Friction modeling, identification,_and_compensation_based_on_friction_hysteresis_yoon_mechatronics2014

Friction modeling, identification, and compensation based on friction
hysteresis and Dahl resonance
Jun Young Yoon ⇑
, David L. Trumper **
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
a r t i c l e i n f o
Article history:
Received 27 June 2013
Revised 24 January 2014
Accepted 3 February 2014
Available online 4 March 2014
Keywords:
Friction
Pre-sliding regime
Dahl resonance
Generalized Maxwell-Slip model
Hysteresis
a b s t r a c t
This paper studies effects of friction on control systems and utilizes the observed frictional behavior to
develop a parameter identification method for a friction model using frequency domain measurements.
Friction exists in a wide range of drive systems due to physical contacts in bearing elements, transmis-
sions, or motion guides. Friction in a control system can deteriorate performance by causing limit
cycles or stick–slip, as well as larger tracking errors. Friction compensation can help to reduce following
errors, but requires physical understanding and a reliable model of friction in both the gross- and the
pre-sliding regimes. In this paper, we adopt the Generalized Maxwell-Slip (GMS) model and develop a
frequency-domain method to identify the model parameters based on the frictional resonances, which
occur due to the elastic behavior of friction at small amplitudes. With the experimentally identified
parameters, the friction model is utilized to compensate the friction effects in a motion control system.
The resulting system performance of a compensated and uncompensated control system is then com-
pared in both the frequency and time domains to demonstrate the Dahl resonance identification
method for a GMS model.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Friction behavior can be divided into two regimes: gross-sliding
and pre-sliding [1]. In the gross-sliding or simply sliding regime,
friction is a function of the relative velocity of two sliding objects
and is well established as the Stribeck curve containing the ele-
ments of static, Coulomb, and viscous friction [2]. Friction in the
pre-sliding regime, on the other hand, is not directly a function
of velocity but depends upon the history of displacement of objects
in contact, acting as a nonlinear hysteretic spring [3].
In this paper, we experimentally explore friction behavior in the
frequency domain, and use that perspective to identify parameters
of the Generalized Maxwell-Slip (GMS) friction model [4]. A fre-
quency domain identification was introduced by Hensen et al. [5]
for an early dynamic friction model, called the LuGre model [6].
The identification approach developed in this paper extends the
frequency-domain view to extract the multiple varying stiffnesses
of the pre-sliding friction in the GMS model based on the frictional
resonance, which is a frequency-domain reflection of the hysteretic
nonlinear behavior of the pre-sliding friction. We experimentally
validate the fidelity of our identification method in both the fre-
quency and the time domains by comparing system performance
with and without a model-based friction compensator.
Prior to an in-depth discussion of empirical friction models, fric-
tion model parameter identification, and friction compensation, we
present a case study using a servomotor to help physically under-
stand friction behaviors.
2. Case study: Motor spin free response
2.1. Experimental setup
Fig. 1 shows a friction experimental testbed including a servo-
motor and a high-resolution encoder. We use a slotless and brush-
less servomotor, BMS 60, by Aerotech so as to minimize unwanted
nonlinear effects including brush friction and cogging. A high per-
formance rotary encoder implemented at the back of the motor
keeps track of the position displacement with an interpolated res-
olution of 106
counts per revolution. The position information goes
through an FPGA encoder counter into a controller which is imple-
mented on a National Instruments (NI) PXI 8110 real-time control-
http://dx.doi.org/10.1016/j.mechatronics.2014.02.006
0957-4158/Ó 2014 Elsevier Ltd. All rights reserved.
⇑ Corresponding author.
⇑⇑ Principal corresponding author.
E-mail addresses: jy_yoon@mit.edu (J.Y. Yoon), trumper@mit.edu (D.L. Trumper).
Mechatronics 24 (2014) 734–741
Contents lists available at ScienceDirect
Mechatronics
journal homepage: www.elsevier.com/locate/mechatronics

ler connecting to the host computer via ethernet. We use this
experimental setup for the case study, and for the rest of the exper-
iments in this paper as well.
2.2. Free response of motor in rotation
Two simple experiments demonstrate the effects of bearing fric-
tion. In the first test, we manually spin the motor shaft by hand with
the motor coils disconnected (open circuit) while reading motor
position and velocity. The resultant time responses are shown in
Fig. 2(a) and (b). The position response is measured at a sampling
rate of 2 kHz, and the velocity is calculated in the servo controller
by the backward difference of the measured position [7]. For the
second test, the motor is excited by a current pulse with the ampli-
tude of 0.05 A and a duration of 70 ms. Fig. 2(c) and (d) show the
resultant responses of the motor driven open-loop test. The com-
parison between the manual and the open-loop tests indicates that
the resulting oscillatory behavior is not caused by any electrical
components of the system, but by friction from the direct physical
contact of the mechanical parts, which is the bearing in this setup.
The spins illustrated in Fig. 2(a) and (c) are roughly 0.55 and
1.17 rad in amplitude with peak velocities of 7.6 and 15.4 rad/s
respectively. For both spins, the friction of the motor ball bearing
acts as a near-constant drag torque during continuous rotation.
However, for the region where the velocity becomes low, the fric-
tion shows elastic behavior, which causes oscillations as seen in
the magnified views. Note that the frequency of the oscillation is
on the order of tens of Hertz, thereby implying that it cannot be
due to mechanical eigenfrequencies such as due to motor
flexibility.
The results of these tests show key aspects of the friction in
both the gross- and pre-sliding regimes. In the gross-sliding
regime, the friction opposes the relative velocity with nearly con-
stant torque. In the pre-sliding regime as the rotor comes to rest,
friction acts as an elastic element which resonates with the rotor
inertia. Note that the oscillation period decreases as the shaft
comes to the stop, as illustrated in Fig. 2(b), which indicates that
the stiffness of the friction elasticity increases for smaller displace-
ments. This observation agrees with the nonlinear hysteretic
behavior of pre-sliding friction as reported in many references
including [1,3] and with the GMS model adopted in this paper. Also
note that the open-loop test shows a bit more lightly-damped
response, but given that the two tests were conducted several
months apart at different shaft positions with different environ-
mental conditions including temperature and humidity, this
discrepancy seems insignificant.
2.3. Paper outline
Both regimes should be taken into account in constructing a
friction model and compensating the friction, especially the pre-
sliding regime for high precision applications, since a positioning
system enters this regime frequently and can be dominated by
the pre-sliding friction. In Section 3, we discuss the classic and
modern empirical friction models, which are efforts to mathemat-
ically represent the friction in both regimes. In this paper we use
the Generalized Maxwell-Slip (GMS) model which has advantages
of high fidelity and relatively simple implementation for real-time
control purposes [4,8]. We develop and discuss a frequency-
domain method to identify parameters of this model in Section 4.
A model-based friction compensation technique is studied and
the resultant performance is shown in both the frequency and
the time domains in Section 5. Section 6 concludes the paper with
final comments.
3. Prior art
In this section, we review representative aspects of friction to-
gether with several empirical friction models which are introduced
in the literature and relevant to our study in this paper.
3.1. Classic friction model
The simplest model of the friction drag force is F ¼ lN where l
is the friction coefficient and N is the normal force. This model is
often referred to as Coulomb friction. When the model is aug-
mented with static friction and a linear viscous drag term, the
force–velocity graph appears as plotted in Fig. 3(a). A model with
continuous velocity dependency in the sliding regime was devel-
oped by Stribeck [9] in the form of
FðvÞ ¼ sðvÞ þ rv ¼ sgnðvÞ Fc þ ðFs À FcÞ exp
v
Vs

d
!
þ rv;
and is illustrated in Fig. 3(b). This model includes viscous friction,
where Fc; Fs; Vs; d and r represent the Coulomb friction, static fric-
tion, Stribeck velocity, shape factor, and viscous friction coefficient
respectively. Note that in the Stribeck curve, FðvÞ has both a veloc-
ity weakening curve, sðvÞ, and a velocity strengthening curve, rv
[10].
The Stribeck curve is often called static since the curve equation
is only a function of the relative velocity of the sliding objects, and
can be experimentally obtained by measuring friction force over a
range of constant velocities. The model’s biggest drawback is the
discontinuity at zero velocity, which is the pre-sliding regime. This
limitation means that the Stribeck model cannot encompass
dynamical phenomena including friction lag [11], rate-dependent
breakaway force [12,13], and hysteresis with non-local memory
[3,14,15].
3.2. More recent empirical friction models
There have been many efforts to develop friction models which
effectively cover the sliding and pre-sliding friction regimes, and a
variety of empirical models have been introduced in the literature
attempting to represent the hysteretic elastic behavior of the pre-
sliding regime depicted in Fig. 3(d) as well as the classic aspects of
sliding friction.
Dahl [16,17] introduced a model describing the pre-sliding
regime friction, in 1968, based on the speculation that the relation-
ship of friction force and displacement resembles the stress–strain
behavior of ductile materials. Introducing the LuGre model in 1995,
Canudas de Wit et al. [6] constructed a state evolving equation
Fig. 1. Experimental setup including an Aerotech servomotor (BMS 60), encoder,
and motor driver (Soloist Hpe 10).
J.Y. Yoon, D.L. Trumper / Mechatronics 24 (2014) 734–741 735

with the pre-sliding displacement as the internal state to model
the friction in both regimes with a smooth transition from the
pre-sliding to sliding regime.
In order to describe the non-local memory characteristic,
observed in the friction hysteresis when there are multiple velocity
reversals in the displacement as illustrated in Fig. 3(c) and (d), the
LuGre model was modified to develop the Leuven model by
Swevers et al. in 2000 [15]. This model adds a hysteresis function
to represent the non-local memory, and the hysteresis function is
constructed with the Maxwell-Slip (MS) model [18] of parallel-
connected elementary blocks as shown in Fig. 4(a), each of which
has the elastic behavior ki of Fig. 4(b) with the Coulomb slip law
of force limit aiFc.
Instead of the constant Coulomb slip law, Al-bender et al. [8]
imposed sliding dynamics as a weighted Stribeck velocity-weaken-
ing curve aisðvÞ on the slipping phase of each Maxwell-Slip block
and constructed a friction model, called the Generalized
Maxwell-Slip (GMS) model. The behavior of each GMS block is
illustrated in Fig. 4(c). In the GMS model, two states of either stick
or slip determine the characteristic behavior of each elementary
block with the rate-state law, which are mathematically
represented as
dFi
dt
¼ kiv; if stick
dFi
dt
¼ sgnðvÞC ai À
Fi
sðvÞ

; if slip
Ff ¼
XN
i¼1
FiðtÞ þ rvðtÞ:
Here C indicates an attraction factor determining how fast the
slipping dynamics follows the velocity weakening curve, sðvÞ. Note
that the elementary block’s behavior illustrated in Fig. 4(c) is exag-
gerated with a large value of C so that it can clearly show how the
slipping dynamics provides a smooth transition from the pre-
sliding to the gross-sliding regime. From the parallel structure,
we obtain the total estimation of friction force simply by summing
the resulting force from each block as shown in the formula as Ff
where N represents the number of elementary blocks. Owing to
its straightforward structure, the GMS model provides a relatively
simple implementation for the purpose of real-time control, and it
has proven to describe many frictional phenomena experimentally
discovered. In this paper, therefore, we work with the GMS model
to estimate and compensate the friction of the experimental setup
introduced in Section 2.1.
(a)
(b)
(c)
(d)
Fig. 2. Responses of a motor shaft. (a) Manual test: angular position and velocity when spun by hand. (b) Manual test: magnified view of oscillation in angular position and
velocity due to friction as the rotor comes to rest. (c) Motor driven open-loop test: angular position and velocity when excited by the phase current. (d) Motor driven open-
loop test: Magnified view of the oscillation showing the similar behavior as the manual test.
736 J.Y. Yoon, D.L. Trumper / Mechatronics 24 (2014) 734–741

4. Friction parameter ID via resonance
The GMS model, despite the advantages of high fidelity and
straightforward implementation, requires nontrivial effort to iden-
tify the many associated parameters. Here we discuss a frequency
domain approach for parameter identification using frictional res-
onances. We call this the Dahl resonance method in this paper.
The parameters of the GMS friction model can be identified
directly in the frequency domain since the nonlinear elastic effects
of the pre-sliding friction are well-observed in the frequency
responses. As illustrated in Fig. 6, the open-loop plant of the servo-
motor setup has varying frequency responses depending on the
current command input amplitude. The response has a smaller
DC gain and a resonance bump moves toward high frequency as
the applied input amplitude decreases. This is consistent with
the GMS nonlinear spring with increasing stiffness for smaller in-
puts. The resonant bump, which we call Dahl resonance named
after Dahl who first investigated the pre-sliding friction [16,17],
can be considered as the resonance of the nonlinear elastic behav-
ior of the pre-sliding friction against the moving inertia. That is, the
nonlinearity in the frequency responses is due to the hysteretic
elasticity of the pre-sliding friction, and the information on its
changing stiffness is contained in the Dahl resonances. This is the
principle of the approach of the frequency domain parameteriza-
tion method developed in this paper. The Dahl resonance method
provides a straightforward way to identify friction model parame-
ters in the frequency domain, comparable to the conventional time
domain method of hysteretic profile fitting [19–21].
4.1. Measurements of Dahl resonance
In order to identify the correct parameters for the friction mod-
el, we need a proper configuration for Dahl resonance frequency
response measurements. What is usually conducted, as shown in
Fig. 5(a), is to directly excite the plant with a sinusoidal input
and measure both the excitation and the response signals while
sweeping a frequency range of interest. A frequency response is
then obtained by computing the magnitude ratio and phase shift
of these signals at each frequency. This simple configuration, how-
ever, often causes drift at the output for inertia-dominated systems
such as servomotors since any DC offset at the input is integrated.
It is this drift that is problematic when it comes to measuring cor-
rect frequency responses in the pre-sliding regime because the
drift keeps pushing the motor away from pre-sliding.
While it may be possible to limit the drift issue using white or
colored noise as the excitation signal, a position controller is gen-
erally implemented in order to control the output of the plant so as
not to drift away. By closing a control loop, however, it might cause
bias in the system identification, especially when the system is dis-
turbed [22]. To address this, the closed loop configuration depicted
in Fig. 5(b) is utilized herein. In this case, we use a low-bandwidth
(’1 Hz) controller so as to both eliminate the low frequency drift
and reduce the possible bias. The drift is eliminated in the loop
due to an infinite loop gain at zero frequency, even with a low-
bandwidth controller. The possible bias is kept small since the con-
trol law is ignorable for higher frequencies than the bandwidth,
which is set to approximately 1 Hz in the measurement
configuration.
The low-bandwidth closed-loop configuration additionally
helps to not affect the current command by large control effort.
By connecting the excitation signal of a dynamic signal analyzer
as a disturbance to the plant, therefore, we can control the ampli-
tude of the applied current command in the frequency range of
interest above 1 Hz. Also note in Fig. 5(b) that a negative sign is re-
quired to have a negative feedback for the system stability.
4.2. Dahl resonance ID method
The frequency responses obtained in the low-bandwidth
closed-loop configuration are shown in Fig. 6. Here we observe
(a)
(b)
(c)
(d)
Fig. 3. Representative frictional behavior and simulation of a motion trajectory and
resulting frictional force. (a) Stiction Fs, Coulomb friction Fc, and viscous friction rv.
(b) Friction model of the Stribeck curve in the sliding regime with function sðvÞ and
rv. (c) Bidirectional position trajectory with small motions and turnarounds. (d)
Hysteretic elastic behavior of the pre-sliding friction with non-local memory
characteristics associated with motion (c).
(a) (b)
(c)
Fig. 4. Schematic of the Maxwell-Slip model with stiffnesses ki, normalized
saturation limits ai, and friction forces Fi. (a) A series of Maxwell-Slip elementary
blocks in parallel where z is the displacement in the pre-sliding regime. (b) Behavior
of a Maxwell-Slip block with the Coulomb slip law. (c) Behavior of a Generalized
Maxwell-Slip block with the slipping dynamics.
(a)
(b)
Fig. 5. Schematic configurations to obtain frequency responses of the servo system.
(a) Open-loop setup allowing drift. (b) Low-bandwidth closed-loop configuration
preventing low frequency drift.

the Dahl resonances, which correspond to varying input amplitude
levels of the current command, Acmd. From this Dahl resonant fre-
quency, fDi
, we can identify the stiffness, ki and the saturation limit,
ai of each of the GMS blocks. The stiffness information is directly
obtained from the Dahl resonant frequencies using the mass mo-
ment of inertia of the rotor J and proper scaling for the unit conver-
sions. The inertia of the motor provided by manufacturer is
J ¼ 1:96 Â 10À5
½kg m2
Š:
Note that the inertia J can be also obtained from the measured
frequency responses. The response curve for the relatively large in-
put amplitude of 0.030 A shows an approximately straight magni-
tude plot with a À40 dB/decade slope, indicating a pure mass
response in the sliding regime. By fitting this measured curve with
an inertia model of
1
Jexps2
Â Kt Â
180
p

s¼jw
;
we can experimentally estimate the value of the inertia. Note that
the measured responses are in the unit of [A/deg], so the inertia
model is scaled by the factor of 180Kt
p to convert it to [Nm/rad] and
calculate the mass moment of inertia in the unit of [kg m2
]. The tor-
que constant Kt is a given value of 0.2 [Nm/A], and it relates the
phase currents of our three phase motor to the torque. Fig. 7 shows
the measured responses together with the simulation curve of
experimentally-obtained inertia Jexp ¼ 2:05 Â 10À5
½kg m2
Š, which
is closely equivalent to the value provided by the manufacturer.
For the rest of the calculations, we use the nominal value of
J ¼ 1:96 Â 10À5
½kg m2
Š.
We then calculate the effective torsional stiffnesses by using the
inertia J and the Dahl resonant frequencies fDi
as
Ktorsi
¼ 2pfDi
À Á2
Â J ½Nm=radŠ
Keffi
¼
Ktorsi
Kt
Â
p
180
½A= degŠ:
The stiffness of the pre-sliding friction changes according to dif-
ferent input levels, and the effective torsional stiffness Ktorsi
is de-
fined as the representative stiffness that the system experiences
at each input level. Keffi
is a scaled value to match the unit to our
measurements. These definitions are illustrated in Fig. 8.
The accumulated stiffnesses of GMS blocks in the ratio of the
current to the angle are then obtained via
K1 ¼ Keff1
½A= degŠ;
Kiþ1 ¼
Aiþ1 À Ai
hiþ1 À hi
½A= degŠ;
where i ¼ 1; 2; . . . ; N À 1 is the number of the GMS blocks engaged
in the slipping phase. We can now calculate the N stiffnesses ki’s
by taking the difference of the adjacent accumulated stiffnesses
Ki’s as
ki ¼ Ki À Kiþ1 ½A= degŠ
kN ¼ KN ½A= degŠ:
The saturation limits ai’s are then obtained using the values of
the calculated stiffness and the break points as
aj ¼ kj Â hj ½AŠ;
where j ¼ 1; 2; . . . ; N is the index of each GMS block. To fit our
experiments, we choose N = 8. The relevant variables and the iden-
tified parameters for the eight elementary blocks are organized in
Table 1. We utilize these values for the friction compensation dis-
cussed in Section 5.
The measured frequency responses in Figs. 6 and 7 can also give
an estimate of the static friction torque. As shown in the figures,
two responses are measured for the current input amplitude of
0.03 A. The first measurement shows notable jumps at around
17 Hz in both the magnitude and phase plots, implying the transi-
tion from the pre-sliding regime to the sliding regime with increas-
ing frequency. The second measurement conducted immediately
after the first one shows a different response as if the system is
in the sliding regime from the lowest frequencies. This transitional
behavior of consecutive frequency responses implies that the cur-
rent value of 0.03 A is on the verge of the breakaway point, and so
the static friction has a very close value of 6 Â 10À3
Nm for the ser-
vomotor setup with the torque constant of 0.2 Nm/A.
Note that Fig. 6 shows an interesting behavior at low frequencies
in both the pre-sliding and sliding regimes. For input levels within
the pre-sliding regime, the responses show phase offsets of approx-
imately À25° as apposed to near zero for a standard mass-spring
system. This is believed to be due to the nonlinear hysteretic behav-
ior of the pre-sliding friction. That is, the spring-like behavior of the
friction dominates over the inertia at low frequencies, but since the
Fig. 6. Input-amplitude-dependent frequency responses of the open-loop plant
with Dahl resonances. Curves are parameterized with the amplitude of the drive
current output of the motor driver, which corresponds to motor torque. Magnitude
is in units of [A/deg].
101 10
2
−10
0
10
20
30
40
50
Magnitude(dB)
Frequency (Hz)
0.030A
0.030A
Inertia Model
Fig. 7. Measured frequency responses and fitted simulation curve of an inertia
model with Jexp ¼ 2:05 Â 10À5
½kg m2
Š. Magnitude is in dB units of [A/deg].
Fig. 8. Schematic describing the effective stiffnesses of the pre-sliding friction Ktorsi
and Keffi
with the scaling factors due to unit conversion. The left figure relates
torque s to h in radians. The right figure relates current command amplitude Acmd to
h in degrees.

frictional elasticity is not linear and has hysteresis, it causes a delay
in the displacement following the input force. Similar behavior is
also observed in the sliding regime response to the input level of
0.030 A. The Coulomb model for the sliding friction acts as a satura-
tion block, adding a nonlinear damping to the system. This can
cause input-level-dependent delay in the displacement at low
frequencies where we expect a low-frequency phase of À90° for a
standard mass-damper system. The delay becomes larger with
increasing input level, and eventually for a sufficiently large input,
the damping effect becomes negligible and the response would
show a pure mass behavior.
To clarify these issues, we conducted a simulation using our
friction model. We used the model to generate simulated fre-
quency response curves which are in general agreement with the
measured data, as shown in Fig. 9. Note that for an input level
(0.001 A) small enough to consider the frictional stiffness to be lin-
ear (i.e. no GMS model blocks are slipping), the phase starts almost
at 0°, and for a large input (0.300 A) the phase begins at almost
À180° at low frequencies, showing the behavior of a nearly pure
mass.
Another thing to note in the measured responses in Fig. 6 is a
large phase delay. It seems that there are calculation delays and
IO delays at certain points in the measurement loop, most likely
in the motor driver. We need to consider this phase delay for the
controller design described in Section 5.1. However, it is not prob-
lematic for the Dahl resonance ID method since we are interested
solely in the Dahl resonant peaks to determine the necessary
parameters for the friction model.
In summary, to identify the parameters via Dahl resonances, we
need to:
(1) Measure Dahl resonances as the highest magnitude points in
the frequency responses where each resonance curve corre-
sponds to varying input levels of the drive current command
to the servo.
(2) Extract the effective stiffness information Keffi
using the
measured Dahl resonant frequencies fDi
and the system iner-
tia J.
(3) Calculate the accumulated stiffnesses Ki of the Generalized
Maxwell-Slip (GMS) blocks using values of input levels and
break points.
(4) Obtain the stiffness ki and the saturation limit ai of each
GMS block through the calculations illustrated previously.
This Dahl resonance method is naturally compatible with
frequency-domain measurements for loop shaping control tech-
niques in that it does not require additional position-controlled
experiments in the time domain as long as only the pre-sliding fric-
tion is considered. To create the sliding function curve requires
additional experiments at various steady velocities. In addition,
the Dahl resonance method can be readily used because such
resonances in the frequency domain are observed in a number of
systems including servomotors [23], linear stages [23,24], hard-
disk drives [25–28], fast tool servos [29,30], and lithographic
machines [31].
5. Friction compensation and experimental results
There have been various methods introduced in the literature to
address the effects of the friction in control systems, which include
both model-based and non-model-based compensation techniques
[24,32–34]. A simple model-based feedforward compensation is
used in this paper so as to effectively observe the performance of
the friction model identified by the Dahl resonance method dis-
cussed earlier. The fidelity of the identified friction model and
the control system performance are tested with the experimental
setup, and the results are presented in both the frequency and
the time domains.
5.1. Model-based feedforward control
The identified GMS friction model based on Dahl resonances is
used in the control algorithm shown in Fig. 10 to compensate the
frictional effects of the experimental setup. In the figure, the
right-hand dashed box represents a hardware plant including the
real friction which we try to estimate with the model. The left-
hand box indicates the overall control algorithm containing a lin-
ear position controller and a model-based feedforward friction
compensator.
Table 1
Relevant variables and identified parameters for the Dahl resonance method.
i Acmd [A] fDi
[Hz] ki [A/deg] ai [A]
1 0.003 60.74 0.0508 0.0306
2 0.005 57.85 0.0693 0.0767
3 0.007 52.49 0.0603 0.1134
4 0.010 42.69 0.0030 0.0123
5 0.013 38.95 0.0266 0.1686
6 0.015 35.33 0.0059 0.0525
7 0.018 31.53 0.0164 0.2196
8 0.020 27.68 0.0169 0.3264
10
0
10
1
10
2
10
3
−20
0
20
40
60
80
Magnitude(dB)
10
0
10
1
10
2
10
3
−225
−180
−135
−90
−45
0
Frequency (Hz)
Phase(Deg)
0.300 A
0.100 A
0.050 A
0.030 A
0.020 A
0.015 A
0.013 A
0.010 A
0.007 A
0.005 A
0.003 A
0.001 A
Fig. 9. Simulated GMS model frequency responses of the servo plant. Note similar
responses to the measured data as shown in Fig. 6.
Fig. 10. Control algorithm schematic with a linear controller and a model-based
feedforward compensator: real hardware plant (right-hand) and software controller
in NI PXI chassis (left-hand). Trajectory xref and velocity vref are generated within
the controller.

The linear position controller for the setup is designed by loop-
shaping to have a crossover frequency of 50 Hz and a phase margin
of 30°. The crossover frequency must be chosen carefully for a non-
linear system with such input-amplitude-dependent frequency re-
sponses. This is because the Dahl resonances move depending on
the input amplitudes. We choose the crossover so as to minimize
the Dahl resonance excitation as well as not to excite higher-
frequency potential flexible modes of the system. We need to also
consider the large excess phase delay in our choice of the crossover
to guarantee a positive phase margin. The feedforward compensa-
tor utilizes the identified GMS model to estimate the real friction
using the velocity command input and thereby reduce the effects
of friction in the system.
5.2. Performance in the frequency domain
The nonlinear hysteretic elasticity of the pre-sliding friction is
the cause of the input-dependent frequency responses. This effect
deteriorates the closed-loop system bandwidth as experimentally
confirmed with the closed-loop frequency responses in Fig. 11(a).
The designed bandwidth is not achieved, with a significant droop
in each response for position commands in the pre-sliding regime,
approximately up to the amplitude of 0.4° for our servomotor set-
up. This undesired outcome is resolved to a significant extent in
the feedforward compensated system as shown in Fig. 11(b). The
droop in the closed-loop frequency responses is notably reduced
with the friction compensation, and the responses of the compen-
sated system become less dependent on the input level. These re-
sults in the frequency domain show that the Dahl resonance
identification method provides reasonable values for the parame-
ters of the friction model.
5.3. Performance in the time domain
The system performance improvements are also observed in the
time domain via tracking performance. When a control system is
commanded to follow a position trajectory, the servo system must
travel the pre-sliding regime where the nonlinear frictional stiff-
ness becomes dominant. Without a proper compensation, there-
fore, this dominant nonlinear effect would not be overcome by
any linear controller, resulting in poor tracking, especially notice-
able in high-precision machines. We ran tracking experiments to
observe the limitation of a linear controller and the improvement
by the friction compensator. In the experiments, we commanded
sinusoidal position references with the frequency of 0.5 Hz and
the amplitude of 7° peak for the sliding regime and 0.12° for the
pre-sliding regime. The results are compared in Fig. 12. The track-
ing performance is significantly improved both in the pre-sliding
and the gross-sliding regimes in both RMS error reduction and
peak error reduction, as presented in Table 2. The peak error reduc-
tion is more important in the sliding regime experiments rather
than the RMS error since the largest tracking errors occur in the
velocity reversals and the friction compensation aims to reduce
those peak errors by eliminating the pre-sliding frictional effects.
There still remains some response distortion in the frequency
domain and position error in the time domain shown in Figs. 11
and 12 respectively. This might be because the linear controller
implemented may not be able to provide a perfect trajectory fol-
lowing, thereby causing discrepancy between the position com-
mand and the actual output while the model-based friction
compensator uses the derivative of the position reference instead
of the real velocity. Within these limitations, however, the identi-
fied friction model shows significant improvements, indicating
the high fidelity of the friction model identified by the Dahl reso-
nance method.
6. Conclusions and future work
This paper studies aspects of friction both in the sliding and pre-
sliding regimes, develops an identification method in the fre-
quency domain, and discusses a friction compensation technique
using the GMS friction model. The model-based compensation
shows promising results when the model parameters are identified
with the Dahl resonance method performed in the frequency do-
main. This frequency domain approach significantly improves the
performance of the friction-compensated system, proving that it
can be readily used with the equivalent benefits of the conven-
tional method of the hysteresis profile fitting. The performance re-
sults experimentally compared in both the frequency and the time
domains for both friction regimes help understand how friction af-
fects a control system and how we can compensate it to achieve
higher accuracy for high-precision motion control systems.
As future work, we suggest that the experimentally identified
friction model can be incorporated with a linear finite element
method (FEM) model of any structural modes to yield a full system
analysis. Connecting the nonlinear friction model block to the FEM
10
0
10
1
10
2
10
3
−30
−20
−10
0
10
20
Magnitude(dB)
10
0
10
1
10
2
10
3
−270
−180
−90
0
Frequency (Hz)
Phase(Deg)
3.00
o
0.40
o
0.20o
0.10
o
0.02
o
10
0
10
1
10
2
10
3
−30
−20
−10
0
10
Magnitude(dB)
10
0
10
1
10
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3
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−180
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0
Frequency (Hz)
Phase(Deg)
3.00
o
0.40
o
0.20o
0.10
o
0.02
o
(a) (b)
Fig. 11. Measured closed-loop frequency responses for varying amplitude of angular displacement. (a) Friction-uncompensated system. (b) Friction feedforward
compensated system.

and observing the behavior of friction model prediction in both
simulations and experiments can be considered as a future work.
In addition, it would be also an interesting research effort to pre-
dict the frictional stiffness of a system from first principles, using
an accurate model for mechanical components in contact, such as
bearings, thereby allowing the use of such models in the machine
design phase.
Acknowledgements
This work was conducted with support from ASML, Wilton, CT.
We particularly appreciate fruitful discussions with Ruvinda Guna-
wardana, an engineer at ASML. Also we would like to thank Aero-
tech Inc. for donating the motor and drive system, and National
Instruments Corporation for contributing the real-time control
system.
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4 4.5 5 5.5 6 6.5 7 7.5 8
−0.020
−0.015
−0.010
−0.005
0
0.005
0.010
0.015
0.020
Time (sec)
TrackingError(deg)
Compensated
Uncompensated
4 4.5 5 5.5 6 6.5 7 7.5 8
−0.15
−0.10
−0.05
0
0.05
0.10
0.15
Time (sec)
TrackingError(deg)
Compensated
Uncompensated
(a) (b)
Fig. 12. Tracking performance comparison in position error between friction-uncompensated system (blue) and feedforward compensated system (red). (a) In the pre-sliding
regime. (b) In the gross-sliding regime. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 2
Tracking error reduction of the friction compensated system.
RMS error reduction (%) Peak error reduction (%)
Pre-sliding regime 68.0 53.4
Sliding regime 56.8 73.2

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