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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

as Coulomb friction effect. With the controller proposed

Proportional Integral type (GPI), Controller is

no estimation of this nonlinear phenomena

designed for the control of an uncertain flexible robotic

is therefore required. This is a substantial improvement

arm with unknown internal parameters in the motor

over existing control schemes based on the on-line

dynamics. The GPI controller is designed using a two-

estimation of the friction parameters. Developments for

stage design procedure entitling an outer loop,

the controller are based on two concepts namely flatness

designed under the singular perturbation assumption

based exact feed forward linearization and Generalized

of no motor dynamics; and subsequently an inner loop

Proportional Integral (GPI) control. As a result using the

which forces the motor response to track the control

GPI control method, a classical compensating second

input position reference trajectory derived in the

order network with a ”roll-off” zero bestowing a rather

previous design stage. For both, the inner and the

beneficial integral control action with respect to constant

outer loop, the GPI controller design method is easily

load perturbations is obtained. The control scheme

implemented to achieve the desired tracking control.

proposed in this article is truly an output feedback controller since it uses only the position of the motor.

Key words—Flexible Arm manipulator, trajectory

Velocity measurements, which always introduce errors in

tracking, generalized proportional integral (GPI)

the signals and noises and makes it necessary the use of

control.

suitable low pass filters, are not required. Furthermore, the only measured variables are the motor and tip position.

I. INTRODUCTION

The goal of this work is to control a very fast system, only

In this paper, a two stage GPI controller is proposed for

knowing the stiffness of the bar, without knowing the rest

the regulation of an uncertain single-link flexible arm with

of the parameters in the system .A brief outline of this

unknown mass parameter at the tip, motor inertia, viscous

work is the following: Section II explains the system and

friction and electromechanical constant in the motor where

the

the tracking of a trajectory must be too precise. As in Feliu

proposed. An on-line closed loop algebraic identifier is

and Ramos [1] a two stage design procedure is used but

presented in the Section III, yielding expressions for DC

using now a GPI controller viewpoint the particular

motor and flexible bar parameters. This results will be

requirement of robustness with respect to unknown

verified via simulation in Section IV. Finally, the last

constant torque on the motor dynamics, this is known

section is devoted to main conclusions and further works.

generalized

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proportional

integrator

controller

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II. MODEL DESCRIPTION Consider the following simplified model of a very

III. GENERALIZED PROPORTIONAL

lightight flexible link, with all its mass concentrated at the

INTEGRATOR CONTROLLER

tip, actuated by a DC motor, as shown in Fig. 1. The

In Laplace transforms notation, the flexible bar transfer

dynamics of the system is given by:

function, obtained from (1), can be written as follows,

mL2t  c m   t  

(1) 



ku  J  m  V  m  T c  T coup 

T coup 

 s   Gb s   t  2 0 2  m s  s   0 2



(2)

c  m   t  n

(4)

(3) Where





c mL

the

unknown

natural

where m and L are the mass in the tip position and the

frequency of the bar due to the lack of precise knowledge

length of the flexible arm, respectively, assumed to be

of m and L. It is assumed that the constant c is perfectly

unknown, and c is the stiffness of the bar, which is

known. As it was done in [1] the coupling torque can be

assumed to be perfectly known, J is the inertia of the

compensated in the motor by means of a feed-forward



motor,

term which allows to decouple the two dynamics, the

V the viscous friction coefficient, T c is the 

unknown Coulomb friction torque, T

coup

flexible link dynamics and the bar dynamics, which allows

is the

an easier design task for the controller since the two

measured coupling torque between the motor and the link,

dynamics can be studied separately. In this case the

k is the known electromechanical constant of the motor,

voltage applied to the motor is of the form,

 

u is the motor input voltage, 



stands for the acceleration

T coup u  uc  K



of the motor gear,  m is the velocity of the motor gear. The constant factor gea.Thus, 

n is the Reduction ratio of the motor

 m

 

(5)





where uc is the voltage applied before the feed-forward

is the angular position of

term. The system in (2) is then given by:

the motor and  t is the unmeasured angular position of the tip.

 





ku c  J  m  V  m  T c

(6)



Where T c 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:

ku c  Jm  Vm

(7)

Fig. 1. Diagram of a single link flexible arm

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System (9) is a second order system in which it is desired

A  K / J,B  v/ J

To simplify the developments, let

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to regulate the tip position of the flexible bar

The DC motor transfer function is then written as:

a given smooth reference trajectory

 m s  A  u c s  s s  B 

proposed

 * t t  with  m

acting as a an auxiliary Control input. Clearly, if there

(8)

exists an auxiliary open loop control input, ideally achieves the tracking of

The

feed-forward

technique

has

 t , towards

 * m t  , that

 *t t  for suitable initial

conditions, it satisfies then the second order dynamics, in

been

reduction gear terms (10).

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 reference trajectory 

 * m t  

 t t  to track a given smooth

t 

mL2 *  t t    * t t  c

(10)

For the synthesis of the Subtracting (10) from (9), obtain an expression in terms

feedback law only the measured motor position

 m and

of the angular tracking errors:



the measured coupling torque T coup are used. Desired

e t 

controller should be robust with respect to unknown constant

torque

disturbances

affecting

the

motor

c e m  e t  mL2

(11)

dynamics. One of the prevailing restrictions throughout *

t , e t

  t   * t t  . For this

our treatment of the problem is our desire of not to

Where e m   m  

measure, or compute on the basis samplings, angular

part of the design, view e m as an incremental control

velocities neither of the motor shaft nor of the load.

input for the links dynamics. Suppose for a moment it is possible to measure the angular position velocity tracking

A. Outer loop controller

error e t , then the outer loop feedback incremental controller could be proposed to be the following PID

Consider the model of the flexible link, given in (1). This Sub system is flat, with flat output given by

controller,

 t t  . This

means that all variables of the unperturbed system may be

em  et 

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

 m t 

 m t  is given, in reduction gear terms, by:

m 

mL2  t  t c

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

(12)

Integrating the expression (11) once, to obtain:

e t t   e t 0  

(9)

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c mL2

 e    e  d   m

(13)

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Disregarding the constant error due to the tracking error

ISBN: 378 - 26 - 138420 - 5

1 t   m   c

velocity initial conditions, the estimated error velocity can

(17)

be computed in the following form: e t 

c mL2

t m

In Fig. 2 depicts the feedback control scheme under which

(14)

 e    e  d   t

the outer loop controller would be actually implemented in

The integral reconstructor neglects the possibly nonzero

practice. The closed outer loop system in Fig. 2 is

initial condition et 0 and, hence, it exhibits a constant

asymptotically exponentially stable. To specify the parameters,

 0 ,  1 ,  2  choose to locate the closed loop

poles in the left half of the complex plane. All three poles estimation error. When the reconstructor is used in the

can be located in the same point of the real line, s = −a,

derivative part of the PID controller, the constant error is using the following polynomial equation,

suitably compensated thanks to the integral control action

s 3  3as 2  3a 2 s  a 3  0

of the PID controller. The use of the integral reconstructor

(18)

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

Where the parameter a represents the desired location of the poles. The characteristic equation of the closed loop

due to the use of the integral reconstructor. Substituting

system is,



the integral reconstructor e  t

s 3  k 2 s 2   0 1  k1 s   0 k 2  k 0   0 2

(14) by into the PID

controller (12) and after some rearrangements :

Identifying each term of the expression (18) with those of (19), the design parameters



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





(15)



 2 ,  1 ,  0  can be uniquely

specified.

(15)

B. Inner loop controller The tip angular position cannot be measured, but it certainly can be computed from the expression relating the

The angular position

tip position with the motor position and the coupling

input in the previous controller design step, is now

torque. The implementation may then be based on the use

regarded as a reference trajectory for the motor controller.

of the coupling torque measurement. Denote the coupling

Denote this reference trajectory by

torque by  it is known to be given by:

 * mr .

The dynamics of the DC motor, including the Coulomb



  c m   t   mL2  n  coup

 m generated as an auxiliary control

(16)

friction term, is given by (6). It is desired to design the controller to be robust with respect to this torque

Thus, the angular position is readily expressed as,

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. Fig. 2. Flexible link dc motor system controlled by a two

The following feedback controller is proposed.

stage GPI controller design t  vˆ J ˆ ev  em   k3em  k2em  k1  em ( )d( ) K K 0  (20)

The closed inner loop system in Fig. 2 is asymptotically exponentially

t t

stable.

design

the

parameters

 3 ,  2 ,  1 ,  0  choose to place the closed loop poles in

 k0  (em ( 2 ))d( 2 )d(1) 00

a desired location of the left half of the complex plane. All

In order to avoid tracking error velocity measurements

poles can be located at the same real value, using the

again obtain an integral reconstructor for the angular

following polynomial equation,

velocity error signal

( s  p ) 4  s 4  4 ps 3  6 p 2 s 2  4 p 3 s  p 4  0 (24) t

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

(21) Where the parameter p represents the common location of all the closed loop poles. The characteristic equation of the

Replacing

eˆ m (21)

into

(17)

and

after

closed loop system is,

some

rearrangements the feedback control law is obtained as:

s 4  (3  B)s3  (3 B  2 A)s 2  1 As  0 A  0 (25)

  2 s 2  1 s   0  * (uc  u c )    ( mr   m )  s( s   3 )  *

(22)

Identifying the corresponding terms of the equations (24) and (25), the parameters

 3 ,  2 , 1 ,  0 

may be

uniquely obtained.

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

u *c (t ) 

1 * B  m (t )  m (t ) A A

(23)

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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  0  9

(rad/sec), the transfer

function of the controller is given by the following expression

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

Fig.3 Simulink file for GPI controller

The open loop reference control input

IV. SIMULATIONS

(28)

 * m t  in (10) is

given by: A. Inner loop

 * m t   0.123* t t    * t t 

(29)

The parameters used for the motor are given by an initial, Arbitrary, estimate of:

C. Results

A = 61.14(N/(Vkgs)), B = 15.15((Ns)/(kgm)) The system should be as fast as possible, but taking care of

The desired reference trajectory used for the tracking

possible saturations of the motor which occur at 2 (V).

problem of the flexible arm is specified as a delayed

The poles can be located in a reasonable location of the

exponential function.The controlled arm response clearly

negative real axis. If closed loop poles are located in, say,

shows a highly oscillatory response. Nevertheless, the

−60, the transfer function of the controller results in the

controller tries to track the trajectory and locate the arm in

following expression:

the required steady state position.

uc  u * c 798s 2  56000s  1300000 (26)  ss  365  *m  m The feed-forward term in (23) is computed in accordance with,

u * c  0.02* m  0.25 * m

(27)

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Fig.6 Trajectory Tracking with GPI controller

Fig.4 Trajectory Tracking with integral 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 Fig.5 Trajectory Tracking with PI controller

trajectory is achieved.

D. Some remarks

The Coulomb’s friction torque in our system is given by . 

 sign( ) where  is the Coulomb’s friction coefficient m

and

. 

m

is the motor velocity. The compensation voltage

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that it would be required to compensate the friction torque,

REFERENCES

. 

[1] V. Feliu and F. Ramos., “Strain gauge based control

as it was proposed in [2] is of about 0.36sign( m ) Volts. In

of single-link flexible very lightweight robots robust to

our feedback control scheme there is no need of a nonlinear

compensation

term

since

the

ISBN: 378 - 26 - 138420 - 5

payload changes.” Mechatronics., vol. 15, pp. 547–571,

controller

2004.

automatically takes care of the piecewise constant

[2] H. Olsson, K. Amstr¨om, and C. C. de Wit., “Friction

perturbation arising from the Coulomb friction coefficient.

models and friction compensation,” European Journal of Control, vol. 4, pp. 176–195, 1998.

V. CONCLUSIONS

[3]

M. Fliess and H. Sira-Ram´ırez, “An algebraic

framework for linear identification,” ESAIM Contr. Optim. A two stage GPI controller design scheme has been

and Calc. of Variat., vol. 9, pp. 151–168, 2003.

proposed and reference trajectory tracking of a single-link

[4]

flexible arm with unknown mass at the tip and parameters

M. Fliess, M. Mboup, H. Mounier, and H. Sira

Ram´ırez,

of the motor. The GPI control scheme here proposed only

Questioning

some

paradigms

signal

processing via concrete examples. Editorial Lagares,

requires the measurement of the angular position of the

M´exico City., 2003, ch. 1 in Algebraic methods in

motor and that of the tip. For this second needed

flatness, signal processing and state estimation, H. Sira-

measurement, a replacement can be carried out in terms of

Ramirez and G. Silva-Navarro (eds).

a linear combination of the motor position and the

[5] Fractional-order Systems and Controls Fundamentals

measured coupling torque provided by strain gauges

and Applications Concepción A. Monje · YangQuan Chen

conveniently located at the bottom of the flexible arm. The

Blas M. Vinagre · Dingyü Xue · Vicente Feliu.

GPI feedback control scheme here proposed is quite robust

[6] ——, “Modeling and control of single-link flexible

with respect to the torque produced by the friction term

arms with lumped

and its estimation is not really necessary. Finally, since the

masses.” J Dyn Syst, Meas Control, vol. 114(3), pp. 59–

parameters of the GPI controller depend on the natural

69, 1992.

frequency of the flexible bar and the parameters of the

[7] V. Feliu, J. A. Somolinos, C. Cerrada, and J. A.

motor A and B, this method is robust respect to zero mean

Cerrada, “A new control

high frequency noises and yields good results, as seen

scheme of single-link flexible manipulators robust to

from digital computer based simulations. Motivated by the

payload changes,” J.

encouraging simulation results presented in this article, is

Intell Robotic Sys, vol. 20, pp. 349–373, 1997.

proposed that by use of an on-line, non asymptotic,

[8] V. Feliu, J. A. Somolinos, and A. Garc´ıa, “Inverse

algebraic parameter estimator for this parameters will be

dynamics based control

the topic of a forthcoming publication.

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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