IAETSD

1. 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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2. 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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3. 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:
)(2
tmt 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.1
θtand 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ˆ 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
mcoul
mc sign






(3)
And when 0ˆ 












)0)(,max(
)0)(,min(
)(.ˆ
uku
uku
usign
coup
coup
c
(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
ˆ And m
ˆ 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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4. 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
c
n
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
)(
)(
)(






ss
s
Gb
m
t
s
(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
nk
uu 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
)()(
)(
)(
Bss
A
su
s
sGm
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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5.  tm in terms of  tm is given, in reduction
gear terms, by:
ttm
c
mL
  
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).
     tt
c
mL
t ttm
**
2
*
  
(14)
Subtracting (10) from (9), we obtain an
expression in terms of the angular tracking
errors:
 tmt ee
mL
c
e   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
c
mL
ee t
t
tttm
0
012
2

(16)
We proceed by integrating the expression (11)
once, to obtain:
           dee
mL
c
ete
t
tmtt  
0
2
0
(17)
Disregarding the constant error due to the
tracking error velocity initial conditions, the
estimated error velocity can be computed in the
following form:
       dee
mL
c
e
t
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
s
s



 







 *
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,

c
mt
1
 (21)
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6. 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 02
2
01
2
0
2
2
3
 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
K
J
e
K
v
e
m
mmm
m
0
12
0
20
0
123
)()())((
)()(ˆ
ˆ







(24)
the following integral reconstructor for the
angular-velocity error signal m
e
ˆ is obtained:
mm
e
J
v
de
J
K
e
t
v    0
)()(ˆ . (25)
Replacing m
e
ˆ in (25) into (24) and, after some
rearrangements, the feedback control law is
obtained
)(
)(
)(
*
3
01
2
2*
mmrcc
ss
ss
uu 










 . (26)
The open-loop control )(*
tu c that ideally
achieves the open-loop tracking of the inner loop
is given by
)()(
1
)( **
t
A
B
t
A
tu 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)()( 01
2
23
3
3
4
 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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7. 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
tfor
tfor
td
tn
arbitary
e
eoe (30)
Where )(tne and )(tde are the output of the
time-varying linear unstable filter
1
2
)()( ztttn te   3)( ztde 
)(421 ttzz t 43 zz 
)(22 tz t ))()((2
4 tttz tm   (31)
Proof: consider (5)
)(2
tmt   (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
2
2
2
22
)()()(
ds
d
ds
d
ds
sd tmt 


(34)
Employing the chain rule, we obtain
)
)()(
(24
)(
2
2
2
2
2
2
2
2
ds
d
ds
d
ds
d
s
ds
d
s tm
t
tt 



(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 2
s . After
some rearrangements, we obtain
)
)ˆ()ˆ(
(
24
)(
2
2
2
2
2
21
2
2
2
ds
d
ds
d
s
s
ds
d
s
ds
d
tm
t
tt









(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
dL
0
)/1(1
.))(()(  taking this into
account, we can translate(36) into the time
domain
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8.  
 


 t t t
t
t
m
t t
tt
dddd
dddtt
0 0 0
2
0
2
0 0 0
t
2
2
)()(
)(2)(4)([




(37)
The time realization of (37) can be written via
time-variant linear (unstable) filters
1
2
)()( ztttn te   3)( ztde 
)(421 ttzz t 43 zz 
)(22 tz t ))()((2
4 tttz tm  
The natural frequency estimator of 2
 is given by







),[
),0[
)(
)(2
tfor
tfor
td
tn
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
tdsF
tnsF
td
tn
e
e
f
f
oe 
(40)
Where )(tnf 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 psp
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
tfor
tfor
td
tn
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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9. 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
*
tt
x
t ttm   
(42)
Moreover,(23) is computed as
0)()1( 021
22
2
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
,
,
ttx
ttx
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 ˆ 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
*
*





ss
ssuu
mm
cc

(46) The
feedforward term in (27), which depends on the
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10. 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 9oi [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
*
*





s
s
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
oi
m 

    (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
*
*





s
s
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
oi
m 

    (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
timeinsec
angleinrad
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
timeinsec
angleinrad
input
output
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11. 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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