[ieee 2009 6th international symposium on mechatronics and its applications (isma) - sharjah, united...

Proceeding o/the 6th International Symposium on Mechatronics and its Applications (ISMA09), Sharjah, UAE, March 24-26,2009

TECHNIQUES OF VIBRATION FEEDBACK CONTROL OF A FLEXIBLEROBOT MANIPULATOR

Mohd AshrafAhmad

Universiti Malaysia PahangFaculty of Electrical and

Electronics Engineering, LockedBag 12, 25000, Kuantan,

Pahang, [email protected]

Ahmad Nor Kasruddin Nasir

Universiti Malaysia PahangFaculty of Electrical and

Electronics Engineering, LockedBag 12, 25000, Kuantan,

Pahang, [email protected]

Najidah Hambali

Universiti Malaysia PahangFaculty of Electrical andElectronics Engineering,Locked Bag 12, 25000,

Kuantan, Pahang, [email protected]

ABSTRACT

This paper presents the use of angular position control approachesfor a flexible robot manipulator with disturbances effect in thedynamic system. Delayed Feedback Signal (DFS) andproportional-derivative (PD)-type fuzzy logic controller are thetechniques used in this investigation to actively control thevibrations of flexible structure. A constrained planar single-linkflexible manipulator is considered and the dynamic model of thesystem is derived using the assume mode method. A completeanalysis of simulation results for each technique is presented intime domain and frequency domain respectively. Performances ofboth controllers are examined in terms of vibration suppressionand disturbances cancellation. Finally, a comparative assessmentof the impact of each controller on the system performance ispresented and discussed.

Keywords: Flexible robot manipulator, vibration control, delayedfeedback signal, PD-type Fuzzy Logic controller.

1. INTRODUCTION

Flexible robot manipulators exhibit many advantages over therigid link manipulators as they require less material, are lighter inweight, have higher manipulation speed, lower powerconsumption, require smaller actuators, are more manoeuvrableand transportable, have less overall cost and higher payload torobot weight ratio [1]. However, the control of flexiblemanipulators to maintain accurate positioning is a challengingproblem. A flexible manipulator is a distributed parameter systemand has infinitely many degrees of freedom. Moreover, thedynamics are highly non-linear and complex. Problems arise dueto precise positioning requirements, system flexibility leading tovibration, the difficulty in obtaining accurate model of the systemand non-minimum phase characteristics of the system [2]. Toattain end-point positional accuracy, a control mechanism thataccounts for both the rigid body and flexural motions of thesystem is required. If the advantages associated with lightness arenot to be sacrificed, precise models and efficient controlstrategies for flexible manipulators have to be developed.

Research on the control methods that will eliminate vibrationfrom wide range of physical systems has found a great deal ofinterest for many years. The methods used to solve the problems

arising due to unwanted structural vibrations include passive andactive control. The passive control method consists of mountingpassive material on the structure in order to change its dynamiccharacteristics such as stiffness and damping coefficient. Thismethod is efficient at high frequencies but expensive and bulky atlow frequencies. Active vibration control consists of artificiallygenerating sources that absorb the energy caused by the unwantedvibrations in order to cancel or reduce their effect on the overallsystem. Lueg in 1930 [3], is among the first who used activevibration control in order to cancel noise vibration.

The control strategies for flexible manipulator systems can beclassified as feedforward and feedback control. A number oftechniques have been proposed as feedforward control strategiesfor control of vibration. These include the development ofcomputed torque based on a dynamic model of the system [4],utilization of single and multiple-switch bang-bang controlfunctions [5], construction of input functions from rampedsinusoids or versine functions [6]. Moreover, feedforward controlschemes with command shaping techniques have also beeninvestigated in reducing system vibration. These include filteringtechniques based on low-pass, band-stop and notch filters [7] andinput shaping [8]. On the other hand, several approaches utilizingclosed-loop control strategies have been reported for control offlexible manipulators. These include linear state feedback control[9], adaptive control [10], robust control techniques based on Hinfinity [11] and intelligent control based on neural networks [12]and fuzzy logic control schemes [13].

This paper presents investigations of angular position controlapproach in order to eliminate the effect of disturbances applied tothe single-link flexible robot manipulator. A simulationenvironment is developed within Simulink and Matlab forevaluation of the control strategies. In this work, the dynamicmodel of the flexible manipulator is derived using the assumemode method (AMM). To demonstrate the effectiveness of theproposed control strategy, the disturbances effect is applied at thetip of the flexible link. This is then extended to develop afeedback control strategy for vibration reduction and disturbancesrejection. Two feedback control strategies which are DFS and PDtype fuzzy logic controller are developed in this simulation work.Performances of each controller are examined in terms ofvibration suppression and disturbances rejection. Finally, acomparative assessment of the impact of each controller on thesystem performance is presented and discussed.

ISMA'09-1

978-1-4244-3481-7/09/$25.00 ©2009 IEEE

Proceeding ofthe 6th International Symposium on Mechatronics and its Applications (ISMA09), Sharjah, UAE, March 24-26,2009

This expression states the internal energy due to the elasticdeformation of the link as it bends. The potential energy due togravity is not accounted for since only motion in the planeperpendicular to the gravitational field is considered.

To obtain a closed-form dynamic model of the manipulator,the energy expressions in (1) and (2) are used to formulate theLagrangian L =T - U. Assembling the mass and stiffnessmatrices and utilizing the Euler-Lagrange equation of motion, thedynamic equation of motion of the flexible manipulator systemcan be obtained as

2. THE FLEXIBLE ROBOT MANIPULATOR SYSTEM

The single-link flexible robot manipulator system considered inthis work is shown in Figure 1, where XoOYo and XOY representthe stationary and moving coordinates frames respectively, T

represents the applied torque at the hub. E, L p, A, Ih, v(x, t) and

8(t) represent the Young modulus, area moment of inertia, massdensity per unit volume, cross-sectional area, hub inertia,displacement and hub angle of the manipulator respectively. Inthis study, an aluminium type flexible manipulator of dimensions900 x 19.008 x 3.2004 mm3

, E= 71 x 109 N/m2, 1= 5.1924 x lOll

m4 ,p = 2710 kg/m3 and IH = 5.8598 x 10-4 kgm2 is considered... .

M Q(t) + D Q(t) + KQ(t) = F(t) (3)

Here, qn is the modal amplitude of the i th clamped-free mode

considered in the assumed modes method procedure and nrepresents the total number of assumed modes. The model of theuncontrolled system can be represented in a state-space form as

where M, D and K are global mass, damping and stiffnessmatrices of the manipulator respectively. The damping matrix isobtained by assuming the manipulator exhibit the characteristic ofRayleigh damping. F(t) is a vector of external forces and Q(t) is amodal displacement vector given as

(5)

(4)

F (t) = [1" 0 0 ... 0]T

R . Hh(IH, r)

Rexiblelirk ( p, E, I, L)

Figure 1. Description ofthe flexible robot manipulator system.

3. MODELLING OF THE FLEXIBLE ROBOTMANIPULATOR

•x=Ax+Bu

y=Cx(6)

This section provides a brief description on the modeling of theflexible robot manipulator system, as a basis of a simulationenvironment for development and assessment of the input shapingcontrol techniques. The assume mode method with two modaldisplacement is considered in characterizing the dynamic behaviorof the manipulator incorporating structural damping. The dynamicmodel has been validated with experimental exercises where aclose agreement between both theoretical and experimental resultshas been achieved [1].

Considering revolute joints and motion of the manipulator ona two-dimensional plane, the kinetic energy of the system can thusbe formulated as

with the vector x = [0 iJ ql q2 ql q2]T and the matrices A and

B are given by

(7)

D= [0]

4. CONTROLLER DESIGN

where I b is the beam rotation inertia about the origin 0 0 as if it

were rigid. The potential energy of the beam can be formulated as

L1 '2 1 J 2 .T =-(IH +Ib )8 +-p (v +2vx8)d:x2 2

o

(1) In this section, two feedback control strategies (DFS and PD-typefuzzy logic controller) are proposed and described in detail. Themain objective of the feedback controller in this study is tomaintain the angular position of flexible manipulator whilesuppressing the vibration due to disturbances effect. All thefeedback control strategies are incorporated in the closed-loopsystem in order to eliminate the effect of disturbances.

L 2

U =.!. fEll a2

vJdx2 ax2

o(2)

4.1. Delayed Feedback Signal Controller

In this section, the control signal is calculated based on thedelayed position feedback approach described in (8) andillustrated by the block diagram shown in Figure 2.

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By setting the real and imaginary parts of Equation (13) to zero,the equation is rearranged as below

(14)

where P=WT.Solving for sin Pand cos Pgives

+

Delayed by t

Figure 2. The delayed/eedback signal controller structure.The critical values of time delay can be determined as follows: if apositive root of Equation (13) exists, the corresponding time delayT can be found by

Substituting Equation (8) into Equation (6) and taking the Laplacetransform gives

The stability of the system given in (9) depends on the roots of thecharacteristic equation

Equation (10) is transcendental and results in an infinite numberof characteristic roots. Several approaches dealing with solvingretarded differential equations have been widely explored. In thisstudy, the approach described in [14] will be used on determiningthe critical values of the time delay T that result in characteristicroots of crossing the imaginary axes. This approach suggests thatEquation (10) can be written in the form

4.2. PD-type Fuzzy Logic Controller

(15)P 2k1!

'ik =-+--OJ OJ

uDFS (t) = -55(8(t) - 8(t - 0.005))

where f3 E [0 21l]. At these time delays, the root loci of the

closed-loop system are crossing the imaginary axis of the s-plane.This crossing can be from stable to unstable or from unstable tostable. In order to investigate the above method further, the timedelayed feedback controller is applied to the single-link flexiblemanipulator. Practically, the control signal for the DFS controllerrequires only one position sensor and uses only the current outputof this sensor and the output T second in past. There is only twocontrol parameter: k and T that needs to be set. Using the stabilityanalysis described in [15], the gain and time-delayed of the systemis set at k=55 and T=0.005. The control signal of DFS controllercan be written as below

(9)

(8)

(10)Ll(s, T) =1sI - A +kBC(I-e-ST) 1= 0

sIx(s) = Ax(s) - kBC(I- e-ST )x(s)

u(t) = k(y(t) - y(t - T))

Ll(jOJ, T) =PR(OJ) + jPj (OJ)

+ (QR (liJ) + jQ/ (liJ))(eos(liJ1') - j sineliJT)) (12)

The characteristic equation Ll(s, T) = 0 has roots on the imaginary

axis for some values of T ~ 0 if Equation (12) has positive realroots. A solution of Ll(jOJ, T) = 0 exists if the

magnitude 1Ll(jOJ, T) 1= O. Taking the square of the magnitude of

Ll(jOJ, T) and setting it to zero lead to the following equation

P(s) and Q(s) are polynomials in s with real coefficients anddeg(P(s)=n>deg(Q(s)) where n is the order of the system. In orderto find the critical time delay 't that leads to marginal stability, thecharacteristic equation is evaluated at s=jw. Separating thepolynomials P(s) and Q(s) into real and imaginary parts andreplacing e-}OJi by cos(wT)-jsin(wT), Equation (11) can be written as

A PD-type fuzzy logic controller utilizing hub angle and hubvelocity feedback is developed to control the rigid body motion ofthe system [16,17]. The hybrid fuzzy control system proposed in

this work is shown in Figure 3, where Band iJ are the hub angleand hub velocity of the flexible robot manipulator, whereas kJ, k2

and k3 are scaling factors for two inputs and one output of thefuzzy logic controller used with the normalised universe ofdiscourse for the fuzzy membership functions.

In this paper, the hub velocity is measured from the systeminstead of deriving it with the equation above. Triangularmembership functions are chosen for hub angle, hub velocity, andtorque input with 50% overlap. Normalized universes of discourseare used for both hub angle and velocity and output torque.Scaling factors k] and k2 are chosen in such a way as to convertthe two inputs within the universe of discourse and activate therule base effectively, whereas k3 is selected such that it activatesthe system to generate the desired output. Initially all these scalingfactors are chosen based on trial and error. To construct a rulebase, the hub angle, hub velocity, and torque input are partitionedinto five primary fuzzy sets as

(13)

(11)Ll(S,T) = P(s)+Q(s)e-ST

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Figure 4. Control surface ofthe Fuzzy Logic controller.

o

Hub angle

-0.5

_ -..... 0.5

-1 -1

.....- -........ ..-"- ..-..-"- ..................-.-................ .- '>-...:........

-0.5

o

Hub \elocity

0.5

Q)

E- 0o~

eCt

eCt)

Hub angle A = {NM NS ZE PS PM},Hub velocity V= {NM NS ZE PS PM},Torque U = {NM NS ZE PS PM},

where A, V, and U are the universes of discourse for hub angle,hub velocity and torque input, respectively. The nth rule of therule base for the FLC, with angle and angular velocity as inputs, isgiven by

Rn: IF( B is Ai) AND (iJ is Vj) THEN (u is Uk),where, Rm n=1, 2, ...Nmax, is the nth fuzzy rule, Ai, Vj, and Uk, fori,j, k = 1,2,... ,5, are the primary fuzzy sets.

FuzzyLogic

Controller

Figure 3. The PD-type Fuzzy Logic controller structure.

A PD-type fuzzy logic controller was designed with 11 rulesas a closed loop component of the control strategy for maintainingthe angular position of flexible manipulator while suppressing thevibration due to disturbances effect. The rule base was extractedbased on underdamped system response and is shown in Table 1.The control surface is shown in Figure 4. The three scalingfactors, kJ, k2 and k3 were chosen heuristically to achieve asatisfactory set of time domain parameters. These values wererecorded as, k] = 1.02 k2 = 0.30 and k3 = 1.0.

Table 1. Linguistic rules ofthe Fuzzy Logic Controller.

No. Rules

5. SIMULATION RESULTS

In this section, the proposed control schemes are implemented andtested within the simulation environment of the flexiblemanipulator and the corresponding results are presented. Thecontrol strategies were designed by undertaking a computersimulation using the fourth-order Runge-Kutta integration methodat a sampling frequency of 1 kHz. The system responses namelymodal displacement and its corresponding power spectral density(PSD) are obtained. In all simulations, the initial

condition Xo =[0 0 1xl0-3 0 1xl0-5 O]T was used. This initial

condition is considered as the disturbances applied to the flexiblemanipulator system. The first two modes of vibration of thesystem are considered, as these dominate the dynamic of thesystem. Two criteria are used to evaluate the performances of thecontrol strategies:

(1) Level of vibration reduction at the natural frequencies. Thisis accomplished by comparing the responses of the controllerwith the response to the open loop system.

(2) Disturbance cancellation. The capability of the controller toachieved zero modal displacement.

1. If( B is NM) and (iJ is ZE) then (u is PM)

2. If ( B is NS) and ( iJ is ZE) then (u is PS)

3. If ( B is NS) and ( iJ is PS) then (u is ZE)

4. If ( B is ZE) and ( iJ is NM) then (u is PM)

5. If ( B is ZE) and ( iJ is NS) then (u is PS)

6. If ( B is ZE) and ( iJ is ZE) then (u is ZE)

7. If ( B is ZE) and ( iJ is PS) then (u is NS)

8. If( B is ZE) and (iJ is PM) then (u is NM)

9. If ( B is PS) and ( iJ is NS) then (u is ZE)

10. If( B is PS) and (iJ is ZE) then (u is NS)

11. If( B is PM) and (iJ is ZE) then (u is NM)

The open loop responses of the free end of the flexible armwere considered as the system response with disturbances effectand will be used to evaluate the performance of feedback controlstrategies. It is noted that, in open loop configuration, the modaldisplacement oscillate between ±0.03 m and the vibrationfrequencies of the flexible manipulator system under disturbanceseffect were obtained as 16 and 55 Hz for the first two modes ofvibration.

The system responses of the flexible manipulator with thedelayed feedback signal controller (DFS) are shown in Figure 5and 6. It shows that, with the gain and time delay of 55 and 0.005srespectively, the effect of the disturbances has been successfullyeliminated. This is evidenced in modal displacement response asshown in Figure 5 whereas the amplitudes of vibration werereduced in a very fast response as compared to the open loopresponse. The modal displacement settled down at 0.25 s withmaximum oscillation of -0.010 m to 0.027 m. This can be clearly

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3.0

0.25

2.5

0.2

--PO type Fuzzy------ Open loop

2.0

I 0.05 p.10

1.0 1.5Time (8)

o

0.5

-0.02

'EQ)

EQ)

~c.UJis

-0.01

0.01

Figure 7. Modal displacement response with PD-type Fuzzy Logiccontroller.

0.02

0.03.----------,---,---------.------.-----.-----------,

3.0

0.20 I 0.25

2.5

--OFS Controller

0.16

2.0

0.10

1.5Time (8)

- ~ it - -:- - - - ~ - - - -: - - - -I • ' ....., I /'\1 !,,~

I ,'- _ ~\ _ _ ,~I _ '~ __

• \!.../ 1\...,1 I ~....l' \ : I 'i ".I I I- -r: I - - - - 1- - - - T - - - -I - - - -

-0.02

g 0.01'EQ)

EQ)

~ 0c.UJ:e(ij

~ -0.01

0.02

0.031---r---~--~------;::J======r:::::==::::::1

demonstrated in frequency domain results as the magnitudes ofthe PSD at the natural frequencies were significantly reduced.

Figure 5. Modal displacement response with DFS controller.

Figure 6. PSD 0/Modal displacement response with DFScontroller.

100

I I I I I I

10.4

- - ~ -I} ~ -- ~ -- -: -- -:- -- ~I '\ I I I I II : I I I I I I

I : \ I I I I II, II I I I I

I: \1 I I I I I I I- - +- ~ -1+ - - -l - - -I - - - 1- - - I- - - +- - - -+ - - -l - -

I: \1 I I I" I I I I

I' \1 I I I I \ I I I III I I I I" I I I III \ I I I' \1 I I I

• 1\ I I \ I I I- - f - - -+,- - -l - - -I - - I'- - - +- - - -+ - - -l - -

Ji I " I I /I 1\ I I I,I I" I I II \ I I I

\ I I' I I \ I I II \ I I I I I" I I I

I I I \ " I I I I I

10.10 f - ~ -- ~ --'\-- -:7'- -:- - - ~ - - L - - ~ - - ~ - -

I I I I ~" "f I I I I It I I I'" I I I I '\ I

: : : : : : : : ',,~10.12 ,

o 10 20 30 40 50 60 70 80 90Frequency (Hz)

Figure 8. PSD o/Modal displacement response with PD-type/uzzylogic controller.

For comparative assessment, the levels of vibration reduction withthe modal displacement using DFS and PD-type fuzzy logiccontroller are shown with the bar graphs in Figure 9. The resultshows that the DFS controller achieved highest level of vibrationreduction with the value of 123.36 dB and 66.94 dB for the firsttwo modes of vibration respectively. While for PD-type fuzzylogic controller, the level of vibration reduction was obtained at102.24 dB and 23.19 dB for the first two modes of vibrationrespectively. Therefore, it can be concluded that overall thedelayed feedback signal (DFS) provide better performance invibration reduction as compared to the PD-type fuzzy logiccontroller.

I -- DFS Controller- - +- -----, Open Loop

I II II II I

__ 1 __ J I _

" I I'\ I II \ I I

II I

- - _I 1 _

I II II II I

40 50 60 70 80 90 100Frequency (Hz)

302010

Figure 7 and 8 show the closed loop system responses of theflexible manipulator under PD-type fuzzy logic controller for themodal displacement and its PSD results respectively. The modaldisplacement response shows a similar pattern as the case DFScontroller with the maximum oscillation of -0.006 m to 0.025 mand settled down at 0.25 s. The results also demonstrated that theartificial PD-type fuzzy logic controller can eliminate the effect ofdisturbances in the system. The vibration amplitude in the modaldisplacement responses were reduced as compared to the openloop response. The PSD result shows that the magnitudes ofvibration were significantly reduced especially for the first twomodes of vibration.

ISMA'09-5


140~------------------,

7. REFERENCES

6. CONCLUSIONS

techniques," Proceedings of IMechE-I: Journal of Systemsand Control Engineering, 2002, 2162, pp. 191-210.

[9] Hasting, G. G. and Book, W. J., "A linear dynamic model forflexible robot manipulators," IEEE Control Systems Mag.,1990, 10(2), 29-33.

[10] Feliu, V., Rattan, K. S. and Brown, H. B., "Adaptive controlof a single-link flexible manipulator," IEEE Control SystemsMag., 1987, 7, 61-64.

[11] Moser, A. N., "Designing controllers for flexible structureswith H-infinity/jl-synthesis," IEEE Control Systems Mag.,1993, 13(2), 79-89.

[12] Gutierrez, L.B., Lewis, P.L. and Lowe, J.A.,"Implementation of a neural network tracking controller for asingle flexible link: comparison with PD and PIDcontrollers," IEEE Trans. Ind Electronics, 1998, 45(3), 307318.

[13] Moudgal, V. G., Passino, K. M. and Yurkovich, S. "Rulebased control for a flexible-link robot," IEEE Trans. ControlSystems Technol., 1994, 2(4), 392-405.

[14] Wang, Z.H., Hu, H.Y. "Stability switches of time-delayeddynamic systems with unknown parameters," J. SoundVibration 233 (2) (2000) 215-233.

[15] Ramesh, M. and Narayanan, S. "Controlling chaotic motionsin a two-dimensional airfoil using time-delayed feedback", J.Sound Vibration, 239(5), pp. 1037-1049 (2001).

[16] Siddique, M.N.H., Tokhi, M.a. "Collocated PD-like fuzzylogic controller for flexible-link manipulator", Proceedingsof A-PVC-99: Asia-Pacific Vibration Conference, 13-15December, Singapore 1, pp. 359-364 (1999).

[17] Siddique, M.N.H. "Intelligent control of flexible-linkmanipulator systems", Ph.D. Thesis. Department ofAutomatic Control and Systems Engineering, University ofSheffield, UK (2002).

EJPD-Fuzzy

Mode 2

Mode of vibration

Mode 1

o

20

120

iii 100~c::fi 80::J'0~'0 60Q)>~ 40

Figure 9. Level ofvibration reduction using DFS and PD-typeFuzzy Logic Controller.

Investigations into vibration suppression of a flexible robotmanipulator with disturbances effect using the DFS and PD-typefuzzy logic controller have been presented. Performances of thecontroller are examined in terms of vibration suppression anddisturbances cancellation. The results demonstrated that the effectof the disturbances in the system can successfully be handled byDFS and PD-type fuzzy logic controller. A significant reduction inthe system vibration has been achieved with the DFS controller ascompared to the PD-type fuzzy logic controller.

[1] Martins, J.M., Mohamed, Z., Tokhi, M.a., Sa da Costa, J.and Botto, M.A. "Approaches for dynamic modelling offlexible manipulator systems", lEE Proceedings-ControlTheory and Application, 150(4),401-411 (2003).

[2] Yurkovich, S. "Flexibility effects on performance andcontrol", Robot Control (Eds.: M.W. Spong, F.L. Lewis andC.T. Abdallah, IEEE Press, 1992), Part 8, pp. 321-323(1992).

[3] Lueg, P. "Process of silencing sound oscillations," US Patent2 043 416, 1936.

[4] Moulin, H. and Bayo, E. "On the accuracy of end-pointtrajectory tracking for flexible arms by non-causal inversedynamic solution", Transactions of ASME: Journal ofDynamic Systems, Measurement and Control, 113, 320-324(1991).

[5] Onsay, T. and Akay, A. "Vibration reduction of a flexiblearm by time optimal open-loop control", Journal of Soundand Vibration, 147(2), 283-300 (1991).

[6] Tokhi, M.a. and Azad, A.K.M. "Control of flexiblemanipulator systems", Proceedings of IMechE-I: Journal ofSystems and Control Engineering, 210, 283-292 (1996).

[7] Singer, N.C. and Seering, W.P. "Preshaping command inputsto reduce system vibration", Transactions ofASME: JournalofDynamic Systems, Measurement and Control, 112(1), 7682 (1990).

[8] Mohamed, Z. and Tokhi, M.a., "Vibration control of asingle-link flexible manipulator using command shaping

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