Friction Problems in Servomechanisms�
Modeling and Compensation Techniques
Jan Tommy Gravdahl�
Department of Engineering Cybernetics
Norwegian University of Science and Technology
Trondheim
Outline of this presentation❏ Introduction
❏ Friction models
�� Static models
�� Models with time delay
�� Dynamic models
❏ Friction compensation
�� Non�model based compensation
�� Compensation based on static friction models
�� Compensation based on dynamic friction models
�� Comparision of compensation schemes
❏ Concluding remarks
Friction
❏ Friction is the tangential reaction force between two surfaces
in contact
❏ Friction depends on contact geometry and topology� proper�
ties of the surface materials� displacement� relative velocity
and lubrication
❏ Very complex phenomenon� composed of several physical
phonomena in combination� Modeling most often empirical�
❏ Friction in servomechanisms can cause limit oscillations�
known as stick�slip� and regulationtracking errors�
❏ Friction causes wear in the system and reduces lifetime�
❏ Friction is dissipative� that is� it can only extract energy from
the system
Magnied section of a photo of a highly polished steel surface�
Halliday and Resnick ��� ��
Schematic drawing of two surfaces in contact� G�afvert �������
Surfaces built up by asperities�
True contact occurs between asperities� asperity junction
Asperity widht� typically ���m� slope� �� ��� �steel��
Friction � de�nition of terms
❏ Static friction �stiction�� The force �torque� needed to initi�
ate motion from rest
❏ Dynamic �Coulumb� friction� A friction component indepen�
dent of velocity
❏ Viscous friction� Velocity dependent friction between solid
and lubricant
❏ Break�away� The transition from rest �stiction� to motion
�dynamic friction�
❏ Break�Away force� The amount of force needed to overcome
static friction
❏ Dahl�e�ect� Elastic deformation of asperity junctions behaves
like a linear spring for small displacements
❏ Stribeck e�ect� Decreasing friction with increasing velocity
at low velocities� Caused by �uid lubricants�
Friction models for constant velocity� Static models
Classic results for constant velocity�
❏ Coulomb friction
Friction force proportional to normal load�
F � Fcsgn�v�� Fc � �FN � Known by L� da Vinci �������
rediscovered by Amontons ������ and developed by Coulomb
��� ��� Not neccesarily symmetric�❏ Static friction�stiction
Intruduced by Morin �� ���� Might be greater than Coulumb
friction
❏ Viscous friction
Velocity dependent friction� Ex� Friction force proportional
to velocity�
F � Fvv� Reynolds �� ��� Caused by viscosity of lubricants�
❏ Negative viscous friction
Introduced by Stribeck ������� The Stribeck e�ect�
Other static models
❏ Karnopp ��� ��� Stribeck friction with a dead zone around
zero velocity to make simuations less time consuming�
❏ Hess and Soom ������� F �v� ��
�Fc� �FS�Fc�
���v�vs���Fvv
�A sgn�v�
❏ Armstrong�H�elvoury�������
F �v� ��
�FC � �Fs � FC�e��v�vs��
� Fvv�
A sgn�v�
❏ The �v�vs���term model the Stribeck e�ect
❏ Canudas de Wit et�al �������
F �v� � �Fc � ��jvj��� � ��v�sgn�v�
Modeling for adaption� linear in parameters�
❏ All these models are discontinious for v � �� An approxima�
tion with a nite slope through the origin would not re�ect
the physical phenomena� Karnopp ��� ���
Various static modelsF
v
f)
F
v
e)F
F F F
v
v v v
a) b) c)
d)
a� Coulumb d� Stribeck e�ect
b� Coulumb�viscous e� Karnopp
c� Coulumb�viscous�stiction f� Hess and Soom
Armstrong� etc
The generalized Stribeck curve
F
v
II III IV
I
I No sliding� elastic deformation
II Boundary lubrication
III Partial �uid lubriacation
IV Full �uid lubrication
Models with time delay
These models include the phenomenon known as frictional mem�
ory �or lag� by using Ff�v� t� � Fvel�v�t� �l��
velocity
FrictionFriction
velocity
lag
time
Hess and Soom ������� F �v� t� ��Fc �
�FS�Fc�
���v�t��l��vs��� Fvv
�sgn�v�
The Armstrong ������ seven parameter model�
F �x� v� t� �����
����x� if v � � �pre sliding displacement��
FC � Fs��� td�
�
���v�t��l��vs���sgn�v� � Fvv� ifv �� �
Includes stiction� Stribeck e�ect� Dahl e�ect and lag� but re�
quires switching and many parameters
Dynamic friction models
Static models do not capture observed friction phenomena like
� the hysteresis observed experimentally by Hess and Soom
������� Low velocity � time delay not accurate enough
� position dependence like the Dahl e�ect� Asperity junctions
behave like linear springs before break�away�
� variations in the break�way force�Break-away force
Force rate
Friction
Dispalcement
� Friction models involving dynamics are neccessary to describe
the friction phenomena accurately
The Dahl model
Inspired by the stress�strain characteristic from solid mechanics�
Dahl ���� � proposed the model�
dFdx
� ��
BB���F
FCsgn�v�
�CCA
��
where x is displacement� Friction depends only on position� In
the time domain �� � ���F � �z
�z � v ��jvj
FCz
A generalization of Coulomb friction�
dFdx
� � � F � Fcsgn�v��
The Dahl model models pre�sliding displacement and frictional
lag� but not stiction or the Stribeck e�ect�
The bristle model �Norw�� bust�
Proposed by Hessig and Friedland ������� Models the micro�
scopic contact points of the asperity junctions�
(x_i-b_i)
Sliding body
Stationary surface
N bonded bristles
Uses an algorithm to calculate
F �
NXi��
���xi � bi�
WhereN is the number of bristles� �� is the sti�ness and �xi�bi�
is the de�ection�
As jxi � bij � �s� the bound snaps� and a new is formed�
Ine�cient due to complexity
The reset integrator model
❏ Proposed by Hessig and Friedland ������ to make the bristle
model computationally feasible
❏ Instead of snapping a bristle� the bond is kept constant at
the point of rupture
❏ Strain in bond�
dzdt
�����
��� if �v � � and z � z�� or �v � and z � �z��
v otherwise
❏ Friction force� F � �� � a�z�����v�z � ��dzdt
❏ Stiction achieved by a�z� �����
��a if jzj z�
� otherwise
❏ Much easier to simulate than the bristle model� but care must
be taken in handling the discontinuities
The models of Bliman and Sorine
❏ Bliman and Sorine ����������� stress rate independence
❏ F depends on sgn�v� and s��
Rt� jv�� �jd�
❏ F a function of path and not velocity
❏ Model given by
F � CTxs
dxsds
� Axs �Bvs
❏ �� order model can be reduced to Dahl model and further to
Coulomb
❏ �� order� A � IR���� B� C� xs � IR��
Correctly models stiction� Emulates Stribeck e�ect by using
two Dahl models in parallel� Olson et�al ���� �� not true
Stribeck e�ect
The LuGre �Lund�Grenoble� dynamic friction model
❏ Introduced by Canudas de Wit et�al� ������❏ An extension of the Dahl model
❏ Based on bristle de�ection in an average sense
❏ Models both Stribeck e�ect� stiction� frictional lag and vary�
ing break�away force
F � ��z � ��dz
dt� ��v
dzdt
� v �
jvj
g�v�z
g�v� � FC � �FS � FC�e�� vv���
❏ Only one rst order di�� equation
Other friction modeling techniques
❏ Neural networks
Dominguez et�al ������ model the dynamic friction of a ser�
vomotor using neural networks� The resulting model includes
Stribeck e�ect and frictional lag� but failed to model other
known friction phenomena� Du and Nair ��������� � more
promising�
❏ Spectral analysis
Popovi�c and Goldenberg ���� � use spectral analysis to
model the position� and velocity dependent friction of a servo
motor� Friction force is represented by a Fourier series�
Ff�q� �q� � A�� �q� �
NXj��
Aj� �q� sin�
BBB�
Cj�Bj� �q�
�CCCA
Veried experimentally� Accuracy can be improved by in�
creasing N � the number of DFT components
Comparative studies of friction models
Haessig and Friedland ������ compare the bristle model� the re�
set integrator model� the Dahl model� the static Karnopp model
and the classical Stribeck model� Results�
� Dahl� No stiction
� Karnopp fast� bristle and classical slow �in simulations�
� The classical model wrongly predicts limit cycles
� Implementation� Karnopp hard� Dahl and reset integrator
easy
G�afvert ������ compares the Bliman and Sorine models to the
LuGre model� Conclusion� LuGre includes more friction phenom�
ena than Bliman and Sorine�
Conclusion� The LuGre model is probably the most accurate
dynamic friction model avaliable
Friction compensation
Tasks in servomechanisms that require friction compensation�
Precision positioning� Velocity reversal and Velocity tracking
Approaches to solve the friction problem�
❏ Friction avoidance� design for control
� Lubricant selection� �uid �oil� grease� or dry �te�on� dia�
mond�
� �Ball� bearings� active control� magnetic� piezoelectric
� Redesign of physical system� inertia reduction
❏ Non�model based friction compensation
� PDPID
� Dither�
❏ Model based friction compensation
� Estimating the friction force F by F using a friction model
and compensating for friction by adding F to the control
� The estimate F can be xed �identied o�ine� or adaptive
Non model based compensation techniques❏ Dither
� Introduction of a high freq� oscillation keeps the system in
motion� avoiding sticktion� �in use on e�g� gun mounts�
� Analysis with describing functions �Balchen ����� or av�
eraging �Mossaheb �� ��
� Normal dither �external vibrator� modies friction� tangen�
tial dither �control input� modies the in�uence of friction
❏ Impulsive control
� Achieve high precision positioning by applying a series of
small impacts� when in stick� Yang and Tomizuka ��� ��
Adaptive pulse width control�
❏ PDPID�
� The regulator problem is stable under PD control�
� Tracking may lead to stick�slip limit cycles�
� Integral action reduces steady�state errors � hunting
OverviewServoMotors
ServoMotors
Method
Problem
Application
FrictionModel
Dynamic
Other N DOFManipulator
PID adaptive
Regulation Tracking
Static
Reg Track Reg Track Reg Track
N DOFManipulator
estimation RobustPassivity Nonlinear
System models
The study of compensation techniques will be restricted to the
following two applications
�� Servomotors driving a load with friction�
J �� � u� Ff
where J is the moment of inertia� � is the angular velocity�
u is the input torque and F f is the friction torque�
�� Robotic manipulators in N DOF with friction in the joints�
D�q��q � C�q� �q� �q � g�q� � u� Ff �
where q � IRn� joint angles� D�q�� inertia matrix� C�q� �q� �q�
vector of Coriolis and centrifugal terms� g�q� � gravity� u �
IRn� control torques and Ff � IRn� friction torques� Only
friction in joints included� Can also have friction when in
contact with environment �force control��
Position regulation for �DOF mass system
Southward et�al �������
❏ System has similar eq� of motion as a servomotor
❏ Uses static Stribeck and Karnopp friction models
❏ Control law� PD � nonlinear �discontinuous� friction com�
pensation�
K_p x+F_c(x)
x
Fs
Fs
F
❏ Global asymptotic stability proved by LaSalle�s theorem using
Dini deriviatives
❏ Position regulation conrmed by experimental results
Adaptive position tracking for servo
Friedland and Park ������������
❏ Position tracking of servo with static Coulomb friction
❏ Adaption of unknown Coulomb friction�
Fc � z � kjvj�
�z � k�jvj����u� F �v� Fc�
�sgn�v�
� Fc � Fc asymptotically�
❏ Control law� u � PD � F �v� Fc�
❏ Position tracking conrmed in simulations� also when includ�
ing viscous friction�
❏ Experimentally conrmed by Mentzelopoulou and Friedland
������ for Coulomb friction� and by Amin et�al ������ for
viscous friction❏ Extended to two DOF manipulator by Yazdizadeh and
Khoasani ������
Friction compensation with static friction models
Adaptive velocity tracking for a tracking telescope
Gilbart and Winston �������
❏ The rst result on adaptive friction compensation
❏ System� a motor driving an optical telescope
❏ Friction modeled as classical Coulomb friction
❏ MRAC� u � K��t� ��p �K��t����m � � ��m� �K��t�sgn� ��p� �z �
Friction est�
❏ GAS proven by Lyapunov
❏ Controller was implemented on a ��in optical telescope used
for tracking satellites
❏ The application requires velocity of motor to pass through
zero
❏ Adaption eliminated dead zone encountered in zero crossings
❏ RMS tracking error reduced by a factor of six due to friction
compensation�
High precision position control
❏ Kim et�al� �����a� study a servo motor driving a xy table
❏ The friction model used is the Armstrong ��parameter model
❏ A �tracking controller� brings the system within a
small distance � from the reference� Then fuzzy�PD
ym yPlant
Fuzzy PD
Trackin contr.
❏ Tracking controller� Adaptive sliding mode � friction comp�
Paramters in friction model found using Evolution Strategies�
❏ Fuzzy rules tuned by Experimental Evolutionary Progr�
❏ Experiments conrm position error less than ��m�
❏ Position error in an area dominated by the Dahl e�ect� em�
phasizing the need for accurate friction models
❏ Extended by Kim et�al� �����b� to tracking
Adaptive friction compensation in manipulators
Canudas de Wit et�al �������
❏ Low velocity tracking of the last link in �DOF manipulator
❏ Uses the static friction model
F �v� � �Fc � ��jvj��� � ��v�sgn�v�
Linear in unknown parameters❏ Controller structure
u � � m� I�v � mgr cos�q� � Ff
v � feed forward � PID
Ff � �T��t�� � estimated
estimation algorithm� Exponentially weighted least squares�
❏ � used to avoid friction overcompensation� which is
shown to cause oscillations
Position regulation of N DOF robot manipulator
Cai and Song �������
❏ Position regulation of manipulator with Coloumb friction and
stiction
❏ Uses the Karnopp friction model� with zero dead band in
stability analysis
❏ Control law� PD�adaptive gravity compensation and robust
friction compensation�
u � �Kv �q �Kpq �G�q�� non
non�i � �msi tanh�iqi
❏ Similar in spirit as Southward et�al ������� but continuous�
❏ Convergence to a set by LaSalle�s theorem� Size of the set
depends on �i�
❏ Conrmed by simulations
Friction compensation with dynamic friction models
Position tracking of airborn servo
Walrath ��� ���
❏ Studied stabilization of airborn pointing and tracking tele�
scope
❏ A servomotor produces a corrective torque to compensate for
gimbal bearing friction
❏ Observed that friction responds continuously to velocity re�
versal� Static model not su�cient � Dahl�s model
❏ Probably the rst reference to employ a dynamic friction
model in control design❏ Controller � u � Proportional � F
❏ The estimate F was calculated using the Dahl model� Adap�
tion on model parameter�
❏ Experimentally veried� Reported of a factor ve improve�
ment in RMS pointing error
Position control of servomotor
Khorrami et�al �������
❏ Considers a servomotor driving a load with friction
❏ Dynamic friction modeled with LuGre model
❏ Uses a robust adaptive variable structure controller�
u � ��BTPx� �BTPx� T tanh��a� bt�BTPx��
with update law
�� � kBTPxk�� � �
❏ The unmeasurable friction states are treated as bounded dis�
turbances
❏ Globally asymptotically stable
❏ Also give similar results using backstepping when considering
compliant transmission with friction at both motor and load
side�
❏ Result extended in Sankaranarayanan and Khorrami ������
to the low velocity tracking problem
Position tracking for servomotor
Canudas de Wit and Lischinsky �������
❏ Study position tracking for a servomotor
❏ Use dynamic LuGre model
❏ The parameters of the LuGre model estimated numerically
from experiments
❏ Fixed friction compensation� u � Js�xr�JH�s�e� F where
H�s� depends on controller choise �PD� ltered PID�
❏ Adapting normal force variations�
u � Js�xr � JH�s�e� ��z � ��dz
dt� ��v
dzdt
� v � ���jvj
g�v�z � ke �
d�dt
� ����jvj
g�v�z�zm � z�
❏ Adaption also for temperature changes
❏ Convergence e� � by Lyapunov� Conrmed experimentally�
Position tracking for manipulators
Vedagarbha et�al �������
❏ Consider the problem of position tracking in a manipulator
with dynamic friction in the joints
❏ Use the dynamic LuGre model in each joint
❏ Employ nonlinear observers to estimate the unmeasurable
friction states� and presents convergence results both for
adaptive and non�adaptive controllers�
❏ Control law� u � w � �� �q�z � kcr� where w is represents
dynamics to be cancelled out� �� �q�z is the estimated friction
torque and r is the ltered tracking error
❏ Adaption is studied for unknown �linear� parameters in LuGre
model and for variations in the normal force�
Position tracking of manipulator
Panteley et�al ��������� ��
❏ Study position tracking in manipulators using LuGre
❏ Main point� Treat the friction compensation problem as a
disturbance rejection problem
❏ The part of the friction force dependent on z is regarded a
disturbance� and the linear �viscous� component is compen�
sated by the adaptive controller❏ Designs an adaptive controller using passivity arguments�
❏ An adaptive Slotine and Li controller strictly passies the
system and an outer loop �tanh� rejects the disturbance�
u � �Kds���� � � �Y��q� �q� qr� �qr�� Y�� �q�� Y�� �q� s��
�� � �Ts� � T � �
❏ Results conrmed experimentally
Comparative studies of compensation schemes
Leonard and Krisnaprasad ������� position tracking of servo
❏ Compares � di�erent controllers�
�� PID
�� Dither
�� MRAC based on Gilbart and Winston ������� asymmetric
Coulumb�Stribeck� AS
�� Computed torque � int��friction comp�� four di�erent
static models of friction
�� Based on Walrath ��� ��� Computed torque � int�� dy�
namic Dahl friction comp�
❏ Compared experimentally
❏ Problem� track a sinusoidal reference trajectory
❏ Conclusions�
�� Model based controllers better than PID and dither
�� The Dahl based controller outperformed the other
Comparasions� contn�
❏ Du and Nair ������
Compare their NN controller with the adaptive schemes of
Canudas de vit et�al� ������ and Friedland and Park �������
NN perform better� but computationally intensive
❏ Canudas de Wit and Lischinsky ������
compare their adaptive LuGre based controller with a PID�
No contest�
❏ Panteley et�al ���� �
Passivity approach compared to Amin et�al� ������� based on
adaptive scheme of Friedland and Park ������� Performance
similar for sinusoidal trajectories� passivity based better for
complicated trajectories� Amin requires knowledge of J
Concluding remarks
❏ Friction is a complicated phenomenon� Many approaches to
model friction� Classic� Ff � Ff�v��
❏ The trend is toward using dynamic friction models� where the
state of the art is the LuGre model�❏ Dynamics are required to explain observed friction phenom�
ena�
❏ Two main approaches to friction compensation� I� Non model
based and II� Model based
❏ Model based � u � unom � Ff � where Ff can be calculated
e�g� by the use of adaption and estimation� Recent publica�
tions also employ disturbance rejection schemes�
References on modelling and control of friction
Updated May ��� ����
The literature was collected during the PhD�study of�
Jan Tommy Gravdahl
Dept� of Engineering Cybernetics
Norwegian University of Science and Technology
N����� Trondheim� Norway
Telephone ��� ��������Fax� ��� ��������Email� TommyGravdahl�itkntnuno
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