NONLINEAR MODELING, IDENTIFICATION, AND COMPENSATION FOR FRICTIONAL DISTURBANCES By CHARU MAKKAR A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2006
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NONLINEAR MODELING, IDENTIFICATION, AND COMPENSATION FORFRICTIONAL DISTURBANCES
By
CHARU MAKKAR
A THESIS PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2006
Copyright 2006
by
Charu Makkar
This work is dedicated to my parents for their unconditional love, unquestioned
support and unshaken belief in me.
ACKNOWLEDGMENTS
I would like to express sincere gratitude to my advisor, Dr. Warren E. Dixon,
who is a person with remarkable affability. As an advisor, he provided the neces-
sary guidance and allowed me to develop my own ideas. As a mentor, he helped
me understand the intricacies of working in a professional environment and helped
develop my professional skills. I feel fortunate in getting the opportunity to work
with him.
I would also like to extend my gratitude to Dr. W. G. Sawyer for his valuable
discussions. I also appreciate my committee members Dr. Carl D. Crane III, Dr. T.
F. Burks, and Dr. J. Hammer for the time and help they provided.
I would like to thank all my friends for their support and encouragement. I
especially thank Vikas for being my strength and a pillar of support for the last
two years. I would also like to thank my colleague Keith Dupree for helping me
out on those difficuilt days when I was doing my experiements, and otherwise
aquainting me with American culture almost everyday.
Finally I would like to thank my parents for their love and inspiration, my
sisters Sonia and Madhvi, my brothers-in-law Anil and Rohit and my darling nieces
Kovida, Adya and Hia for keeping up wih me and loving me unconditionally.
2—22 The proposed friction model very closely approximates the experimen-tally identified friction term obtained from the adaptive controller devel-oped in Makkar et al. [41]. Top plot depicts the experimentally obtainedfriction torque, middle plot depitcs the friction plot obtained from theproposed model and the bottom plot depicts a comparison of the twowith solid line indicating experimentally obtained friction torque anddashed line indicating friction torque obtained from the proposed fric-tion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3—19 Identified friction from the adaptive term in the proposed controller. . . . 49
3—20 Position tracking error with the proposed controller when the circulardisk was lubricated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3—21 Torque input by the proposed controller when the circular disk was lu-bricated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3—22 Identified friction from the adaptive term in the proposed controller whenthe circular disk was lubricated. . . . . . . . . . . . . . . . . . . . . . . . 51
3—23 Net external friction induced with no lubrication. The net friction wascalculated by subracting the identified friction term in Experiment 1 fromthe identified friction term in Experiment 2. . . . . . . . . . . . . . . . . 52
3—24 Net external friction induced with lubrication. The net friction was cal-culated by subracting the identified friction term in Experiment 1 fromthe identified friction term in Experiment 3. . . . . . . . . . . . . . . . . 52
3—25 The friction torque calculated from the model in (2—1) approximates theexperimentally identified friction torque in (3—9). . . . . . . . . . . . . . 53
3—26 Wearing of the Nylon block where it rubbed against the circular disk. . . 56
4—1 The experimental testbed consists of a 1-link robot mounted on a NSKdirect-drive switched reluctance motor. . . . . . . . . . . . . . . . . . . . 68
4—2 Desired trajectory used for the experiment. . . . . . . . . . . . . . . . . . 70
4—3 Position tracking error when the adaptive gain is zero. . . . . . . . . . . 71
4—4 Torque input when the adaptive gain is zero. . . . . . . . . . . . . . . . . 71
4—5 Position tracking error for the control structure that includes the adap-tive update law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4—6 Torque input for the control structure that includes the adaptive updatelaw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4—7 Parameter estimate for the mass of the link assembly. . . . . . . . . . . . 73
x
Abstract of Thesis Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Master of Science
NONLINEAR MODELING, IDENTIFICATION, AND COMPENSATION FORFRICTIONAL DISTURBANCES
By
Charu Makkar
May 2006
Chair: Warren E. DixonMajor Department: Mechanical and Aerospace Engineering
For high-performance engineering systems, model-based controllers are
typically required to accommodate for the system nonlinearities. Unfortunately,
developing accurate models for friction has been historically challenging. Despite
open debates in Tribology regarding the continuity of friction, typical models
developed so far are piecewise continuous or discontinuous. Motivated by the fact
that discontinuous and piecewise continuous friction models can be problematic
for the development of high-performance controllers, a new model for friction is
proposed. This simple continuously differentiable model represents a foundation
that captures the major effects reported and discussed in friction modeling and
experimentation. The proposed model is generic enough that other subtleties such
as frictional anisotropy with sliding direction can be addressed by mathematically
distorting this model without compromising the continuous differentiability. From
literature, it is known that if the friction effects in the system can be accurately
modeled, there is an improved potential to design controllers that can cancel
the effects, whereas excessive steady-state tracking errors, oscillations, and limit
cycles can result from controllers that do not accurately compensate for friction.
A tracking controller is developed in Chapter 3 for a general Euler-Lagrange
system based on the developed continuously differentiable friction model with
uncertain nonlinear parameterizable terms. To achieve the semi-global asymptotic
tracking result, a recently developed integral feedback compensation strategy is
used to identify the friction effects on-line, assuming exact model knowledge of the
remaining dynamics. A Lyapunov-based stability analysis is provided to conclude
the tracking and friction identification results. Experimental results illustrate the
tracking and friction identification performance of the developed controller.
The tracking result in Chapter 3 is further extended to include systems with
unstructured uncertainties while eliminating the known dynamics assumption.
The general trend for previous control strategies developed for uncertain dynamics
in nonlinear systems is that the more unstructured the system uncertainty, the
more control effort (i.e., high gain or high frequency feedback) is required to reject
the uncertainty, and the resulting stability and performance of the system are
diminished (e.g., uniformly ultimately bounded stability). The result in Chapter
4 is the first result that illustrates how the amalgamation of an adaptive model-
based feedforward term with a high gain integral feedback term can be used to
yield an asymptotic tracking result for systems that have mixed unstructured and
structured uncertainty. Experimental results are provided that illustrate a reduced
root mean squared tracking error.
xii
CHAPTER 1INTRODUCTION
The class of Euler-Lagrange systems considered in this thesis are described by
the following nonlinear dynamic model:
M(q)q + Vm(q, q)q +G(q) + f(q) = τ(t). (1—1)
In (1—1), M(q) ∈ Rn×n denotes the inertia matrix, Vm(q, q) ∈ Rn×n denotes the
where γi ∈ R ∀i = 1, 2, ...6 denote unknown positive constants1 . The friction model
in (2—1) has the following properties.
• It is continuously differentiable and not linear parameterizable.
• It is symmetric about the origin.
• The static coefficient of friction can be approximated by the term γ1 + γ4.
• The term tanh(γ2q)− tanh(γ3q) captures the Stribeck effect where the friction
coefficient decreases from the static coefficient of friction with increasing slip
velocity near the origin.
• A viscous dissipation term is given by γ6q.
• The Coulombic friction is present in the absence of viscous dissipation and is
modeled by the term γ4 tanh(γ5q).
1 These parameters could also be time-varying.
11
• The friction model is dissipative in the sense that a passive operator q(t) →
f(·) satisfies the following integral inequality [4]Z t
t0
q(τ)f(q(τ))dτ ≥ −c2
where c is a positive constant, provided q(t) is bounded.
Figures 2—1 and 2—2 illustrate the sum of the different effects and characteris-
tics of the friction model. Figure 2—3 shows the flexibility of such a model.
2.2 Stick-Slip Simulation
The qualitative mechanisms of friction are well-understood. To illustrate
how the friction model presented in (2—1) exhibits these effects, various numerical
simulations are presented in this section. The system considered in Figure 2—4
is a simple mass-spring system, in which a unit mass M is attached to a spring
with stiffness k resting on a plate moving with a velocity.xp(t) in the positive
X direction, which causes the block to move with a velocity.xb(t) in the same
direction. The modeled system can be compared to a mass attached to a fixed
spring moving on a conveyor belt. The plate is moving with a velocity that slowly
increases and saturates, given by the following relation:
.xp = 1− e−0.1t.
The system described by Figure 2—4 is modelled as follows:
M..xb(t) + kxb(t)−Mgf(
.xp(t)−
.xb(t)) = 0
where the term.xp(t)−
.xb(t) represents the slip velocity, (i.e., the difference between
the plate velocity and block velocity at any instant of time). To demonstrate the
flexibility of the model, model parameters were varied in order to capture the
Stribeck effect, Coulombic friction effect and viscous dissipation. For example,
12
Figure 2—1: Friction model as a composition of different effects including: a)Stribeck effect, b) viscous dissipation, c) Coulomb effect, and d) the combinedmodel.
Figure 2—2: Characteristics of the friction model.
13
Figure 2—3: Modular ability of the model to selectively model different frictionregimes: top plot-viscous regime (e.g., hydrodynamic lubrication), middle plot-Coulombic friction regime (e.g., solid lubricant coatings at moderate slidingspeeds), and bottom plot-abrupt change from static to kinetic friction (e.g., non-lubricous polymers).
14
Figure 2—4: Mass-spring system for demonstrating stick-slip friction.
hydrodynamic lubrication in many operating regimes is viscous, lacking the other
effects, which are easily set to zero in the model. Simple Coulombic friction models
are often good for solid lubricant coatings at moderate sliding speeds. To capture
this effect, the static and viscous terms can be set to zero. For some sticky or non-
lubricous polymers, there exists an abrupt change from static to kinetic friction,
which is captured by making the Stribeck decay very rapid.
A Coulombic friction regime is displayed in Figure 2—5 where the friction
model parameters in (2—1) were set as follows: γ1 = 0, γ2 = 0, γ3 = 0, γ4 = 0.1,
γ5 = 100, γ6 = 0. The Coulombic friction coefficient is a constant, opposing
the motion of the block as seen in Figures 2—5 and 2—8. The block velocity, slip
velocity, and the friction force as a function of time are depicted in Figures. 2—6 -
2—8. These figures indicate that the block velocity slowly rises, reaches a maximum
and then begins to oscillate. The slip velocity also rises and then oscillates after
reaching a maximum value. These figures indicate that the friction force causes
the block to move along with the plate until the spring force overcomes the friction
force; hence, the block begins to slip in an opposite direction of the plate velocity
15
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
−0.1
−0.05
0
0.05
0.1
[m/sec]
Figure 2—5: Friction coefficient vs slip velocity.
causing the spring to compress. As the spring releases energy back into the system,
the block velocity exceeds the plate velocity. The magnitude of the constant
friction coefficient results in a constant oscillation between the friction force and
the spring force.
The viscous friction plot in Figure (2—9) is obtained by adjusting the parame-
ters as follows: γ1 = 0, γ2 = 0, γ3 = 0, γ4 = 0, γ5 = 0, γ6 = 0.01. The block
velocity, slip velocity and the friction force are given in Figures 2—10 - 2—12. The
block velocity in Figure 2—10 slowly decreases as the viscous friction increases as
displayed in Figure 2—9. The viscous coefficient of friction is an order of magnitude
smaller in comparison to the Coulombic friction coefficient, as a result the friction
force is not sufficient enough to sustain the oscillations of the block. The block
eventually comes to rest and constantly slips on the moving plate.
The Stribeck effect in (2—13) is modeled using the following friction model
Figure 2—22 shows a plot of the experimentally determined friction torque, a plot
of friction torque developed from (2—1) and a plot comparing the experimentally
determined friction torque with the developed friction model overlaid.
The experimentally obtained friction torque in Figure 2—22 has viscous and
static friction components and exhibits the Stribeck effect. These facts were taken
into consideration while choosing the constants in (2—1). Values for γ1 and γ4 were
chosen to account for the static friction, and γ6 was chosen to capture the viscous
friction component. As seen in Figure 2—22, the analytical model approximates the
experimental data with the exception of some overshoot. The experimental origin
of the directional frictional anisotropy is discussed in detail in Schmitz et al. [48]
and is attributed to small misalignment between the loading axis and the motor
axis.
2.4 Concluding Remarks
A new continuously differentiable friction model with nonlinear parameter-
izable terms is proposed. This model captures a number of essential aspects of
friction without involving discontinuous or piecewise continuous functions and can
26
0 5 10 15 20 25 30 35 40−200
−100
0
100
200
Time [sec]
Fric
tion
Torq
ue [N
m]
Experimentally identified friction
0 5 10 15 20 25 30 35 40−200
−100
0
100
200
Time [sec]
Fric
tion
Torq
ue [N
m]
Identified friction from model
0 5 10 15 20 25 30 35 40−200
−100
0
100
200
Time [sec]
Fric
tion
Torq
ue [N
m]
Comparison
Figure 2—22: The proposed friction model very closely approximates the experi-mentally identified friction term obtained from the adaptive controller developedin Makkar et al. [41]. Top plot depicts the experimentally obtained friction torque,middle plot depitcs the friction plot obtained from the proposed model and thebottom plot depicts a comparison of the two with solid line indicating experimen-tally obtained friction torque and dashed line indicating friction torque obtainedfrom the proposed friction model.
27
be modified to include additional effects. The continuously differentiable property
of the proposed model provides a foundation to develop continuous controllers that
can identify and compensate for nonlinear frictional effects. The development of
one such controller is discussed in next chapter.
CHAPTER 3IDENTIFICATION AND COMPENSATION FOR FRICTION BY HIGH GAIN
FEEDBACK
Motivated by the desire to include dynamics friction models in the control
design, a new tracking controller is developed in this chapter that contains the new
continuously differentiable friction model with uncertain nonlinear parameterizable
terms that was developed in Chapter 2. Friction models are often based on the
assumption that the friction coefficient is constant with sliding speed and have a
singularity at the onset of slip. Such models typically include a signum function of
the velocity to assign the direction of friction force (e.g., see Lampaert et al. [34],
and Swevers et al. [52]), and many other models are only piecewise continuous
(e.g., the LuGre model in [8]). In Makkar et al. [42], Makkar and Dixon et al.
[43], and Chapter 2, a new friction model is proposed that captures a number of
essential aspects of friction without involving discontinuous or piecewise continuous
functions. The simple continuously differentiable model represents a foundation
that captures the major effects reported and discussed in friction modeling and
experimentation and the model is generic enough that other subtleties such as
frictional anisotropy with sliding direction can be addressed by mathematically
distorting this model without compromising the continuous differentiability.
Based on the fact that the developed model is continuously differentiable, a
new integral feedback compensation term originally proposed by Xian et al.
[57] is exploited to enable a semi-global tracking result while identifying the
friction on-line, assuming exact model knowledge of the remaining dynamics.
A Lyapunov-based stability analysis is provided to conclude the tracking and
friction identification results. Experimental results show two orders of magnitude
28
29
improvement in tracking control over a proportional derivative (PD) controller,
and a one order of magnitude improvement over the model-based controller.
Experimental results are also used to illustrate that the experimentally identified
friction can be approximated by the model in Makkar et al. [42] and Makkar and
Dixon et al. [43].
This chapter is organized as follows. The dynamic model and the associated
properties are provided in Section 3.1. Section 3.2 describes the development of
errorsystem followed by the stability analysis in Section 3.3. Section 3.4 describes
the experimental set up and results that indicate improved performance obtained
by implementing the proposed controller followed by discussion in Section 3.5 and
conclusion in Section 3.6.
3.1 Dynamic Model and Properties
The class of nonlinear dynamic systems considered are assumed to be modeled
by the general Euler-Lagrange formulation in (1—1) where the friction term f(q) is
assumed to have the form in (2—1) as in Makkar et al. [42] and Makkar et al. [43].
The subsequent development is based on the assumption that q(t) and q(t)
are measurable and that M(q), Vm(q, q), G(q) are known. Moreover, the following
properties and assumptions will be exploited in the subsequent development:
Property 3.1: The inertia matrix M(q) is symmetric, positive definite, and
satisfies the following inequality ∀ y(t) ∈ Rn:
m1 kyk2 ≤ yTM(q)y ≤ m(q) kyk2 (3—1)
where m1 ∈ R is a known positive constant, m(q) ∈ R is a known positive function,
and k·k denotes the standard Euclidean norm.
Property 3.2: If q(t) ∈ L∞, then∂M(q)
∂q, and
∂2M(q)
∂q2exist and are bounded.
Moreover, if q(t), q(t) ∈ L∞ then Vm(q, q) and G(q) are bounded.
30
Property 3.3: Based on the structure of f(q) given in (2—1), f(q), f(q, q), and
f(q, q,...q ) exist and are bounded provided q(t), q(t), q(t),
...q (t) ∈ L∞.
3.2 Error System Development
The control objective is to ensure that the system tracks a desired trajectory,
denoted by qd(t), that is assumed to be designed such that qd(t), qd(t), qd(t),...q d(t) ∈ Rn exist and are bounded. A position tracking error, denoted by e1(t) ∈
Rn, is defined as follows to quantify the control objective:
e1 , qd − q. (3—2)
The following filtered tracking errors, denoted by e2(t), r(t) ∈ Rn, are defined to
facilitate the subsequent design and analysis:
e2 , e1 + α1e1 (3—3)
r , e2 + α2e2 (3—4)
where α1, α2 ∈ R denote positive constants. The filtered tracking error r(t) is not
measurable since the expression in (3—4) depends on q(t).
After premultiplying (3—4) by M(q), the following expression can be obtained:
M(q)r =M(q)qd + Vm(q, q)q +G(q) (3—5)
+ f(q)− τ(t) +M(q)α1e1 +M(q)α2e2
where (1—1), (3—2), and (3—3) were utilized. Based on the expression in (3—5) the
Figure 3—14: Comparison of position tracking errors from the three controlschemes.
0 5 10 15 20 25 30 35 40−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time [sec]
Pos
ition
Tra
ckin
g E
rror
[Deg
rees
]
Model−based controllerProposed controller
Figure 3—15: Comparison of position tracking errors from the model-based con-troller and the proposed controller.
48
0 5 10 15 20 25 30 35 40−200
−150
−100
−50
0
50
100
150
200
Time (sec)
Torq
ue (N
)
Figure 3—16: Torque input by the PD controller.
0 5 10 15 20 25 30 35 40−250
−200
−150
−100
−50
0
50
100
150
Time [sec]
Torq
ue [N
m]
Figure 3—17: Torque input by the model-based controller with friction feedforwardterms as described in (3—40).
49
0 5 10 15 20 25 30 35 40−250
−200
−150
−100
−50
0
50
100
150
200
Time [sec]
Torq
ue [N
m]
Figure 3—18: Torque input by the proposed controller.
0 5 10 15 20 25 30 35 40−200
−150
−100
−50
0
50
100
150
Time [sec]
Iden
tifie
d Fr
ictio
n [N
m]
Figure 3—19: Identified friction from the adaptive term in the proposed controller.
50
0 5 10 15 20 25 30 35 40−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
Time [sec]
Pos
ition
Tra
ckin
g E
rror
[Deg
rees
]
Figure 3—20: Position tracking error with the proposed controller when the circulardisk was lubricated.
0 5 10 15 20 25 30 35 40−200
−150
−100
−50
0
50
100
150
200
Time [sec]
Torq
ue [N
m]
Figure 3—21: Torque input by the proposed controller when the circular disk waslubricated.
51
0 5 10 15 20 25 30 35 40−150
−100
−50
0
50
100
150
Time [sec]
Iden
tifie
d Fr
ictio
n [N
m]
Figure 3—22: Identified friction from the adaptive term in the proposed controllerwhen the circular disk was lubricated.
3.4.4 Experiment 4
In the fourth experiment, the net external friction induced on the system as
a result of external load applied to the circular disk by the linear thruster was
identified. The friction in the testbed under no-load conditions was identified as in
Experiment 1 using the control gains of Experiment 2. This identified friction term
was subtracted from the identified friction terms obtained from Experiment 2 and
Experiment 3 respectively. The friction between the circular disk and Nylon block
when the disk was not lubricated can be seen in Figure 3—23, and the friction when
the disk was lubricated can be seen in Figure 3—24.
3.4.5 Experiment 5
In the fifth experiment, the experimentally identified friction torque using the
adaptive term in (3—9) was compared with the friction torque model in (2—1). The
friction identified in Figure 3—22 was compared with the model parameters in (2—1)
that were adjusted to match the experimental data. The rotor velocity signal was
obtained by applying a standard backwards difference algorithm to the position
signal, and the friction torque in (2—1) was calculated as a function of this rotor
52
0 5 10 15 20 25 30 35 40−120
−100
−80
−60
−40
−20
0
20
40
60
80
Time [sec]
Iden
tifie
d Fr
ictio
n [N
m]
Figure 3—23: Net external friction induced with no lubrication. The net frictionwas calculated by subracting the identified friction term in Experiment 1 from theidentified friction term in Experiment 2.
0 5 10 15 20 25 30 35 40−20
−15
−10
−5
0
5
10
15
20
Time(sec)
Iden
tifie
d Fr
ictio
n (N
)
Figure 3—24: Net external friction induced with lubrication. The net friction wascalculated by subracting the identified friction term in Experiment 1 from theidentified friction term in Experiment 3.
53
velocity with the constants chosen as
γ1 = 34.8 γ2 = 650 γ3 = 1
γ4 = 26 γ5 = 200 γ6 = 19.5.
The matching of the friction torque with the experimental data is plotted in Figure
3—25.
0 5 10 15 20 25 30 35 40−150
−100
−50
0
50
100
150
Time [sec]
Fric
tion
Torq
ue [N
m]
Comparison
Figure 3—25: The friction torque calculated from the model in (2—1) approximatesthe experimentally identified friction torque in (3—9).
3.5 Discussion
In Experiment 1, it was observed that the position tracking error obtained
from the PD controller in (3—39) deviated around 0.7337 degrees (as seen in Figure
3—2) compared to 0.0506 degrees (as in Figure 3—3) in the model-based controller
in (3—40). This error was further reduced to about 0.0116 degrees when the
proposed controller in (3—37) was implemented (as seen in Figure 3—4). A detailed
comparison of the position tracking errors (in degrees) and control torque (in Nm)
can be seen in Table I.
Similar difference in the order of the magnitude of position tracking errors
was also observed in Experiment 2, when an external friction was applied to the
54
Table 3—1: Comparison of tracking results when no external load was applied to thecircular disk.
where ρ (kzk) is some positive globally invertible nondecreasing function.
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BIOGRAPHICAL SKETCH
Charu Makkar was born in New Delhi, the capital city of India. She completed
her Bachelor in Technology in the year 2003 from GGS Indraprastha University,
Delhi. She worked for a year in Continental Device India Ltd., the leading semi-
conductor manufacturing firm in India as a Process RD Engineer. She then joined
the nonlinear controls and robotics research group in the University of Florida for
Master of Science in mechanical and aerospace engineering in the year 2004.
She will join Deloitte Consulting as a Systems Analyst in July 2006.