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Tip position control of a lightweight flexiblemanipulator using a fractional order controller

C.A. Monje, F. Ramos, V. Feliu and B.M. Vinagre

Abstract: A new method to control single-link lightweight flexible manipulators in the presence ofpayload changes is proposed. Undoubtedly, the control of this kind of structures is nowadays one ofthe most challenging and attractive research areas, being remarkable its application to the aero-space industry, among others. One of the interesting features of the design method presentedhere is that the overshoot of the controlled system is independent of the tip mass. This allows aconstant safety zone to be delimited for any given placement task of the arm, independent of theload being carried, thereby making it easier to plan collision avoidance. Other considerationsabout noise and motor saturation issues are also presented. To satisfy this performance, theoverall control scheme proposed consists of three nested control loops. Once the friction andother nonlinear effects have been compensated, the inner loop is designed to give a fast motorresponse. The middle loop simplifies the dynamics of the system and reduces its transfer functionto a double integrator. A fractional derivative controller is used to shape the outer loop into the formof a fractional order integrator. The result is a constant phase system with, in the time domain, step

responses exhibiting constant overshoot, independent of variations in the load, and robust, in a stab-ility sense, to spillover effects. Experimental results are shown, when controlling the flexiblemanipulator with this fractional order derivator, that prove the good performance of the system.

1 Introduction

During the last two decades, a considerable interest hasbeen attracted to the control of lightweight flexible manip-ulators, which is becoming one of the most challengingresearch areas of robotic control. The necessity of thiskind of robots arises from new robotic applications thatrequire lighter robots that can be driven using smalleramounts of energy, such as the aerospace industry, whereweight has to be minimised, or mobile robotics, wherepower limitations imposed by battery autonomy have tobe taken into account. In addition, collisions of this typeof robots present remarkably less destructive effects thanthose caused by traditional robots, as the kinetic energy ofthe movement is transformed into potential energy of defor-mation at the moment of impact. This fact allows us toperform some control strategy over the actuator beforeany damage takes place or, at least, to minimise it, whichmay lead, in a not very far future, to a robot human

cooperation without the actual dangers in case of malfunc-tion, which is an emergent topic of research interest [1].

The challenge of controlling vibrations in flexible struc-tures, such as robotic manipulators, has been approachedwith very different methods, from classical schemes [2],to nonlinear methods such as sliding control [3] or neuralnetworks [4]. A recent published survey [5] gives a com-plete, detailed overview of all the work developed in this

field since the late 1970s. However, in spite of all theresearch devoted to modelling and controlling these kindof robots, there is no universal solution for the control,which is clearly demonstrated by the number of recentpapers presenting new improved solutions for vibrationcontrol.

The present work shows a new and simple controlscheme for single-link flexible arms with a variablepayload, based on the use of a fractional order derivativecontroller. This scheme uses measurement of the linkdeflection provided by a strain gauge placed at the base ofthe link. This sensorial system lets us construct controlschemes that are more robust than those based on acceler-ometer measurements (as was demonstrated in [6], too),being strain gauges simpler to instrument.

The general control scheme proposed in this paper con-sists of three nested loops (see Fig. 1):

1. An inner loop that controls the position of the motor.

This loop uses a classical proportional derivative (PD) con-troller to give a closed loop transfer function close to unity.2. A simplifying loop using positive unity-gain feedback.The purpose of this loop is to reduce the dynamics of thesystem to that of a double integrator.3. An outer loop that uses a fractional order derivative con-troller to shape the loop and to give an overshoot indepen-dent of payload changes.

In Fig. 1, um is the motor angle, ut the tip-position angle,Gb(s) the transfer function of the beam and Ri(s) and Re(s)the inner and outer loop controllers, respectively. Thedesign of the first two loops follows [7]. The fractional

order control strategy of the outer loop, which is based onthe operators of fractional calculus (see [8]), is presentedin this paper. A former use of this strategy was discussedin [9] considering a more ideal case and only simulation

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-cta:20060477

Paper first received 8th November 2006 and in revised form 14th February 2007

C.A. Monje and B.M. Vinagre are with the Industrial Engineering School,

University of Extremadura, Badajoz, SpainF. Ramos and V. Feliu are with the Superior Technical School of IndustrialEngineers, University of Castilla-La Mancha, Ciudad Real, Spain

E-mail: [email protected]

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results. The state of the art in fractional order control can befound in [1012].

For a better understanding of this work, it is organised asfollows. First, the physical model of the single-link flexiblemanipulator is presented in Section 2, followed by adescription of the general control scheme and the threenested loops in Section. 3 The design of the outer loop isexplained in detail in Section 4 and the effect of higher-order dynamics is discussed in Section 5. Next, Section 6presents the results obtained from the test of the controlstrategy in an experimental platform. Finally, some relevant

concluding remarks are drawn in Section 7.

2 Modelling of the single-link flexiblemanipulator

The single-link lumped mass model that will be used in thispaper is well known [6]. Particularly, a single tip mass thatcan rotate freely (no torque is produced at the tip) will beadopted for the description of the link dynamics [6]. Theeffect of the gravity is assumed negligible as the armmoves in a horizontal plane. The motor has a reductiongear with a reduction relation n. The magnitudes seenfrom the motor side of the gear will be written with anupper hat, whereas the magnitudes seen from the link side

will be denoted by standard letters.The dynamics of the link is described by

c(um ut) ml2 ut (1)where m is the mass at the end, landc are the length and thestiffness of the bar, respectively, um is the angle of the motorandut is the angle of the tip.

The dynamics of the motor with a closed-loop currentcontrol system (where the voltage V is proportional to thecurrent output) is given by

KV Jum n_um ^Gcoup ^GCoul (2)

where K is the motor constant, V the voltage signal that

controls the motor, Jthe motor inertia, n the viscous frictioncoefficient, ^GCoul the coupling torque between the motorand the link and ^GCoul the Coulomb friction. From nowon, we will suppose that the Coulomb friction is negligibleor is compensated by a term (see [13]) of the form

VCoul GCoulK

sign(_um) (3)

as shown in Fig. 2a, where ^GCoul is an estimation of theCoulomb friction value.

On the other hand, the coupling torque equation betweenthe motor and the link is

Gcoup c(um ut) (4)

and finally, the conversion equations u nu and ^G G=ncomplete the dynamic model.

Laplace transform is applied to (1) leading to the follow-ing transfer function

Gb(s) ut(s)

um(s) v

20

s2 v20(5)

where v0 is the natural frequency of the link v02

c/ml2

,which is mass-dependent.Combining all the previous equations, the transfer func-

tions of the robot are

um(s)

V(s) Kn(s

2 v20)s[Jn2s3 nn2s2 (Jn2v20 c)s nn2v20]

(6)

ut(s)

V(s) Knv

20

s[Jn2s3 nn2s2 (Jn2v20 c)s nn2v20](7)

It is evident that these robot equations are mass-dependent (so is v0) and therefore changes in the masswill affect the system behaviour.

3 General control scheme

The general control scheme is shown in Fig. 1, where it canbe seen that it is composed of three nested loops: inner loop,simplifying loop and outer loop, as commented previously.The features of the inner and outer loops have beenpreviously detailed in [6], whereas the simplifying loop isnow included to cope with the fractional order control strat-egy. Basically, this scheme allows us to design the loopsseparately, making the control problem simpler and mini-mising the effects of the inaccuracies in the estimation ofCoulomb and viscous frictions in control performance (as

shown in [13]).

Fig. 2 Block diagram of the loops

a Block diagram for the inner loopb Block diagram for the simplifying loop

Fig. 1 Proposed general control scheme

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3.1 Inner loop

The inner control loop, shown in Fig. 2a, fastens thedynamic behaviour of the motor. This purpose is achievedby means of a standard PD controller with proportional con-stant Kp and derivative constant Kv, which is tuned to makethe motor dynamics critically damped. The second-ordercritically damped expression obtained for the motor loop is

M(s) um(s)

urm(s) 1

(1 gs)2 (8)

where umr (s) is the reference angle for the motor and g

the motor dynamics constant, which is given byg ffiffi(pJ=KpK). Theoretically, it is possible to make themotor dynamics as fast as desired by simply making g 0,but a very demanding speed would saturate the motor, withthe subsequent malfunction of the controlled system. Thisfact implies that, although the motor dynamics can be madequite fast, we cannot consider it negligible in general.

Effects of this dynamics can be studied by dividing theoverall system into two time-scale subsystems: the fastdynamics subsystem defined by the inner control loop andthe slow dynamics subsystem defined by the flexible link.Then, interactions between both subsystems are studied.Singular perturbation techniques [14] define the generalframework to carry on this study. Application of these tech-niques to flexible arms can be found in [15], where the armdynamics is divided into a slow subsystem (which includesmotor dynamics) and a fast subsystem (which includeshigh-order vibration modes). A recent work based on thisapproach is presented in [16], where the dynamics of thehighest vibration modes are neglected in the controllerdesign but the maximum possible gain in the L2 sense ofthese removed dynamics is taken into account in such adesign. It is possible to modify these techniques in order

to cope with the flexible arm control problem defined inthis paper. In this sense, the effects of the servocontrolledmotor fast dynamics on the slow dynamics defined bythe flexible link and the outer controller can be quantifiedand used to adequately tune the outer controller. But themain purpose of this paper is to show the feasibility of arobust fractional controller designed by using a verysimple technique that does not take into account this fastdynamics. We will show in Section 4.5 that the effects ofthe inner loop fast dynamics can be reduced by adding aproportional term (tuned easily by simulations) to the frac-tional controller designed with our robust control technique.Then, a more elaborate analysis is not needed to achieveacceptable results with our arm.

Tables 1 and2 show the parameters of the motor-gear setand the flexible link used for the experimental tests,respectively.

It is important to remark that the coupling torque is com-pensated within the inner loop by a term of the form

Vcoup 1

KnGcoup (9)

Previous experimental works have proven the correctnessof this assumption in direct driven motors, and motors withreduction gears as well [7]. It has also been demonstratedthat, in the case of motors with gears, the effect of the coup-ling torque is very small compared with the motor inertia

and friction, as its value is divided by n [6].

3.2 Simplifying loop

As commented previously, the response of the inner loop(position control of the motor) is significantly faster thanthe response of the outer loop (position control of the tip).The motor position is first supposed to track the referenceposition with negligible error and the motor dynamics willbe considered later. That is, the dynamics of the innerloop can be approximated by 1 when designing the outerloop controller. Taking this into account, a strategy for sim-plifying the dynamics of the arm, shown in Fig. 2b, is

proposed.For the case of a beam with only one vibrational mode, asimplifying loop can be implemented that reduces thedynamics of the system to a double integrator by simplyclosing a positive unity-gain feedback loop around the tipposition (b 1). Then, the equation relating the outputand input of the loop is

ut(s) v20s2

u(s) 1s2

P(s) (10)

where P(s) represents disturbances with the form of a first-order polynomial in s, which models initial deviations in tipposition and tip velocity [6]. In (10), the dynamics of the

arm has been reduced to a double integrator dynamics, sim-plifying the control strategy proposed in this work, as willbe seen later.

The stability study by using Nyquist diagrams shows thatthe condition b 1 is not critical to obtain stable controlsystems, being sufficient to implement a feedback gainclose to 1.

3.3 Outer loop

The block diagram for the outer loop used in this work isshown in Fig. 3. As it is observed in the scheme, an esti-mation of the tip position, ut

e, is used to close the loop.We actually feed back the deformation measurements ofthe two strain gauges, placed at the base of the link in a half-bridge, 2-active-gauges configuration, to control the arm.These sensors provide the value of the coupling torque

Table 1: Data of the motor-gear set

g Kp Kv J, (kg m2) n, (N m s) K, (N m/V) n

0.022 1 0.025 24.24 . 1024 51.66 . 1024 3.399 50

Table 2: Data of the flexible link

c, (N m) m, (kg) l, (m) r, (m) E, (GPa) I, (m4) Ks

443.597 1.9 0.866 0.008 68.9 1.86e-9 2.11

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Gcoup between the arm and the motor by means ofexpression

Gcoup 1

2

2:00

Ks

EI

re (11)

where Ks is the gauge factor provided by the manufacturer,r the outer beam radius, E is Youngs modulus of the beammaterial and I the cross-section inertia of the beam. Thesevalues can be found in Table 2. On the other hand, e rep-resents the deformation measured by the strain gauges,which is proportional to a voltage signal and calibratedwith a dynamic strain amplifier. The 1=2 gain is due tothe chosen configuration, which doubles the sensed valuewhile remains insensitive to temperature variations andcancels the compressive/tensile strain.

Coupling torque is used to decouple motor and linkdynamics and estimate tip position [6]. Combining (1), (4)and the fundamental frequency definition, we can obtainthe relation between Gcoup and the output um, which yields

GG(s) cs2

s2 v20(12)

Given the values of um and Gcoup, experimentallymeasured, the value of ut can be estimated according to

(4) by ute

um2

Gcoup/c. This estimated tip angle is usedto close the control loop, as shown in Fig. 3, whereH(s) 21/c.

The main purpose of this work is to design the outer con-trollerRe(s) (see Fig. 3) so that the time response of the con-trolled system has an overshoot independent of the tip massand the effects of disturbances are removed. This will leadto the use of a fractional derivative controller, as will bedetailed in the next section.

Besides, in this particular case, the outer controller willbe designed in the frequency domain for the specificationsof phase margin (damping of the response) and crossoverfrequency (speed of the response). In order to guarantee acritically damped response (overshoot Mp 0), a phase

margin wm 76.58 is selected. Besides, the response isdesired to have a rise time around 0.3s, so the crossover fre-quency is fixed to vcg 6 rad/s. The crossover frequencydefines the speed of response of the closed-loop system.The practical constraint that limits the speed of responseof the arm, and hence the value of vcg, is the maximumtorque provided by the motor. This maximum torquelimits the speed of response of the inner motor loop andhence its bandwidth. The control scheme proposed hereworks ideally if the dynamics of the inner loop is negligible.Consequently, its bandwidth must be much larger than thedesired crossover frequency vcg. However, it must betaken into account that the experimental results to be pre-

sented in this work show the behaviour of the arm assumingnon-negligible inner loop dynamics, since the value of thetorque provided by the motor limits the speed of responseof this loop. This fact may change slightly the final

frequency specifications found for the system, as will beshown later.

4 Design of the outer loop controller Re(s)

With the inner and simplifying loops closed, the reduceddiagram of Fig. 4 is obtained, which is based on (10).From this diagram, the expression for the tip position is

ut(s) 1

1

s2=Re

(s)v2

0

urt (s) 1

1

s2=Re

(s)v2

0

P(s)

Re

(s)v2

0(13)

The controllerRe(s) has a 2-fold purpose. One objective isto obtain a constant phase margin in the frequency response;in other words, a constant overshoot in time response to astep reference for varying payloads. The other is to removethe effects of the disturbance, represented by the initial con-ditions polynomial, on the steady state. To attain these objec-tives, most authors propose the use of some kind of adaptivecontrol scheme (see [7]). We propose here a fractional orderderivative controller with enhanced robustness properties toachieve the above two objectives, without needing anykind of adaptive algorithm. Some methods have been devel-

oped in the last years to tune fractional propotional integra-tive derivative (PID) controllers with robustness propertiesto changes in one parameter (typically the plant gain) [17,18]. These methods are based on optimisation procedures.Our approach is much simpler as it is specifically tailoredto our particular arm dynamics, leading to very straightfor-ward tuning rules.

4.1 Condition for constant phase margin

The condition for a constant phase margin can beexpressed as

arg Re(jv)v2

0(jv)2

constant 8v (14)and the resulting phase margin wm is

wm arg[Re(jv)] (15)For a constant phase margin 0 , wm , p/2, the control-

ler that achieves this must be of the form

Re(s) Kcsa, a 2

pwm (16)

so that 0 , a , 1. This Re(s) is a fractional derivative con-troller of ordera. The two definitions used for the general

fractional integro-differential operation are the GrunwaldLetnikov (GL) definition and the RiemannLiouville (RL)definition [8]. The GL definition is

aDat f(t) lim

h0haX[tah ]

j0(1)j a

j

f(tjh) (17)

Fig. 4 Reduced diagram for the outer loop

Fig. 3 Basic scheme of the outer control loop


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where [.] means the integer part, whereas the RLdefinition is

aDat f(t)

1

G(n a)dn

dtn

ta

f(t)

(t t)an1 dt (18)

for (n2 1 , a , n) and where G(.) is the Eulers gammafunction.

For convenience, Laplace domain notion is usually usedto describe the fractional integro-differential operation. TheLaplace transform of the RL fractional derivative/integral(18) under zero initial conditions for order a (0 , a , 1)is given by

L{aD+at f(t)} s+aF(s) (19)Taking this notation into account, Re(s) corresponds to a

fractional derivative controller of ordera.

4.2 Condition for removing the effects ofdisturbance

From the final value theorem, the condition to remove theeffects of the disturbance is

lims0

11 (s2=Re(s)v20)

P(s)Re(s)v

20

0 (20)

Substituting Re(s) Kcsa and P(s) as b (initial tip

position and velocity errors different from zero), this con-dition becomes

lims0

1

1 (s2a=Kcv20)b

Kcv20

sa 0 (21)

lims0

1

1 (s2a=Kcv20)a

Kcv20

s1a 0 (22)

which implies that a , 1.

4.3 Ideal response to a step command

Assuming that the dynamics of the inner loop can beapproximated by unity and that disturbances are absent,the closed-loop transfer function with controller (16) is

Fcl(s) uturt 1

1 (s2=Re(s)v20) Kcv

20

s2a Kcv20(23)

which exhibits the form of Bodes ideal loop transfer func-tion [19]. The corresponding step response is

ut(t) L

1 Kcv20

s(s2a Kcv20) Kcv20t2aE2a,3a(Kcv20t2a) (24)

where Ed,d1(2Atd) is the two-parameter MittagLeffler

function [8]. The overshoot is fixed by 22 a, which is inde-pendent of the payload, and the speed by Kcv0

2, that is, bythe payload and the controller gain. In fact, notice thatthis expression can be normalised with respect to time by

ut(tn) t2an E2a,3a( t2an ) (25)where tn t(Kcv0

2)1/(22a). This equation shows that theeffect of a change in the payload implies a change in v0

that only means a time scaling of the response ut(t).To obtain a required step response, it is then necessary toselect the values of two parameters. The first one is the ordera to adjust the overshoot between 0 (a 1) and 1 (a 0),

or, equivalently, a phase margin between 908 and 08. Thesecond one is the gain Kc to adjust the crossover frequency,or, equivalently, the speed of the response for a nominalpayload. Note that increasing a decreases the overshootbut increases the time required to correct the disturbanceeffects (see [20]).

4.4 Controller design

As commented above, the design of the controller thus

involves the selection of two parameters:

a, the order of the derivative, which determines: (a) theovershoot of the step response, (b) the phase margin or (c)the damping. Kc, the controller gain, which determines for a given a:(a) the speed of the step response or (b) the crossoverfrequency.

These parameters can be selected by working in thecomplex plane, the frequency domain or the time domain.In the frequency domain, the selection of the parametersof the fractional order derivative controller can be regardedas choosing a fixed phase margin by selecting a, and choos-

ing a crossover frequency vcg, by selecting Kc for a given a.That is

a 2pwm, Kc

vcg

v20(26)

According to Table 2, where the parameters of the flex-ible manipulator are presented, the fundamental frequencyof the system is v0 17.7 rad/s. The frequency specifica-tions required for the controlled system, commented pre-viously, are: phase margin wm 76.58, and crossoverfrequency aroundvcg 6 rad/s. Therefore the parametersof the fractional derivative controller are a 0.85 and

Kc 0.02. With this controller, and under the assumption

of negligible inner loop dynamics, the Bode plots obtainedfor the open-loop system are shown in Fig. 5a, where it canbe observed that at the crossover frequency vcg 6 rad/sthe phase margin is wm 76.58, fulfilling the designspecifications.

The simulated step responses of the controlled system form 0.6, 1.9, 3.2 and 6 kg are shown in Fig. 5b. I t isobserved that the overshoot of the response remains con-stant to payload changes, being Mp 0, fulfilling therobustness purpose. For the nominal mass (m 1.9 kg), arise time tr 0.3 s is obtained.

This controller has been implemented as described inSection 4.6, except for the constant k 0.25, which hasbeen introduced later to compensate the effects of the non-negligible inner loop dynamics, as explained next.

4.5 Effect of the non-negligible inner loopdynamics

In the practical case presented in this work, the dynamics ofthe inner loop is not negligible, being given by the transferfunction

M(s) um(s)urm(s)

1(1 gs)2 (27)

with g 0.022. Notice that M(s) is independent of the value

of the tip payload as its effects on the motor dynamics areremoved by the compensation term Vcoup in (9) based onthe measurement of the motor-beam coupling torque. Theintroduction of M(s) implies that the response of the


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controlled system will be affected by this dynamics, as thesimplifying loop does not result in a double integrator

anymore.Besides, it is important to remark that step inputs are notvery appropriate for robotic systems, being more suitablethe use of smoother references to avoid surpassing the phys-ical limitations of the robot, such as the maximum torqueallowed to the links before reaching the elastic limit orthe maximum feasible control signal value (V for a DCmotor-amplifier set). For that reason, a fourth order poly-nomial reference ut

r has been used in our case.It has been observed that with the introduction of M(s),

the settling time of the response gets longer. To reduce it,a proportional part k is introduced in the controller tomake the output converge faster to its reference.However, it must be taken into account that the introduction

of this constant affects the frequency response of thesystem, changing the specifications. Therefore there mustbe a trade-off between the fulfillment of the frequency spe-cifications and the settling time required, resulting k 0.25in our case. Then, the final controller is

Re(s) 0:25 0:02s0:85 (28)

Only a slight modification of the frequency specificationsis obtained with this controller, resulting vcg 6.6 rad/sand wm 708. The time responses of the system forpayload changes are shown in Fig. 6. Masses greater than3.2 kg have not been considered as they could cause the

beam to reach its elastic limit and, hence, they will beneither simulated nor experimented. For the nominalmass, an overshoot Mp 0% is obtained. As far as therobustness is concerned, a slight change in the overshootof the response appears when the payload changes, due tothe effect of the non-negligible inner loop dynamics.However, only a 0.59% variation in the overshoot isobtained for the different masses.

4.6 Fractional order controller implementation

No physical devices are available to perform the fractionalderivatives, and then approximations are needed to imple-

ment fractional controllers. These approximate implemen-tations of fractional order controller (FOC) can beclassified into either continuous [2123] or discretemethods [22, 24]. In this particular case, an indirect

discretisation method is used. That is, first a finite-dimensional continuous approximation is obtained and sec-

ondly the resulting s-transfer function is discretised.It must be taken into account that the fractional derivatives0.85 has been implemented as s0.85 s . s20.15 (s1/s0.15),that is, an integer derivative plus a fractional integrator. Thisway, the resulting open-loop system in the ideal case wouldbe Re(s)(v0

2/s2) (v02/s)s20.15, guaranteeing the cancella-

tion of the steady-state position error due to the effect ofthe pure (integer) integral part. Therefore only the fractionalpart Rd(s) s

20.15 has been approximated.To obtain a finite-dimensional continuous approximation

of the fractional integrator, a frequency domain identifi-cation technique is used, provided by the Matlab functioninvfreqs. An integer order transfer function that fits thefrequency response of the fractional order integrator Rd in

the range v[ (1022, 102) is obtained. Later, the discretisa-tion of this continuous approximation is made by using theTustin rule with prewarp frequency vcg and sample periodTs 0.002 s, obtaining a fifth-order digital infiniteimpulse response (IIR) filter

Rd(z) 0:1124z5 0:7740z4 2:0182z3 2:5363z2 1:5523z1 0:3725

0:4332z5 2:6488z4 6:3441z3 7:4747z2 4:3462z1 1

(29)

Fig. 6 Time responses of the system with controller Re(s), con-sidering non-negligible inner loop dynamics

Fig. 5 Bode plots and simulated responses with fractional order derivative

a Bode plots of the open-loop system with the fractional order derivative, considering negligible inner loop dynamicsb Simulated time responses of the system with the fractional order derivator to a step input for different payloads, considering negligible inner loopdynamics


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Therefore the resulting total fractional order controller isa sixth-order digital IIR filter given by

Re(z) 0:25 0:021 z1

Ts

Rd(z) (30)

5 Robustness to higher vibration modes

This section studies the robustness of the developed frac-tional controller to non-modelled higher vibration modes.

These modes can influence the closed-loop system in twoways: (1) they are fed back to the controller and, if theywere not taken into account in the controller design, theglobal system can become unstable and (2) the estimatorof the tip position based on expression (4) no longerremains correct, exhibiting high-frequency estimationerrors that are fed back to the closed-loop system. Inorder to avoid these destabilising effects, we propose alemma based on the next dynamic property.

Let us consider the transfer function GG(s) between themotor angle and the motor-beam coupling torque (theother measured variable). Then we say that this transferfunction exhibits the interlacing property of the poles andzeros on the imaginary axis if it verifies that

GG(s) s2(s2 421) (s2 42i )

(s2 42n)(s2 v20)(s2 v21) (s2 v2i )

(s2 v2n)(31)

where vi21 ,4i , vi, 1 i n, andc . 0. This propertyis verified by uniform single-link flexible manipulators withdistributed mass and a payload at the tip as illustrated next.

The governing equation of a flexible link (EulerBernoulli equation) can be normalised by definingT ffiffi(prl4=EI), where r is the mass per unit length andintroducing the dimensionless time tn t/T.

Consequently, the tip payload is also normalised withrespect to the beam mass: mn rl/m. Transfer functionsGG(s) are obtained for the normalised beam for differentmn ratios and the poles and zeros associated to the first sixmodes are calculated. We assume that modelling sixmodes is enough to study the spillover effects in most flex-ible manipulators. Fig. 7 shows the values of these poles

and zeros for mass ratios ranging from negligible linkmass (mn 0.01) to the case of a link mass 10 times largerthan the tip payload (mn 10). This figure shows that theaforementioned interlacing property is verified by anyuniform beam at least in the specified range of variation ofthe link mass.

The following lemma proves that if this property is veri-fied, the controller proposed in the previous section is robustto non-modelled higher vibration modes (spillover).

Lemma 1: Assume that our flexible arm verifies the interla-cing property (31) and that the inner loop dynamics is neg-ligible (M(s) 1). Then any outer loop controller ofthe form

Re(s) kKcsa, Kc . 0, 0 a 1 (32)combined with a tip position estimator of the form given by

uet um Gcoup

c(33)

Fig. 7 Interlacing property between poles (dashed lines) andzeroes (solid lines) of transfer function GG(s) is numericallydemonstrated for a wide range of beam masses

Fig. 8 Nyquist plots used in the study of spillover

a GG(s) proves to be marginally stable when interlacing property is fulfilledb Addition of the controller Re(s) and the simplifying loop subtracts phase, achieving a stable behaviour for controlled system


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where parameter c verifies

c ! c (34)keeps stable the closed loop system.

Proof: Fig. 3 shows the block diagram of the outer controlloop. Operating this block diagram, we obtain the equival-ent transfer function

Gcoup(s)

Qrt (s) GG(s)

1 1=cGG(s)(1 R1e (s)) (35)If GG(s) verifies the above interlacing property, the alter-

nation between poles and zeros of expression (31) producesa Nyquist diagram of GG(jv) of the form shown in Fig. 8a.It exhibits as many half-turns in the infinity as vibrationmodes has the transfer function [as many as terms(s2 vi2) are in the denominator of this transfer function].This plot shows that the closed-loop system associated toGG(s) is marginally stable. The product 1=c(1 R1e (s))subtracts phase to the system from zero, when frequencyis very small, to 1808, when frequency tends to infinity,hence progressively rotating the Nyquist of GG(jv), but

never crossing the negative x-axis, as Fig. 8b shows for athree vibrational modes example. In order to guarantee

that the Nyquist plot does not embrace the point (21,0),it must be verified that

limv1

1

cGG(s)(1 R1e (s)) ! 1 (36)

Assuming that Re(s) is of the form (32) and taking intoaccount (31), it easily follows that condition (36) becomesc ! c, and expression (34) is proven.

In addition, if Re(s) is of the form (32), after some oper-

ations we have that

j(jv) 1c

(1 R1e ( jv))

k(1 k) Kcva(1 2k)cos(p=2)aK2cv2

jKcva sinp=2ac(k2 2Kckva cos(p=2)aK2cv2a)

(37)

The imaginary component of this expression is negative8v! 0 provided that 0 a 1 and K! 0. Then/j(jv) 0 8v! 0, and it subtracts phase from GG(jv)at all frequencies, as the Nyquist stability conditionrequires. A

Fig. 9 Photo of the experimental platform

Fig. 10 Experimental results obtained using controller Re(s) for m 3.2 kg

a Experimental tip angle ut and motor angle um obtainedb Comparison between the experimental and simulated motor voltage V


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Remark: Conditions (32) (34) make the closed-loopsystem stable for any single-link flexible arm that fulfilsthe interlacing property (31), independent of the numberof high-frequency modes considered.

6 Experimental results

The control strategy proposed here, with the use of the outerloop controller in (30), has been tested in the experimentalplatform of the picture in Fig. 9, whose dynamics corre-

sponds to the one described previously for the single-linkflexible manipulator. In this section, the experimentalresults obtained are presented. The robustness of thesystem has been tested by changing the payload at the tip.Motor and tip position records are shown. Simulatedcontrol signals are plotted together with the experimentalmotor control signals for comparison purposes. Note thatthe simulations neglect Coulomb friction, whereas in theexperimental platform a compensation term, 0.3 V forpositive motor velocities and20.25 V for negative ones,has been added to the control signal. These compensationvalues have been found by a trial-and-error process.

Fig. 10a shows the measurements of the tip angle ut and

the motor angle um, whereas Fig. 10b shows the motorvoltage V, both figures corresponding to a massm 3.2 kg. A relay type control appears in the transientand steady states due to Coulomb friction compensation.

For this reason, the experimental voltage signal obtainedpresents quick oscillations and is not zero in the steady state.

Fig.11 shows themeasurements of the tipand motor anglesand motor voltage obtained for the nominal mass m 1.9 kg.And finally, Fig. 12 shows the results when m 0.6 kg.

Through Figs. 10b, 11b and 12b, it can be observed thatthe peak of the control signal keeps lower than the satur-ation limit and remains almost constant in the presence ofpayload changes, with a value around 0.65 V, making thiscontrol strategy very suitable for motor saturation problems.

Another important aspect to remark is that the fractionalderivator part of the controller, s0.85, is implemented by

s0.85 s(1/s0.15). That is, the fractional integrator partacts like a low-pass filter of the signal that enters the deriva-tive operator and reduces the noise introduced through thecontrol loop. Therefore with the fractional controller, thesystem is not only more robust to payload changes, butalso to noise presence.

7 Conclusions

A new method to control single-link lightweight flexiblearms in the presence of payload changes has been presented

in this work. The overall controller consists of three nestedcontrol loops. Once the Coulomb friction and the motor-beam coupling torque have been compensated, the innerloop is designed to give a fast motor response. The






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simplifying loop reduces the system transfer function to adouble integrator. The fractional order derivative controlleris used to shape the outer loop into the form of a fractionalorder integrator. The result is an open-loop constant-phasesystem whose closed loop responses to a step commandexhibit constant overshoot, independent of variations inthe load. A study of the effect of spillover has beencarried out, where system stability to any non-modelledhigher-order dynamics has been proven.

The fractional order controller has been tested in an exper-

imental platform by using discrete implementations. Fromthe results obtained, it can be concluded that an interestingfeature of the fractional control scheme is that the overshootis independent of the tip mass. This allows a constant safetyzone to be delimited for any given placement task of the arm,independent of the load being carried, thereby making iteasier to plan collision avoidance. It must be remarked thatwith the fractional order controller, the control signal isless noisy than with a standard PD controller, as the frac-tional integrator acts like a low-pass filter and reduces theeffects of the noise introduced in the control loop. Besides,with this control strategy, changes in the payload implyonly slight variations in the maximum value of the controlsignal, avoiding possible saturation issues.

8 Acknowledgments

This work was financially supported by the SpanishGovernment Research Program via ProjectDPI-2003-03326 (M.E.C.) and by the Spanish ResearchGrant 2PR02A024 of the Junta de Extremadura.

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