Iaetsd modelling of one link flexible arm manipulator using

MODELLING OF ONE LINK FLEXIBLE ARM MANIPULATOR USING
TWO STAGE GPI CONTROLLER
B Sudeep Dr.K.Rama Sudha
ME Control systems Professor
Department Electrical & Electronics Engineering,
Andhra University,
Visakhapatnam,
Andhra Pradesh
Abstract—In this article, a two stage Generalized
Proportional Integral type (GPI), Controller is
designed for the control of an uncertain flexible robotic
arm with unknown internal parameters in the motor
dynamics. The GPI controller is designed using a two-
stage design procedure entitling an outer loop,
designed under the singular perturbation assumption
of no motor dynamics; and subsequently an inner loop
which forces the motor response to track the control
input position reference trajectory derived in the
previous design stage. For both, the inner and the
outer loop, the GPI controller design method is easily
implemented to achieve the desired tracking control.
Key words—Flexible Arm manipulator, trajectory
tracking, generalized proportional integral (GPI)
control.
I. INTRODUCTION
In this paper, a two stage GPI controller is proposed for
the regulation of an uncertain single-link flexible arm with
unknown mass parameter at the tip, motor inertia, viscous
friction and electromechanical constant in the motor where
the tracking of a trajectory must be too precise. As in Feliu
and Ramos [1] a two stage design procedure is used but
using now a GPI controller viewpoint the particular
requirement of robustness with respect to unknown
constant torque on the motor dynamics, this is known
as Coulomb friction effect. With the controller proposed
no estimation of this nonlinear phenomena
is therefore required. This is a substantial improvement
over existing control schemes based on the on-line
estimation of the friction parameters. Developments for
the controller are based on two concepts namely flatness
based exact feed forward linearization and Generalized
Proportional Integral (GPI) control. As a result using the
GPI control method, a classical compensating second
order network with a ”roll-off” zero bestowing a rather
beneficial integral control action with respect to constant
load perturbations is obtained. The control scheme
proposed in this article is truly an output feedback
controller since it uses only the position of the motor.
Velocity measurements, which always introduce errors in
the signals and noises and makes it necessary the use of
suitable low pass filters, are not required. Furthermore, the
only measured variables are the motor and tip position.
The goal of this work is to control a very fast system, only
knowing the stiffness of the bar, without knowing the rest
of the parameters in the system .A brief outline of this
work is the following: Section II explains the system and
the generalized proportional integrator controller
proposed. An on-line closed loop algebraic identifier is
presented in the Section III, yielding expressions for DC
motor and flexible bar parameters. This results will be
verified via simulation in Section IV. Finally, the last
section is devoted to main conclusions and further works.
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II. MODEL DESCRIPTION
Consider the following simplified model of a very
lightight flexible link, with all its mass concentrated at the
tip, actuated by a DC motor, as shown in Fig. 1. The
dynamics of the system is given by:
 tmt cmL  2
(1)
coupcmm TTVJku



 (2)
 tmcoup
n
c
T  

(3)
where m and L are the mass in the tip position and the
length of the flexible arm, respectively, assumed to be
unknown, and c is the stiffness of the bar, which is
assumed to be perfectly known, J is the inertia of the
motor, V the viscous friction coefficient, cT

is the
unknown Coulomb friction torque, coupT

is the
measured coupling torque between the motor and the link,
k is the known electromechanical constant of the motor,
u is the motor input voltage, m

 stands for the acceleration
of the motor gear, m

 is the velocity of the motor gear.
The constant factor n is the Reduction ratio of the motor
gea.Thus, nmm /

  , m

 is the angular position of
the motor and t is the unmeasured angular position of
the tip.
Fig. 1. Diagram of a single link flexible arm
III. GENERALIZED PROPORTIONAL
INTEGRATOR CONTROLLER
In Laplace transforms notation, the flexible bar transfer
function, obtained from (1), can be written as follows,
 
 
  2
0
2
2
0






ss
s
sG
m
t
b (4)
Where
20
mL
c
 is the unknown natural
frequency of the bar due to the lack of precise knowledge
of m and L. It is assumed that the constant c is perfectly
known. As it was done in [1] the coupling torque can be
compensated in the motor by means of a feed-forward
term which allows to decouple the two dynamics, the
flexible link dynamics and the bar dynamics, which allows
an easier design task for the controller since the two
dynamics can be studied separately. In this case the
voltage applied to the motor is of the form,
K
T
uu
coup
c

 (5)
where uc is the voltage applied before the feed-forward
term. The system in (2) is then given by:
cmmc TVJku



 (6)
Where cT

is a perturbation, depending only on the sign of
the angular velocity. It is produced by the Coulomb‘s
friction phenomenon. The controller to be designed will be
robust with respect to unknown piecewise constant torque
disturbances affecting the motor dynamics. Then the
perturbation free system to be considered is the following:
mmc VJku    (7)
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To simplify the developments, let JvBJKA /,/ 
The DC motor transfer function is then written as:
 
   Bss
A
su
s
c
m



(8)
The proposed feed-forward technique has been
successfully tested in previous experimental works where
it was implemented with direct driven motors [5], [6], and
in motors with reduction gears [1], [7], [8]. It is desired to
regulate the load position  tt to track a given smooth
reference trajectory  tm
*
 For the synthesis of the
feedback law only the measured motor position m and
the measured coupling torque coupT

are used. Desired
controller should be robust with respect to unknown
constant torque disturbances affecting the motor
dynamics. 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 neither of the motor shaft nor of the load.
A. Outer loop controller
Consider the model of the flexible link, given in (1). This
Sub system 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  tm
in terms of  tm is given, in reduction gear terms, by:
ttm
c
mL
  
2
(9)
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 order dynamics, in
reduction gear terms (10).
     tt
c
mL
t ttm
**
2
*
   (10)
Subtracting (10) from (9), obtain an expression in terms
of the angular tracking errors:
 tmt ee
mL
c
e   2
 (11)
Where    tete tttmmm
**
,   . For this
part of the design, view me as an incremental control
input for the links dynamics. Suppose for a moment it is
possible to measure the angular position velocity tracking
error te , then the outer loop feedback incremental
controller could be proposed to be the following PID
controller,
  





   deKeKeK
c
mL
ee t
t
tttm
0
012
2

(12)
Integrating the expression (11) once, to obtain:
           dee
mL
c
ete
t
tmtt  
0
2
0 (13)
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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
 (14)
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) by into the PID
controller (12) and after some rearrangements :
   ttmm
s
s



 







 *
2
01*
(15)
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
 (16)
Thus, the angular position is readily expressed as,

c
mt
1
 (17)
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 ,,  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 (18)
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  (15)
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.
Denote this reference trajectory by mr
*
 .
The dynamics of the DC motor, including the Coulomb
friction term, is given by (6). It is desired to design the
controller to be robust with respect to this torque
disturbance. A controller for the system should then
include a double integral compensation action which is
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capable of overcoming ramp tracking errors. The ramp
error is mainly due to the integral angular velocity
reconstructor, performed in the presence of constant,or
piece-wise constant, torque perturbations characteristicof
the Coulomb phenomenon before stabilization around zero
velocity. The integral reconstructor is hidden in the GPI
control scheme.
The following feedback controller is proposed.










t t
t
v
ddek
dekekek
K
J
e
K
v
e
m
mmmm
0
12
0
20
0
123
)()())((
)()(ˆˆ



 
(20)
In order to avoid tracking error velocity measurements
again obtain an integral reconstructor for the angular
velocity error signal
mm
e
J
v
de
J
K
e
t
v    0
)()(ˆ (21)
Replacing m
e
ˆ (21) into (17) and after some
rearrangements the feedback control law is obtained as:
)(
)(
)(
*
3
01
2
2*
mmrcc
ss
ss
uu 










 (22)
The new controller clearly exhibits an integral action
which is characteristic of compensator networks that
robustly perform against unknown constant perturbation
inputs. The open loop control that ideally achieves the
open loop tracking of the inner loop is given by
)()(
1
)( **
t
A
B
t
A
tu mmc    (23)
Fig. 2. Flexible link dc motor system controlled by a two
stage GPI controller design
The closed inner loop system in Fig. 2 is asymptotically
exponentially stable. To design the parameters
 0123 ,,,  choose to place the closed loop poles in
a desired location of the left half of the complex plane. All
poles can be located at the same real value, using the
following polynomial equation,
0464)( 4322344
 pspsppssps (24)
Where the parameter p represents the common location of
all the closed loop poles. The characteristic equation of the
closed loop system is,
0)()( 01
2
23
3
3
4
 AAssABsBs  (25)
Identifying the corresponding terms of the equations (24)
and (25), the parameters  0123 ,,,  may be
uniquely obtained.
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Fig.3 Simulink file for GPI controller
IV. SIMULATIONS
A. Inner loop
The parameters used for the motor are given by an initial,
Arbitrary, estimate of:
A = 61.14(N/(Vkgs)), B = 15.15((Ns)/(kgm))
The system should be as fast as possible, but taking care of
possible saturations of the motor which occur at 2 (V).
The poles can be located in a reasonable location of the
negative real axis. If closed loop poles are located in, say,
−60, the transfer function of the controller results in the
following expression:
 365
130000056000798 2
*
*





ss
ssuu
mm
cc

(26)
The feed-forward term in (23) is computed in accordance
with,
mmcu ***
25.002.0    (27)
B. Outer loop
The parameter used for the flexible arm is c = 1.584 (Nm),
being unknown the mass (m) and the length (L). The poles
for the outer loop design are located at −35 in the real axis.
With a natural frequency of the bar given by an initial,
arbitrary, estimate of 90  (rad/sec), the transfer
function of the controller is given by the following
expression
30
7.177.2
*
*





s
s
tt
mm


(28)
The open loop reference control input  tm
*
 in (10) is
given by:
     ttt ttm
***
123.0    (29)
C. Results
The desired reference trajectory used for the tracking
problem of the flexible arm is specified as a delayed
exponential function.The controlled arm response clearly
shows a highly oscillatory response. Nevertheless, the
controller tries to track the trajectory and locate the arm in
the required steady state position.
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Fig.4 Trajectory Tracking with integral controller
Fig.5 Trajectory Tracking with PI controller
Fig.6 Trajectory Tracking with GPI controller
Fig.7 Comparison with different controllers
It can be noticed from Fig.7 that the reference trajectory
tracking error rapidly converges to zero by using a GPI
controller, and thus a quite precise tracking of the desired
trajectory is achieved.
D. Some remarks
The Coulomb’s friction torque in our system is given by
 sign( m
.

 ) where  is the Coulomb’s friction coefficient
and m
.

 is the motor velocity. The compensation voltage
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that it would be required to compensate the friction torque,
as it was proposed in [2] is of about 0.36sign( m
.

 ) Volts. In
our feedback control scheme there is no need of a
nonlinear compensation term since the controller
automatically takes care of the piecewise constant
perturbation arising from the Coulomb friction coefficient.
V. CONCLUSIONS
A two stage GPI controller design scheme has been
proposed and reference trajectory tracking of a single-link
flexible arm with unknown mass at the tip and parameters
of the motor. The GPI control scheme here proposed only
requires the measurement of the angular position of the
motor and that of the tip. For this second needed
measurement, a replacement can be carried out in terms of
a linear combination of the motor position and the
measured coupling torque provided by strain gauges
conveniently located at the bottom of the flexible arm. The
GPI feedback control scheme here proposed is quite robust
with respect to the torque produced by the friction term
and its estimation is not really necessary. Finally, since the
parameters of the GPI controller depend on the natural
frequency of the flexible bar and the parameters of the
motor A and B, this method is robust respect to zero mean
high frequency noises and yields good results, as seen
from digital computer based simulations. Motivated by the
encouraging simulation results presented in this article, is
proposed that by use of an on-line, non asymptotic,
algebraic parameter estimator for this parameters will be
the topic of a forthcoming publication.
REFERENCES
[1] V. Feliu and F. Ramos., “Strain gauge based control
of single-link flexible very lightweight robots robust to
payload changes.” Mechatronics., vol. 15, pp. 547–571,
2004.
[2] H. Olsson, K. Amstr¨om, and C. C. de Wit., “Friction
models and friction compensation,” European Journal of
Control, vol. 4, pp. 176–195, 1998.
[3] M. Fliess and H. Sira-Ram´ırez, “An algebraic
framework for linear identification,” ESAIM Contr. Optim.
and Calc. of Variat., vol. 9, pp. 151–168, 2003.
[4] M. Fliess, M. Mboup, H. Mounier, and H. Sira
Ram´ırez, Questioning some paradigms of signal
processing via concrete examples. Editorial Lagares,
M´exico City., 2003, ch. 1 in Algebraic methods in
flatness, signal processing and state estimation, H. Sira-
Ramirez and G. Silva-Navarro (eds).
[5] Fractional-order Systems and Controls Fundamentals
and Applications Concepción A. Monje · YangQuan Chen
Blas M. Vinagre · Dingyü Xue · Vicente Feliu.
[6] ——, “Modeling and control of single-link flexible
arms with lumped
masses.” J Dyn Syst, Meas Control, vol. 114(3), pp. 59–
69, 1992.
[7] V. Feliu, J. A. Somolinos, C. Cerrada, and J. A.
Cerrada, “A new control
scheme of single-link flexible manipulators robust to
payload changes,” J.
Intell Robotic Sys, vol. 20, pp. 349–373, 1997.
[8] V. Feliu, J. A. Somolinos, and A. Garc´ıa, “Inverse
dynamics based control
system for a three-degree-of-freedom flexible arm,” IEEE
Trans Robotics
Automat, vol. 19(6) pp. 1007-1014, 2003.
[9] H. Sira-Ram´ırez and S. Agrawal, “Differentially flat
systems,” Marcel
Dekker, 2004.
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