Proceedings of the 2002 IEEE/RSJ Intl. Conference on Intelligent Robots and Systems EPFL, Lausanne, Switzerland • October 2002

Compensation of Stick-Slip effect Actuator

in

Electrical

&n

R. Merzouki, J. C. Cadiou and N. K M'Sirdi Laboratoire de Robotique de Versailles 10-12, avenue de l'Europe 78140 Vdizy e-mail: merzouki~robot.uvsq, fr Abstract--In some mechanical systems which make mom e n t l y a v e r y low m o t i o n s like a n i n v o l v e d i n d u s t r i a l w a g ons or constructor robots, the dispositive of their control presents usually some imperfections, concerning a success i o n o f j u m p s a n d s t o p s , w h e n t h e s t a t i c f r i c t i o n f o r c e is b i g g e r t h a n t h e d y n a m i c f r i c t i o n f o r c e . T h i s e f f e c t is c a l l e d the Stick-Slip phenomenon. In this paper we develop a non linear observer in order to estimate the friction force of the contact during the motion, and to compensate its effects which causes the Stick-Slip phenomenon.

I.

INTRODUCTION

In almost mechanical motion, friction appears in the surface of a body to prevents slipping on another body (static friction) or t h a t dissipates the mechanical energy in case of slip (dynamic friction). This friction produces a tangential force to the surface of contact, between rigid bodies which are tighten one against the other. In a first time, the rigid bodies remain in almost relative immobility; beyond a certain intensity of the applied force, they slip. The slip dissipates the energy and worn the surfaces. Except for t h a t phenomenon, the presence of a dead zone due to the backlash effects introduces an hysteresis behaviour. It causes a non stability to the system. Among research dealing with the friction effects: Friedland in [4] developed a new algorithm in order to estimate the constant parametric in a dynamic system. This algorithm is a reduced order observer containing two non linear functions, one is the Jacobean of the second. A good choice of the non linear function in the observer allows an asymptotic stability of the error. Then, Amin in [1] has developed two types of observers: The first one estimates the friction force as a constant of the time, the second one is used to estimate the relative velocity of the motion between surfaces in contact. Canudas in [3] proposed a model of friction t h a t includes different effects like: Hysteretic behaviour and stiction.effect. In [5], [7] and [8], an adaptive control to compensate the friction force has been developed, by using a non linear friction observer. In this paper, we use an adaptive control applied on an electrical actuator of Figure(l-left) based on the estimation of a friction force, in order to compensate the Stick-Slip elfect, which causes different disturbances to the mechanical system. Our experimental actuator is given by Figure (l-left), it is defined by two parts: the motoring part is representing by six DC motors, and the reducer part which is rich in

0-7803-7398-7/02/$17.00 ©2002 IEEE

static friction and backlash imperfections. A simulation and an experimental results are given in this paper. They describ an asymptotic convergence of the system to the equilibrium state and a good performence of this latter after applying the friction observer compensation. II.

DESCRIPTION

OF T H E S T I C K - S L I P

MOTION

In order to describ the physical meaning of the Stick-Slip phenomenon in our electrical actuator of Figure(l-left), we can consider the following mechanism of Figure(lright), where the theoretical translation Xl of the motorizing p a r t ( l ) is t r a n s m i t t e d to the wagon (2) by a string with rigidity k, which represents the elastic deformations of the involved mechanism. The normal load N generate a friction force F against the motion. A dash-pot complete the model in order to consider the dissipative phenomenon

[9]

,,-v,

2x Fig. 1. (left : T h e Electrical A c t u a t o r m a s s sliding on an h o r i z e n t a l plan

(right) M o d e l of an involved

At the first time, we neglect the damping coefficient (i.e c = 0), then the string is relaxed and the wagon is immobile. If we stretch the string with a quantity of x0, the involved force k.xo will hit the static friction force c~0. The mass will translate and the friction coefficient will take the dynamic value Fa, generally less t h a n the static friction. The global system is now driven by the force F0 = c~0 of the string and with a friction force >. So, their acceleration is given by: c~o -

Fa

=

m.

Jc

(1)

The mass moves in a front bound before it stops. This is the phenomenon of Stick-Slip, t h a t ' s mean when the motion control is known, then the motion of the slipping mass

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is given by a succession of stops and slipping periods. This physical description shows that the Stick-Slip motion is caused when the static friction is bigger than the dynamic one. Let's now translate the motorizing part (I) with a constant velocity vl. We choose the time origin when the mass start moving, t - O, x - 0 and x l - x0. The string apply the force FR:

We obtain the velocity by derivation of the last equation (s)-

cto

1

2:)

The dynamic equation, after starting motion, will be written as follows:

)ft. X -~-~.Y - - (t o -- F d . S i g T t ( 2 ) -Jr- ~ . V l . t

(4)

Fd

k

~o

.--)] vl

P.

c~o --

Fd.SigIt(2)

@ Vlt

k

(6)

"

t1

oF

t2

t3

t'-

Fig. 3. Translation and velocity with stick-slip

(5)

1

2

*A~ t

with the pulsation in a non-dumping and free regime:

co~.2 + z

(10)

X

~

So, we obtain:

(9)

position X

(3)

sin(co0.()

ao -

coo

(2)

(t o -~- ]g.(Vl.t -- 2:) -- F d . S i g I t ( 2 ) - - Irt. X

Fd

-

The maximal velocity will be hit as shown in Figure(3) at time: t I -- --[7l- @ a r c t a n ( -

F R - - (t o -~- ~ . ( V l . t -

-

2 = Vl(1 --cos(w0.()) -~- ~ . w 0 .

It is equal to zero at t2 = 2.tl. In this time, the motion of the wagon stops and the mass stays immobile until the string will be tighten so that it apply again the driving force F0 at time t3.

t3. Friction variable with the velocity This differential equation is not linear; now, we check the case when the dynamic friction coefficient is constant and the case when it depends on the slipping velocity.

A. Dynamic firction coefficient is constant Let's suppose that the dynamic friction coefficient Fa is constant [9], but already against the motion Figure(2-1eft). In the first part of the motion, as long as 2 > 0, the solution of equation (6) is:

Now, we suppose that the dynamic friction varies linearly with the velocity Figure (2-right) and it's equal to: Fa + I.I [9] Let's consider in this case the viscous dumping, then we obtain the followsing differential equation of the system (1-right):

7yt. :~ -FC(2 - - V l ) - - [ c t 0 -F ] ~ . ( V l . t -

x)]-

- ( F d - F C t 2.

Ixl) (11)

The string elongation is given by: F Thus"

F

~- + V l . t - x.

c 0~0

z - - p + - 7 + vl.t

(12)

2-- Vl-- @

(13)

tc

2

-C

/q

2= Fig. 2. (left)" D y n a m i c Friction coefficient is constant - (right)" D y n a m i c Friction coefficient is variable

-

(14)

~

In the case of 2 > 0, we have:

+

- ed +

2Vl

W i t h a relative dumping coefficient"

cto

-

Fd

-~- V l . t -~- A. cos(coo.t)+/3, sin(coo.t)

(7)

The constants A a n d / 3 are given by the initial conditions z-0and2=0int-0. Then:

ct 0 -- F d 2? - - V l . t @ ~ . ( 1

k

Vl

-- COS(CO0.()) -- - - . sin(co0.() w0

c d- c~2

(15)

q - - 2 ~/-k. m

then we obtain the differential equation of the string elongation"

1 (8)

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2.q

-

1

+

(16)

III. C O N T R O L W I T H S T I C K - S L I P P H E N O M E N O N

we put:

Starting with (11) where the friction force F in this case is given by:

7r.Po .a 2

+

(25)

2

t hus:

(17) with:

F - ~.~ 7v.Po.a 2

Fa

-------~-- .#

(18)

Now, the friction force describs the T u s t i n model presenting in [6],[7] and [8]. P0 is the nominal contact pressure of the wagon applied on the ground, a is the length of the wagon and a2 corresponds to the stiction friction. All these last parameters are supposed known. # is the dynamic friction force which is unknown and is estimated on line. By replacing expressions (12), (13), (14) and (17) in (11) we find: m. ~ +c. @ + k . F = U - F

Finally, the equation describing the system in the closed loop is given as follows:

(,~.p~ + ( ~ - K,).p ~ + ( k - K , ) . p - K,)~ = -~.p.p (27) B. The stability of the global system in a closed loop

We give a state representation of the system in a closed loop described in equation (27) and Figure (4), by choosing of the followsing states:

(19)

1 K1

gl

Now we apply a control law U on system (1-right), describing by the followsing equation:

c2

(28)

--(--).e

p -

m

1

~.~

p

m

-(cl-t

P

1

U -- K p . e + KD. e + K I / e.dt + m. ~d +c. ~d +k'Pd + p d

(26)

c-

p

(2o)

k-/~p.e~ m

]

KD

m

t hus:

with: Kp, KD et KI are respectively the coefficients of the P I D controller, and Fd corresponds to the desired elongation of the string, given by the followsing equation:

(00

1 0

l

~-~

G1

)

[2

"

,,

@

m

(~0 ~ d - - T -~- v l ' t -- :;cd

0

(21)

xd is the desired translation. Using the complexe representation after replacing equation (20)into (19), we obtain:

e

z

(00

1

c -mK ,

1).

c3

c2 £3

it could be described by this form:

(,~.p~ + ( ~ - K,).p ~ + ( k - K , ) . p - K,)~ = - ? . p

(22)

{ k=A.e+B.~ e--

where:

A. The non linear friction observer

£T__(

Let's take the last expression (22): The non linear friction observer is studied in [6],[7], [8] and is given by the following system: +

~.~o.~

2.a2 .Po. }st~i

sign(x) - p.e "

(23)

7r.Po.a

(29)

C.£

(00 (o) (0010 )

with e - p - Pd the elongation error and /~ - F - / ~ represents the estimated friction force error.

£1

£2

£3 ) , A -

m

1

0

k--Kp

0

1

c--KD m

m

From (27) and (24), we can represent the global system by the following schema: Then, we deduce the following characteristic equation:

2st~i is the Stribeck velocity, I and p are a positive constants, supposed known, also given in [6],[7] and [8]. Then the estimated friction error is given as follows:

7r.Po.a

2

(24)

(30) with e (~) the n order derivation of the position error e.

"/*

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After applying our friction observer, we notice that, the position tracking has been improved as shown in Figure(6left), where the dead zones have been eliminated and the asymptotic convergence of the position error in the permanent mode to zero. We can notice also t h a t the wagon doesn't stop during the motion as shown in Figure(6-right). In Figure(7-1eft), we see the estimated friction force and for Figure(7-right), we show the tracking of a slope input with the different values of the dumping coefficient (i.e for: c = 1 N . s / m , c = 2 N . s / m and c = 3 N . s / m ) and without applying our friction observer. We can notice by comparing with the observer compensation case, t h a t a good tracking is obtained in the transient mode t h a n the permanent one, when we increase the value of the damping coefficient. It could be explained by the increasing of the string rigidity when we increase the dumping coefficient.

¢P.p

m.p 3 +(c + KD).p 2 +(K + Kp).p+ K,

Fig. 4.

We put: A~A B

c+KD

-+

~

Global System in bloc

,.

X.(~+K~) ~ K+K~.

Xstri.?rt

?rt

A.(K+Kp)

D =

K i +d2.

K~.X 5

We define now, the imposed conditions on the controller constants for what the global system converge to the equilibrium state (e ~ O, @ ~ 0), by applying the _~outh criterium. So, we find: A>0 A.B>C C 2 + A2.D - A.B.C < 0 D>0

I£p

>

KD >

3--a2"x+a'x2.rrt - - / £ x A.rn xst~i

C

0. 0

C -0A

-0.3

-5 0 10 20 30 ..... Input Position, ~

-0.5 40

50 60 70 80 90 100 O u t p u t P o s i t i o n ( m ) - T i m e is)

0 10 20 30 40 50 60 70 80 90 100 ..... Input Velocity,~ O u t p u t V e l o c i t y ( m / s ) - T i m e is)

(31) Fig. 5. Simulation results: (left): Input and output Wagon positions before friction compensation - (right): Input and output Wagon velocities before friction compensation.

Where:

I

0.5

0.2

(32)

5

0.5

0.3

K~>0

0

with:

.

1

~

...................................

0 -0.1

Ct X

7

X xst,~ c+KD m

-0.3

-0.5

(33)

0 10 20 30 40 50 60 70 80 90 100 ..... Input Position, ~ O u t p u t P o s i t i o n ( m ) - T i m e is)

K1 +~.p

IV. SIMULATION RESULTS The simulation constants are given as follows: = 17N, m = 1Kg, c = I N . s / m , I£p = 5 N / m , Kz) = 2 . 5 N . s / m , K1 = .5N, k = 1 N / m , ao = 8 N , 32 = 1 2 N . s / m and 2st~i= 5 . 1 0 - a m / s . Figure(5-1eft) illustrates the tracking of the input signal corresponding to an increasing slope of the position with the presence of a Stick-Slip effect relative to the friction forces due to the contact. The curve which is describing by a doted line (output position) shows the stick-slip motion, which followss the input signal with a stairs configuration. The dead zones in this case, i.e, zones where the position is constant for example: between 30s and 35s, the translation velocity is equal to zero Figure(5-right). This last Figure shows the variation of the output velocity relatively to an input constant of the velocity: . 0 5 m / s . The stick zones are defined when the output vilocity is equal to zero.

0 10 20 30 40 50 60 70 80 90 100 ..... Input Velocity,~ O u t p u t V e l o c i t y ( m / s ) - T i m e is)

Fig. 6. Simulation results: (left): Input and output Wagon positions after friction compensation - (right): Input and output Wagon velocities after friction compensation.

20

/

.........................

o

"

20 40

T

1

/

i:;......

.......

The estimated friction force (N) - Velocity (m/s) ~

20

40 ;0:0 ;0

;0 Joo

I n p u t p o s i t i o n , -. . . . O u t p u t p o s i t i o n s (m) - T e m p s i s )

Fig. 7. (left) The estimated friction force- (right) the displacements of the wagon with different values of the dumping coefficient.

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V.

E X P E R I M E N T A L RESULTS

[4]

The experimental constants are defined in out line and given as follows: (I) = 153N, rn = 9Kg, c = O.O01Ns/rn, K p = 5N/rn, KD = 3N.s/rn, K1 = 1.5N, k = lO00N/rn, c~o = 8N, c~2 = 12N.s/rn and 2st~i= 5.10-arn/s. Thus, we have choosen an increasing slope for the input signal to our system of Figure(I-left). The superposition of the input and output signals are represented in Figure(8left), where we notice t h a t the tracking is characterized by the Stick-Slip effects. The traking has been improved when we apply the friction observer compensation as descring in Figure(8-right), where the Stick-Slip phenomenon has been reduced and the position error is asymptotically converging to zero. .

.

.

.

.

.

..j

B.Friedland, ' A non Linear Observer for Estimating Parameters in Dynamic Systems', Automatica Vol.33, N°8, P P 1525-1530, 1997. [5] R.Merzouki, J.C.Cadiou, N.K.M'Sirdi, and S.Femmam, 'A Non Linear Observer and Adaptive Compensation Control of an Electrical Actuator with Backlash and Friction', Paper accepted in the 16th IMACS World Congress, Lausanne, August 21-25, 2000. [6] R.Merzouki, H. Elhadri, J.C.Cadiou, N.K.M'Sirdi, 'An adaptive Compensation of Friction', Paper accepted in the SCI2000ISAS2000 International conference, Orlando, USA from July 23 26, 2000. [7] R.Merzouki, J.C.Cadiou, N.K.M'Sirdi, 'Compesation of backlash effects in an electrical actuator', Paper accepted in the IASTED International conference on Intelligent Systems and Control (IASTED2000), Hawai, USA from August 14-16, 2000. [8] R. Merzouki, J.C. Cadiou, N.K. M'Sirdi, 'Adaptive Control of an Electrical actiator', the ICAR the IEEE International conference on Intelligent Systems and Control, Budapest, Hungary August 2001. [9] Spinnler, 'Conception des Machines', Tomes 1 et 2, 'Presses Polytechniques et Universitaires Romandes, F6vrier 1997. [10] G.Tao, P.V. Kokotovic,'Adaptive Control of Systems with Backlash', Automatica, Vol. 29, No. 2, pp. 323-335, 1993.

\;

-200

'2

,

,

,

-20 4 6 8 10 1 2 3 4 5 6 7 8 9 1)0 - - - - I n p u t P o s i t i o n , - . . . . O u t p u t P o s i t i o n ( d e g ) - T i m e (s) - - - - I n p u t P o s i t i o n , - . . . . O u t p u t P o s i t i o n ( d e g ) - T i m e (

Fig. 8. Experimental results: (left)" Input and output Electrical actuator positions before friction compensation - (right)" Input and output Electrical actuator positions after friction compensation

VI. CONCLUSION The Stick-Slip is a p e r t u r b e d effet issue from the presence of the static friction during the motion. This fluctuant movement which make difficult the position control of mechanical systems with a big precision, could be eliminated by a control manner. A good estimation of the present friction force during the motion can be introduced in the law control to compensate all the generated forces of the Stick-Slip effects as shown in the last analysis. This adaptive control includes a non linear observer to estimate the necessary friction force which has given a good improvement of the position traking for our electrical actuator and an asymptotic convergence of the position error. We can also a t t e n u a t e the Stick-Slip effect mechanically, by increasing the dumping coefficient. In this case, the results could be interesting in the transient mode. REFERENCES [1]

[2]

[3]

J.Amin, B.Friedland, and A.Harnoy, 'Implementation of Friction Estimation and Compensation Technique', 5th IEEE International Conference on control applications, Dearborn, Sept, 15-18, 1996. G. Brandenburg, U. Schafer, 'Influence and partial compensation of simultaneously acting backlash and coulomb friction in a speed and position controlled elastic two-mass system', ICED 88, Proceedings of International Conference on electrical drives, Romania, pp. 1-12, 1988. C.Canudas de Wit, H.Olsson, K.J.Astrom, and P.Lischinsky, ' A new Model for Control of Systems with Friction', IEEE Transactions on Automatic Control, Vol.40, N°3, P P 419-424, March 1995.

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