Comparison of EKBF-based and
Classical Friction Compensation
Ashok Ramasubramanian1
e-mail: [email protected]
Laura R. Ray
e-mail: [email protected]
Thayer School of Engineering,
Dartmouth College,
Hanover, NH 03755
In servo control, traditionally, models that attempt to capture the
friction-velocity curve and interactions at contacting surfaces
have been used to compensate for friction-introduced tracking
errors. Recently, however, extended Kalman-Bucy filter (EKBF)based approaches that do not use a phenomenological or structured model for friction have been proposed. In addition to being
cast as a friction estimator, the EKBF can also be used to provide
parameter adaptation for simple friction models. In this paper, a
traditional motor-driven inertia experiment is used to demonstrate
the usefulness of EKBF in friction compensation. In addition, a
numerical simulation is used to test the robustness of the new
methods to normal force variations. Using root mean square position tracking error as the performance metric, comparisons to
traditional model-based approaches are provided.
关DOI: 10.1115/1.2431817兴
1
Introduction
The motion control precision of many mechanical systems is
severely compromised due to friction. Precise positioning systems
that involve repeated velocity reversals are particularly vulnerable. Examples of such systems include airborne navigation systems, systems used in computer controlled manufacturing, semiconductor manufacturing and inspection systems, robotics, and
automated surgical instruments. A primary feature of these applications is low velocity, bidirectional position tracking.
In cases where it is feasible to measure or estimate friction
accurately in real time, it is possible to use the measurement or
estimate to provide instantaneous feedforward compensation.
Friction is thus monitored in real time, and corrective action is
taken simultaneously. Optimized proportional-integral-derivative
共PID兲 control combined with feedforward friction compensation
provides high tracking accuracy with smaller control gains. In
general, real-time friction measurement is difficult and an observer is used to estimate friction using measured quantities, such
as position, velocity, and input torque. This estimate is then used
to provide a compensating torque at the input of the machine 共Fig.
1兲. This topology is considered in a number of previous studies
关1–5兴.
Friction observers are typically model-based, i.e., phenomenological or empirical modeling is used to characterize the surface
interactions leading to friction. Dahl developed the first successful
friction model for use in servo machines and used it to cancel
bearing friction 关1兴. This model accounts for presliding displacement 共sometimes referred to as the “Dahl effect”兲. Bliman studied
the Dahl model as a map between the displacement or velocity
1
Corresponding author.
Contributed by the Dynamic Systems, Measurement, and Control Division of
ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received October 18, 2005; final manuscript received June 23,
2006. Assoc. Editor: George Chiu.
236 / Vol. 129, MARCH 2007
and friction force and showed the existence of bounded hysteresis
loops 关6兴. A simplified version of the Dahl model was used by
Walrath to compensate for bearing friction in an airborne tracking
system 关2兴. Friction modeling in motion control is a widely studied field with a rich body of literature. Reference 关7兴 gives a
comprehensive survey.
The most significant recent development in model-based friction compensation for servo control has been the development of
the LuGre model 关8兴 and various schemes to adapt its six parameters 关4,5兴. This model improves Dahl’s work and includes the
Stribeck effect. Common friction phenomena, such as hysteresis,
presliding displacement, frictional lag, and stick-slip motion, are
also captured 关8兴. It has been applied successfully to cancel friction in a DC motor-driven inertia 关5兴 and is also used in industrial
applications 关9兴.
Following their successful application in estimating aerodynamic forces 关10兴 and tire forces 关11兴, extended Kalman-Bucy
filter 共EKBF兲-based methods have been introduced as an addition
to model-based friction compensation 关12,13兴. These methods append the friction state to the system state using a Gauss-Markov
共GM兲 formulation, and no attempt is made to capture the surface
level interactions leading to friction. Instead, friction is calculated
from simple Newtonian dynamics using the known plant model
and motion measurements 共Sec. 2.1兲.
An earlier comparative study performed by the authors 关12兴
demonstrated the effectiveness of the EKBF as a friction estimator. In this paper, we investigate this method further and address
the following issues: 共i兲 Whereas the earlier study used a second
order GM formulation for friction, it is possible that higher-order
formulations give better performance at an increased computational cost. First-, second-, and third-order formulations are tested
in this study, and the optimal order is selected. 共ii兲 Using a numerical simulation, we test the robustness of EKBF-based methods to friction level uncertainty. Also, the simulation uses a twobody system with dynamics significantly different compared to
the traditional DC motor-driven inertia test bed and we test how
the EKBF performs in this case. 共iii兲 At least two previous comparative studies that involve experimental validation of friction
compensation have reported that the simple and elegant Dahl friction compensator either had the best performance or was statistically similar to the best compensator for a wide array of operating
conditions 关3,12兴. However, both these studies point out the difficulty in tuning the free parameters of the Dahl model: a time
constant ␶ and the level of Coulomb friction, Tc. In this paper, we
consider an adaptive Dahl model where the EKBF is used to estimate Tc in real time. 共iv兲 Because of continued interest in the
EKBF-based methods, certain implementation aspects are covered
in more detail in this paper.
In order to compare the various friction compensation algorithms, a sinusoidal position tracking experiment is used. The angular position of a load inertia is used as the controled variable in
a PID-controlled system, where friction-related performance
losses are significant. Such position tracking experiments have
been used in previous comparative studies 关3,12,14兴. Following
the example set by these prior studies, root mean square 共RMS兲
position error is used as the performance metric. In addition, a
numerical simulation tests the efficacy of friction compensation
when friction levels change over time.
2
Friction Compensation Methodologies
2.1 EKBF-Based Friction Compensation (KF Estimator).
In many mechanical systems, the rigid-body dynamics are well
known, whereas certain external forces acting on the system are
poorly understood. Often, the well established rigid-body dynamics, along with motion measurements, can be exploited to recover
these unknown external forces. The basic idea behind EKBFbased friction estimation can be summarized as follows: Extract
friction force by exploiting the well-known motion equations and
using the information from motion measurements. This funda-
Copyright © 2007 by ASME
Transactions of the ASME
Fig. 1 Block diagram of a position control system with friction compensation
mental idea is illustrated using friction compensation in a DC
motor-driven inertia as an example. The system equations typically take the form
J␪¨ + B␪˙ = Uin − T f
共1兲
where J is the moment of inertia, B is the viscous damping, Uin is
the input torque, and T f is the opposing friction torque. ␪˙ and ␪¨
denote the angular velocity and acceleration, respectively, and are
time derivatives of the angular position ␪. Usually, it is simple to
determine J and a nominal value for B from the physical properties 共density and geometry of the inertia兲 of the system along with
a measured response to a step input torque. For a position control
system, Uin, is the output of a PID-type controller and is also
known. If velocity and acceleration are measured perfectly, friction in is the only unknown in Eq. 共1兲 and a time history of T f may
be extracted. In practice, however, measurements are prone to
measurement noise, system parameters are not perfectly known
and multiple sensors 共acceleration and velocity兲 are expensive.
When an accurate position measurement is available, state estimation techniques are used to extract friction from Eq. 共1兲 in real
time.
A Gauss-Markov 共GM兲 formulation is used to append the friction torque, and the state estimator is written as
冤 冥 冤 冥冤 冥 冤 冥
ˆ
␪¨
ˆ
␪˙
ˆ
Ṫ f
ˆ
␪˙
␪ˆ
1
B
0
0 −
−
J
J
= 1 0 0 0
0 0 0 1
0
ˆ
T̈ f
+
0
0
0
T̂ f
0
ˆ
Ṫ f
0
冤 冥冤 冥
1 0 0 0
w1
0 1 0 0
w2
0 0 1 0
w3
0 0 0 1
w4
0
trix associated with T f and Ṫ f therefore become tuning parameters.
The tuned parameters used in this study are as follows:
Experimental apparatus: Qc共T f 兲 = Qc共Ṫ f 兲 = 2 ⫻ 10−4. All other
Qc matrix entries= 0.
1
J
+
lation 共T̈ f = 0, used in Eq. 共2兲兲 treats the derivative of the friction
torque as a random constant during each sampling interval. The
optimal order is chosen based on the computational cost and performance improvement over the uncompensated case for a DC
motor-driven inertia. The GM formulation is sometimes referred
to as a “random walk model” and has been used in previous studies to append the system states to estimate aerodynamic forces
关10,15兴 and tire forces 关11兴.
A distinction needs to made between the process noise sequences w1, w2 and the process noise sequences w3, w4. The first
two sequences represent actual unknown disturbances that affect
the position and velocity. For instance, the input torque, Uin, could
contain electrical noise that could affect position and velocity. The
latter two sequences are not physical entities but are required to
drive the two friction state elements. These noise sources are internal to the EKBF and only their covariance needs to be
specified.2 Used in this way, these do not represent unknown external inputs, but rather quantify the amount of uncertainty in the
augmented states. The entries of the process noise covariance ma-
Simulated apparatus: Qc共T f 兲 = Qc共Ṫ f 兲 = 300. All other Qc matrix
entries= 0.
Uin
where Qc共T f 兲 is the entry in the process noise covariance matrix
共Qc兲 corresponding to T f . This method is denoted by the term “KF
estimator” to differentiate it from the adaptive Dahl model, which
also utilizes the EKBF.
共2兲
where the first two states are from the system dynamic model 共Eq.
共1兲兲, whereas the next two denote the friction states. w1, w2, w3,
w4 are zero-mean Gaussian distributed random variables included
to represent the uncertainty in each state element 共see discussion
below兲.
A first-order GM formulation 共Ṫ f = 0兲 treats the friction torque
as a random constant during each sampling interval. The EKBF is
a state estimator that estimates this constant, along with the other
states, during each time increment. It is noted that using a GaussMarkov formulation in this manner is merely a method to append
the system state for estimation. It does not mean that we are
constraining the first time derivative of the friction torque to be
zero during the entire time history. The second-order GM formuJournal of Dynamic Systems, Measurement, and Control
2.2
Classical Friction Compensation Methods.
2.2.1 Adaptive LuGre Model (ALM). The LuGre model 关8兴 is
a system of differential equations that captures several observed
friction traits, such as presliding displacement, stick-slip, and friction hysteresis. The contact surfaces are thought to contain
bristles, the interaction between which leads to friction. A firstorder nonlinear differential equation governs the average bristle
deflection z, and a function g共␪˙ 兲 characterizes the different friction
regimes and the Stribeck effect:
2
Note that the wk are not injected into the system in the manner persistent excitation is injected in some adaptive methods 共e.g., 关4兴兲.
MARCH 2007, Vol. 129 / 237
兩␪˙ 兩
dz ˙
=␪−
z,
dt
g共␪˙ 兲
˙ ˙ 2
␴0g共␪˙ 兲 = Tc + 共Ts − Tc兲e−共␪/␪s兲
共3兲
where Tc and Ts are the levels of Coulomb and static friction and
␪˙ s is the Stribeck velocity. The friction torque is given by
T f = ␴ 0z + ␴ 1
dz
+ ␴2␪˙
dt
共4兲
where ␴0 is the bristle stiffness, ␴1 is the bristle damping coefficient, and ␴2 is the viscous friction coefficient. The procedure to
estimate the static parameters 共Tc, Ts, ␴2, and ␪˙ s兲, and the dynamic
parameters 共␴0 and ␴1兲 is described in Ref. 5.
Several algorithms have been proposed to adapt the LuGre
model parameters, and in this paper, the Canudas de Wit and
Lischinsky method 关5兴 is chosen as a representative method. This
method addresses variations in Tc and Ts when there is a change in
the normal load. Equation 共3兲 is modified as
dẑ ˙ ˙ ␴0兩␪˙ 兩
=␪−⌰
ẑ − ke,
dt
g共␪˙ 兲
ˆ
␴0兩␪˙ 兩
d⌰
ẑ共zm − ẑ兲,
=−␥
dt
g共␪˙ 兲
k⬎0
zm = u f − a f
共5兲
where ẑ denotes that the average bristle deflection is now being
ˆ is the parameter to be adapted that denotes a simulestimated. ⌰
taneous uniform change in Ts and Tc with time. ␥ and k are tunable gains, and e is the position error. u f and a f are the filtered
signals that depend on the system inertia 关5兴.
This compensator was used only in the simulation study. The
parameters that are not estimated 共␪˙ s, ␴0, ␴1, and ␴2兲 are set to be
the same as that of the LuGre model used to inject friction 共see
Sec. 3.2兲. In addition, the two parameters to be estimated, Tc and
Ts, are initialized using their true values. The two gains in the
adaptation algorithm 共␥ = 20,000 and k = 1.0兲 are tuned to minimize rms position tracking error for a 0.8 Hz sinusoidal trajectory.
Fig. 2 Schematic of experimental and simulation apparatus
„not to scale…: The experimental apparatus „a… consists of a
motor driven disk under an optional normal load, which contacts the disk through an aluminum rider mounted at the end of
an I-beam. The simulation apparatus „b… features two linked
gear wheels. Friction-velocity curve for the experimental apparatus „a⬘… and the simulated apparatus „b⬘… are also shown.
append the system state, which is estimated in real time by the
EKBF. The state equations 共process noise sequences not shown兲
now become
冤冥冤
ˆ
␪¨
ˆ
␪˙
ˆ
Ṫ f
2.2.2 Dahl Model (DM). In this paper, we consider Walrath’s
implementation of the Dahl model 关2兴, according to which
Tf + ␶
dT f
= sgn共␪˙ 兲Tc
dt
共6兲
where T f is the friction torque, Tc is the level of Coulomb friction,
␪˙ is the angular velocity, and the time constant ␶ is a tuning
parameter.
The Dahl model is used only in the experimental study. The
value of Tc during the loaded and unloaded conditions is given in
Sec. 3. The parameter ␶ is tuned for each operating condition by
minimizing the rms tracking error for a 0.3 Hz driving frequency:
Unloaded: ␶ = 1/24 s 共0.1 Hz兲,
␶ = 1/16 s 共0.3 Hz兲,
␶ = 1/20 s 共0.3 Hz兲,
␶ = 1/36 s 共1.0 Hz兲
The Dahl model is said to be “quasi adaptive” as the parameters
are tuned offline for each loading condition and driving frequency.
It is also possible to use the EKBF to provide parameter adaptation for the Dahl model 共see below兲.
2.2.3 Adaptive Dahl Model (ADM). The Dahl model is used to
formulate the friction state and a first order Gauss-Markov formulation 共Ṫc = 0, see Sec. 2.1 for more information兲 is now used to
238 / Vol. 129, MARCH 2007
−
1
B ˙ˆ
␪−
T̂ f
J
J
ˆ
␪˙
+
ˆ
sgn共␪˙ 兲
1
T̂c − T̂ f
␶
␶
0
冥冤 冥
冉冊
1
Uin
J
0
共7兲
0
0
where again the first two state equations are from the system
dynamic model 共Eq. 共1兲兲 and the third equation is the Dahl friction
model 共Sec. 2.2.2, Eq. 共6兲兲. The parameter ␶ is not adapted, but
also not changed with operating frequency as in the nonadaptive
Dahl model 共Sec. 2.2.2兲. This model was used in both the experimental and the simulation studies with the following parameters:
Experimental apparatus: Qc共T f 兲 = 0.1, Qc共Tc兲 = 0.0001. All other
Qc matrix entries= 0, ␶ = 1 / 16 s.
Simulated apparatus: Qc共T f 兲 = 3 ⫻ 106, Qc共Tc兲 = 3 ⫻ 106. All
other Qc matrix entries= 0, ␶ = 1 / 20 s.
3
␶ = 1/40 s 共1.0 Hz兲
Loaded: ␶ = 1/10 s 共0.1 Hz兲,
ˆ
Ṫc
=
冉 冊 冉冊
Test Methods for Comparative Study
3.1 Experimental Procedure. The experimental apparatus is
very similar to those used in earlier studies involving position
tracking 关3,5,16兴. It consists of a DC motor that drives a heavy
disk. The schematic is shown in Fig. 2共a兲. Two ball bearings support the shaft on which the disk is mounted. The disk was made of
52100 steel 共density= 7800 kg m−3兲 with a Brinell hardness of
192. It was ground to be concentric in its shaft to 2.54
⫻ 10−4 cm. The disk radius is 6.99 cm, and its thickness is
1.91 cm. The disk mass is 2.28 kg, and its moment of inertia is
0.0056 kg m2. In this apparatus, friction exists in two shaft support bearings, as well as in the bearing internal to the motor itself.
A cylindrical rider in contact with the disk and under static normal
Transactions of the ASME
load provides an additional source of friction 共Fig. 2共a兲兲. The
motivation behind using additional sources of friction is to elicit a
variety of different friction traits in order to thoroughly test each
friction compensation scheme. The rider forms a line contact with
the disk, and a normal load of up to 2.3 kg 共5 lbs兲 can be placed
on the contact. The rider is made of aluminum and is mounted at
the end of an l-beam 共Fig. 2共a兲兲. The use of a rider in this fashion
is similar to the use of a mechanical brake in 关5兴. The apparatus
can be used with and without the rider in contact, and friction
levels are higher when the loaded line contact exists. To differentiate between these two cases, we use the term unloaded to refer
to the case when the external rider is not in contact with the disk
and the term loaded to refer to the case when the loaded external
rider is used to introduce additional friction.
The apparatus is modeled as a second-order system driven by a
known input torque Uin,
J␪¨ + B␪˙ = Uin − T f
共8兲
where ␪˙ and ␪¨ represent the shaft angular velocity and acceleration respectively, J is the moment of inertia of the disk and motor,
and B is the viscous damping in the bearings. T f , the unknown
friction torque, may include the unmodeled component of viscous
damping. An optical encoder 共resolution= 6.3⫻ 10−3 rad 共or兲 1000
points per revolution兲 is used to measure ␪. The plant parameters,
J = 0.0056 kg m2 and B = 0.0067 kg m2 s−1, are determined using
the disk material properties and geometry. The friction characteristics of the apparatus were determined using velocity control experiments as outlined in 关5兴. The following are the friction parameters of the experimental apparatus:
Unloaded: Tc = 0.128 Nm,
Ts = 0.22 Nm,
and ␪˙ s = 0.01 rad/s
␴2 = 0.0276 Nms/rad,
Loaded: Tc = 0.26 Nm,
Ts = 0.80 Nm,
␴2 = 0.0344 Nms/rad,
and ␪˙ s = 2.4 rad/s
The friction-velocity curve is shown in Fig. 2共a⬘兲.
3.1.1 Sinusoidal Position Tracking. Position tracking experiments were performed using the apparatus with the feedback control architecture shown in Fig. 1. The position command for the
rotating inertia is a sinusoidal tracking reference, ␪d共t兲
= 0.1745 sin共2␲ ft兲. This corresponds to a 20 deg peak-to-peak
sinusoidal command trajectory. Three different frequencies, f
= 0.1 Hz, 0.3 Hz, and 1.0 Hz are considered to evoke a wide variety of friction traits. A well-tuned PID controller provides basic
position tracking while different friction observers provide a compensating torque to cancel the actual friction torque 共Fig. 1兲. RMS
of the tracking error between the commanded trajectory and the
actual trajectory is used as the performance metric.
The total input torque to the motor, Utotal is given by
Utotal = UPID + UFF + T̂ f − T f
共9兲
where UPID is the PID control signal, T̂ f is the estimated friction
torque, and T f is the actual friction torque, which is unknown. A
feedforward acceleration term, UFF = J␪¨ is added to the input
torque to maintain consistency with the earlier comparative studies 关3,12兴. The PID controller is designed to maximize closed-loop
bandwidth. The controller gains are K P = 40, KI = 5, and KD = 2,
leading to a closed-loop bandwidth of ⬃20 Hz, which is significantly higher than the maximum driving frequency 共1.0 Hz兲.
Performance of each compensator is examined for each of the
three frequencies noted above and for the two contact conditions,
loaded and unloaded. For each frequency, contact condition, and
compensator, three trials are taken over a period of five days. Four
complete cycles of data 共sampling rate= 100 Hz兲 are taken for the
Journal of Dynamic Systems, Measurement, and Control
0.1 Hz driving frequency 共40 s of data兲, 6.66 cycles for 0.3 Hz
共20 s of data兲 and 10 cycles for 1.0 Hz 共10 s of data兲.
3.2 Simulation Procedure. It is of interest to test the robustness of the various friction compensation methodologies when
friction levels change over time 共e.g., due to a time varying normal load兲. A numerical simulation in which friction level variations can be precisely controlled is employed to test robustness of
friction compensation to varying friction magnitudes. Friction is
injected in the simulation using the LuGre model. The simulation
also allows the testing of friction compensation in a two-body
system, whose dynamics vary significantly compared to the standard DC motor-driven inertia, which has been used in almost all
previous studies 关3,5,12,14兴.
The simulated system is a gear train modeling a gun turret. The
system is described in detail by Teolis 关17兴, and the frictional
interface is modeled by Tao 关18兴. It features a coupled two-body
system linked by gear teeth 共Fig. 2共b兲兲. The input torque is fed to
the lighter gear wheel, and the position of the heavier gear wheel
共load兲 is the controled variable. Both wheels have the same radius
and hence the gear ratio is one. All simulations are performed in
SIMULINK 共The MathWorks, Inc.兲 using varying time-step integration. The dynamic equations governing the two-body system are
given by
Jm␪¨ m共t兲 + bm␪˙ m共t兲 = u共t兲 − uk共t兲
共10兲
Jl␪¨ l共t兲 + T f 共t兲 = uk共t兲
共11兲
where Jm is the driven inertia with instantaneous position, ␪m共t兲.
bm is the viscous damping in the motor, and u共t兲 is the input
torque. The load inertia is Jl and its position, ␪l共t兲, is the controled
variable. T f 共t兲 is the friction torque at the gear teeth. ␪˙ m and ␪¨ m
denote the driven inertia angular velocity and acceleration, respectively. Similar remarks apply for the load inertia.
The term uk共t兲 denotes the force coupling the two bodies and is
given by
uk共t兲 = k␦共t兲 + bk␦˙ 共t兲
共12兲
␦共t兲 = ␪m共t兲 − ␪l共t兲
共13兲
where k is the stiffness constant of the gear teeth and bk is the
damping coefficient. The plant parameters are Jm = 0.016 kg m2,
Jl = 0.182 kg m2, bm = 1.7 Nm s / rad, bk = 0.6155 Nm s / rad, and k
= 246.2 Nm/ rad. The frictional interface at the coupling is simulated using the LuGre model 共Sec. 2.2.1兲 with the following parameters: Tc = 0.47 Nm, Ts = 0.52 Nm, ␴2 = 1.7 Nm s / rad, ␪˙ s
= 0.0016 rad/ s, ␴0 = 5781 Nm/ rad, and ␴1 = 67.7 Nm s / rad. The
friction-velocity curve for this apparatus is shown in Fig. 2共b⬘兲.
3.2.1 Robustness Study. As in the experimental study, a baseline PID-controlled system is used with feedforward friction compensation to compare the efficacy of various friction compensation schemes. The PID control gains are K P = 200, KI = 100, and
KD = 20, leading to a closed-loop bandwidth of ⬃25 Hz, which is
much higher than the driving frequency used in simulation
共0.8 Hz兲.
The level of Coulomb friction Tc and the level of static friction
Ts were allowed to vary with time
Tcvar共t兲 =
Tc Tc
+
sin共2␲0.2t兲
2
2
共14兲
Tsvar共t兲 =
Ts Ts
+ sin共2␲0.2t兲
2
2
共15兲
where Tcvar and Tsvar are the time-varying Coulomb and static
friction levels used in the plant. Each parameter varies sinusoidally about its nominal value by 50%. This corresponds to a sinuMARCH 2007, Vol. 129 / 239
Table 1 Improvements „percentages… over uncompensated
PID system for different GM orders for the KF estimator
EKBF Order
I
II
III
Unloaded
0.1 Hz
0.3 Hz
1.0 Hz
46.97
65.15
65.15
41.90
62.86
67.62
42.37
56.78
62.65
25.46
54.76
55.49
16.18
43.60
13.93
Loaded
I
II
III
34.82
59.66
59.87
soidal variation of the parameter ⌰ in the adaptive LuGre compensator 共Eq. 共5兲兲, i.e.,
⌰共t兲 = 0.5 + 0.5 sin共2␲0.2t兲
共16兲
Note that the frequency of variation 共0.2 Hz兲 is small compared to
the driving frequency 共0.8 Hz兲.
4
Results
4.1
Experimental Results.
4.1.1 Optimal GM Order for KF Estimator. Table 1 shows the
percent improvements in RMS tracking error over the uncompensated PID controller for the a first-order 共Ṫ f = 0兲, second-order
共T̈ f = 0兲, and third-order 共T̈ f = 0兲 Gauss-Markov formulations for
the EKBF estimator. For each contact condition and frequency,
five different data points were taken and a mean was computed.
Note that for most cases, the improvement going from first to
second order is significantly higher compared to the improvement
gained by going from second to third order. Given the computational cost that increases with the EKBF order, the second-order
GM formulation is chosen for comparison to other methods.
4.1.2 Comparative Study. Figures 3共a兲–3共c兲 show the RMS
tracking error data points for each of the four compensators tested
under the unloaded condition. The following four methods are
tested: no friction compensation 共PID兲, nonadaptive Dahl model
共DM兲, adaptive Dahl model 共ADM兲, and KF estimator 共KF兲. It is
noted that all three compensators provide significant reduction in
the tracking error over the uncompensated case. For instance, at
the 0.1 Hz driving frequency, the Dahl model reduces the tracking
error by as much as 73%. At this frequency, performance of all
compensators are statistically similar. At higher frequencies, performance of the nonadaptive Dahl model 共DM兲 deteriorates compared to that of adaptive Dahl model 共ADM兲 and the KF estimator
共KF兲 共Figs. 3共b兲 and 3共c兲兲. The two EKBF-based approaches,
ADM and KF estimator, have statistically similar performance for
all cases.
Figures 3共a⬘兲–3共c⬘兲 show the RMS tracking error data points
for the loaded condition. Note that the data show a larger variance
than the unloaded case. As a consequence of friction levels being
higher, the RMS error values are also higher under this condition.
It is noted that ADM and KF improve performance compared to
baseline PID for all three driving frequencies. For instance, at
0.1 Hz, the KF estimator reduces the baseline tracking error by as
much as 70%. Also, as in the unloaded case, all compensators
perform statistically similar for the lowest frequency and DM performance deteriorates at higher frequencies, with performance
dropping below baseline PID at 1.0 Hz. The two EKBF-based
compensators have statistically similar performance for the lower
two frequencies. ADM has slightly better performance at 1.0 Hz.
Note that for all cases, parameter adaptation improves DM performance.
240 / Vol. 129, MARCH 2007
Fig. 3 Results from the experimental study: rms error data
points from three trials for each of the four compensators
tested. The first column of plots „a…-„c… contain data from the
unloaded condition and the second column of plots „a⬘…-„c⬘…
contain data from the loaded condition. The following abbreviations are used: PID= baseline PID compensator „no friction
compensation…, DM= nonadaptive Dahl model, ADM= adaptive
Dahl model, KF= Kalman filter estimator.
4.2 Simulation Results. Figure 4共a兲 shows the results from
the simulation to test compensator performance when friction levels change with time. The dotted line in Fig. 4共a兲 shows the actual
friction torque and the solid line shows the friction torque estimated by the adaptive LuGre compensator. The estimation algorithm takes ⬃1 s to converge. It is observed that this compensator
is unable to properly track the time varying friction level. The
estimate does not deviate significantly from the initial value of
0.52 Nm. Note also the high-frequency content in the estimate
共arrows in Fig. 4共a兲兲. The RMS tracking error for this case was
8.72⫻ 10−4 rad.
In contrast, the adaptive Dahl model is able to accurately track
the time varying friction level 共Fig. 4共b兲兲. The adaptation algorithm converges rapidly and the tracking error 共1.92⫻ 10−4 rad兲 is
much smaller compared to the adaptive LuGre compensator. The
error is, in fact, actually very close to tracking with no friction
共1.21⫻ 10−4 rad兲. With such a small error, it is difficult to differentiate between the actual and estimated friction curves at this
scale 共Fig. 4共b兲兲. Similar results were obtained with the KF estimator.
5
Discussion
Although EKBF-based methods have been used previously to
estimate aerodynamic forces 关10,15兴 and tire forces 关11兴, their use
has been limited to gathering data for off-line modeling. In a
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Fig. 4 Results from the simulation study: Actual friction
torque „solid line… and estimated friction torque „dotted line…:
„a… Adaptive LuGre model, „b… Adaptive Dahl model. Arrows in
„a… indicate high-frequency content in the friction estimate.
complicated modeling of this time-varying nonlinear phenomenon
by using simple Newtonian dynamics, accurate position measurements, and the time-tested Kalman filter.
All compensators have tunable parameters that need to be adjusted for optimal performance. These were set to minimize tracking error for a specific operating condition 共0.3 Hz unloaded for
the experimental study and 0.8 Hz for the simulation兲 and were
not readjusted for other operating conditions, even though doing
so results in substantial performance gains. Overall, the EKBFbased methods were significantly easier to tune. It was relatively
easy to select optimal values for entries in the process noise covariance matrix corresponding to the estimated states, and there
were no significant performance losses for reasonable perturbations in tuned parameters. Note also that EKBF-based methods do
not require the level of Coulomb friction and other friction parameters and, therefore, do not require a separate set of experiments to
determine them.
Closed-loop stability for a system under EKBF-based friction
compensation was formally investigated in 关20兴. Using eigenanalysis, a gain margin for the closed-loop system was calculated
and verified experimentally. The results indicate that EKBF-based
friction compensation is stable under a wide range of operating
conditions. Informally, 关12兴 tested the stability characteristics by
randomly perturbing the shaft during the experiment. It was found
that model-based methods were much more susceptible to such
perturbations.
6
previous study, EKBF-based friction estimates have been used to
estimate unknown normal loads 关19兴. In the current study, EKBFbased friction estimates are applied to provide friction compensation in real time.
Using a standard DC motor-driven inertia experiment, we have
shown that EKBF-based approaches provide excellent reduction
in friction-related tracking errors for a wide range of operating
conditions 共Fig. 3兲. In most cases, EKBF-based methods outperform traditional model-based methods. In a previous study, where
a direct friction measurement was available, we have also shown
that the friction torque estimated by the Kalman filter was very
similar to measured friction 关12兴. In addition, results from the
numerical simulation indicate that EKBF-based methods are able
to better track time varying friction 共Fig. 4兲. This is significant
because the adaptive Dahl model, which has no knowledge of the
LuGre model used to inject friction, is able to outperform the
adaptive LuGre model, which has the same structure as the LuGre
model used to inject friction, and is initialized with true parameters.
We have also demonstrated that a second-order GM formulation adequately captures the time-varying friction torque 共KF estimator兲, whereas a first-order GM formulation adequately captures variations in the Coulomb friction level 共adaptive Dahl
model兲. The success of EKBF-based methods is attributed to the
following factors.
Nonadaptive friction models use a fixed set of parameters that
are no longer valid when the operating conditions change. No
friction model is used in the KF estimator, and therefore, it is not
affected by parameter drift. In the adaptive Dahl model, the level
of Coulomb friction is estimated by the EKBF and we have found
that this type of parameter estimation is superior compared traditional methods, many of which require persistent excitation and
carry assumptions such as slowly time-varying parameters 关4兴.
The simulation also shows that EKBF-based estimation converges
faster 共Fig. 4兲.
By posing no structure at all 共KF estimator兲 or using a simple
friction model structure 共adaptive Dahl model兲, many of the problems associated with models using complex structure, such as the
need to estimate multiple parameters in real time, are avoided. In
spite of recent advances, friction remains notoriously hard to
model. The KF estimator shows that it is possible to circumvent
Journal of Dynamic Systems, Measurement, and Control
Conclusion
The main conclusion of this paper is that EKBF-based methods
are able to provide robust friction compensation under a wide
range of operating conditions. It is, however, noted that EKBFbased approaches are presented as an addition to and not as a
replacement of traditional model-based approaches. We have
found that simple models, such as the Dahl model, do perform
very well when well tuned and when the underlying assumptions
are satisfied. EKBF-based methods work on the assumption that
the plant dynamics are very well known. When this is not the
case, friction models may be more useful. In addition, we have
found that the LuGre model is a very efficient tool to model frictional contact in a numerical simulation.
Acknowledgment
This work was supported by the National Science Foundation,
under Contract No. CMS9721468. This support is gratefully acknowledged. The authors wish to thank Jennifer Townsend for her
help with building the experimental apparatus.
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