Download control of systems with friction

CONTROL OF SYSTEMS WITH FRICTION
Karl J. Åström
Department of Automatic Control
Lund Institute of Technology, Lund, SWEDEN
E-mail:[email protected], WWW: http://www.control.lth.se
ABSTRACT
Friction appears in all mechanical systems and
has a significant impact on control. Successful design of mechatronic systems requires an understanding of the effects of friction as well as techniques to compensation. Friction phenomena are
complicated because they are caused by many different physical mechanisms. Friction can cause a
substantial deterioration of the performance of a
control system. Typical effects are steady state errors and oscillations. Many attempts have been
made to compensate for friction. Early efforts include introduction of dither signals. Other ideas
are to based on model based control. This paper
reviews several models for friction that have been
useful to model friction in motion control systems
and to compensate for effects of friction.
INTRODUCTION
Friction is important in all motion control systems.
It leads to deterioration in precisions and can
generate limit cycles. It is therefore important to
understand friction phenomena to understand and
improve the behavior of the systems.
Friction has been investigated for a long time.
Leonardo da Vinci investigated the motion of
rectangular blocks sliding over flat surfaces. The
French physicist Amonton [1] made similar investigations and concluded that the friction force at
a sliding interface is proportional to the normal
load. He also discovered that the the friction force
does not depend on the apparent area of contact. A
small block sliding on a surface thus experiences
the same friction as a large block if the normal
forces are the same. The French physicist Coulomb
[15] found that once the motion starts the friction is independent of the velocity, resulting in
the famous Coulomb’s friction law. Friction phenomena have been extensively studied in tribology see [11], [12] and [19]. Lately there have been
∗ This work has been partially supported by the Swedish
Research Council for Engineering Science, contract 95-759.
a resurgence of interest in friction in many scientific communities [42], [26]. This has been driven
by scientific curiosity, engineering needs and improved instrumentation such as laser interferometry and atomic force microscopes. Physicists have
investigated friction on an atomic scale [6]. Geologists have been investigating friction to better understand earth quakes [40], [25]. Friction plays an
important role in micro-mechanical systems. Control engineers have tried to come up with better
ways of dealing with frictions in mechatronic systems see [32], [29] and [35].
Even if the phenomenological properties of friction are reasonably well understood the fundamental physical mechanism that cause friction are not
well known. For example there are no strong correlation between surface roughness and friction.
There are cases where friction is less if a surface
is rougher. Experiments with mica surfaces have
shown that there is friction even if there are no
asperities in the surfaces. In this case the friction
is explained in terms of complicated molecular adhesion. The issue of lubrication is also poorly understood. Experiments on some surfaces have indicated that friction is less when surfaces are dry.
FRICTION MODELS
Before going presenting detailed mathematical
model we give a phenomenological description of
sliding friction. For this purpose we will consider
a rectangular block over a flat surface. Friction is
the tangential reaction force between the surfaces.
If a small tangential force is applied there will
initially be a small deflection of the block but the
block will return to its original position if the force
is reset to zero. There may be a small residual off
set. The residual off-set increases with increasing
force. If the force increases further the block starts
to move. The force required to do this is called the
break-away force. Its value depends on how long
the block has been at rest. The force is also varying
depending on many factors. The force required to
keep the block moving depends on the velocity and
a)
b)
F
v
c)
v
d)
F
where the force Fe is the force applied to the block.
The model can be thought of as the composition
of two models one for v 0 and a Coulomb
friction model for v 6 0. The model for v 0
says that the friction opposes motion as long as
the force applied is less than the stiction force
FS . The model with Coulomb and viscous friction
can also be augmented with a stiction model as is
illustrated in Figure 1 c).
F
F
v
v
Figure 1 Static friction models gives the friction
forcea as a function of the velocity.
many other factors.
Static Models
The simplest models describe friction as a function
of the difference in the velocities of the sliding surfaces. Such models are often called static models.
Coulomb friction which is described by
F FC sgn(v) µ F N sgn(v),
( 1)
where FC is the friction force FN the normal
load and µ the friction coefficient. This model is
illustrated in Figure 1 a).
The Coulomb friction model is very useful in spite
of its simplicity. It can explain several phenomena
associated with friction and it has is commonly
used for friction compensation [21, 5]. Many motion control boards have incorporated it in the software. The Coulomb friction model can be improved
by adding viscous friction F R v as illustrated in Figure 1 b) [39].
Experiments have shown that the parameters F C
and F R often depend on the direction of the
velocity. The model with Coulomb and viscous
friction has been used very successfully for friction
compensation in velocity drives [13].
The model (1) does not describe what happens
when the velocity is zero because the sgn function
is not defined for v 0. As mentioned in the introduction of this section friction acts like a spring
when a small force is applied. This phenomena is
often referred to as stiction [30]. A simple way to
capture this is to augment the Coulomb friction
model 1 with a specification of the friction force at
zero velocity. This results in the model
(
F
Fe
if v 0 and t Fet < FS
FS sgn( F e)
if v 0 and t Fet ≥ FS
There are several difficulties with the stiction
models. One problem is that the friction force is
not a function of velocity when v 0. This leads
to mathematical complications [10]. The model
does not capture the spring like behavior. The
stiction model is however one of the simplest
model that will describe stick-slip motion. There
are also practical difficulties in using the model
for simulation because it is always difficult to
determine precisely when the velocity is zero.
The Karnopp model [27] is similar to (2) but the
condition v 0 is replaced by tvt < ε . This
model is widely used in simulation programs for
mechatronic system. The model is an improvement
over the stiction model but there are difficulties
because the behavior may depend strongly on the
choice of the parameter ε and the tolerances in the
numerical integration routines. Modern software
for numerical integration permits test when the a
variable such as velocity changes sign and a proper
implementation of the stiction model can then be
obtained.
There is experimental evidence that the friction
force does not drop suddenly when velocity increases. Stribeck [43] observed that the friction
force had the shape shown in Figure 1 d) with
a minimum in the friction force. The velocity vS
where the friction force is minimal is called the
Stribeck velocity. Stribeck’s model can be described
by

F (v)
if v 6 0



if v 0 and t Fet < FS (3)
F Fe



FS sgn( F e)
otherwise.
The function F (v) can be determined by measuring the force required to maintain a constant velocity. It is often asymmetrical. One form of F (v)
that has been suggested is
δS
F (v) FC + ( FS − FC ) e−tv/v S t
+ F R v.
(4)
Dynamic models
(2)
Strictly speaking neither the stiction model (2),
the Karnopp model nor the model (3) are static
F
never be larger than F c if its initial value is such
that t F (0)t < F c .
Fc
x˙ < 0
x˙ > 0
x
slopeσ 0
−Fc
Figure 2
Stress-strain curve.
models in the sense that the friction force is a function of the velocity. Because of this the models have
both fundamental and practical drawbacks. They
can be avoided by recognizing that friction is indeed a dynamic phenomenon which should be modeled as dynamical systems. This a natural viewpoint for a control engineer but it is interesting
to see that the need has also been recognized in
other communities. The geologist are for example
talking about “rate and state models” [40].
Dahl’s model
Dahl [16], [17] and [18], developed a simple model
for the simulating control systems with friction.
His starting point was experiments on friction in
servo systems with ball bearings. One of his findings was that bearing friction behaves like solid
friction. Dahl’s experiments indicate that there are
metal contacts between the surfaces. This is agrees
very well with the results in [11], [12] which show
that friction is caused by microscopic irregularities
(asperities) that are pressed into the surfaces in
contact. This also explained Amontons observation
that friction was independent of the apparent surface. Friction is, however, proportional to the true
contact area of the surfaces.
The starting point for Dahl’s model is the stressstrain curve in classical solid mechanics [38],
[41], see Figure 2. When subject to stress the
friction force increases gradually until rupture
occurs. Dahl modeled the stress-strain curve by a
differential equation. Let x be the displacement,
F the friction force, and Fc the Coulomb friction
force. Then Dahl’s model has the form
α
dF
F
σ 1−
sgn v
dx
Fc
where σ is the stiffness coefficient and α is a
parameter that determines the shape of the stressstrain curve. The value α 1 is most commonly
used. Higher values will give a stress strain curve
with a sharper bend. The friction force t F t will
Notice that in this model the friction force is only
a function of the displacement and the sign of
the velocity. This so called rate independence is
an important property of the model. It makes it
possible to use the theory of hysteresis operators
[28]. It is also used in extensions of the model [7].
To obtain a time domain model Dahl observed that
α
dF
dF dx
dF
F
sgn v v. (5)
v σ 1−
dt
dx dt
dx
Fc
For α 1 the Dahl model (5) reduces to
F
dF
σv−
tvt.
dt
Fc
Introducing F σ z the model can be written as
dz
σ t vt
v−
z,
dt
Fc
F σ z.
(6)
Dahl’s model has been used extensively to simulate
systems with ball bearing friction. For a long
time it was the standard simulation model in
the aerospace industry. The model avoids the
numerical problems with the stiction model and
Karnopps models. The model has also been used
for adaptive friction compensation, see [46] and
[20]. The drawbacks of Dahl’s model are that it
does not describe stick slip motion and that it does
not captures the Stribeck effect.
The Bliman-Sorine Model
Bliman and Sorine have developed a family of
dynamic models in a series of papers [8, 9, 10].
It is based on the experimental investigations by
Rabinowicz, see [37]. It is assumed that friction
only dependsR on the sign of the velocity and the
t
variable s 0 tv(τ )t dτ . The model is given by
dxs
Axs + Bvs
ds
F C xs
(7)
Because of the rate independence it possible to
use the elegant theory of hysteresis operators
developed in [28, 45].
The complexity of the models is given by the
dimension of the state space. The first order model
is identical to Dahl’s model. This model does not
give stiction, nor does it give a friction peak at a
specific break-away distance as. The second order
model
−1/(η ε f )
0
,
A
−1/ε f
0
(8)
f 1/(η ε f )
B
and C ( 1 1 ) ,
− f 2/ε f
The parameter σ 0 is the stiffness of the bristles,
and σ 1 (v) the damping. The sum α 0 + α 1 then
corresponds to stiction force and α 0 to Coulomb
friction force. The parameter v0 determines how
g (v) vary within its bounds α 0 < g (v) ≤ α 0 + α 1 .
A common choice of f (v) is linear viscous friction
f (v) α 2v.
will, however, give stiction. This model can be
viewed as a parallel connection of a fast and
a slow Dahl model. The fast model has higher
steady state friction than the slow model. The force
from the slow model is subtracted from the fast
model, which results in a stiction peak. Both the
first and second order models can be shown to be
dissipative. Bliman and Sorine also show that, as
ε f goes to zero, the first order model behaves as a
classical Coulomb friction model, and the second
order model as a classical model with Coulomb
friction and stiction.
It is useful to let the damping σ 1 decrease with
increasing velocity, e.g.
The LuGre Model
The LuGre model [14] is another generalization
of Dahl’s model. This model is given by
tvt
dz
v − σ0
z,
dt
g ( v)
dz
F σ 0 z + σ 1(v)
+ f (v),
dt
(9)
The state variable z can be interpreted as the
average deflection of the asperities.
Extensive analysis of the model and its application are be found in [32]. The model captures many
properties of friction such as stiction rate dependent friction an frictional lag.
The model also includes rate dependent friction
phenomena such as varying break-away force and
frictional lag. For constant velocity the steady
state friction force is
F g (v) sgn(v) + f (v).
(10)
Any velocity dependence can thus be obtained by
a proper choice of the function g (v). One possible
choice which gives the Stribeck effect is
g (v) α 0 + α 1 e−(v/v0) ,
2
(11)
compare with (4).
Linearization of (9) around zero velocity and zero
state gives
d(δ z)
δ v,
dt
δ F σ 0δ z + (σ 1(0) + f T(0))δ v.
σ 1(v) σ 1 e−(v/vd) .
2
(12)
Physically this is motivated by the change of the
damping characteristics as velocity increases, due
to more lubricant being forced into the interface.
Another reason for using (12) is that it gives a
model which is dissipative, see [32].
A more detailed description of these and many
other friction models is given in [35].
EFFECTS OF FRICTION
In this section we will illustrate behaviors caused
by friction in motion control systems. We will also
illustrate how the models presented in Section 2
can be used to explain what happens.
Stick-slip motion
Stick-slip motion is a classic common problem
phenomena that occurs in system with friction. It
can be observed experimentally simply by pulling
a spring attached to a block on a surface. The
block starts to move when the spring is extended
so much that the string force exceeds the breakaway force. The motion stops when the spring
is compressed. The net result is a jerky motion
which switches between sticking and slipping. The
phenomenon also occurs when a chalk is drawn at
a proper speed on a black board creating a squeaky
noise. Stick-slip motion occurs in in motion control
system with slow motion and in metal processing.
It is of course highly undesirable is of course highly
undesirable in a motion control system.
Stick slip motion cannot be explained using a
Coulomb friction model (1). The stiction model (2)
is a simple model that captures stick-slip motion.
This model also explains that stick-slip does not
occur if the velocity of the spring is sufficiently
high. This is illustrated in the simulation in
Figure 3. The system simulated is described by
y, x
a)
1
0
0
0.4
x
10
Theta from the real pendulum
Phi from the real pendulum
b)
0.1
2
0.05
20
Time
v dx/dt
1
Angle (rad)
y
Angle (rad)
2
0
-0.05
0
-1
0.2
-0.1
0
0
0
20
Time
F, k( x − y)
c)
-2
0
10
Theta from simulations
d)
0.1
0
0
10
20
Time
Figure 3 Simulation of stick-slip motion. The upper
curve shows the positions of mass and the spring. The
curve in the middle shows the velocity of the mass
and the bottom curve shows the friction force and the
spring force.
0
-0.05
-0.1
0
5
Time (s)
10
Phi from simulations
2
1
Angle (rad)
0.05
Angle (rad)
1
10
5
Time (s)
0
-1
5
Time (s)
10
-2
0
5
Time (s)
10
Figure 4 Limit cycles caused by friction when controlling an inverted pendulum. The variable θ correspond to the tilt of the pendulum and the variable ϕ
correspond to cart position.
the equations
m
d2 x
k( y − x) − F
dt2
dy
vre f
dt
(13)
The friction force is given by (2) with FS 1.5
and FC 1. Linear viscous friction with F R 0.4 is added to the linear process model and the
parameters are vre f 0.1, m 1, and k 2. The
system starts with the mass is at rest. The force
from the spring increases linearly as y increases.
The spring force is balance by the friction force and
the mass remains still. The mass starts to slide
when the spring force exceeds the stiction force
FS and the friction force drops to the Coulomb
level FC . The mass accelerates but as it moves the
spring contracts and the spring force decreases.
The mass slows down and finally the motion stops.
The phenomenon then repeats itself in another
cycle. is no limit cycle if F S FC . Notice the rapid
change of the friction force when the mass stops.
Some care has to be exercised in a simulation of
this type.
Simulation with the LuGre model is more robust
since no switching occurs [14]. This gives a better
description of what happens when the velocity is
close to zero. Notice that stick-slip motion cannot
be explained using Dahl’s model.
Creation of stick-slip motion requires that there
is friction that can keep the mass at rest and
that there is a mechanism that creates a force on
the mass that increases when it is at rest. In the
classical stick slip motion the force is created by
pulling the string. In many motion control system
the force is instead created by the integral action
in a controller. This is called hunting [31].
Unstable modes in a system can also generate the
increasing force. This happens for example in control of an inverted pendulum on a cart. The cart
sticks because of friction when the pendulum is
close to the upright position and the cart is close
to its desired position. The pendulum starts to fall
and the control signal then increases until break
away occurs. Figure 4 shows comparisons of simulations with the LuGre model and experiments
results from experiments using a rotating pendulum [44].
Analysis Methods
It is in general difficult to analyze systems with
friction because of the the friction models are
strongly nonlinear. The Coulomb friction model being so simple is one exception. If the rest of the
system is linear the system can be represented as
a feedback connection of a linear system and a
relay representing the Coulomb friction. Approximate analysis of limit cycles can then be made
using describing functions [3]. By using special
techniques it is also possible to analyze limit cycles with Coulomb friction and stiction exactly [33].
With more complicated friction models we have to
resort to simulation. Simulation can also be difficult for models with switches and care must be
taken to obtain reliable results.
Both experiments and simulations have shown
that the tracking error for systems with friction
is far from Gaussian [34]. This means that traditional stochastic analysis is not appropriate. It also
Tracking reference
Fˆ
Friction
observer
4
2
xr
+
e
Σ
−
Linear
Controller
+
Σ
−
1
Js
v
1
s
[rad]
+
x
0
-2
-4
F
Real
Friction
-6
0
10
20
30
40
50
60
Time [sec]
70
80
90
100
Tracking error
Figure 5 Block diagram of the model-based friction
compensation scheme.
0.1
[rad]
0.05
0
-0.05
means that traditional quality measures such as
standard deviations can be very misleading [34].
FRICTION COMPENSATION
Since friction occurs in all motion control systems
it is of course very interesting to investigate to
methods for compensating for friction. Several
different schemes have been attempted.
Dither
Dither is a simple way to eliminate some effects
of friction. A mechanical vibrator was the earliest
form of dither [36] which was used in gyroscopes
in the 1940s. Electrical dither signals can easily
be superimposed on the signals generated by the
controller. This is used to improve conventional
control valves [23].
Acceleration Feedback
The advances in micro-electronics have given several new cheap components, micro-mechanical accelerators is one example. By closing a feedback
loop around an accelerometer it is possible to obtain a high gain loop that controls the acceleration
of a mass directly. This is perhaps the simplest an
most robust way of eliminating friction.
Model Based Friction Compensation
Many schemes form model based friction compensation have been proposed. The key idea is shown
in Figure 5. The system is provided with a friction
observer which tries to estimate the friction based
on available measurements and a friction model.
The estimated friction F̂ is then added as an input
to the system. Effective friction compensation requires that good velocity measurements are available and that the input is introduced in such a way
that there is little dynamics between the injection
point and the point where friction acts.
There are many schemes of this type that mainly
differ in the complexity of the friction model used.
-0.1
without friction compensation
0
10
20
30
40
with fixed friction compensation
50
60
Time [sec]
70
80
90
100
Figure 6 Experiments that show the effect of model
based friction compensation [29]. The upper curve
shows the tracking signal and the lower curve shows
the tracking error.
Some schemes use a simple Coulomb friction models. They are available in many standard system
for motion control. Since the properties of friction
changes significantly it is essential to adapt the
friction model [2]. The paper [22] describes a model
reference adaptive system and [13] describes a
self-tuning controller for velocity control. A static
friction model with an asymmetric Coulomb and
viscous friction model is used. There are other
schemes that use more elaborate dynamic friction
models [32], [29]. Design of friction compensation
is difficult because of the strong nonlinearities.
Passivity is one technique that has been used very
successfully to design compensators with guaranteed stability [34]. The technique is based on the
idea that a system with friction observer can be
represented as a feedback connection of a linear
system and a passive system. The dynamics of the
linear system depends on the controller. In some
cases it is possible to design the controller so that
the linear system is passive and the stability of the
closed loop system then follows from the passivity
theorem, see [47] and [24]. The LuGre model has
passivity properties, [14], [4].
Figure (6) illustrates what can be achieved with
model based friction compensation for a tracking
servo. The time segment 0-50 shows a typical behavior for a system with friction. Large errors occur when the system sticks at the velocity reversals. Friction compensation based on the LuGre
model is applied for the time segment 50-100. Notice that the large spikes in the tracking error are
drastically reduced.
CONCLUSIONS
Friction is important in all motion control systems. There is currently much research on friction in many different fields which is generating
much knowledge and insight. This can be used to
increase our understanding of friction phenomena
and to design better friction compensators. This
paper has reviewed models for friction and techniques for friction compensation. There are two
promising methods for friction compensation, acceleration feedback and adaptive model based friction compensation. Development of dynamic models for friction and adaptive friction compensation
schemes are interesting research areas which have
good application potential for motion control.
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