2.
The borderline distinguishing the two regimes is the “pre-
sliding distance” and the “breakaway force” or the “static
friction force” threshold.
The friction is modelled separately in pre-sliding regime
and sliding regime because of the different characters in two
regimes.
A. Modeling of static fiction
Generally, there is a certain extent roughness at the
contact surface. Borderline lubrication is the main lubrication
mode. Fig. 1 shows the borderline lubrication model. During
the borderline lubrication, the oil film is very thin; the
thickness of film can even decrease to one or two molecule-
deep. At the same time, the distance between friction surfaces
reduces and the mutual function strengthens between the
rough peaks. The friction characteristics are fully decided on
the physical chemistry function of the surface film and the
contact mechanics of the rough peaks [19].
base borderline lubrication film lubricant oil
Fig. 1. Borderline lubrication model
Taking into account the borderline lubrication mechanism
and properties of static friction, we can analyse the friction
contact surface as the viscoelasticity body. So the Kelvin
chain model of viscoelasticity body is adopted to describe the
static friction model, which is shown in Fig. 2.
This model consists of two elements: a nonlinear spring
module (k) and a viscous camper (c) [20]. These elements are
massless and, hence, do not really exist. They are
phenomenological elements. Each element exhibits a special
mechanical property and is described by a simple
mathematical expression. The combined result then matches
the complete pre-sliding behaviour. For convenience, this
model is described as a system in linear motion and the
following analysis that is based on this model gives the
relations between force, mass, linear displacement and
velocity [21].
Considering the modules shown in Fig.2, in which F is
applied force and x is the elongation of the spring module, at
time t, the governing equation is described as follows
)()( txktxcF iiii (1)
If F(t) is input, we have
FtJtx ii )()( (2)
where
iiii kctktJ /)/exp(1)(
principle of superposition and using heredity integral, the final
pre-sliding displacement can be represented as follows
(3)
From the characteristic of linear viscoelasticity body, we
know that the contribution of each stage exerted force to the
final deformation is independent. According to the Boltzmann
K1
C1
F
X
K2
C2
Kn
Cn
…
nxxx1 2
Fig. 2. Kelvin chain model
d
d
dFt
tJtJFt iii
0
)(
)()()0()( (4)
Equation (4) can be expressed as the convolut rm:
x
ion fo
)(*)()()0()( tdFtJtJFtx iii (5)
The total deformation is
n
i 1
(6)
From above equ tween applied force
and the pre-sliding displacement can be obtained at time t, in
whi
ynamic friction regime, the friction force is a
namic friction behaviour,
the
i txtx )()(
ations, the relation be
ch there are only two kinds of parameters: stiffness
coefficient (ki) and viscous coefficient (ci). So if the applied
force has been known, it is convenient to identify the
parameter from the experiment data with curve fitting
technology.
B. Modeling of dynamic fiction
In the d
function of velocity. To describe dy
LuGre friction model proposed by Canudas et al. [10] was
adopted.
At the constant velocity, the general friction force can be
written as
vvvFFFvs scsc 2/exp)()( (6)
Wit he Coulomb friction, the static frich cF t sF tion, s the
Stri elbeck v ocity, 1 an shape fact and 2or, viscous fr ion
II EXPERIMENTAL SET-UP
ict
coefficient.
I.
A. X-Y Table
Fig. 3(a) shows the XY AC servo table, in which the
riments were performed with a computer
con
friction expe
trolled system. The XY table is connected to ac servo
motors through ball-screws with a 5mm pitch. The servo
motor is MSMA082AIG permanent synchronous motor of
Panasonic, equipped with a MSMA083AIA driver. It
generates 10,000 pulses per revolution. The input is given as a
velocity command by the computer through a motion
controller of Googol Technology Ltd. The computer
exchanges information with the controller through a host
(a) (b)
Fig. 3. Th perimental set-upe ex
619
3.
computer com sending motion
surement
earch on friction compensation and
TION
munication interface, including
commands to the controller and acquiring the present state and
control parameters. Current measurement was carried out with
a K25 current-type Hall current senor. Its full range is 25A,
while one phase rated current of ac motor is 4.3A, so with
allowance considered, the lead was wound four loops around
the Hall device. The measured current values are read in the
computer by a 12-bit A/D converter. When the computer
sends out the slope velocity command, servo motor will speed
up from rest to the given velocity with the given acceleration.
The linear grating resolution was employed to measure
displacement of the table dynamically. Actual acceleration
measurement was performed with an accelerator of
Brül&Kjær, by adjustment; the reading 1V is equal to
8.92857m/s2
.
B. Torque Mea
At present, most res
control for servo table is based on input and output signal.
Generally, input signal is the output torque or servo current of
motor. For DC motor, it is easy to measure and control
current. So the servo current is always treated as the input
signal in most research. However, in AC servo system, the
angle between excitation magnetic field and armature
magnetomotive force is not fixed. The method of vector
control and magnetic field direction detection is usually
adopted. So it is more difficult to control output torque of
motor through adjusting armature current. In this situation, it
is a good idea to adopt the output torque of AC servo system
as the input signal. To deal with this problem, measurement of
motor output torque is the first work to do.
To measure the output torque of motor, a newly
developed torque sensor based on strain gauge is designed,
show as Fig. 3(b). Motor and basement of XY table are
connected with a sleeve. Four semiconductor strain gauges
with high sensitivities are glued at the sleeve, which compose
the Huygens' construction electric bridge. When the table is
driven by motor, a torque signal will be produced in the sleeve
whose size is equal and direction is opposite to the motor
output torque signal. This strain signal may transmit directly
through the dynamic strain gauge to computer. The strain
voltage signal is obtained through processing. Compared with
classical sensor using carbon brush and collecting ring, this
torque sensor can obtain torque signal more precisely.
The measured torque data is processed to obtain the
friction with related dynamical equation.
IV. PARAMETERS IDENTIFICA
(a) (b)
Fig. 4. The measured and estimated curve
It is also no rts contribute
the
city
V. SIMULATION OF FRICTION COMPENSATION
The oposed
mod
THE ESTIMATED VALUES OF PARAMETERS
ted that, in this system, many pa
friction force. They include the friction of ball bearings,
the friction between brushes and commutator bars and the
friction in the encoders. These frictions can be divided into
two categories, rolling friction and sliding friction. However,
the former is of an order much smaller than that of the latter
and can be neglected. In this research we did not try to
distinguish the individual contribution of each friction source
but treated them as a combined result as what happens in most
machines. Hsieh C. et al.[21] refers that there is no difference
between such kind of friction and those frictions in other cases
as reported in the literature as long as sliding friction is
concerned.
The parameter set for sliding regime depends on the
choice of )(vS . The parameters are determined based on
constant velo tests over the full velocity range of the table.
Fig. 4(a) shows the mean measured friction for different
constant positive and negative velocities, in which the friction
force corresponding to each measurement point is obtained by
averaging data samples, measured during five experiments.
Table 1 shows the identified parameters for positive and
negative velocities. The parameters are identified using a
nonlinear least squares identification algorithm in the Matlab
optimization toolbox. The full line in Fig. 4(a) shows the
estimated Stribeck curve.
To estimate the parameters of pre-sliding regime, a sine
displacement signal with small amplitude and low frequency
is applied to the system. The amplitude of this reference signal
is chosen in such a way that the system operates in the pre-
sliding regime. The low frequency of signal can guarantee the
movement inertia is minute. The displacement of worktable is
recorded. At the same time, the current of servo motor is
measured.
To model the hysteresis curve a static friction model
containing six units of nonlinear spring module and viscous
camper is used. So the numbers of elements in Kelvin chain
are chosen to be 6. Stiffness coefficient (ki) and viscous
coefficient (ci) of each element is estimated using Matlab
identification tools. The estimated curve which represents the
relation of pre-sliding displacement and friction force is
shown in Fig. 4(b). The figure shows the hysteresis effect
clearly.
effect of friction compensation using the pr
el is simulated in this section.
TABLE I
Parameter Positive Negative
Fs (N) 240.6770 -290.2720
Fc (N) 152.9965 -200.8617
Vs (m/s) 0.0448 -0.2868
1.9954 2.0561
2 (N·s/m) 803.9993 831.7143
620
4.
This friction el discu ve is a n model.
The transitio ecides wh part o n model
should be use t is neces cert lue of the
transition tim research nce [ s that the
transition tim rsely p nal to are-root of
the acceleration at the start of h has been
verified in their experiments is rela used in our
simulation.
mod
n time d
ssed abo
ich
partitio
f frictio
d. So i sary to as ain the va
e. The of refere 22] show
e is inve roportio the squ
the worktabl
. So th
e, whic
tion is
The method of PID-controller and feed forward control is
applied to compensate the errors caused by friction. The
friction behaviour is compensated using a feed forward
control. The real position is fed back and the tracking error is
used a PID-controller. The form of PID-controller is
dtxxKxxKxxKu didddp )()()(
The total control scheme and simulation programme are
shown in Fig.5.
The reference signal of displacement used in this
simulation is )5.0sin(25)sin(50 ttxd (mm). In Fig.
6, the curve of ‘line 1’ shows the tracking errors using friction
compensation based on the proposed model in this paper. The
curve of ‘line 2’ shows the tracking errors obtained only with
the Stribeck friction model. Obviously, the tracking effect of
form
he velocity reversal point. The reason is the
disc
e latter model.
proposed in this paper has a
more outstanding per l the tracking errors.
Bec
er is better than that of the latter. The largest position
errors appear in t
ontinuity in the friction model at velocity reversal,
especially in th
VI. CONCLUSIONS
This paper analyses the friction characteristics in two
regimes: pre-sliding and dynamic. A partition friction model is
built. Pre-sliding friction is modelled by means of hysteresis
model with nonlocal memory based on the viscoelasticity
theory. Two friction models are used to compensation tracking
error in which the effects are compared. The simulation result
shows that the new friction model
formance to contro
ause it reflects the static friction characteristic in pre-
sliding stage. In this way, the developed model is a effective
friction model for description of experimentally observed
friction behaviour.
Being flexible, the obtained model structure should
include other friction effects such as position dependence of
friction behaviour. This will be the subject of future work that
will also include practical application of the model in friction
compensation of machine tools.
Fig. 5. The control scheme of friction compensation
Fig. 6. Tracking errors using two models
ACKNOWLEDGMENT
The part of this research was supported by the Key
Project of Chinese Ministry of Education (No.104111). The
authors would like to express sincere gratitude to these
financial supports.
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