iaetsd estimation of frequency for a single link-flexible

ESTIMATION OF FREQUENCY FOR A SINGLE LINK-FLEXIBLE MANIPULATOR USING ADAPTIVE CONTROLLER

KAZA JASMITHA Dr K RAMA SUDHA M.E (CONTROL SYSTEMS) PROFESSOR

DEPARTMENT OF ELECTRICAL ENGINEERING Andhra University,

Visakhapatnam, Andhra Pradesh.

Abstract- In this paper, it is proposed an adaptive control procedure for an uncertain flexible robotic arm. It is employed with fast online closed loop identification method combined with an output –feedback controller of Generalized Proportional Integral (GPI).To identify the unknown system parameter and to update the designed certainty equivalence, GPI controller a fast non asymptotic algebraic identification method is used. In order to examine this method, simulations are done and results shows the robustness of the adaptive controller. Index Terms- Adaptive control, algebraic estimation, flexible robots, generalized proportional integral (GPI) control. I.INTRODUCTION FLEXIBLE arm manipulators mainly applied: space robots, nuclear maintenance, microsurgery, collision control, contouring control, pattern recognition, and many others. Surveys of the literature dealing with applications and challenging problems related to flexible manipulators may be found in [1] and [2]. The system, which is partial differential equations (PDEs), is a distributed-parameter system of infinite dimensions. It makes difficult to achieve high- level performance for nonminimum phase behavior. To deal with the control of flexible manipulators and the modeling based on a truncated (finite dimensional) model obtained from either the

finite-element method (FEM) or assumed modes methods, linear control [3], optimal control [4]. Adaptive control [5], sliding-mode control [6], neural networks [7], or fuzzy logic [8] are the control techniques. To obtain an accurate trajectory tracking these methods requires several sensors, for all of these we need to know the system parameters to design a proper controller. Here we propose a new method, briefly explained in [9], an online identification technique with a control scheme is used to cancel the vibrations of the flexible beam, motor angle obtained from an encoder and the coupling torque obtained from a pair of strain gauges are measured as done in the work in [10]. Coulomb friction torque requires a compensation term as they are the nonlinearities effects of the motor proposed in [11]. To minimize this effect robust control schemes are used [12]. However, this problem persists nowadays. Cicero et al. [13] used neural network to compensate this friction effect. In this paper, we proposed an output-feedback control scheme with generalized proportional integral (GPI) is found to be robust with respect to the effects of the unknown friction torque. Hence, compensation is not required for these friction models. Marquez et al was first untaken this controller. For asymptotically sable closed-loop system which is internally unstable the velocity of a DC motor should be controlled.For this we propose, by further manipulation of the integral

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reconstructor, an internally stable control scheme of the form of a classical compensator for an angular-position trajectory task of an un-certain flexible robot, which is found to be robust with respect to nonlinearities in the motor. Output-feedback controller is the control scheme here; velocity measurements, errors and noise are produced which are not required. Some specifications of the bar are enough for the GPI control. Unknown parameters should not affect the payload changes. Hence this allows us to estimate these parameter uncertainties, is necessary. The objective of this paper is unknown parameter of a flexible bar of fast online closed-loop identification with GPI controller. The author Fliess et al. [15] (see also [16]) feedback-control system s are reliable for state and constant parameters estimation in a fast (see also [17], [18]). As these are not asymptotic and do need any statistical knowledge of the noise corrupting data i.e. we don’t require any assume Gaussian as noise. This assumption is common in other methods like maximum likelihood or minimum least squares. Furthermore, for signal processing [19] and [20] this methodology has successfully applied.

Fig. 1. Diagram of a single-link flexible arm This paper is organized as follows. Section II describes the flexible-manipulator model. Section III is devoted to explain the GPI-controller design. In Section IV, the algebraic estimator mathematical development is explained. Section V describes the

adaptive-control procedure. In Section VI, simulations of the adaptive-control system are shown. . Finally, Section VII is devoted to remark the main conclusions.

II. MODEL DESCRIPTION

A. Flexible-Beam Dynamics PDE describes the behavior of flexible slewing beam which is considered as Euler-Bernoulli beam. Infinite vibration modes involves in this dynamics. So, reduced modes can be used where only the low frequencies, usually more significant, are considered. They are several approaches to reduce model. Here we proposed: 1) Distributed parameters model where the infinite dimension is truncated to a finite number of vibration modes [3] 2) Lumped parameters models where a spatial discretization leads to a finite-dimensional model. In this sense, the spatial discretization can be done by both a FEM [22] and a lumped-mass model [23]. As we developed in [23] a single-link flexible manipulator with tip mass is modeled, it rotates Z-axis perpendicular to the paper, as shown in Fig. 1. Gravitational effect and the axial deformation are neglected. As the mass of the flexible beam is floating over an air table itallows us to cancel the gravitational effect and the friction with the surface of the table. Stability margin of the system, increases structural damping, a design without considering damping may provide a valid but conservative result [24]. In this paper the real structure is studied is made up of carbon fiber, with high mechanical resistance and very small density. We considered the studies is under the hypothesis of small deformation as the total mass is concentrated at the tip position because the mass of the load is bigger than that of the bar, then the mass of the beam can be

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neglected. I.e., the flexible beam vibrates with the fundamental mode; as the rest modes are very far from the first so they can be neglected. Therefore we can consider only one mode of vibration. Here after the main characteristics of this model will influence the load changes in very easy manner, that make us to apply adaptive controller easily.

Based on these considerations, we propose the following model for the flexible beam:

)(2tmt cmL (1)

Where m is the unknown mass at the tip position. L and c = (3EI/L) are the length of the flexible arm and the stiffness of the bar, respectively, assumed to be perfectly known. The stiffness depends on the flexural rigidity EI and on the length of the bar L. θm is the angular position of the motor gear.1θtand t are the unmeasured angular position and angular acceleration of the tip, respectively.

B. DC-Motor Dynamics In many control systems a common electromechanical actuator is constituted by the DC motor [25]. A servo amplifier with a current inner loop is supplied for a DC motor. The dynamics of the system is given by Newton’s law

n

vjku cmm ˆˆˆ (2)

Where J is the inertia of the motor [in kilograms square meters], ν is the viscous friction coefficient [in Newton meters seconds], and c is the unknown Coulomb friction torque which affects the motor dynamics [in Newton meters]. This nonlinear-friction term is considered as a perturbation, depending only on the sign of the motor angular velocity. As a consequence,

Coulomb’s friction, when 0ˆ , follow the model:

)0ˆ(ˆ

)0ˆ(ˆ)ˆ(.ˆ

mcoul

mcoulmc sign

(3)

And when 0ˆ

)0)(,max()0)(,min(

)(.ˆuku

ukuusign

coup

coupc

(4)

For motor torque must exceed to begin the movement coul is used as static friction value. Motor servo amplifier system [in Newton meter per volts] electromechanical constant is defined by

parameter k . m And m

are the angular acceleration of the motor [in radians per seconds squared] and the angular velocity of the motor [in radians per second], respectively. Γ is the coupling torque measured in the hub [in Newton meters], and n is the reduction ratio of the motor gear. u is the motor input voltage [in volts]. This is the control variable of the system. It is given as the input to a servo amplifier which controls the input current to the motor by means of an internally PI current controller [see Fig. 2(a)]. As it is faster than the mechanical dynamics electrical dynamics are rejected. Here the servo amplifier can be considered as a constant relation ek between the voltage and current to the motor: em Vki [see Fig.2 (b)], where imis the armature circuit current and keincludes the gain of the amplifier k~ and R as the input resistance of the amplifier circuit.

Fig.2.(a)Complete amplifier scheme. (b) Equivalent amplifier scheme

The total torque given to the motor ΓT is

directly proportional to the armature circuit in the form ΓT = kmim, where kmis the electromechanical constant of the motor. Thus, theelectromechanical constant of the motor servo amplifier system is k = kekm.

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C. Complete-System Dynamics

The dynamics of the complete system, actuated by a DC motor are formulated by given simplified model:

)(

ˆˆˆ

)(2

tm

cmm

tmt

cn

vJku

cmL

(5,6,7)

Equation (5) represents the dynamics of the flexible beam; (6) expresses the dynamics of the dc motor; and (7) stands for the coupling torque measured in the hub and produced by the translation of the flexible beam, which is directly proportional to the stiffness of the beam and the difference between the angles of the motor and the tip position, restively. III. GPI CONTROLLER The flexible-bar transfer function in Laplace notation from (5) can be re written as

22

2

)( )()(

sss

Gbm

ts

(8)

Where ω = (c/(mL2))1/2 is the unknown natural frequency of the bar due to the lack of precise knowledge of m. as done in [10], the coupling torque can be canceled in the motor by means of the a compensation term. In the case, the voltage applied to the motor is of the form

nkuu c .

(9)

Fig.3 compensation of the coupling torque measured in the hub Where cu is the voltage applied before the compensation term. The system in (6) is then given

by

cmmc vJku ˆˆˆ (10)

The controller to be designed will be robust with respect to the unknown piecewise constant torque disturbances affecting the motor dynamics. Then the perturbation-free system to be considered is the following: mmc vjku (11)

Where nkK / .To specifies the developments, let JKA / and JvB / .The transfer function of a DC motor is written as

)()()()(

BssA

sussGm

c

m

(12)

Fig.3 shows the compensation scheme of the coupling torque measured in the hub. The regulation of the load position θt(t) to track a given smooth reference trajectory )(* tt is desired. To synthesis the feedback-control law, we will use only the measured coupling motor position m and coupling torque . One of the prevailing restrictions throughout our treatment of the problem is our desire of not to measure, or compute on the basis samplings, angular velocities of the motor shaft or of the load.

Fig. 4. Flexible-link dc-motor system controlled by a two-stage GPI-controller design

A. Outer loop controller Consider the model of the flexible link, given in (1). This subsystem is flat, with flat output given by tt . This means that all variables of the

unperturbed system may be written in terms of the flat output and a finite number of its time derivatives (See [9]). The parameterization of

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tm in terms of tm is given, in reduction gear terms, by:

ttm cmL

2

(13)

System (9) is a second order system in which it is desired to regulate the tip position of the flexible bar t , towards a given smooth

reference trajectory tt* with m acting as a

an auxiliary Control input. Clearly, if there exists an auxiliary open loop control input,

tm* , that ideally achieves the tracking of

tt* for suitable initial conditions, it satisfies

then the second orderdynamics, in reduction gear terms (10).

ttc

mLt ttm**

2*

(14)

Subtracting (10) from (9), we obtain an expression in terms of the angular tracking errors:

tmt eemL

ce 2

(15)

Where tete tttmmm** , . For

this part of the design, we view me as an incremental control input for the linksdynamics. Suppose for a moment we are able to measure theangular position velocity tracking error te , then the outer loopfeedback incremental controller could be proposed to be the followingPID controller,

deKeKeK

cmLee t

t

tttm0

012

2

(16) We proceed by integrating the expression (11) once, to obtain:

deemL

cetet

tmtt 0

20

(17)

Disregarding the constant error due to the tracking error velocity initial conditions, the estimated error velocity can be computed in the following form:

deemL

cet

tmt 0

2 (18)

The integral reconstructor neglects the possibly nonzero initial condition 0te and, hence, it exhibits a constant estimation error. When the reconstructor is used in the derivative part of the PID controller, the constant error is suitably compensated thanks to the integral control action of the PID controller. The use of the integral reconstructor does not change the closed loop features of the proposed PID controller and, in fact, the resulting characteristic polynomial obtained in both cases is just the same. The design gains 2,10 , kkk need to be changed due to the use of

the integral reconstructor. Substituting the

integral reconstructor te (14) byinto the PID

controller (12) and after some rearrangements we obtain:

ttmm ss

*

2

01* (19)

The tip angular position cannot be measured, but it certainly can be computed from the expression relating the tip position with the motor position and the coupling torque. The implementation may then be based on the use of the coupling torque measurement. Denote the coupling torque by it is known to be given by:

couptm nmLc

2 (20) Thus, the angular position is readily expressed as,

cmt1 (21)

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In Fig. 2 depicts the feedback control scheme under which the outer loop controller would be actually implemented in practice. The closed outer loop system in Fig. 2 is asymptotically exponentially stable. To specify the parameters, 210 ,, we can choose to locate the closed loop poles in the left half of the complex plane. All three poles can be located in the same point of the real line, s = −a, using the following polynomial equation, 033 3223 asaass (22) Where the parameter a represents the desired location of the poles. The characteristic equation of the closed loop system is,

01 022

012

02

23 kksksks (23)

Identifying each term of the expression (18) with those of (19), the design parameters 012 ,, can be uniquely specified.

B. Inner loop controller

The angular position m , generated as an auxiliary control input in the previous controller design step, is now regarded as a reference trajectory for the motor controller. We denote this reference trajectory by mr

* . The dynamics of the DC motor, including the Coulomb friction term, is given by (10). The design of the controller to be robust with respect to this torque disturbance is desired. The following feedback controller is proposed:

t t

t

v

ddek

dekekek

KJe

Kve

m

mmm

m

012

020

0123

)()())((

)()(ˆˆ

(24) the following integral reconstructor for the angular-velocity error signal

me is obtained:

mme

Jvde

JKe

t

v 0

)()( . (25)

Replacing m

e in (25) into (24) and, after some

rearrangements, the feedback control law is obtained

)()(

)( *

3

012

2*mmrcc ss

ssuu

. (26)

The open-loop control )(* tu c that ideally achieves the open-loop tracking of the inner loop is given by

)()(1)( ** tABt

Atu mmc . (27)

The inner loop system in Fig. 4 is exponentially stable. We can choose to place all the closed-loop poles in a desired location of the left half of the complex plane to design the parameters { 0123 ,,, }. As done with the outer loop, all poles can be located at the same real value,

123 ,, , and 0 can be uniquely obtained by equalizing the terms of the two following polynomials:

0464)( 4322344 pspsppssps (28)

0)()( 012

233

34 AAssABsBs

(29)

Where the parameter p represents the common location of all the closed-loop poles, this being strictly positive.

IV. IDENTIFICATION

As explained in the previous section, the control performance depends on the knowledge of the parameter ω. In order to do this task, in this section, we analyze the identification issue, as well as the reasons of choosing the algebraic derivative method as estimator. Identification of continuous-time system parameters has been studied from different points of view. The surveys led by Young in [27] and Unbehauen and Rao in [29] and [30], respectively, describe most of the available techniques. The different approaches are usually classified into two categories:1) Indirect approaches: An equivalent discrete-time model to fit the date is

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needed. After that, the estimated discrete-time parameters are transferred to continuous time.2) Direct approaches: In the continuous-time model, the original continuous-time parameters from the discrete-time data are estimated via approximations for the signals and operators. In the case of the indirect method, a classical well-known theory is developed (see [31]). Nevertheless, these approaches have several disadvantages: 1) They require computationally costly minimization algorithms without even guaranteeing con-vergence. 2) The estimated parameters may not be correlated with the physical properties of the system. 3) At fast sampling rates, poles and zeros cluster near the −1 point in the z-plane.

Therefore, many researchers are doing a big effort following direct approaches (see [32]–[35], among others). Unfortunately, identification of robotic systems is generally focused on indirect approaches (see [36], [37]), and as a consequence, the references using direct approaches are scarce. On the other hand, the existing identification techniques, included in the direct approach, suffer from poor speed performance.Additionally, it is well known that the closed-loop identification is more complicated than its open-loop counterpart (see [31]). These reasons have motivated the application of the algebraic derivative technique previously presented in the introduction. In the next point, algebraic manipulations will be shownto develop an estimator which stems from the differentialequations, analyzed in the model description, incorporating the measured signals in a suitable manner.

A. Algebraic Estimation of the Natural Frequency.

In order to make more understandable the equation deduction, we suppose that signals are noise free. The main goal is to obtain an estimation of 2 as fast as possible, which we will denote by oe .

Proposition 4.1: The constant parameter 2 of the noise free system described by (5)–(7) can be exactly computed, in a nonasymptotic fashion, at

some arbitrarily small time t = Δ >0, by means of the expression.

),[

),0[

)()(2

tfortfor

tdtn

arbitary

e

eoe (30)

Where )(tne and )(tde are the output of the time-varying linear unstable filter

12 )()( ztttn te 3)( ztde

)(421 ttzz t 43 zz

)(22 tz t ))()((24 tttz tm (31)

Proof: consider (5)

)(2tmt (32)

The Laplace transform of (32) is ))()(()0()0()( 22 sssss tmttt (33)

Taking two derivatives with respect to the complex variable s, the initial conditions are cancelled

2

2

2

22

2

22 )()()(ds

dds

ddssd tmt

(34)

Employing the chain rule, we obtain

))()((24)(2

2

2

22

2

22

dsd

dsd

dsds

dsds tm

ttt

(35)

Consequently, in order to avoid multiplications by positive powers of s, which are translated as undesirable time derivatives in the time domain, we multiply the earlier expression by 2s . After some rearrangements, we obtain

))ˆ()ˆ((

24)(

2

2

2

22

212

2

2

dsd

dsds

sds

ds

dsd

tm

ttt

(36)

Let L denote the usual operational calculus transform acting on exponentially bounded signals with bounded left support (see[38]). Recall that

),(.)1())(/(),)(/()( 11 vvvv tdsdLdtdsL

and t

s dL0

)/1(1 .))(()( taking this into

account, we can translate(36) into the time domain

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t t t

t

t

m

t t

tt

dddd

dddtt

0 0 0

2

0

2

0 0 0t

2

2

)()(

)(2)(4)([

(37) The time realization of (37) can be written via time-variant linear (unstable) filters

12 )()( ztttn te 3)( ztde

)(421 ttzz t 43 zz

)(22 tz t ))()((24 tttz tm

The natural frequency estimator of 2 is given by

),[

),0[

)()(2

tfortfor

tdtn

arbitary

e

eoe

(38)

Where ia an arbitrary small real number. Note that, for the time t=o,ne(t) and de(t) are both zero. Therefore, the quotient is undefined for a small period of time. After a time t = Δ >0, the quotient is reliably computed. Note that t = Δ depends on the arithmetic processor precision and on the data acquisition card. The unstable nature of the linear systems in perturbed Brunovsky’s form (38) is of no practical consequence on the determination of the unknown parameters since we have the following reasons: 1) Resetting of the unstable time-varying systems and of the entire estimation scheme is always possible and, specially, needed when the unknown parameters are known to undergo sudden changes to adopt new constant values. 2) Once the parameter estimation is reliably accomplished, after some time instant t = Δ >0, the whole estimation process may be safely turned off. Note that we only need to measure θm and Γ, since θt is available according to (21). Unfortunately, the available signals θm and Γ are noisy. Thus, the estimation precision yielded by the estimator in (30) and (31) will depend on the signal-to-noise ratio3 (SNR).

B. Unstructured Noise We assume that θm and Γ are perturbed by an added noise with unknown statistical properties. In order to enhance theSNR, we simultaneously filter the numerator and denominator by the same low-pass filter.

Taking advantage of the estimator rational form in (37), the quotient will not be affected by the filters. This invariance is emphasized with the use of the different notations in frequency and time domain such as

)()()()(

)()(2

tdsFtnsF

tdtn

e

e

f

foe

(40)

Where )(tn f and )(td f are the filtered

numerator and denominator, respectively, and F(s) is the filter used. The choice of this filter depends on the a priori available knowledge of the system. Nevertheless, if such a knowledge does not exist, pure integrations of the form ,

1,/1 ps p may be utilized, where high-frequency noise has been assumed. This hypothesis has been motivated by recent developments in nonstandardanalysis toward a new nonstochastic noise theory (more details in [39]). Finally, the parameter 2 is obtained by

),[

),0[

)()(2

tfortfor

tdtn

arbitary

e

eoe

(41)

V.ADAPTIVE CONTROL PROCEDURE Fig. 4 shows the adaptive-control system implemented in practice in our laboratory. The estimator is linked up, from time ][00 st , to

the signals coming from the encoder m and the pair of strain gauges Γ. Thus, the estimator begins to estimate when the closed loop begins to work, and then, we can obtain immediately the estimate of the parameter. When the natural frequency of the system is estimated at time t1, the switch s1 is switched on, and the control system is updated with this new parameter estimate. This is done in closed loop and at real time in a very short period of time. The updating of the control system is carried out by substituting ω by the estimated parameter ω0e in (14) and (19). In fact, the feedforward term which ideally controls in open loop the inner loop subsystem cu* [see (27)] also depends on

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the ω value because the variable *m is obtained

from the knowledge of the system natural frequency. Obviously, until the estimator obtains the true value of the natural frequency, the control system begins to work with an initial arbitrary value which we select ω0i. Taking these considerations into account, the adaptive controller canbe defined as follows. For the outer loop, (14) is computed as

)()(1)( **2

* ttx

t ttm

(42)

Moreover,(23) is computed as 0)()1( 021

222

3 xsxss (43) For the inner loop only changes the feedforward term in (27) which depends on the bounded derivatives of the new )(* tm in (42). The variable x is defined as

1

1

,,

ttxttx

oe

oi

(44, 45)

VI. SIMULATIONS The major problems associated with the control of flexible structures arise from the structure is a distributed parameter system with many modes, and there are likely to be many actuators [40]–[44]. We propose to control a flexible beam whose second mode is far away from the first one, with the only actuator being the motor and the only sensors as follows: an encoder to measure the motor position and a pair of strain gauges to estimate the tip position. The problem is that the high modal densities give rise to the well-known phenomenon of spillover [45], where contributions from the unmodeled modes affect the control of the modes of interest. Nevertheless, with the simulations as follows, we demonstrate that the hypothesis proposed before is valid, and the spillover effect is negligible. In the simulations, we consider a saturation in the motor input voltage in a range of [−10, 10] [in volts]. The parameters used in the simulations are as follows: inertia J = 6.87 ·10−5 [kg · m2], viscous friction ν = 1.041 ·

10−3 [N · m · s], and electromechanical constant k = 0.21 [(N · m)/V]. With these parameters, A and B of the transfer function of the dc motor in (12) can be computed as follows: A = 61.14 [N/(V · kg · s)] and B=15.15 [(N · s)/(kg · m)]. The mass used to simulate the flexible-beam behavior is m=0.03 [kg], the length L=0.5 [m], and the flexural rigidity is EI =0.264 [N · m2]. According to these parameters, the stiffness is c=1.584 [N · m], and the natural frequency is ω = 14.5 [rad/s]. Note that we consider that the stiffness of the flexible beam is perfectly known; thus, the real natural frequency of the beam will be estimated as well as the tip position of the flexible bar. Nevertheless, it may occur that the value of the stiffness varies from the real value, and an approximation is then included in the control scheme. Such approximated value is denoted by c0. In this case, we consider that the computation of the stiffness fits in with the real value, i.e., cc 0 . A meticulous stability analysis of the control system under variations of the stiffness c is carried out in Appendix, where a study of the error in the estimation of the natural frequency is also achieved. The sample time used in the simulations is 1 · 10−3 [s]. The value of 119.7 · 10−3 [N · m] taken in simulations is the true value estimated in real experiments. In voltage terms is

][57.0/ˆ Vkc In order to design the gains of the inner loop controller, the poles can be located in a reasonable location of the negative real axis. If closed-loop poles are located in, for example, −95, the transfer function of the controller from (26), that depends on the location of the poles in closed loop of the inner loop and the values of the motor parameters A and B as shown in (28) and (29), respectively, results in the following expression:

)365(10.3.110.6.5789 642

*

*

ssssuu

mm

cc

(46) The

feedforward term in (27), which depends on the

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values of the motor parameters, is computed in accordance with )(25.0)(02.0 *** ttu mmc (47) with a natural frequency of the bar given by an initial arbitrary estimate of 9oi [rad/s]. The transfer function of the controller (19), which depends on the location of the closedloop poles of the outer loop, −10 in this case, and the natural frequency of the bar as shown in (22) and (23), respectively, is given by the following expression.

30

7.177.2*

*

ss

tt

mm

(48)

The open-loop reference control input from (14) in terms of the initial arbitrary estimate of

oi is given by

)()(10.3.12)()(1)( **3**2

* ttttt tttt

oim

(49)

The desired reference trajectory used for the tracking problem of the flexible arm is specified as a Bezier’s eighth-order polynomial. The online algebraic estimation of the unknown parameter ω, in accordance with (31), (40), and (41), is carried out in Δ = 0.26 s [see Fig. 5(a)]. At the end of this small time interval, the controller is immediately replaced or updated by switching on the interruptor 1s(see Fig. 4), with the accurate parameter estimate, given by oe = 14.5 [rad/s]. When the controller is updated, s1 is switched off. Fig. 5(b) shows the trajectory tracking with the adaptive controller. Note that the trajectory of the tip and the reference are superimposed. The tip position t tracks the desired trajectory t

*

with no steady-state error [see Fig. 5(c)]. In this figure, the tracking error tt * is shown. The corresponding transfer function of the new updated controller is then found to be

30

7.253.0*

*

ss

tt

mm

(50)

The open-loop reference control input )(* tm

from (14) in terms of the new estimate oe is given by

)()(10.7.4)()(1)( **3**2

* ttttt tttt

oim

(51)

The input control voltage to the dc motor is shown in Fig. 5(d), the coupling torque is shown in Fig. 5(e), and the Coulomb friction effect in Fig. 5(f). In Fig. 6, the motor angle θm is shown RESULTS

Without using estimation

For 14.5[ / sec], A 61.14, B 15.15rad

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

time in sec

angl

e in

rad

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time in sec

angl

e in

rad

inputoutput

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Error

CONCLUSION A two-stage GPI-controller design scheme is proposed in connection with a fast online closed-loop continuous-time estimator of the natural frequency of a flexible robot. This methodology only requires the measurement of the angular positionof the motor and the coupling torque. Thus, the computation of angular velocities and bounded derivatives, which always introduces noise in the system and makes necessary the use of suitable low-pass filters, is not required. Among the advantages of this technique, we find the following advantages: 1) a control robust with respect to the Coulomb friction; 2) a direct estimation of the parameters without an undesired translation between discrete- and continuous-time domains; and 3) independent statistical hypothesis of the signal is not required, so closedloop operation is easier to implement. This methodology is well suited to face the important problem of control degradation in flexible arms as a consequence of payload changes. Its versatility and easy implementation make the controller suitable to be applied in more than 1-DOF flexible beams by applying the control law to each separated dynamics which constitute the complete system. The method proposed establishes the basis of this original adaptive control to be applied in more complex problems of flexible robotics.

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