Transcript
Page 1: [IEEE 2005 IEEE Conference on Control Applications, 2005. CCA 2005. - Toronto, Canada (Aug. 29-31, 2005)] Proceedings of 2005 IEEE Conference on Control Applications, 2005. CCA 2005

Friction Identification in Robotic Manipulators: Case Studies

M.R. Kermani†, R.V. Patel‡, M. Moallem§†[email protected], ‡[email protected], §[email protected]

Department of Electrical and Computer EngineeringUniversity of Western Ontario

London,Ontario,Canada

Abstract— In this paper, friction identification in robotic ma-nipulators are studied. A single state dynamic model of thefriction force is utilized for the friction compensation algorithm.A practical method for obtaining parameters of the modelwhich pertains to robotic manipulators is presented. In orderto evaluate the competence of this method, two differentmanipulators with different friction characteristics are exam-ined. A 2-DOF manipulator used for high-speed, micro-meterprecision manipulation and a 4-DOF macro-manipulator usedfor long-reach positioning tasks are considered. It is shown thatdespite the different nature of the two manipulators, the samemethod can effectively improve the speed and performance ofmanipulation in both cases.

I. INTRODUCTION

Friction compensation can play an important role in diverse

control applications. This includes applications that involve

high-precision and fast motion control as well as those

containing heavy and sluggish servomechanisms with slow

tracking velocities. In each case, the friction force needs to be

adequately compensated for in order to improve the transient

performance and reduce the steady-state tracking errors. This

also ensures a smooth control signal without resorting to

high feedback gains. However, the nature of friction force is

not completely understood and finding a model that captures

all characteristics of friction such as Stribeck, stiction and

sliding, is difficult. In addition, friction also includes dynamic

effects which are highly nonlinear and depend on time,

position and velocity. In this regard, many researchers have

studied this phenomenon and suggested a number of models.

Dahl was the first to present a systematic model for Coulomb

and sliding friction [5]. This model has been widely used

for dynamic and adaptive friction compensation by other

researchers. A dynamic model of friction was proposed by

Olsson et al. [10]. The model, also known as LuGre model,

incorporates a single continuous state to model pre-sliding

displacement. Ehrich at al. [7] studied an adaptive model of

friction for low-velocity position tracking systems. Dupont

at al. [6] introduced a drift-free model of friction for precise

motion control tasks.

In order to implement any of the aforementioned models, the

parameters of the model need to be identified experimentally.

Friction identification is also another challenging part of the

†This research was supported by grants STPGP215729-98, RGPIN1345and RGPIN227612 from the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada.

friction compensation process. It can be achieved either off-

line or online as part of a system’s operation. In off-line

identification, the motion through which the data is gathered

can be deliberately specified. Online identification must use

data from normal system operations. In this paper, off-line

identification is studied and a practical method for measuring

the friction force, particularly in robotic manipulators is

presented. It is shown that a dynamic model of friction

force based on initial measurements results in very accurate

motion control without the necessity of resorting to high

feedback gains. The main motivation for the present work

arose after unsuccessful attempts at controlling industrial

manipulators within a certain accuracy. Friction can cause

more than 50% error in some heavy industrial manipulators

while any changes in control parameters in order to improve

the performance of the systems is usually unsuccessful.

Adding a steady-state model of friction, provided that such

a model is obtainable, can alleviate the problem to a certain

extent. The final remedy for this problem involves utilizing a

dynamic model of the friction force. In order to identify the

parameters of the friction force, the suggested methods in the

current literature are not feasible for robotic manipulators [3],

[9]. This is due to the fact that in most robotic manipula-

tors with limited workspace, running a joint with constant

velocity may not be possible. In this regard, a new method

for obtaining the model parameters using a low frequency

sinusoidal torque is suggested.

The organization of this paper is as follows. In section II, a

dynamic model of friction force is studied. In section III, a

generalized method for measuring friction force in manipula-

tors is discussed. Finally, in section IV, experimental results

are presented for two different types of manipulators.

II. MODELING

Consider the following dynamics of a robot manipulator:

M(q)q + C(q, q)q + G(q) = τ − τF (1)

where q = (q1, q2, . . . , qn)T is the joint angle vector, M(q)is the mass matrix, C(q, q)q is the Coriolis and centrifugal

term, G(q) is the gravity term, and τ and τF are vectors

of applied forces/torques and the friction forces/torques,

respectively. Henceforth, the term force is used to refer to

both force and torque. Each element of τF is the friction

force between two adjacent joints of the manipulator which

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

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are moving with relative velocity q with respect to each

other. The friction force at each joint, e.g., joint j of the

manipulator, can be dynamically modeled as follows [4]:

dz

dt= qj − σ0|qj |

s(qj)z (2)

τFj= σ1

dz

dt+ σ0z + σ2qj

where z is assumed as an average deflection of bristles

between the two surfaces with friction, τFjis the joint j

friction force, σ0, σ1 and σ2 are the stiffness, damping

and viscous coefficients, respectively, and s(qj) is a scalar

function given by

s(qj) = Fc + (Fs − Fc)e−α|qj |. (3)

In equation (3), Fc is the Coulomb force, Fs is the stiction

force and the α determines the variation of s(qj) between

Fs and Fc. As seen, the friction force is characterized

by six parameters, namely Fc, Fs, σ0, σ1, σ2 and α, from

which Fc, Fs and α describe the steady-state part of the

model whereas, σ0 and σ1 describe dynamic properties of

the friction force. The values of these parameters, for each

joint of the manipulator can be estimated experimentally. In

a system with viscous friction, the necessary and sufficient

condition for passivity [1] of the model is given by: ε2 �ε1(1 + σ2

σ1), where ε1 = 1

s(qj)|qj=0and ε2 = 1

s(qj)|qj→∞.

III. FRICTION IDENTIFICATION

The first step in identifying friction parameters of a manipu-

lator’s joint is to obtain a map between the friction force and

the joint velocity. However, obtaining such friction-velocity

map by running the joint at different constant velocities and

measuring friction force is not always feasible [3], [9]. Thus,

an alternative method in which the joint angle is kept within

the manipulator’s workspace is desirable. To this end, a low

frequency sinusoidal torque is applied to each joint of the

manipulator while other joints are locked and the correspond-

ing velocity is measured. The amplitude and the frequency of

the applied signal should be deliberately selected so that the

effect of Coriolis and centrifugal forces are negligible, and

the joint angles remain within the manipulator’s workspace.

Nonetheless, if the manipulator is moved too slowly it may

not be possible to capture all friction characteristics as the

system always remains in pre-sliding state. In the end, finding

an appropriate signal is a matter of trial and error. In the

sequel, this notion will be exemplified. Provided that the

above conditions are met, the equation of motion for each

joint of the manipulator (e.g., joint j) can be rewritten as

follows:

Mjj qj + Gj = τj − τFj(4)

where Mjj is the moment of inertia of joint j around its zaxis, τj = τMj

sin(ωjt) + Gj is the applied torque and τFj

is the joint friction force. By calculating qj after numerical

differentiation of joint velocity and assuming perfect gravity

cancelation, τFjcan be obtained from (4) as follows:

τFj= τMj

sin(ωjt) − Mjj qj (5)

The value of the friction force (τFj) is next plotted versus

joint velocity in order to obtain a typical friction-velocity

map as depicted in Fig. 1. All parameters in (3) including σ1

and σ2 can be derived from this graph. Fig. 1 illustrates the

relationship between the desired parameters and the graph.

In most applications, the exact value of Mjj depends on the

Fig. 1. Friction force/torque versus velocity.

manipulator configuration, and using an estimated value of

this parameter for measuring the friction force is inevitable.

Small values of qj can reduce the sensitivity of the measure-

ment (friction force) to the exact value of Mjj . However, as

mentioned earlier, moving a manipulator too slowly does not

reveal all friction characteristics. Thus, finding an appropriate

input signal for an unknown system requires a few trials.

To show the effect of Mjj on the friction-velocity graph

we followed the above procedure of applying a sinusoidal

signal to simulate the friction-velocity map of a system given

by (4). The friction force was simulated according to (2) for

the nominal values Fc = 1.2, Fs = 2.0, σ0 = 103, σ1 =400, σ2 = 0.4, α = 0.2 and Mjj = 1. To plot the friction-

velocity graph, instead of using the simulated values of the

friction force, we calculated the friction force using (5),

similar to a practical case. The results are summarized in

Fig. 2. The solid line shows the friction-velocity graph for

nominal values of the parameters and the dashed lines show

the graphs for the perturbed values of Mjj from its nominal

value. It is observed that the value of Mjj not only affects

the shape of the graph but it also changes the direction in

which the graph is followed. A similar effect on the friction-

velocity graph is observed for overestimated values of Mjj ,

except that the direction of the graph remains unchanged.

In the following, the relationship between parameters of the

model, i.e., Fc, Fs, α, σ2, and the above graph is discussed.

First, it should be pointed out that the starting point of the

model (2), assuming that the system is in pre-sliding mode,

represents the stress-strain curvature of the average bristle

deflections. Thus, the value of σ0 can be measured from the

slope of the friction force graph versus displacement at the

origin (see for example [2], [10]).

For a large value of the joint velocity the average bristle

deflections can be assumed constant. This is due to the fact

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Fig. 2. The effect of mass/inertia perturbation on the friction-velocitygraph.

that in lubricated contacts, a fluid layer of lubricant is built

up at high velocities and consequently, friction is determined

by the viscous characteristics of the lubricants [10]. Because

of the small value of the lubricant viscous coefficient, the

average deflection of bristles can be considered constant or

equivalently dzdt � 0. In this case the friction force is given

by,

0 = qj − σ0|qj |s(qj)

z (6)

⇒ z = sgn(qj)s(qj)σ0

⇒ τFj= sgn(qj)s(qj) + σ2qj

Without losing generality, let us assume that qj is positive.

Additionally, for the large values of the joint velocity where

αqj � 1, (3) yields

s(qj) = Fc + (Fs − Fc)e−α|qj | � Fc (7)

Now, substituting (7) in (6) renders the friction force as,

τFj= Fc + σ2qj (8)

Thus, for the large values of the joint velocity, the friction

force is given by a straight line whose slope and y-intercept

are equal to σ2 and Fc respectively. This is illustrated in

Fig. 1.

On the other hand, while dzdt � 0, small values of the joint

velocity for which αqj � 1 results in,

τFj � Fs + σ2qj � Fs (9)

The stated conditions (αqj � 1 and dzdt � 0) represent

the pre-sliding condition in the system and depending on

the system’s direction, i.e., leaving or approaching the pre-

sliding condition, two different situations can be observed.

Let us first look at the behavior of the system qualitatively.

Consider the system in pre-sliding condition. As the input

signal increases the value of the friction force is increased.

At this so called elastic stage the friction behaves as a spring.

After rupturing this elastic bond, the system enters the sliding

mode. Upon transition of the system from pre-sliding to

sliding condition, because of the assumed friction dynamics,

the value of the friction force may exceed the stiction force

(static friction) value. In this case, the friction-velocity graph

as depicted Fig. 1 has an overshoot. The amplitude of the

overshoot depends on the stiffness and damping coefficients

i.e., σ0 and σ1, as well as the amplitude of the input signal.

Thus, upon leaving the origin of the friction-velocity graph

(entering sliding condition), the value of the friction force

does not necessarily represent the stiction force. On the other

hand, prior to approaching the origin of the friction-velocity

graph, the system is in sliding condition and, as a result, the

average bristle deflection is relatively constant and dzdt � 0

(the stated conditions). Hence, the value of the friction force

represents the stiction force. In order to illustrate this fact

quantitatively, let us consider the linearized model of the

system in (2) with the dynamics (4) around the equilibrium

point of the system as follows [8]:

qj(s)τj(s)

=1

Mjjs2 + (σ1 + σ2)s + σ0(10)

τFj= σ0qj + (σ1 + σ2)qj , qj = z

To study the transition of the system from pre-sliding to slid-

ing condition, the response of the the linearized model (10)

to the input τMjsin(ωjt) � Ajt, where Aj = τMj

ωj , is

calculated. It has been shown that the linearized model,

despite the non-smooth behavior of the system at the origin,

is valid for both positive and negative velocities [8]. Thus,

the complete response of the system is given by,

qj(t) = wj(t) + hj(t) (11)

where wj(t) and hj(t) are the zero state and zero input

responses as follows:

wj(t) = K1es1t + K2e

s2t + at + b (12)

hj(t) = K3es1t + K4e

s2t

s1,2 = −ζ ±√

ζ2 − ω20︸ ︷︷ ︸

ωd

, ζ =σ1 + σ2

2Mjj, ω2

0 =σ0

Mjj

K1,2 = ±Aj

ω40

(ζ ± ωd)2

2ωd, a =

Aj

ω20

, b =−2ζAj

ω40

K3,4 = ± qj(0) + (ζ ± ωd)qj(0)2ωd

It is clear that the response of the system in pre-sliding

condition is characterized by the values σ0, σ1, σ2 and Mjj .

For the large values of ζ (ζ � ω0), the roots of the system,

s1,2 tend to 0− and −∞, respectively. As a result, qj , qj and

τFjgiven by (10), approach their final values at + b, a and

σ0(at+b)+(σ1 +σ2)a, more gradually. Now, depending on

the values of s1,2 two different situations are recognizable.

Let us call the time that the input signal reaches Fs by

ts = Fs

Aj. Calculating τFj

at ts yields,

τFj = σ0qj(ts) + (σ1 + σ2)qj(ts) (13)

Now, if ζ is such that, at ts the exponential terms in qj and

qj expressions are near zero then,

τFj= σ0(ats + b) + (σ1 + σ2)a (14)

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Substituting from (12) in (14) yields,

τFj = Fs

Therefore, upon transition of the system from pre-sliding to

sliding condition the friction force is equal to Fs. However

if due to the large values of ζ the exponential terms in qj

and qj are nonzero, then

τFj = Fs + �

where

� � σ0(K1es1ts+K3e

s1ts)+(σ1 + σ2)(K1s1e

s1ts + K3s1es1ts)

In this case, the friction force is larger than Fs prior to

entering the sliding condition.

The above discussion indicates that the value of the friction

force in the friction-velocity graph while entering the sliding

condition, may be different than the stiction force depending

on the values of σ0, σ1 and input signal. The transitions of

the friction-velocity graph between pre-sliding and sliding

conditions are depicted in Fig. 3 for three different values of

ζ and the nominal values used for plotting Fig. 2.

Fig. 3. Transition between pre-sliding and sliding conditions.

As mentioned earlier, in the sliding condition and just before

entering the pre-sliding condition, dzdt � 0, and the low values

of the velocity guarantees that the friction force is equal to

Fs as given in (9). Thus, the values of Fs and σ1 can be

measured from the friction force graph as illustrated in Fig. 1.

To obtain the coefficient α, let us calculate the slope of the

friction force graph in the vicinity of the origin. From (6)

we have,

∂τFj

∂qj= −α(Fs − Fc)e−αqj + σ2 (15)

Having measured the values of Fs, Fc and σ2, α can be

obtained from (15) by measuring the slope of the graph near

the origin, (κ1 in Fig. 1) as follows:

α � −κ1 − σ2

Fs − Fc(16)

In Fig. 3, all three graphs have slope κ1 � 0 at qj = 0.2,

which, using (16), yields α = 0.5. Note that finding an exact

value of the velocity at which α can be calculated is a tedious

task. Having obtained an estimated value of α, the fine tuning

of this parameter can be achieved by finding a good match

between the friction-velocity map from the model and the

measured one.

IV. CASE STUDIES

In order to evaluate the validity of the results presented in

the previous section, experiments were performed on two

different robotic manipulators. In each case, following the

identification of the parameters of the friction model, the

closed-loop performance for tracking a desired trajectory

using a PD controller was studied.

A. Micro-Manipulator

A 2-DOF manipulator which is used for high-speed and

accurate manipulation, was studied. This manipulator con-

sists of a a linear stage carrying a revolute joint. Both

joints are actuated with direct drive motors. This makes the

identification results more accurate in the sense that no error

is introduced in measurements because of the gearbox. The

system has micro-meter accuracy and is capable of moving

as fast as 1.98 m/s linearly and 2.5 rad/s angularly. The

mass and inertia of the linear stage and the revolute joint

are known. Moreover, their values do not depend on the

manipulator configuration.

In order to identify the friction force at each joint, a

sinusoidal signal was applied to the motors one at a time

while the other motor was locked. The friction-velocity map

for each joint was obtained, from which the parameters of

the friction force model were extracted. Table I lists the

estimated parameters of the friction force for each motor.

The evaluation procedure for the estimated results included

TABLE I

FRICTION FORCE PARAMETERS

Fc Fs α σ0 σ1 σ2

Linear 2.4 4.9 6.7 104 500 11.1

Units [N] [N] [ sm ] [ N

m ] [ N.sm ] [ N.s

m ]

Rotary 0.12 0.15 0.1 102 2 0.075

Units [Nm] [Nm] [ srad ] [ Nm

rad ] [ Nm.srad ] [ Nm.s

rad ]

two parts. In the first part, a friction force model based on the

identified parameters was simulated and its friction-velocity

map was compared with that from the experiment. Fig. 4

illustrates the experimental friction-velocity map superim-

posed on its simulation counterpart.

It is worth noting that in cases similar to the rotary motor

in which the values of the static and the Coulomb frictions

(Fs and Fc) are close, it is difficult to calculate the value of

α from (16). Fortunately, in such cases the incorrect values

of α do not affect the results since the variation of friction

force between its two values, i.e., Fs and Fc is very small. It

is clear that despite different friction force characteristics of

the two motors, accurate selection of the model parameters

creates an accurate friction-velocity map.

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Fig. 4. Simulation and experimental friction-velocity map.

In the second part, the friction model was evaluated by

observing the closed-loop performance of the manipulator for

tracking a desired trajectory. In this regard a PD controller

was utilized and the estimated values of the friction force

were added to the control signal to compensate for the actual

friction of the system. Fig. 5 compares the experimental

results for the position and tracking error of each motor with

friction compensation, to those without compensation.

Fig. 5. Tracking performance of linear and rotary motors with and withoutfriction compensation.

B. Macro-Manipulator

In the second case, a 4-DOF macro-manipulator was studied

(Fig. 6). Unlike the previous case, the macro-manipulator

cannot move fast. It has a much larger load capacity and

workspace. The macro-manipulator includes two sections,

each of which consists of a revolute joint with a vertical

axis in series with a∏

-joint, producing a 2 DOF motion.

The whole system has 4-DOF and is used for positioning a

platform at the end point of the manipulator. The platform

is capable of carrying a load of up to 70 Kg (e.g., a

PUMA 560). The macro-manipulator has a reach of 3.8 m

in its fully stretched configuration. The manipulator is a

relatively heavy mechanism and the maximum speed of its

joints are less than 0.4 rad/sec. The joints of the manipulator

are attached to electric servomotors using gearboxes with

gear ratios, J1 1:489.14, J2 1:600, J3 1:310 and J4 1:324.2,

respectively. The presence of gears makes the friction force

measurement very difficult. It has been shown that gear

transmission systems can be a source of chaotic behavior in

multibody systems [11]. In fact, this can be a failure mode

in an identification process. However, a good understanding

of friction helps to arrive at reasonably accurate results. The

presence of such chaotic motion in the fiction-velocity map

of the macro-manipulator is noticeable.

(A) (B)

Fig. 6. (A) The schematic of the 4-DOF macro-manipulator, (B)The upper section of the macro manipulator.

The macro-manipulator system is quite different than previ-

ous example. The inertias of the manipulator’s joints depend

on the joint angles. The joints are under gravity force. A gear

transmission system has been used at each joint. Moreover,

at low velocity, the gear pitches show considerable mismatch

(backlash) which results in a sudden change of velocity

(chaos). At higher velocity this effect is reduced and allows

for measurement of friction parameters. The procedure of

applying a sinusoidal torque was followed in order to identify

the friction force at each joint of the manipulator. Except

for α, which specifically requires low velocity information

of the system, the other parameters were identified using

the same procedures. The final value of α was selected to

produce a good match between the friction-velocity map

from experiments and simulations. In showing the chaotic

effect of the gear, Fig. 7 depicts the friction-velocity map

for the first joint for two velocity levels. Note that the first

joint of the manipulator is not under gravity force and has a

constant inertia. The discrepancies between the two graphs

are merely due to the use of gearbox.

As observed, by selecting an appropriate input signal, a

friction-velocity graph whose parameters are measurable can

be obtained. The process was repeated for all four joints.

The results of the identification are summarized in Table II,

and the friction-velocity map for the second joint of the

manipulator is shown in Fig. 8. Note that this joint has the

maximum gravity force and its inertia depends on the joint

angle.

The closed-loop performance of the manipulator for tracking

a desired trajectory was also examined after including the

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Fig. 7. Friction-velocity map for two velocity levels at joint 1.

Fig. 8. Simulation and experimental friction-velocity map at joint 2.

TABLE II

FRICTION FORCE PARAMETERS

Fc[Nm] Fs[Nm] α[ srad ] σ0[

Nmrad ] σ1[

Nm.srad ] σ2[

Nm.srad ]

Joint 1 1500 1614 18 106 8900 11262

Joint 2 1380 1546 11 106 8500 10914

Joint 3 105 116 5 105 800 1878

Joint 4 564 610 10 105 800 3450

friction model in the control algorithm. A PD controller

was utilized for controlling each joint and in addition to

the gravity force, the estimated values of the friction force

were added to the control feedback signal. Fig. 9 compares

the experimental results for the second joint of manipulator,

in tracking a desired trajectory with and without friction

compensation.

V. CONCLUSIONS

In this paper, friction compensation in robotic manipulators

was studied. A single-state model of friction force was

utilized and its parameters were identified. To that end, a

practical method for measuring friction force in manipulators

was discussed. Two different manipulators with different

friction characteristics were examined. The closed loop per-

formance of each system for tracking a desired trajectory

Fig. 9. Tracking performance of joint 2 with and without frictioncompensation .

with and without friction compensation was evaluated. Ex-

perimental results showed that including the friction force in

the control algorithm improved the accuracy of manipulation

to a significant degree, without sacrificing the speed and

smoothness of the manipulation.

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[1] N. Barabanov and R. Ortega, “Necessary and sufficient conditionsfor passivity of the LuGre friction model,” IEEE Transactions onAutomatic Control, vol. 45, no. 4, pp. 830–832, 2000.

[2] P. A. Bliman, “Mathematical study of the Dahl’s friction model,”European Journal of Mechanics, vol. 11, no. 6, pp. 835–848, 1992.

[3] C. Canudas de Wit and P. Lischinsky, “Adaptive frition compensationwith partially known dynamic friction model,” International Journalof Adaptive Control and Signal Processing, vol. 11, pp. 65–80, 1997.

[4] C. Canudas de Wit, H. Olsson, K. J. Astrom, and P. Lischinsky, “Anew model for control of systems with friction,” IEEE Transactionson Automatic Control, vol. 40, no. 3, pp. 419–425, 1995.

[5] P. R. Dahl, “A solid friction model,” The Aerospace Corporation, ElSegundo, CA, Tech. Rep. TOR-158(3107-18), 1968.

[6] P. Dupont, V. Hayward, B. Armstrong, and F. Altpeter, “Singlestate elasto-plastic friction models,” IEEE Transactions on AutomaticControl, vol. 47, no. 5, pp. 787–792, 2002.

[7] N. Ehrich Leonard and P. Krishnaprasad, “Adaptive friction compen-sation for bi-directional low-velocity position tracking,” in Proc. ofIEEE Conference on Decision and Control, Tucson, AZ, 1992, pp.267–273.

[8] R. H. A. Hensen, M. Molengraft, and M. Steinbuch, “Frequencydomain identification of dynamic friction model parameters,” IEEETrans. on Control Systems Technology, vol. 10, no. 2, pp. 191–196,2002.

[9] R. Kelly, J. Llamas, and R. Campa, “A measurement procedure forviscous and Coulomb friction,” IEEE Transactions on Instrumentationand Measurement, vol. 49, no. 4, pp. 857–861, 2000.

[10] H. Olsson, K. J. Astrom, C. Canudas de Wit, M. Gafvert, andP. Lischinsky, “Friction models and friction compensation,” EuropeanJournal of Control, vol. 4, no. 3, pp. 176–195, 1998.

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