Iaetsd estimation of frequency for a single link flexible by Iaetsd Iaetsd

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY

ISBN: 378 - 26 - 138420 - 5

ESTIMATION OF FREQUENCY FOR A SINGLE LINK-FLEXIBLE MANIPULATOR USING ADAPTIVE CONTROLLER KAZA JASMITHA M.E (CONTROL SYSTEMS)

Dr K RAMA SUDHA 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 â&#x20AC;&#x201C;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

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

206

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY

reconstructor, an internally stable control scheme of the form of a classical compensator for an angularposition 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 closedloop 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 donâ&#x20AC;&#x2122;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

ISBN: 378 - 26 - 138420 - 5

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 Zaxis 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

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

207

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY

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: mL2t  c( m   t ) (1)

Motor servo amplifier system [in Newton meter per volts] electromechanical constant is defined by   parameter k . ˆm And ˆm 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 ke between the

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    ku  jˆm  vˆm ˆ c  (2) n Where J is the inertia of the motor [in kilograms square meters], ν is the viscous friction coefficient [in Newton meters seconds], and ˆ c is the

voltage and current to the motor: im  Vk e [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.

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 ˆ  0 , follow the model: ˆ (ˆ  0)    coul m  ˆ c .sign (ˆm )      ˆ coul (ˆm  0) (3)

 And when ˆ  0 min(ku, coup )(u  0)  ˆ c .sign(u )    max(ku,coup )(u  0)

ISBN: 378 - 26 - 138420 - 5

For motor torque must exceed to begin the movement coul is used as static friction value.

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.

(4)

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

208

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY

C. Complete-System Dynamics

ISBN: 378 - 26 - 138420 - 5

  kuc  Jˆm  vˆm  ˆ c

(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: kuc  jm  vm (11)

The dynamics of the complete system, actuated by a DC motor are formulated by given simplified model: mL2t  c( m   t )

   ku  Jˆm  vˆm  ˆ c  n   c( m   t )

(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.

Where K  k / n .To specifies the developments, let A  K / J and B  v / J .The transfer function of a DC motor is written as  ( s) A Gm( s)  m  u c ( s) s( s  B)

(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  * t (t ) is desired.

III. GPI CONTROLLER

To synthesis the feedback-control law, we will use only the measured coupling motor position  m and

The flexible-bar transfer function in Laplace notation from (5) can be re written as  t (s) 2 Gb( s )    m (s) s 2   2 (8) 2 1/2 Where ω = (c/(mL )) 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  u  uc  k .n (9)

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 t t  . This means that all variables of the

Fig.3 compensation of the coupling torque measured in the hub

Where uc is the voltage applied before 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

compensation term. The system in (6) is then given

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

209

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY

 m t  in terms of  m t  is given, in reduction

gear terms, by:

m 

ISBN: 378 - 26 - 138420 - 5

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

mL2  t  t c

e t 

(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

c mL2

(18)

 e    e  d   m

The integral reconstructor neglects the possibly nonzero initial condition e t 0  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 k 0 , k1, k 2  need to be changed due to the use of

reference trajectory  * t t  with  m acting as a an auxiliary Control input. Clearly, if there exists an auxiliary open loop control input,

 * m t  , that ideally achieves the tracking of  *t t  for suitable initial conditions, it satisfies then the second orderdynamics, in reduction gear terms (10).

mL2 *  m t    t t    *t t  c *

(14) Subtracting (10) from (9), we obtain an expression in terms of the angular tracking errors: c e t  e m  e t  mL2 (15)

the integral reconstructor. Substituting the

Where e m   m   * m t , e t   t   * t t  . For

 s   0  *   *m   1 (19)   t  t  s2  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:



integral reconstructor e  t (14) byinto the PID controller (12) and after some rearrangements we obtain:



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

em  et 

t  mL2   K2et  K1et  K0 et  d  c  0 









(16)

  c  m   t   mL2  n  coup

We proceed by integrating the expression (11) once, to obtain: c e t t   e t 0   mL2

(20)

Thus, the angular position is readily expressed as,

 e    e  d   m

1 t   m   c

(21)

(17) INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

210

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY

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,  0 ,  1 ,  2 we can choose to locate

The open-loop control u * c (t ) that ideally achieves the open-loop tracking of the inner loop is given by 1 B u * c (t )  * m (t )  m (t ) . (27) A A 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 {  3 ,  2 , 1 ,  0 }. As done with the outer loop, all

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, (22) s 3  3as 2  3 a 2 s  a 3  0 Where the parameter a represents the desired location of the poles. The characteristic equation of the closed loop system is,

s 3  k 2 s 2   0 1  k1 s   0 k 2  k 0   0 (23) Identifying each term of the expression (18) with those of (19), the design parameters  2 ,  1 ,  0 can be uniquely specified. 2

poles can be located at the same real value,  3 , 2 , 1 , and  0 can be uniquely obtained by equalizing the terms of the two following polynomials:

B. Inner loop controller

(s  p) 4  s 4  4 ps 3  6 p 2 s 2  4 p 3 s  p 4  0 (28) 4 3 2 s  ( 3  B ) s  ( 3 B   2 A) s   1 As   0 A  0 (29) Where the parameter p represents the common location of all the closed-loop poles, this being strictly positive.

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:

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

t   ˆ   k e  k e  k  3 m 2 m 1  e m ( ) d ( )  v J 0  ev  eˆ m   t t   K K  k 0   (e m ( 2 ))d ( 2 )d ( 1 )   0 0  (24) the following integral reconstructor for the angular-velocity error signal eˆ is obtained: m

K v eˆ m   ev ( ) d ( )  e m . J 0 J

ISBN: 378 - 26 - 138420 - 5

rearrangements, the feedback control law is obtained  s 2   1s   0  * (u c  u * c )   2  ( mr   m ) . (26) s( s   3 )  

(25)

Replacing eˆ in (25) into (24) and, after some m

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

211

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY

ISBN: 378 - 26 - 138420 - 5

some arbitrarily small time t = Δ >0, by means of the expression. arbitary for t  [0, )  2  oe   ne (t ) (30) t  [,)  d e (t ) for

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 wellknown 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 .

Where ne (t ) and de (t ) are the output of the time-varying linear unstable filter d e (t )  z 3 ne (t )  t 2 t (t )  z1

z1  z 2  4t t (t ) z2  2t (t )

z3  z 4 (31)

z4  t 2 ( m (t )   t (t ))

Proof: consider (5) t   2 ( m   t )

(32)

The Laplace transform of (32) is s 2 t ( s )  s t (0)  t (0)   2 ( m ( s )   t ( s )) (33) Taking two derivatives with respect to the complex variable s, the initial conditions are cancelled 2 d 2 ( s 2 t ) d 2 ( t )  2  d ( m )      2  ds 2  ds 2 ds   (34) Employing the chain rule, we obtain s2

d 2 ( t ) d d 2 ( m ) d 2 ( t )  4s t  2 t   2 (  ) 2 ds ds ds 2 ds 2

(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 s 2 . After some rearrangements, we obtain d 2 ( t ) d  4 s 1 t  2 s 2 t 2 ds  2  ds 2 ˆ d ( m ) d 2 (ˆt ) s 2 (  ) ds 2 ds 2 (36) Let L denote the usual operational calculus transform acting on exponentially bounded signals with bounded left support (see[38]). Recall that

L1 s()  (d / dt )(), L1 (d v / ds v )()  (1) v .t v (), t

and L1(1 / s ) ()   ()( ) d .

taking

this

into

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

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

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

212

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY t t 

2 

[t  t (t )  4   t ( )d  2    t ( )dd 0

0 0

t t

t t 2

 

( )d d     2 t ( )dd

(37) The time realization of (37) can be written via time-variant linear (unstable) filters d e (t )  z 3 ne (t )  t 2 t (t )  z1 0 0

0 0

z1  z 2  4t t (t ) z2  2t (t )

ISBN: 378 - 26 - 138420 - 5

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 n (t ) F (s )ne (t )  2 oe  f  d f (t ) F ( s)d e (t ) (40) Where n f (t ) and d f (t ) are the filtered

z3  z 4

z 4  t 2 ( m (t )   t (t ))

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 ,

The natural frequency estimator of  2 is given by arbitary for t  [0, )   2 oe   ne (t ) t  [,)  d e (t ) for (38)

1/ s p , p  1 may be utilized, where highfrequency 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 arbitary for t  [0, )  2  oe   ne (t ) t  [,)  d e (t ) for (41)

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 timevarying 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.

V.ADAPTIVE CONTROL PROCEDURE Fig. 4 shows the adaptive-control system implemented in practice in our laboratory. The estimator is linked up, from time t0  0[ s] , 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 u *c [see (27)] also depends on

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

213

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY *

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  m* (t )  2 t * (t )   t * (t ) x (42) Moreover,(23) is computed as s 3  2s 2  x 2 (1   1 ) s  x( 2   0 )  0

(43)

For the inner loop only changes the feedforward term in (27) which depends on the bounded derivatives of the new  *m (t ) in (42). The variable x is defined as x  oi , t  t1

x  oe , t  t1

ISBN: 378 - 26 - 138420 - 5

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., c0  c . A meticulous stability

the ω value because the variable  m is obtained

(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 ·

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 ˆ  119.7 · 10−3 [N · m] taken in simulations is the true value estimated in real experiments. In voltage terms is ˆ c / k  0.57[V ] 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:

uc  u *c 789s 2  5.6.104 s  1.3.106  (46) The  *m   m s( s  365) feedforward term in (27), which depends on the

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

214

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY

values of the motor parameters, is computed in accordance with u *c  0.02*m (t )  0.25 *m (t ) (47)

given by  m * (t ) 

with a natural frequency of the bar given by an initial arbitrary estimate of oi  9 [rad/s]. The

RESULTS

(48) 1.2

The open-loop reference control input from (14) in terms of the initial arbitrary estimate of oi is given by 1 * *  t (t )   t (t )  12.3.10 3 * t (t )   *t (t ) 2  oi

0.8

(49)

0.6

a n g le in ra d

 m (t ) 

0.4

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 s1 (see Fig. 4), with the accurate parameter estimate, given by oe = 14.5 [rad/s]. When the

0.2

-0.2 0

0.5

1.5

2.5 time in sec

3.5

4.5

Without using estimation 1 input output

0.9 0.8 0.7

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

a n g le in r a d

0.6 0.5 0.4 0.3 0.2 0.1

with no steady-state error [see Fig. 5(c)]. In this

figure, the tracking error  t *   t is shown. The corresponding transfer function of the new updated controller is then found to be

 m   * m 0.3s  25.7  * s  30 t   t

1 *  t (t )   t * (t )  4.7.103*t (t )   *t (t ) (51) 2 oi

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

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.

 m   * m 2.7 s  17.7  * s  30 t  t

ISBN: 378 - 26 - 138420 - 5

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

For

0.5

1.5

2.5 time in sec

3.5

4.5

  14.5[rad / sec], A  61.14, B  15.15

(50)

The open-loop reference control input  * m (t )

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

215

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY

REFERENCES

-3

x 10

[1] S. K. Dwivedy and P. Eberhard, “Dynamic analysis of flexible manipulators, a literature review,” Mech.Mach. Theory, vol. 41, no. 7, pp. 749–777, Jul. 2006. [2] V. Feliu, “Robots flexibles: Hacia una generación de robots con nuevas prestaciones,” Revista Iberoamericana de Automática e Informática Industrial, vol. 3, no. 3, pp. 24–41, 2006. [3] R. H. Canon and E. Schmitz, “Initial experiments on the end-point control of a flexible robot,” Int. J. Rob. Res., vol. 3, no. 3, pp. 62–75, 1984. [4] A. Arakawa, T. Fukuda, and F. Hara, “H∞ control of a flexible robotics arm (effect of parameter uncertainties on stability),” in Proc. IEEE/RSJ IROS, 1991, pp. 959–964. [5] T. Yang, J. Yang, and P. Kudva, “Load adaptive control of a single-link flexible manipulator,” IEEE Trans. Syst., Man, Cybern., vol. 22, no. 1, pp. 85–91, Jan./Feb. 1992. [6] Y. P. Chen and H. T. Hsu, “Regulation and vibration control of an FEMbased single-link flexible arm using sliding-mode theory,” J. Vib. Control, vol. 7, no. 5, pp. 741–752, 2001. [7] Z. Su and K. A. Khorasani, “A neural network-based controller for a single-link flexible manipulator using the inverse dynamics approach,” IEEE Trans. Ind. Electron., vol. 48, no. 6, pp. 1074–1086, Dec. 2001. [8] V. G. Moudgal, W. A. Kwong, K. M. Passino, and S. Yurkovich, “Fuzzy learning control for a flexible-link robot,” IEEE Trans. Fuzzy Syst., vol. 3, no. 2, pp. 199–210, May 1995. [9] J. Becedas, J. Trapero, H. Sira-Ramírez, and V. Feliu, “Fast identification method to control a flexible manipulator with parameter uncertainties,” in Proc. ICRA, 2007, pp. 3445– 3450. [10] V. Feliu and F. Ramos, “Strain gauge based control of singlelink flexible very light

a n g le in ra d

-1

-2

-3 0

0.5

1.5

2.5 time in sec

3.5

4.5

ISBN: 378 - 26 - 138420 - 5

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 lowpass 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.

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

216

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY

weight robots robust to payload changes,” Mechatronics, vol. 15, no. 5, pp. 547–571, Jun. 2005. [11] H. Olsson, H. Amström, and C. C. de Wit, “Friction models and friction compensation,” Eur. J. Control, vol. 4, no. 3, pp. 176–195, 1998. [12] V. Feliu, K. S. Rattan, and H. B. Brown, “Control of flexible arms with friction in the joints,” IEEE Trans. Robot. Autom., vol. 9, no. 4, pp. 467–475, Aug. 1993. [13] S. Cicero, V. Santos, and B. de Carvahlo, “Active control to flexible manipulators,” IEEE/ASME Trans. Mechatronics, vol. 11, no. 1, pp. 75–83, Feb. 2006. [14] R. Marquez, E. Delaleau, and M. Fliess, “Commande par pid généralisé d’un moteur électrique sans capteur mecanique,” in Premiére Conférence Internationale Francophone d’Automatique, 2000, pp. 521–526. [15] M. Fliess and H. Sira-Ramírez, “An algebraic framework for linear identification,” ESAIM Control Optim. Calculus Variations, vol. 9, pp. 151–168, 2003. [16] M. Fliess, M. Mboup, H. Mounier, and H. Sira-Ramírez, Questioning Some Paradigms of Signal Processing via Concrete Examples. México, México: Editorial Lagares, 2003, ch. 1. [17] J. Reger, H. Sira-Ramírez, andM. Fliess, “On non-asymptotic observation of nonlinear systems,” in Proc. 44th IEEE Conf. Decision Control, Sevilla, Spain, 2005, pp. 4219–4224. [18] H. Sira-Ramírez and M. Fliess, “On the output feedback control of a synchronous generator,” in Proc. 43rd IEEE Conf. Decision Control, The Bahamas, 2004, pp. 4459–4464. [19] J. R. Trapero, H. Sira-Ramírez, and V. Feliu-Batlle, “An algebraic frequency estimator for a biased and noisy sinusoidal signal,” Signal Process., vol. 87, no. 6, pp. 1188–1201, Jun. 2007. [20] J. R. Trapero, H. Sira-Ramírez, and V. Feliu-Batlle, “A fast on-line frequency

ISBN: 378 - 26 - 138420 - 5

estimator of lightly damped vibrations in flexible structures,” J. Sound Vib., vol. 307, no. 1/2, pp. 365–378, Oct. 2007. [21] F. Bellezza, L. Lanari, and G. Ulivi, “Exact modeling of the flexible slewing link,” in Proc. IEEE Int. Conf. Robot. Autom., 1991, vol. 1, pp. 734–804. [22] E. Bayo, “A finite-element approach to control the end-point motion of a single-link flexible robot,” J. Robot. Syst., vol. 4, no. 1, pp. 63–75, Feb. 1987. [23] V. Feliu, K. S. Rattan, and H. Brown, “Modeling and control of single-link flexible arms with lumped masses,” J. Dyn. Syst. Meas. Control, vol. 114, no. 7, pp. 59–69, 1992. [24] W. Liu and Z. Hou, “A new approach to suppress spillover instability in structural vibration,” Struct. Control Health Monitoring, vol. 11, no. 1, pp. 37–53, Jan.–Mar. 2004. [25] R. D. Begamudre, Electro-Mechanical Energy ConversionWith Dynamics of Machines. New York: Wiley, 1998. [26] H. Sira-Ramírez and S. Agrawal, Differentially Flat Systems. NewYork: Marcel Dekker, 2004. [27] P. C. Young, “Parameter estimation for continuous-time models—A survey,” Automatica, vol. 17, no. 1, pp. 23–29, Jan. 1981. [28] H. Unbehauen and G. P. Rao, “Continuous-time approaches to system identification—A survey,” Automatica, vol. 26, no. 1, pp. 23–35, Jan. 1990. [29] N. Sinha and G. Rao, Identification of Continuous-Time Systems. Dordrecht, The Netherlands: Kluwer, 1991. [30] H. Unbehauen and G. Rao, Identification of Continuous Systems. Amsterdam, The Netherlands: North-Holland, 1987. [31] L. Ljung, System Identification: Theory for the User, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1999. [32] S. Moussaoui, D. Brie, and A. Richard, “Regularization aspects in continuous-time

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

217

www.iaetsd.in

INTERNATIONAL CONFERENCE ON CURRENT INNOVATIONS IN ENGINEERING AND TECHNOLOGY

model identification,” Automatica, vol. 41, no. 2, pp. 197–208, Feb. 2005. [33] T. Söderström and M. Mossberg, “Performance evaluation of methods for identifying continuous-time autoregressive processes,” Automatica, vol. 36, no. 1, pp. 53 59, Jan. 2000. [34] E. K. Larsson and T. Söderström, “Identification of continuous-time AR processes from unevenly sampled data,” Automatica, vol. 38, no. 4, pp. 709–718, Apr. 2002. [35] K. Mahata and M. Fu, “Modeling continuous-time processes via input-tostate filters,” Automatica, vol. 42, no. 7, pp. 1073– 1084, Jul. 2006. [36] R. Johansson, A. Robertsson, K. Nilsson, and M. Verhaegen, “Statespace system identification of robot manipulator dynamics,” Mechatronics, vol. 10, no. 3, pp. 403–418, Apr. 2000. [37] I. Eker, “Open-loop and closed-loop experimental on-line identification of a three mass electromechanical system,”Mechatronics, vol. 14, no. 5, pp. 549–565, Jun. 2004. [38] J. Mikusinski and T. Boehme, Operational Calculus, 2nd ed., vol. I.New York: Pergamon, 1987. [39] M. Fliess, “Analyse non standard du bruit,” C.R. Acad. Sci. Paris, 2006. p. Ser. I 342. [40] T. Önsay and A. Akay, “Vibration reduction of a flexible arm by time optimal open-loop control,” J. Sound Vib., vol. 147, no. 2, pp. 283–300, 1991. [41] D. Sun, J. Shan, Y. Su, H. H. T. Liu, and C. Lam, “Hybrid control of a rotational flexible beam using enhanced pd feedback with a nonlinear differentiator and PZT actuators,” Smart Mater. Struct., vol. 14, no. 1, pp. 69–78, Feb. 2005. [42] J. Shan, H. T. Liu, and D. Shun, “Slewing and vibration control of a single link flexible manipulator by positive position feedback

ISBN: 378 - 26 - 138420 - 5

(ppf),” Mechatronics, vol. 15, no. 4, pp. 487– 503, May 2005. [43] C. M. A. Vasques and J. D. Rodrigues, “Active vibration control of smart piezoelectric beams: Comparison of classical and optimal feedback control strategies,” Comput. Struct., vol. 84, no. 22/23, pp. 1402–1414, Sep. 2006. [44] S. S. Han, S. B. Choi, and H. H. Kim, “Position control of a flexible gantry robot arm using smart material actuators,” J. Robot. Syst., vol. 16, no. 10, pp. 581–595, Oct. 1999. [45] M. J. Balas, “Active control of flexible systems,” J. Optim. Theory Appl., vol. 25, no. 3, pp. 415–436, Jul. 1978. [46] E. G. Christoforou and C. Damaren, “The control of flexible link robots manipulating large payloads: Theory and experiments,” J. Robot. Syst., vol. 17, no. 5, pp. 255–271, May 2000. [47] C. J. Damaren, “Adaptive control of flexible manipulators carrying large uncertain payloads,” J. Robot. Syst., vol. 13, no. 4, pp. 219–288,Apr. 1996. [48] V. Feliu, J. A. Somolinos, and A. García, “Inverse dynamics based control system for a three-degree-of-freedom flexible arm,” IEEE Trans. Robot. Autom., vol. 19, no. 6, pp. 1007 1014, Dec. 2003. [49] K. Ogata, Modern Control Engineering, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.

INTERNATIONAL ASSOCIATION OF ENGINEERING & TECHNOLOGY FOR SKILL DEVELOPMENT

218