Active vibration suppression techniques of a very flexible robot manipulator

8/9/2019 Active vibration suppression techniques of a very flexible robot manipulator

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Int. J. Mechatronics and Manufacturing Systems, Vol. 2, No. 3, 2009 311

Copyright 2009 Inderscience Enterprises Ltd.

Active vibration suppression techniques of a veryflexible robot manipulator

Mohd Ashraf Ahmad

Faculty of Electrical and Electronics Engineering,

Universiti Malaysia Pahang,

Lebuhraya Tun Razak,

26300 Kuantan, Pahang, Malaysia

E-mail: [email protected]

Abstract: This paper presents the use of angular position control approaches

for a flexible robot manipulator with disturbances effect in the dynamic system.Delayed Feedback Signal and Proportional-Derivative (PD)-type fuzzy logiccontroller are the techniques used in this investigation to actively control thevibrations of flexible structure. A complete analysis of simulation results foreach technique is presented in time domain and frequency domain respectively.Performances of both controllers are examined in terms of vibrationsuppression, disturbances cancellation, time response specifications and inputenergy. Finally, a comparative assessment of the impact of each controlleron the system performance is discussed.

Keywords: flexible manipulator; vibration control; DFS; delayed feedbacksignal; PD-type fuzzy logic controller.

Reference to this paper should be made as follows: Ahmad, M.A. (2009)Active vibration suppression techniques of a very flexible robot manipulator,Int. J.Mechatronics and Manufacturing Systems, Vol. 2, No. 3, pp.311330.

Biographical notes: Mohd Ashraf Ahmad obtained his Bachelor ofElectrical-Mechatronic Engineering with Honours (BEng) from the UniversitiTeknologi Malaysia, Johor, in 2006. He received his Master Degree inModelling and Control of Flexible Robot Manipulator from the UniversitiTeknologi Malaysia, Johor, in 2008. He is currently lecturer in Faculty ofElectrical and Electronics Engineering, Universiti Malaysia Pahang (UMP).His research interest are in the area of modelling and control of flexible robotmanipulator, vibration control techniques, command shaping control, modellingand control of gantry crane, robotics and automation. He is a member of IEEE,IAENG and Asia Control Association (ACA).

1 Introduction

Flexible robot manipulators exhibit many advantages over the rigid link manipulators,

as they require less material, are lighter in weight, have higher manipulation speed, lower

power consumption, require smaller actuators, are more manoeuvrable and transportable,

have less overall cost and higher payload to robot weight ratio (Martins et al., 2003).

However, the control of flexible manipulators to maintain accurate positioning is a

challenging problem. A flexible manipulator is a distributed parameter system and has


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312 M.A. Ahmad

infinitely many degrees of freedom. Moreover, the dynamics are highly non-linear

and complex. Problems arise owing to precise positioning requirements, systemflexibility leading to vibration, the difficulty in obtaining accurate model of the system

and non-minimum phase characteristics of the system (Yurkovich, 1992). To attain

end-point positional accuracy, a control mechanism that accounts for both the rigid body

and flexural motions of the system is required. If the advantages associated with lightness

are not to be sacrificed, precise models and efficient control strategies for flexible

manipulators have to be developed.

Research on the control methods that will eliminate vibration from wide range of

physical systems has found a great deal of interest for many years. The methods used to

solve the problems arising owing to unwanted structural vibrations include passive and

active control. The passive control method consists of mounting passive material on the

structure to change its dynamic characteristics such as stiffness and damping coefficient.

This method is efficient at high frequencies but expensive and bulky at low frequencies(Hossain and Tokhi, 1997). Passive vibration control usually leads to an increase in the

overall weight of structure, which makes it less transportable especially for space

applications. Active vibration control consists of artificially generating sources that

absorb the energy caused by the unwanted vibrations to cancel or reduce their effect on

the overall system. Lueg in 1930 (Lueg, 1936) is among the first who used active

vibration control to cancel noise vibration. Since then, a large number of researchers have

concentrated on developing methodologies for the design and implementation of active

vibration control systems in various applications.

The requirement of precise position control of flexible manipulators implies that

residual vibration of the system should be zero or near zero. Over the years,

investigations have been carried out to devise efficient approaches to reduce the vibration

of flexible manipulators. The considered vibration control schemes can be divided into

two main categories: feed-forward control and feedback control techniques. Feed-forward

techniques for vibration suppression involve developing the control input through

consideration of the physical and vibrational properties of the system, so that system

vibrations at dominant response modes are reduced. This method does not require

additional sensors or actuators and does not account for changes in the system once the

input is developed. On the other hand, feedback control techniques use measurement and

estimations of the system states to reduce vibration. Feedback controllers can be designed

to be robust to parameter uncertainty. For flexible manipulators, feed-forward and

feedback control techniques are used for vibration suppression and end-point position

control, respectively. An acceptable system performance without vibration that accounts

for system changes can be achieved by developing a hybrid controller consisting of both

control techniques. Thus, with a properly designed feed-forward controller, the

complexity of the required feedback controller can be reduced.A number of techniques have been proposed as feed-forward control strategies for

control of vibration. These include utilisation of Fourier expansion as the forcing function

to reduce peaks of the frequency spectrum at discrete points (Aspinwall, 1980),

derivation of a shaped torque that minimises vibration and the effect of parameter

variations (Swigert, 1980), development of computed torque based on a dynamic

model of the system (Moulin and Bayo, 1991), utilisation of single and multiple-switch

bangbang control functions (Onsay and Akay, 1991) and construction of input functions

from ramped sinusoids or versine functions (Meckl and Seering, 1990). Moreover,

feed-forward control schemes with command shaping techniques have also been


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Active vibration suppression techniques 313

investigated in reducing the system vibration. These include filtering techniques

based on low-pass, band-stop and notch filters (Singhose et al., 1995; Tokhi andPoerwanto, 1996; Pao, 2000; Tokhi and Azad, 1996) and input shaping (Singer and

Seering, 1990; Mohamed and Tokhi, 2002). In filtering techniques, a filtered torque input

is developed on the basis of extracting the input energy around the natural frequencies

of the system. Previous experimental studies on a single-link flexible manipulator have

shown that higher level of vibration reduction and robustness can be achieved with input

shaping technique than with filtering techniques. However, the major drawback of the

feed-forward control schemes is their limitation in coping with parameter changes and

disturbances to the system (Khorrami et al., 1994). Moreover, this technique requires

relatively precise knowledge of the dynamics of the system.

On the other hand, feedback control techniques use measurements and estimates of

the system states and change the actuator input accordingly for control of rigid body

motion and vibration suppression of the system. Feedback controllers can be designed tobe robust to parameter uncertainty. In general, control of flexible manipulators can be

made easier by locating every sensor exactly at the location of the actuator, as collocation

of sensors and actuators guarantees stable servo control (Gevarter, 1970). In the case of

flexible manipulator systems, the end-point position can be controlled using the

measurement obtained from the hub and end-point of the manipulator. The measurement

is then used as a basis for applying control torque at the hub. Thus, feedback control

strategies can be divided into collocated and non-collocated control techniques. Sensors

that can be utilised are strain gauge and accelerometer.

Several approaches utilising closed-loop control strategies have been reported for

control of flexible manipulators. These include linear state feedback control (Cannon and

Schmitz, 1984), adaptive control (Hasting and Book, 1990), robust control techniques

based onH-infinity (Feliu et al., 1987), variable structure control (Moser, 1993) and

intelligent control based on neural networks (Gutierrez et al., 1998) and fuzzy logic

control schemes (Moudgal et al., 1994).

Another method of controlling flexible structures is based on Time Delay Control

(TDC). In the TDC method, time delay is used to estimate the effects of unknown

dynamics and unpredictable disturbances (Youcef and Ito, 1990; Youcef and Bobbett,

1992; Richard, 1998). The TDC introduces delay terms in the closed loop of the system

to cancel the unwanted dynamics. In Youcef and Wu (1992), time delay has been used to

achieve an input/output linearisation of a class of non-linear systems with a special

application to the position control of a single-link flexible arm. In general, time delays

occur in real systems in several forms. Transport delays and acoustic feedback are

considered to be the main sources. The stability of systems with delay has been dealt with

extensively in the literature (Youcef and Reddy, 1990; Malek-Zavarei and Jamshidi,

1987; Kharitonov, 1979). Recently, a generalised approach to investigate the stability oftime delay systems has been presented in Olgac and Sipahi (2001). This approach

resembles the RouthHurwitz technique for linear systems and can be used to select the

time delay parameters that lead to a stable closed-loop system. More recently, TDC has

been used in the control of aerodynamic systems. In Ramesh and Narayanan (2001),

a time-delayed feedback to control the chaotic motions in a two-dimensional airfoil was

used, and a similar technique to stabilise the motion of helicopter rotor blades was used in

Krodkiewski and Faragher (2000) except that the time delay in this case was selected to

be the period of the motion to be stabilised. A method for determining the stability

switches for time-delayed dynamic systems with unknown parameters has been presented


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314 M.A. Ahmad

in Wang and Hu (2000) and Jnifene (2007). In this paper, the time delay has been

introduced to generate the control signal, and the delay time is considered as the designparameter.

This paper presents investigations of angular position control approach to eliminate

the effect of disturbances applied to the single-link flexible robot manipulator.

A simulation environment is developed within Simulink and Matlab for evaluation

of the control strategies. In this work, the dynamic model of the flexible manipulator is

derived using the AMM. To demonstrate the effectiveness of the proposed control

strategy, the disturbances effect is applied at the tip of the flexible link. This is then

extended to develop a feedback control strategy for vibration reduction and disturbances

rejection. Two feedback control strategies, which are DFS and PD-type FLCs, are

developed in this simulation work. Performances of each controller are examined in

terms of vibration suppression, disturbances rejection, hub angle response specifications

and input energy. Finally, a comparative assessment of the impact of each controller onthe system performance is presented and discussed.

2 The flexible manipulator system

The single-link flexible manipulator system considered in this work is shown in

Figure 1, where XoOYo and XOY represent the stationary and moving coordinates

frames, respectively, and represents the applied torque at the hub. E,I, ,A, Ih and mp

represent the Young modulus, area moment of inertia, mass density per unit volume,

cross-sectional area, hub inertia and payload mass of the manipulator, respectively.

In this work, the motion of the manipulator is confined to the XoOYo plane. Transverse

shear and rotary inertia effects are neglected, since the manipulator is long and slender.

Thus, the BernoulliEuler beam theory is allowed to be used to model the elastic

behaviour of the manipulator. The manipulator is assumed to be stiff in vertical

bending and torsion, allowing it to vibrate dominantly in the horizontal direction, and

thus the gravity effects are neglected. Moreover, the manipulator is considered to have a

constant cross-section and uniform material properties throughout. In this study,

an aluminium-type flexible manipulator of dimensions 900 19.008 3.2004 mm3,

E= 71 109 N/m2, I= 5.1924 1011 m4, = 2710 kg/m3 and Ih= 5.8598 104 kgm2 is

considered.

Figure 1 Description of the flexible manipulator system


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3 Modelling of the flexible manipulator

This section provides a brief description on the modelling of flexible robot manipulator

system, as a basis of a simulation environment for the development and assessment of

control techniques. The AMM with two modal displacements is considered in

characterising the dynamic behaviour of the manipulator incorporating structural

damping and hub inertia. Further details of the description and derivation of the dynamic

model of the system can be found in Subudhi and Morris (2002). The dynamic model

is validated with an actual experimental rig to study the performance of the model in

Martins et al. (2003).

Considering revolute joints and motion of the manipulator on a two-dimensional

plane, the position vector that describes an arbitrary point along the deflected link with

respect to its local inertial coordinate frame {O0,X0, Y0,Z0} is given by

1 2( , ) [ cos ( ) ( , )sin ( )] [ sin ( ) ( , )cos ( )]r x t x t v x t t p x t v x t t p = + + (1)

wherep1 andp2 are the unit vectors along the X0 and Y0 axes, respectively. To simplify

the notation, C and S represent cos and sin, respectively. The velocity vector of thesame infinitesimal element is accordingly given by

1 2( , ) [ ( ) ] [( ) ]r x t x v S v C p x v C v S p = + + + (2)

where the superposed dot has the usual meaning of time derivative. The kinetic energy of

the system can thus be formulated as

2

0

2 2 2 2 2 2

0

1 1( , ) ( , ) d

2 2

1 1 ( 2 ) d .2 2

LT

H

LH

T I r x t r x t x

T I x v vx v x

= +

= + + + +

(3)

A common simplification is usually made at this point. This simplification is motivated

by the simplicity desired from the models used for control purposes and has the

underlying assumption that the displacement (x, t) is small. The comparative values arethe first natural frequency for the rotational velocity and the length of the beam for

transverse displacement. Rearranging the terms in the expression for the kinetic energy

that multiply 2 ( )t yields

2 2 2 2 2

0

2 2 2 2 2

0 0

1 1(( ) 2 ) d

2 2

1 1 1

( ) d ( 2 ) d .2 2 2

L

H

L L

H

T I x v v vx x

I x v x v vx x

= + + + +

= + + + +

(4)

Under the stated assumption, the second term on the right-hand side of the above

equation can be simplified as

2 2 2 2 2 2 2 2

0 0 0( ) d ( ) d ( ) d

L L L

bx v x x x x x I + = = (5)

where Ib is the beam rotation inertia about the origin O0 as if it was rigid. This

simplification is of common practice for control applications, and as is shown further,


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316 M.A. Ahmad

results in much simpler (linear) equations of motion for the system. The simplified form

of the kinetic energy is finally obtained as

2 2

0

1 1( ) ( 2 ) d .

2 2

L

H bT I I v vx x = + + + (6)

The potential energy of the beam can be formulated as

22

20

1d .

2

L vU EI x

x

=

(7)

This expression states the internal energy owing to the elastic deformation of the link

as it bends. The potential energy owing to gravity is not accounted for because only

motion in the plane perpendicular to the gravitational field is considered.

Next, to obtain a closed-form dynamic model of the manipulator, the energyexpressions in equations (6) and (7) are used to formulate the Lagrangian L = TU.

Assembling the mass and stiffness matrices and utilising the EulerLagrange equation of

motion, the dynamic equation of motion of the flexible manipulator system can be

obtained as

q Dq Kq F + + = (8)

where M, D and Kare global mass, damping and stiffness matrices of the manipulator,

respectively. The damping matrix is obtained by assuming that the manipulator exhibits

the characteristic of Rayleigh damping.Fis a vector of external forces and q is a modal

displacement vector given as

[ ]1 2TT T

nq q q q q = =

(9)

[ ]0 0 0 .T

F = (10)

Here, qn is the modal amplitude of the ith clamped-free mode considered in the AMM

procedure and n represents the total number of assumed modes. The model of the

uncontrolled system can be represented in a state-space form as

x x u

y x

= +

=

A B

C

(11)

with the vector1 2 1 2

T

x q q q q = and the matrices A and B are given by

[ ]

3 3 3 3 3 1

1 1 1

1 3 1 3

0 0, ,

0 , 0 .

I

M K M D M

I

= =

= =

A B

C D

(12)

4 Controller design

In this section, two feedback control strategies (DFS and PD-type FLC) are proposed and

described in detail. The main objective of the feedback controller in this study is to


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maintain the angular position of flexible manipulator at the same time as suppressing the

vibration owing to disturbances effect. All the feedback control strategies areincorporated in the closed-loop system to eliminate the effect of disturbances.

4.1 Delayed Feedback Signal controller

In this section, the control signal is calculated based on the delayed position feedback

approach described in equation (13) and illustrated by the block diagram shown in

Figure 2.

( ) ( ( ) ( )).u t k y t y t = (13)

Substituting equation (13) into equation (11) and taking the Laplace transform gives

( ) ( ) (1 e ) ( ).

s

sIx s Ax s kBC x s

=

(14)

Figure 2 The Delayed Feedback Signal controller structure

The stability of the system given in equation (14) depends on the roots of the

characteristic equation

( , ) | (1 e ) | 0.ss sI A kBC = + = (15)

Equation (15) is transcendental and results in an infinite number of characteristic roots

(Olgac and Sipahi, 2001). Several approaches dealing with solving retarded differential

equations have been widely explored. In this study, the approach described in Ramesh

and Narayanan (2001) will be used in determining the critical values of the time delay

that result in characteristic roots of crossing the imaginary axes. This approach suggests

that equation (15) can be written in the form

( , ) ( ) ( )e .ss P s Q s = + (16)

P(s) and Q(s) are polynomials in s with real coefficients and deg(P(s)) = n > deg(Q(s))

where n is the order of the system. To find the critical time delay that leads to marginal

stability, the characteristic equation is evaluated at s = j. Separating the polynomials

P(s) and Q(s) into real and imaginary parts and replacing ej

by cos() jsin(),equation (16) can be written as

( , ) ( ) ( ) ( ( ) ( ))(cos( ) sin( )).R I R I j P jP Q jQ j = + + + (17)


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318 M.A. Ahmad

The characteristic equation (s,) = 0 has roots on the imaginary axis for some values of

0 if equation (17) has positive real roots. A solution of (j, ) = 0 exists if themagnitude | (j,) | = 0. Taking the square of the magnitude of(j, ) and setting it tozero will lead to the following equation

2 2 2 2( ) 0.R I R I P P Q Q+ + = (18)

By setting the real and imaginary parts of equation (18) to zero, the equation is

rearranged as follows

cos,

sin

R I R

I R I

Q Q P

Q Q P

=

(19)

where= .

Solving for sin and cosgives

2 2 2 2

( ) ( )sin( ) and cos( ) .

( ) ( )

R I I R R R I I

R I R I

P Q P Q P Q P Q

Q Q Q Q

+ = =

+ +

The critical values of time delay can be determined as follows: if a positive root of

equation (18) exists, the corresponding time delay can be found by

2k

k

= + (20)

where [0 2]. At these time delays, the root loci of the closed-loop system arecrossing the imaginary axis of thes-plane. This crossing can be from stable to unstable or

from unstable to stable. To investigate the above method further, the time-delayed

feedback controller is applied to the single-link flexible manipulator. Practically, thecontrol signal for the DFS controller requires only one position sensor and uses only the

current position and the position with second delay in past. There are only two control

parameters: kand that need to be set. Using the stability analysis described in Ramesh

and Narayanan (2001), the gain and time-delayed of the system is set at k = 55 and

= 0.005. The control signal of DFS controller can be written as follows:

( ) 55( ( ) ( 0.005)).DFSu t t t =

4.2 PD-type Fuzzy Logic Controller

Fuzzy control can be viewed as a way of converting expert knowledge into an automatic

control strategy without a detailed knowledge of the plant (Zadeh, 1973; Mamdani, 1974;Mamdani and Assilian, 1975). The input is first fuzzified and then processed by the fuzzy

inference engine using heuristic decision rules. FLC uses rules in the form of

IF [condition] THEN [action] to linguistically describe the input/output relationship.

The membership functions convert linguistic terms into precise numeric values.

The output of the fuzzy controller is obtained by a defuzzification process that converts

the fuzzy quantities representing the control signal into a signal that can be used as the

control input to the plant.

A PD-type FLC utilising hub angle and hub velocity feedback is developed to control

the rigid body motion of the system (Siddique and Tokhi, 1999; Siddique, 2002).


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The hybrid fuzzy control system proposed in this work is shown in Figure 3, where and

are the hub angle and hub velocity of the flexible robot manipulator, respectively,whereas k1, k2 and k3 are scaling factors for two inputs and one output of the FLC used

with the normalised universe of discourse for the fuzzy membership functions.

Figure 3 The PD-type Fuzzy Logic Controller structure

In this paper, the hub velocity is measured from the system instead of deriving it with

the equation above. Triangular membership functions are chosen for inputs and output.

The membership functions for hub angle, hub velocity, and torque input are shown in

Figure 4. Normalised universes of discourse are used for both hub angle and velocity and

output torque. Scaling factors k1 and k2 are chosen in such a way as to convert the

two inputs within the universe of discourse and activate the rule base effectively,

whereas k3 is selected such that it activates the system to generate the desired output.

Initially, all these scaling factors are chosen based on trial and error. To construct a rulebase, the hub angle, hub velocity, and torque input are partitioned into five primary fuzzy

sets as follows (Siddique and Tokhi, 1999; Siddique, 2002):

Hub angleA = {NM NS ZE PS PM},

Hub velocity V= {NM NS ZE PS PM},

Torque U= {NM NS ZE PS PM},

whereA, V, and Uare the universes of discourse for hub angle, hub velocity and torque

input, respectively. The nth rule of the rule base for the FLC, with angle and angular

velocity as inputs, is given by

Rn: IF(isAi) AND ( is Vj) THEN (u is Uk),

where Rn, n = 1, 2, ,Nmax, is the nth fuzzy rule, Ai, Vj, and Uk, fori, j, k= 1, 2, , 5,

are the primary fuzzy sets.

A PD-type FLC was designed with 11 rules as a closed-loop component of the control

strategy for maintaining the angular position of flexible manipulator at the same time

as suppressing the vibration owing to disturbances effect. The rule base was extracted

based on underdamped system response and is shown in Table 1. The control surface is

shown in Figure 5. The three scaling factors, k1, k2 and k3, were chosen heuristically to

achieve a satisfactory set of time-domain parameters. These values were recorded as

k1= 1.02, k2= 0.30 and k3= 1.0.


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320 M.A. Ahmad

Figure 4 Membership functions of a Fuzzy Logic Controller: (a) hub angle; (b) hub velocity

and (c) torque

(a) (b)

(c)

Table 1 Linguistic rules of Fuzzy Logic Controller

Rules

If (Hub angle is NM) and (Hub velocity is ZE) then (Torque is PM)

If (Hub angle is NS) and (Hub velocity is ZE) then (Torque is PS)

If (Hub angle is NS) and (Hub velocity is PS) then (Torque is ZE)

If (Hub angle is ZE) and (Hub velocity is NM) then (Torque is PM)

If (Hub angle is ZE) and (Hub velocity is NS) then (Torque is PS)

If (Hub angle is ZE) and (Hub velocity is ZE) then (Torque is ZE)

If (Hub angle is ZE) and (Hub velocity is PS) then (Torque is NS)

If (Hub angle is ZE) and (Hub velocity is PM) then (Torque is NM)

If (Hub angle is PS) and (Hub velocity is NS) then (Torque is ZE)

If (Hub angle is PS) and (Hub velocity is ZE) then (Torque is NS)

If (Hub angle is PM) and (Hub velocity is ZE) then (Torque is NM)


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Figure 5 Control surface of the Fuzzy Logic Controller

5 Simulation results

In this section, the proposed control schemes are implemented and tested within the

simulation environment of the flexible manipulator and the corresponding results are

presented. The control strategies were designed by undertaking a computer simulation

using the fourth-order RungeKutta integration method at a sampling frequency of

1 kHz. Three system responses namely the hub angle, hub velocity and modal

displacement are obtained. Moreover, the Power Spectral Density (PSD) of the modal

displacement is evaluated to investigate the dynamic behaviour of the system in

frequency domain. Four criteria are used to evaluate the performances of the control

strategies:

Level of vibration reduction at the natural frequencies. This is accomplished by

comparing the responses of the controller with the response to the open-loop system.

Disturbance cancellation. The capability of the controller to achieve steady-state

conditions at zero angular position.

The hub angle response specifications. Parameters that are evaluated are settling time

and magnitude of oscillation of the angular position response. The settling time is

calculated on the basis of0.02% of the steady-state value.

Input energy. The magnitude and frequency of oscillation of input torque to theflexible manipulator are observed.

In all simulations, the initial condition x0 = [0 0 1 103 0 1 105 0]T was used. This

initial condition is considered as the disturbances applied to the flexible manipulator

system. The first two modes of vibration of the system are considered, as these dominate

the dynamic of the system.

Figure 6 shows the open-loop response of the free end of the flexible arm, which

consists of hub angle, hub velocity, modal displacement and PSD results. These results

were considered as the system response with disturbances effect and will be used to


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322 M.A. Ahmad

evaluate the performance of feedback control strategies. It is noted that, in open-loop

configuration, the steady-state hub angle for the flexible manipulator system wasachieved at 0.014 rad within the settling times of 1 s. The hub velocity response shows

the maximum oscillation between 4 rad/s and 6 rad/s, whereas the modal displacement

oscillates between 0.03 m. Resonance frequencies of the system were obtained by

transforming the time-domain representation of the system responses into frequency

domain using power spectral analysis. The vibration frequencies of the flexible

manipulator system under disturbances effect were obtained as 16 and 55 Hz for the first

two modes as demonstrated in Figure 6.

Figure 6 Open-loop response of the flexible manipulator: (a) hub angle; (b) hub velocity;(c) modal displacement and (d) PSD of modal displacement

(a) (b)

(c) (d)

The system responses of the flexible manipulator with the DFS controller are shown in

Figure 7. It shows that, with the gain and time delay of 55 and 0.005 s, respectively,

the effect of the disturbances has been successfully eliminated. This is evidenced in hub

angle response, whereas the flexible manipulator system maintained its steady-state

conditions at zero radian in a very fast response. It is noted that the vibrations in the hub

angle, hub velocity and modal displacement responses were reduced when compared with


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the open-loop response. This can be clearly demonstrated in frequency domain results as

the magnitudes of the PSD at the natural frequencies were significantly reduced. Table 2summarises the levels of vibration reduction of the system response at the first two modes

in comparison with the open-loop system. Besides, the corresponding settling time and

oscillation of the hub angle response and input energy of the torque in the case of DFS

controller are also depicted in Table 2. The results demonstrated that the DFS controller

exhibits high oscillation magnitude of input torque to compensate the angular position to

steady-state conditions.

Figure 7 Response of the flexible manipulator with DFS controller: (a) hub angle;(b) hub velocity; (c) modal displacement; (d) PSD of modal displacementand (e) input torque

(a)

(b)


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324 M.A. Ahmad

Figure 7 Response of the flexible manipulator with DFS controller: (a) hub angle;

(b) hub velocity; (c) modal displacement; (d) PSD of modal displacementand (e) input torque (continued)

(c)

(d)

(e)


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Table 2 Level of vibration reduction of the modal displacement, specifications of hub angle

response and oscillation of input torque using DFS and PD-type Fuzzy LogicController

Attenuation (dB)of vibration modal

displacementSpecifications of hub angle

response

Controller Mode 1 Mode 2 Settling time (s)Maximum

oscillation (rad)

Maximum torqueoscillation (Nm)

DFS 123.36 66.94 0.505 0.011 to 0.022 1.921 to 1.706

PD-type Fuzzy 102.24 23.19 0.427 0.008 to 0.014 0.839 to 0.513

Figure 8 shows the response of the closed-loop system using the PD-type FLC.

The angular position result demonstrates that the PD-type FLC can also handle the

effect of disturbances in the system and achieve zero radian steady-state conditions

as similar to the case DFS controller. It is noted that the overall system vibrations

were significantly reduced with the PD-type FLC even though the level of vibration

reduction was less than the case with the DFS controller. Besides, the overall time

response results exhibit small magnitude of oscillation when compared with the

DFS controller. The levels of vibration reduction with the modal displacement at the first

two modes of vibration in comparison with the open-loop system and the hub angle

response specification are summarised in Table 2. The PSD result shows that

the magnitudes of vibration were significantly reduced especially for the first mode

of vibration as demonstrated in Figure 8. In terms of input energy performances,

the PD-type FLC requires a small input torque when compared with the case of the DFS

controller.

Figure 8 Response of the flexible manipulator with PD-type Fuzzy Logic Controller:(a) hub angle; (b) hub velocity; (c) modal displacement; (d) PSD of modal displacementand (e) input torque

(a)


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326 M.A. Ahmad

Figure 8 Response of the flexible manipulator with PD-type Fuzzy Logic Controller:

(a) hub angle; (b) hub velocity; (c) modal displacement; (d) PSD of modal displacementand (e) input torque (continued)

(b)

(c)

(d)


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Figure 8 Response of the flexible manipulator with PD-type Fuzzy Logic Controller:

(a) hub angle; (b) hub velocity; (c) modal displacement; (d) PSD of modal displacementand (e) input torque (continued)

(e)

6 Comparative assessment

By comparing the results presented in Table 2, it is noted that higher performance in the

reduction of vibration of the system is achieved with the DFS control strategies. This is

observed and compared with the PD-type FLC at the first two modes of vibration.

For comparative assessment, the levels of vibration reduction with the modaldisplacement using DFS and PD-type FLC are shown with the bar graphs in Figure 9.

The result shows that higher level of vibration reduction is achieved with the DFS

controller when compared with the PD-type FLC for both modes of vibration. Therefore,

it can be concluded that overall DFS provides better performance in vibration reduction

when compared with both PD-type FLCs.

Figure 9 Level of vibration reduction using DFS and PD-type Fuzzy Logic Controller


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328 M.A. Ahmad

Comparisons of the specifications of the hub angle responses using DFS and PD-type

FLC are summarised in Table 1 for the settling time and magnitude of oscillation of hubangle. It is noted that the settling time of the hub angle response using the PD-type FLC

is faster than the DFS controller. The result reveals that the PD-type FLC provides a

high-speed system response to cater the disturbances effect to the flexible manipulator

system. For the oscillation of hub angle specifications, the magnitude of hub angle tends

to oscillate higher with the DFS controller when compared with the case of PD-type FLC.

Therefore, it can be concluded that the PD-type FLC provides less magnitude of

oscillation of hub angle with a faster-response capability when compared with the DFS

controller.

The performance of the control strategies can also be focused on the input torque to

the flexible manipulator system. The results from Figures 7(e) and 8(e) show that the

input torque for DFS and PD-type FLC are very similar in terms of frequency of

oscillation except for the magnitude of oscillation torque. It is noted that both DFS andPD-type FLC exhibit high-frequency oscillations of input torque, and as a consequence,

both controllers will require an actuator with higher bandwidth to cater wide range

of oscillation. By comparing the results presented in Table 1, it is noted that the

magnitude of oscillation torque using the DFS control strategies is almost two-fold

higher than the case of PD-type FLC. The results reveal that the PD-type FLC can

achieve the same performance as the DFS controller in terms of disturbances rejection

and zero radian steady-state conditions by exhibiting low input energy to the flexible

manipulator system.

7 Conclusion

Investigations into vibration suppression of a flexible robot manipulator with

disturbances effect using the DFS and PD-type FLC have been presented. Performances

of the controller are examined in terms of vibration suppression, disturbances

cancellation, hub angle response specifications and input energy. The results

demonstrated that the effect of the disturbances in the system can successfully be handled

by both DFS and PD-type FLC. A significant reduction in the system vibration has been

achieved with the DFS controller when compared with the PD-type FLC. By using the

PD-type FLC, the speed of the response is slightly faster at the expense of small

magnitude of oscillation in hub angle response specifications. The results show that the

PD-type FLC provides a small input energy to the flexible manipulator system when

compared with the DFS control strategies.

Acknowledgement

This work was supported by Faculty of Electrical and Electronics Engineering, Universiti

Malaysia Pahang, especially Control & Instrumentation (COINS) Research Group.


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Active vibration suppression techniques of a very flexible robot manipulator