39)
= 0,
(C.40)
= 0.
(C.41)
2
3x,
3x
3
3x,
3x,
C.3.2 Axial Velocity
Equation (3.51) describes the axial velocity of the actuator:
95
1
M
A* A^
I
2
J
+
I
0*0*2*4
J
2
1
2
V2
- C T , X
2
+ *5
X
0*1*4*2
•••+•
'
a + «, exp
f2
Y
+
2\\
(C.42)
012X2
r
0
V
J J
sk
V
The state equations are:
dx
= 0.
2
dx,
dx _ 1
0*0*4
2
3x7" M |
0*0*2 * 4
+ * 5
V*2
+ X
(x
2
2
5
3
(C.43)
0 * , - « 2
0'o
+
i*4
'*2 +*ni
2
)
2
C r
a + a exp
0
x
V
V
sk
v
J J
2a (7 cr x x exp
2
x
0
x
4
V
•••+•
sk
v
x
J J
sk
v
(C.44)
'* * ^
2
2 +
a + a exp
0
V
2
5
JJ)
V
dx _ (A + A )
2
x
dx
2
3x7 M|
0" *2
V*2
+
a ax
+•
O
_
5
X
(C.45)
2M
3
3x _ 1
2
Q
x
*2
a + a exp
0
2
dx
*5
x
V
dx
(C.46)
2
0,
V
v
*
JJ)
(C.47)
5
dx
2
dx
(C.48)
A
96
C.3.3 Pressure
Equation (3.52) describes the load pressure of the actuator:
4/?
2V„ + A L
A +A
Kx.
x
2
(C.49)
2
The state equations are:
dx,
dx,
dx _
2
dx,
dx,
(C.51)
2V„ +A L
2
V2/? Kx
3
(C.50)
2/7(/I +A )
3
dx
= 0,
S{x„)
s}
2V +A,L
TJPS
k
S(-xJ
+
X j " ^P + X3
S
(C.52)
J
(C.53)
dx
4
dx
3
dx
5
dx
3
0,
(C.54)
(C.55)
dx
fi
C.3.4 Friction
Equation (3.53) describes the average bristle deflection (friction) of the actuator:
x
4
C T Q X -\JX ~t~ x^
^ x "i" X g
2
4
2
a
0
<
+ a, exp
(C.56)
x\+x ^'
(
2
The state equations are:
dx
4
dx,
= 0,
dx.
dx
(C.57)
CTQ
2
2
2
+x
2
-r A
4l
X
5
+
5
X
a
0
X
2
X
4
(
+ or, exp
\ V
(x\+x ^
2
J
J)
97
2a o~ x x ^x + xj exp
l
0
2
A
2
V V
v
*
J)
(C.58)
a + a, exp|
0
dx = 0.
dx.
(C.59)
4
3x
(C.60)
4
dx
A
a + a, exp
0
dx.
<7 x x
0
3*5
4
5
f
<Jxl + X
x x^
f
r
•yjxl + Xa + «, exp
V V
2
2
2+
0
'
2a (7 x x -yjxl + x] exp
x
Q
4
J
*k
v
J)
'xl+xl^
5
'
f~2
a + a, exp|
0
V V
,
(C.61)
i\W
2
v„.
">k
J))
dx =0.
(C.62)
4
dx.
C.3.5 Rotating Velocity
Equation (3.54) describes the transverse velocity of the actuator:
J_
M
^l 6
°~0 4 S
X
Rcyl
X
^x\
+
X
x]
0~Q(T, X
•c x +[
5
a + cf, exp
Q
4
(C.63)
X$
X2
I
v i
X^
v
J)
The state equations are:
dx
5
dx,
= 0,
(C.64)
98
ox _ 1
5
dx
M
2
2a a a x x x
x
^0 ^ 2 ^4 ^5
(x
2
2
+X
Q
2
A
exp
5
V V
+•
)2
2
5
x
<
a +a
0
ox
(C.65)
x\+x ^
f
exp
x
JJ
sk
v
V
2
J
sk
V
J)
5
(C.66)
dx.
3x _ 1
5
ox
4
I
M
ax
0
5
2
2
<T CT X
5
5
+ —
V * 2 + * 5
2
2
5
'x
a, exp
1
2
2
V
+
x
(C.67)
^
2
5
J J J
sk
V
cr (j x
'
'x
or + a, exp
a +
- cr, -
3
(x +X
5
f
0
ox _ 1
ox
M
l
0
+
2
)
0
1
4
2
x ^
2
+
0
V V
2a <7 <7,x x
l
0
4
2
5
2
X
exp
a +a
0
x
dx
5
exp
_
dx
6
JJ
sk
v
(C.68)
'x +x ^
2
sk
J J
*5
V V
v
"**
V
2
sk
y
K,
J
J)
(C.69)
MR
cyl
C.3.6 Motor Current
Equation (3.55) describes the motor current:
m
mo
Rcyl
(C.70)
The state equations are:
^
= 0,
(C.71)
OX,
f i = o.
(C.72)
OX,
99
3*6
_Q
3-*6
_Q
(C.73)
(C.74)
3x,
K bemf
dx.
3*5
L
R
m
(C.75)
cyl
3*6
(C.76)
dx„
C.3.7 Motor Voltage Input
The state equations with respect to the first input (motor voltage) are:
dx,
dv
= 0,
(C.77)
= 0,
(C.78)
= 0,
(C.79)
= 0,
(C.80)
= 0,
(C.81)
m
2
dx-.
3
3F
m
3*4
3F
m
5
3F
m
3*
1
6
(C.82)
3^
C.3.8 Servo Valve Input
The state equations with respect to the second input (servo valve opening) are:
dx,
3*
0.
(C.83)
= 0,
(C.84)
JV
dx
2
dx..
dx.
dx
sv
4/3K
2V +A L
h
2
+
S(- x.
4
P +x
s
3
(C.85)
100
3x 4
=0,
(C.86)
dx
=0,
dxsv
(C.87)
dx
5
dx,
(C.88)
C.3.9 Jacobians
The Jacobian state transition matrix is:
J, =
dx,
dx.
dx,
dx
3x
3x
2
dx,
3x,
2
3x
3
3x
2
3x
2
3x
dx,
dx
3x
3*3
3
5
3x
2
3*
3x
4
3x
5
3*6
3
3x
3
3*3
3x
5
6
2
2
3x
3
3x
4
3*
4
3*4
3*4
3x
2
3x
3
3x
4
3x
5
3*6
5
3x
5
3x
5
3x
5
3x
4
3x
5
3*6
3x
dx,
dx
dx
5
3x
dx,
dx
3*
3*
3x
3x
2
3
dx
6
4
3x
dx,
dx,
dx,
3x
3x
3*
4
dx
3x,
2
3
4
3*6
5
2
3*3
3x
6
3*6
3*6
3*6
3*6
2
3x
3*4
3x
3*6
3
5
(C.89)
4
and the Jacobian input matrix is:
3x,
W
3x
m
2
w
m
J
T —
u
~
3x
3
3^
3x
4
m
dv
dx,
3*
3x
iV
2
3*,
3x
v
3
3*,
3x
v
3xm
3*,
3x
3*6
3*. .„
3*
5
(C.90)
4
v
5
s
6
3* .
iV
101