Optimal Control with Input Shaping for Input Tracking and Vibration Suppression of a Flexible Joint Manipulator

8/14/2019 Optimal Control with Input Shaping for Input Tracking and Vibration Suppression of a Flexible Joint Manipulator

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European Journal of Scientific Research

ISSN 1450-216X Vol.28 No.4 (2009), pp.583-599

EuroJournals Publishing, Inc. 2009http://www.eurojournals.com/ejsr.htm

Optimal Control with Input Shaping for Input Tracking and

Vibration Suppression of a Flexible Joint Manipulator

Mohd Ashraf Ahmad

Control and Instrumentation Research Group (COINS), Faculty of Electricaland Electronics Engineering, Universiti Malaysia Pahang, Lebuhraya

Tun Razak, 26300, Kuantan, Pahang, MalaysiaE-mail: [email protected]

Tel: +609-5492366; Fax: +609-5492377

Raja Mohd Taufika Raja Ismail

Control and Instrumentation Research Group (COINS), Faculty of Electrical

and Electronics Engineering, Universiti Malaysia Pahang, LebuhrayaTun Razak, 26300, Kuantan, Pahang, Malaysia

E-mail: [email protected]

Tel: +609-5492366; Fax: +609-5492377

Mohd Syakirin Ramli

Control and Instrumentation Research Group (COINS), Faculty of Electrical

and Electronics Engineering, Universiti Malaysia Pahang, Lebuhraya

Tun Razak, 26300, Kuantan, Pahang, MalaysiaE-mail: [email protected]

Tel: +609-5492366; Fax: +609-5492377

Abstract

This paper presents investigations into the development of optimal control for inputtracking and vibration suppression of a flexible joint manipulator. A single-link flexible

joint manipulator is considered and the dynamic model of the system is derived using theEuler-Lagrange formulation. To study the effectiveness of the controllers, a linear-

quadratic regulator (LQR) controller is developed for tip angular position control of a

flexible joint manipulator. This is then extended to incorporate input shaper controlschemes for vibration reduction of the flexible joint system. The positive zero-vibration-

derivative-derivative (ZVDD) and new modified specified negative amplitude zero-

vibration-derivative-derivative (SNA-ZVDD) input shapers are then designed based on theproperties of the system for vibration control. The new SNA-ZVDD is proposed to improve

the robustness capability while increasing the speed of the system response. Simulation

results of the response of the flexible joint manipulator with the controllers are presented in

time and frequency domains. The performances of the LQR with input shaping controlschemes are examined in terms of input tracking capability, level of vibration reduction,

time response specifications and robustness to parameters uncertainty. A comparative

assessment of the positive ZVDD and modified SNA-ZVDD shapers to the hybrid systemperformance is presented and discussed.


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Vibration Suppression of a Flexible Joint Manipulator 584

Keywords: Flexible joint manipulator, vibration control, input shaping, LQR controller.

1. IntroductionIt has been recognized that control of robot link based on rigid body dynamic formulation is not

enough to deal with stringent condition. As a result, the joint flexibility should be taken intoconsideration in order to achieved better performance. In some cases, joint flexibility may lead to

instability of the system if it is neglected in the control design. Rotary flexible joint is considered as anideal representation intended to model a flexible joint in a robot manipulator. The control issue of the

flexible joint is to design the controller so that link of robot can reach a desired position or track a

prescribed trajectory precisely with minimum vibration to the link. In order to achieve these objectives,various methods using different technique have been proposed.

Yim [1], Oh and Lee [2] proposed adaptive output-feedback controller based on a backstepping

design. This technique is proposed to deal with parametric uncertainty in flexible joint. The relevantwork also been done by Ghorbel et al. [3]. Lin and Yuan [4] and Spong et al. [5] introduced non linear

control approach using namely feedback linearization technique and the integral manifold technique

respectively. A robust control design was reported by Tomei [6] by using simple PD control and Yeonand Park [7] by applying robust H control. Among the proposed techniques, the conventional

feedback control design handled by pole placement method and LQR method also have been widelyused due to its simplicity implementation. Particularly in LQR method, the values of QandRmatrices

are pre-specified to determine optimal feedback control gain via Riccati equation [8].

Input shaping control techniques are mainly develop for vibration suppression and involve

developing the control input through consideration of the physical and vibrational properties of thesystem, so that system vibrations at response modes are reduced. This method does not require any

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

developed. Investigations have shown that with the input shaping technique, a system response withdelay is obtained. To reduce the delay and thus increase the speed of the response, negative amplitude

input shapers have been introduced and investigated in vibration control. By allowing the shaper tocontain negative impulses, the shaper duration can be shortened, while satisfying the same robustnessconstraint. A significant number of negative shapers for vibration control have also been proposed.

These include negative unity-magnitude (UM) shaper, specified-negative-amplitude (SNA) shaper,

negative zero-vibration (ZV) shaper, negative zero-vibration-derivative (ZVD) shaper and negativezero-vibration-derivative-derivative (ZVDD) shaper [9,10,11]. Comparisons of positive and negative

input shapers for vibration control of a single-link flexible manipulator have also been reported [11].

This paper presents investigations into the development of hybrid control schemes for input

tracking and vibration control of a flexible joint manipulator system. A flexible joint manipulatorsystem is considered and the dynamic model of the system is derived using the Euler-Lagrange

formulation. Hybrid control schemes based on input shaping with LQR controllers are investigated. In

this work, input shaping with positive ZVDD input shapers and new modified specified negativeamplitude zero-vibration-derivative-derivative (SNA-ZVDD) input shapers are considered. A new

modified shaper from the previous SNA input shapers [11] is proposed where more negative impulses

are added to improve the robustness of the controller while increasing the speed of the systemresponse. To demonstrate the effectiveness of the proposed control schemes, a LQR controller is

developed for control of tip angular position of the flexible joint. This is then extended to incorporate

the proposed input shapers for vibration control of flexible joint. Simulation exercises are performed

within the flexible joint simulation environment. Performances of the developed controllers areexamined in terms of input tracking capability, level of vibration reduction, time response

specifications and robustness to errors in vibration frequency. In this case, the robustness of the hybrid

control schemes is assessed with up to 30% error tolerance in vibration frequencies. Simulation results


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585 Mohd Ashraf Ahmad, Raja Mohd Taufika Raja Ismail and Mohd Syakirin Ramli

in time and frequency domains of the response of the flexible joint to the unshaped input and shapedinputs with positive and modified SNA input shapers are presented. Moreover, a comparative

assessment of the effectiveness of the hybrid controllers with positive and negative input shapers in

suppressing vibration and maintaining the input tracking capability of the flexible joint is discussed.

2. Gantry Crane SystemThe flexible joint manipulator system considered in this work is shown in Figure 1, where is the tipangular position and is the deflection angle of the flexible link. The base of the flexible joint

manipulator which determines the tip angular position of the flexible link is driven by servomotor,while the flexible link will response based on base movement. The deflection of link will be

determined by the flexibility of the spring as their intrinsic physical characteristics.

Figure 1:Description of the Flexible Joint Manipulator System

3. Modelling of the Flexible Joint ManipulatorThis section provides a brief description on the modelling of the flexible joint manipulator system, as a

basis of a simulation environment for development and assessment of the LQR with input shaping

control techniques. The Euler-Lagrange formulation is considered in characterizing the dynamicbehaviour of the system.

Considering the motion of the flexible joint system on a two-dimensional plane, the potential

energy of the spring can be formulated as

2

stiff

2

1KU= (1)

where stiffK is the joint stiffness. The kinetic energies in the system arise from the moving hub and

flexible link can be formulated as

2

link

2

eq )(2

1

2

1 &&& ++= JJT (2)

where eqJ and linkJ are the equivalent inertia and total link inertia, respectively.

To obtain a closed-form dynamic model of the flexible joint, the energy expressions in (1) and(2) are used to formulate the Lagrangian, that is

2

stiff

2

link

2

eq2

1)(

2

1

2

1 KJJUTL ++== &&& (3)


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Let the generalized torque corresponding to the generalized tip angle be &eqoutput B . Using

Lagrangians equation

&

& eqoutputB

LL

dt

d=

(4)

0=

LL

dt

d

& (5)

the equation of motion is obtained as below,

&&&&&&& eqoutputlinkeq )( BJJ =+ (6)

0)( stifflink =++ KJ &&&& (7)

where eqB is the equivalent viscous damping and output is the output torque on the load from

the motor, defined as

m

mgmgtgm

R

KKVKK )(output

&= (8)

where m is the motor efficiency, g is the gearbox efficiency, tK is the motor torque constant, gK is

the high gear ratio, mK is the motor back emf constant and mR is the linkature resistance.

The linear model of the uncontrolled system can be represented in a state-space form as shownin equation (9), that is

CxyBuAxx

=+=&

(9)

with the vector [ ]Tx &&= and the matricesA,Band Care given by

[ ]0001,00

00

00

1000

0100

2

2

=

=

++

+=

CRJ

KK

RJ

KKB

RJ

RBKKK

JJ

)J(JKRJ

RBKKK

J

KA

meq

gtgm

meq

gtgm

meq

meqgmtgm

armeq

armeqstiff

meq

meqgmtgm

eq

stiff

(10)

In equation (9), the input u is the input voltage of the servomotor, mV which determines the

flexible joint manipulator base movement. In this study, the values of the parameters are defined asRm=2.6 , Km=0.00767 V-s/rad, Kt=0.00767 N-m/A, Jlink=0.0035 kg-m

2, Jeq= 0.0026 kg-m

2, Kg=2.8,

Kstiff=1.2485,Beq=0.004 N-m-s/rad, g=0.9 and m=0.69.

4. Linear Quadratic Regulator (LQR) Control SchemeA more common approach in the control of manipulator systems involves the utilization LQR design

[8]. Such an approach is adopted at this stage of the investigation here. Figure 2 illustrates the LQR

control structure. In order to design the LQR controller a linear state-space model of the flexible jointmanipulator was obtained by linearising the equations of motion of the system. For a linear time

invariant (LTI) system


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Figure 2:Delayed feedback signal controller structure

Flexible Joint

Manipulator

K

N

+

_

Desired

input

Output

responses

BuAxx +=& (11)the technique involves choosing a control law )(xu= which stabilizes the origin (i.e., regulates xto

zero) while minimizing the quadratic cost function

+=0

)]()()()([ dttRututQxtxJ TT (12)

where 0=T

QQ and 0>= TRR . The term linear-quadratic refers to the linear system dynamicsand the quadratic cost function.

The matrices Q and R are called the state and control penalty matrices, respectively. If the

components of Q are chosen large relative to those of R, then deviations of x from zero will be

penalized heavily relative to deviations of ufrom zero. On the other hand, if the components ofRarelarge relative to those of Q, then control effort will be more costly and the state will not converge to

zero as quickly.

A famous and somewhat surprising result due to Kalman is that the control law which

minimizes J always takes the form Kxxu == )( . The optimal regulator for a LTI system with

respect to the quadratic cost function above is always a linear control law. With this observation inmind, the closed-loop system takes the form

xBKAx )( =& (13)and the cost functionJtakes the form

+=

+=

0

0

)()()(

))](())(()()([

dttxRKKQtx

dttKxRtKxtQxtxJ

TT

TT

(14)

Assuming that the closed-loop system is internally stable, which is a fundamental requirement

for any feedback controller, the following theorem allows the computation value of the cost functionfor a given control gain matrixK.

5. Input Shaping Control SchemesInput shaping technique is a feed-forward control technique that involves convolving a desiredcommand with a sequence of impulses known as input shaper. The shaped command that results from

the convolution is then used to drive the system. Design objectives are to determine the amplitude and

time locations of the impulses, so that the shaped command reduces the detrimental effects of systemflexibility. These parameters are obtained from the natural frequencies and damping ratios of the

system. Thus, vibration reduction of a flexible joint system can be achieved with the input shaping

technique. Figure 3 illustrates the input shaping process. Several techniques have been investigated to

obtain an efficient input shaper for a particular system. A brief description and derivation of the controltechnique is presented in this section.


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Figure 3:Illustration of input shaping technique.

Am

litude

Time

*

A1

A2

TimeAm

litude

Unshaped Input Input Shaper Shaped input

Generally, a vibratory system of any order can be modelled as a superposition of second ordersystems each with a transfer function

22

2

2)(

++=

sssG

(15)

where is the natural frequency of the vibratory system and is the damping ratio of the system.

Thus, the response of the system in time domain can be obtained as

)](1sin[1)(0

2)(

2

0

tte

A

ty

tt

=

(16)

whereAand 0t are the amplitude and the time location of the impulse respectively. The response to a

sequence of impulses can be obtained by superposition of the impulse responses. Thus, forNimpulses,

with 21 =d , the impulse response can be expressed as

)sin()( += tMty d (17)

where2

1

2

1

sincos

+

=

==

N

i

ii

N

i

ii BBM ,)(

2

0

1

tti

i eA

B

=

, idi t =

andAiand tiare the amplitudes and time locations of the impulses.The residual single mode vibration amplitude of the impulse response is obtained at the time of

the last impulse, Nt as

2

2

2

1 VVV += (18)

where

)cos(1

)(

12

1 id

ttN

i

ni teA

V iNn

=

= ; )sin(

1

)(

12

2 id

ttN

i

ni teA

V iNn

=

=

To achieve zero vibration after the last impulse, it is required that both 1V and 2V in Equation

(18) are independently zero. This is known as the zero residual vibration constraints. In order to ensure

that the shaped command input produces the same rigid body motion as the unshaped referencecommand, it is required that the sum of amplitudes of the impulses is unity. This yields the unity

amplitude summation constraint as

=

=N

i

iA1

1

(19)

In order to avoid response delay, time optimality constraint is utilised. The first impulse is

selected at time t1= 0 and the last impulse must be at the minimum, i.e. )min( Nt . The robustness of the

input shaper to errors in natural frequencies of the system can be increased by taking the derivatives of

1V and 2V to zero. Setting the derivatives to zero is equivalent to producing small changes in vibration

corresponding to the frequency changes. The level of robustness can further be increased by increasing


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the order of derivatives of 1V and 2V and set them to zero. Thus, the robustness constraints can be

obtained as

01 =i

n

i

d

Vd

; 02 =

i

n

i

d

Vd

(20)

Both the positive and modified SNA input shapers are designed by considering the constraints

equations. The following section will further discuss the design of the positive and modified SNA input

shapers.

5.1. Positive Input Shaper

The positive input shapers have been used in most input shaping schemes. The requirement of positive

amplitude for the impulses is to avoid the problem of large amplitude impulses. In this case, each

individual impulse must be less than one to satisfy the unity magnitude constraint. In order to increasethe robustness of the input shaper to errors in natural frequencies, the positive ZVDD input shaper is

designed by setting the second derivatives of 1V and 2V in Equation (18) to zero. Simplifying22

ni dVd yields

=

=

==

N

iid

tt

ii

n

N

iid

tt

ii

ntetAd

Vd

tetAd

VdiNniNn

1

)(2

2

2

2

1

)(2

2

1

2

)cos();sin(

(21)

The positive ZVDD input shaper, i.e. four-impulse sequence is obtained by setting Equations

(18) and (21) to zero and solving with the other constraint equations. Hence, a four-impulse sequencecan be obtained with the parameters as

01=t ,d

t

=2 ,

d

t

23 = ,

dt

34 =

321 331

1

KKKA

+++= ,

322 331

3

KKK

KA

+++= (22)

32

2

3

331

3

KKK

KA

+++

=

,32

3

4

331 KKK

KA

+++

=

where

21

= eK 21 = nd

n and representing the natural frequency and damping ratio respectively. For the impulses, jt and

jA are the time location and amplitude of impulsejrespectively.

5.2. Modified SNA Input Shapers

Input shaping techniques based on positive input shaper have been proved to be able to reduce

vibration of a system. In order to achieve higher robustness, the duration of the shaper is increased andthus, increases the delay in the system response. By allowing the shaper to contain negative impulses,

the shaper duration can be shortened, while satisfying the same robustness constraint.

To include negative impulses in a shaper requires the impulse amplitudes to switch between 1 and 1as

1)1( += iiA ; ni ,,2,1 K= (23)

The constraint in Equation (23) yields useful shapers as they can be used with a wide variety of

inputs. For a unity magnitude (UM) negative zero-vibration (ZV) shaper, i.e. the magnitude of each

impulse is |1|, the shaper duration is one-third of the vibration period of an undamped system, while theshaper duration for the positive shaper is half of the vibration period. However, the increase in the

speed of system response achieved using the SNA input shapers is at the expense of some tradeoffs and


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penalties. The shapers containing negative impulses have tendency to excite unmodeled high modes

and they are slightly less robust as compared to the positive shapers. Besides, negative input shapersrequire more actuator effort than the positive shapers due to high changes in the set-point command at

each new impulse time location.

To overcome the disadvantages, a modified SNA input shaper is introduced, whose negative

amplitudes can be set to any value at the centre between each normal impulse sequences. In this work,

the previous SNA input shaper [11] is modified by locating the negative amplitudes at the centrebetween each positive impulse sequences with even number of total impulses. This will result the

shaper duration as one-fourth of the vibration period of an undamped system as shown in Figure 4. Themodified SNA-ZVDD shaper is proposed and applied in this work to enhance the robustness capability

of the controller while increasing the speed of the system response. By considering the form of

modified SNA-ZVDD shaper shown in Figure 4, the amplitude summation constraints equation can beobtained as

12222 =+ dbca (24)The values of a, b, c anddcan be set to any value that satisfy the constraint in (24). However,

the suggested values of a, b, c anddare less than |1| to avoid the increase of the actuator effort.

Figure 4:Modified SNA-ZVDD shaper.

c c

a a

d

0.5t2 1.5t2 2.5t2 3.5t20 t2 t3 t4

dbb

6. Implementation and ResultIn this section, the proposed control schemes are implemented and tested within the simulationenvironment of the flexible joint manipulator and the corresponding results are presented. In this work,

positive ZVDD and modified SNA-ZVDD are investigated as the input shaping control schemes. The

tip angle position of the flexible joint is required to follow a trajectory within the range of 50 asshown in Figure 5. System responses namely the tip angular position, angular velocity and deflection

angle of the flexible joint are observed. To investigate the vibration of the system in the frequency

domain, power spectral density (PSD) of the response at the deflection angle is obtained. The

performances of the hybrid controllers are assessed in terms of input tracking and vibration suppressionin comparison to the LQR control. Moreover, robustness of the controllers to variations in vibration

frequency is also investigated. In this case, 30% error tolerance in vibration frequency is considered.


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Figure 5:The Trajectory Reference Input

0 5 10 15 20 25 30-60

-40

-20

0

20

40

60

Time (s)

Desiredinput(d

eg)

6.1. LQR Control

In this investigation, the responses of the flexible joint manipulator system to the unshaped trajectoryreference input were analyzed in time-domain and frequency domain (spectral density) as shown in

Figure 6. These results were considered as the system response to the unshaped input under tracking

capability and will be used to evaluate the performance of the input shaping techniques. The steady-state tip angular position trajectory of +50 for the flexible joint was achieved within the rise and

settling times and overshoot of 0.489 s, 1.235 s and 4.08% respectively. It is noted that the flexible

joint reaches the required position from +50 to 50 within 2.3 s, with low overshoot.

However, a noticeable amount of deflection angle occurs during movement of the tip angular. Itis noted from the deflection angle response with a maximum residual of 7.4. Moreover, from the

PSD of the deflection angle response, the vibration frequency is dominated by a single mode, which is

obtained as 6 Hz with magnitude of 0.8668 deg

2

/Hz respectively. The closed loop parameters with theLQR control will subsequently be used to design and evaluate the performance of hybrid controllers

with positive ZVDD and SNA-ZVDD shapers.

Figure 6:Response of the flexible joint manipulator with LQR controller.

0 5 10 15 20 25 30-60

-40

-20

0

20

40

60

Time (s)

Tipangular

position(deg)

(a) Tip angular position.


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0 5 10 15 20 25 30-200

-150

-100

-50

0

50

100

150

200

Time (s)

Angularvelocity(deg/s)

(b) Angular velocity.

0 5 10 15 20 25 30-8

-6

-4

-2

0

2

4

6

8

Time (s)

Deflectionangle(deg)

(c) Deflection angle.

0 10 20 30 40 50 6010-15

10-10

10-5

100

Frequency (Hz)

Magnitude((deg)*(deg)/Hz)

(d) PSD of deflection angle.

6.2. Hybrid Control

Figure 7 shows a block diagram of the proposed hybrid control scheme where the LQR is combined

with the input shaping control schemes. The positive ZVDD and modified SNA-ZVDD shapers weredesigned based on the dynamic behaviour of the closed-loop system obtained using only the LQR

controller. As demonstrated in the previous section, the natural frequency of the deflection angle was

obtained at 6 Hz for the single mode of vibration. With exact natural frequencies, the time locations


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and amplitudes of the impulses for positive ZVDD shaper were obtained by solving Equation (23).

Moreover, the amplitudes of the modified SNA-ZVDD shaper were deduced as [0.3 0.1 0.5 0.2 0.5

0.2 0.3 0.1] and the time locations of the impulses were chosen at the half of the time locations of

positive ZVDD shaper as shown in Figure 4. For evaluation of robustness, input shapers witherroneous natural frequencies were also evaluated. With 30% error in natural frequency, the system

vibration was considered at 7.8 Hz for the single mode of vibration. Similarly, the amplitudes and time

locations of the input shapers with 30% erroneous natural frequencies for both the positive and

modified SNA-ZVDD shapers were calculated.

Figure 7:Block diagram of the hybrid control schemes configuration.

Flexible joint

manipulator

Output

responsesInput

shaperShaped

input

Desired

input

K

+

-

N

LQR Controller

For digital implementation of the input shaper, locations of the impulses were selected at thenearest sampling time. The developed input shaper was then used to pre-process the input reference

shown in Figure 5. Figure 8 shows the shaped inputs using both the positive and modified SNA-ZVDD

shapers with exact natural frequencies. It is noted that the shaped input with the modified SNA shaper

is not as smooth as compared to the positive shaper. This is due to higher number of switching of theactuator.

Figure 8: Shaped inputs with exact natural frequencies using positive ZVDD and modified SNA-ZVDDshapers.

0 5 10 15 20 25 30-60

-40

-20

0

20

40

60

Time (s)

Desiredinput(deg)

Positive ZVDD

Modified SNA ZVDD

0 0.50

20

40

60

Figure 9 shows the system responses of the flexible joint using the hybrid controllers with exactnatural frequencies. Table 1 summarises the levels of vibration reduction of the system responses at the

single mode in comparison to the LQR control. It is noted that the proposed hybrid controllers are

capable of reducing the system vibration while maintaining the input tracking performance of the tipangular position. Similar tip angular position and tip angular velocity responses were observed as

compared to the LQR controller. Moreover, a significant amount of vibration reduction was


12/17



demonstrated at the deflection angle of the flexible joint with both control schemes. With the positive

ZVDD and modified SNA-ZVDD shapers, the maximum deflection angles were obtained at 6.2 and

5.3 respectively. These are three-fold and two-fold improvements as compared to LQR controller.

This is also evidenced in the PSD of the deflection angle residual that shows lower magnitude at theresonance mode. The corresponding rise time, settling time and overshoot of the tip angular response

using LQR controller with positive and modified SNA ZVDD shapers with exact natural frequencies isdepicted in Table 1. The simulation results show that the tip angular position reaches the required

trajectory position of +50 within the settling times of 1.329 s and 1.277 s with positive ZVDD and

modified SNA-ZVDD respectively. It is noted with the input shaping controller, a slower settling timeas compared to the LQR controller was achieved.

To examine the robustness of the hybrid controllers, the shapers with 30% error in vibration

frequency were designed and implemented to the flexible joint manipulator system. Figure 10 shows

the response of the flexible joint with the hybrid controllers with erroneous natural frequency. Table 1summarises the levels of vibration reduction with erroneous natural frequency in comparison to the

LQR controller. The time response specifications of the tip angular position with error in natural

frequency are also summarised in Table 1. Similar to the case with exact frequency, the proposed

hybrid controllers are capable of reducing the system vibration while maintaining the input trackingperformance of the tip angular position. Moreover, the vibration of the system was considerable

reduced as compared to the response with LQR controller.The simulation results show that performance of the hybrid controller with positive ZVDD

control scheme is better than SNA-ZVDD scheme in vibration suppression of the flexible joint for the

exact natural frequency case. This is further evidenced in Figure 11 that demonstrates the level ofdeflection angle reduction of the flexible joint with the hybrid controllers as compared to the LQR

controller. It is noted that higher deflection angle reduction is achieved with positive ZVDD at the

exact natural frequency of 6 Hz. Almost less than two-fold improvement in the vibration reduction was

observed as compared to SNA-ZVDD. Comparisons of the tip angular position responses show that thehybrid controller with SNA-ZVDD shaper is faster than the case using the positive ZVDD shaper. The

result reveals that the speed of the system responses can be improved by using negative impulses inputshaper.Comparison of the results shown in Figure 11 also reveals that both hybrid controllers with the

positive and SNA input shapers can successfully handle errors in natural frequencies. It shows that,

almost three-fold improvement was observed with SNA-ZVDD as compared to positive ZVDDdeflection angle reduction of the flexible joint. As positive ZVDD performs better than SNA-ZVDD

with exact natural frequency, the results demonstrate that, the modified SNA-ZVDD is capable ofimproving the robustness of the controller to uncertainty in vibration frequency. Comparisons of the tip

angular position response with the hybrid controllers show a similar pattern as the case with exact

natural frequency. With the new proposed SNA shaper, it is shown that the robustness of the controllercan be improved while increasing the speed of the response. The work thus developed and reported in

this paper forms the basis of design and development of hybrid control schemes for input tracking andvibration suppression of flexible joint and flexible link manipulator systems and can be extended to

and adopted in practical applications.


13/17


Table 1: Level of deflection angle reduction and specifications of the tip angular position response for thehybrid control schemes

Specifications of tip angular position responseFrequency

Types of

shaper

(ZVDD)

Attenuation (dB) of

deflection angle in

frequency domainRise time (s) Settling time (s) Overshoot (%)

Exact Positive 53.68 0.515 1.329 2.88

Modified SNA 29.14 0.517 1.277 2.76

Error Positive 25.15 0.505 1.313 3.14Modified SNA 69.42 0.506 1.275 2.88

Figure 9:Response of the flexible joint manipulator with hybrid controllers with exact natural frequencies.

0 5 10 15 20 25 30-60

-40

-20

0

20

40

60

Time (s)

Tipangular

position(deg)

Positive ZVDD

Modified SNA ZVDD

0.450.50 0.55 0.6030

35

40

10.610.6510.710.75-40

-35

-30

(a) Tip angular position

0 5 10 15 20 25 30-200

-150

-100

-50

0

50

100

150

200

Time (s)

Angularvelocity(deg/s)

Positive ZVDD

Modified SNA ZVDD

20 21 22

0

50

100



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0 5 10 15 20 25 30-8

-6

-4

-2

0

2

4

6

8

Time (s)

Deflectiona

ngle(deg)

Positive ZVDD

Modified SNA ZVDD

0 0.5 1 1.5 2-2

0

2


0 10 20 30 40 50 6010

-15

10-10

10-5

100

Frequency (Hz)


Positive ZVDD

Modified SNA ZVDD


Figure 10: Response of the flexible joint manipulator with hybrid controllers with erroneous natural

frequencies.

0 5 10 15 20 25 30-60

-40

-20

0

20

40

60

Time (s)

Tipangularposition(deg)

Positive ZVDD

Modified SNA ZVDD

0.45 0.50 0.55 0.6030

35

40

10.5510.6010.6510.70-40

-35

-30

(a) Tip angular position.


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0 5 10 15 20 25 30-200

-150

-100

-50

0

50

100

150

200

Time (s)

Angula

rvelocity(deg/s)

Positive ZVDD

Modified SNA ZVDD

0 5 10 15 20

0

50

100


0 5 10 15 20 25 30-8

-6

-4

-2

0

2

4

6

8

Time (s)

Deflectionangle(deg)

Positive ZVDD

Modified SNA ZVDD

0 0.5 1 1.5 2-2

0

2


0 10 20 30 40 50 6010

-15

10-10

10-5

100

Frequency (Hz)


Positive ZVDD

Modified SNA ZVDD



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Figure 11: Level of Deflection Angle Reduction with Exact and Erroneous Natural Frequencies with HybridControllers.

0

10

20

30

40

50

60

70

80

Exact Error

Natural frequencies

Levelofdeflectionanglereduction(

dB)

PositiveZVDD

Modified SNAZVDD

7. ConclusionThe development of hybrid control schemes based on LQR control with positive and negative input

shapers for input tracking and vibration suppression of a flexible joint manipulator has been presented.

The proposed control schemes have been implemented and tested within simulation environment of aflexible joint manipulator system derived using the Euler-Lagrange formulation. The performances of

the control schemes have been evaluated in terms of input tracking capability, level of vibration

reduction, time response specifications and robustness. Acceptable performance in input tracking

control and vibration suppression has been achieved with both control strategies. Moreover, asignificant reduction in the system vibration has been achieved with the hybrid controllers regardless of

the polarities of the shapers. For exact natural frequency case, a comparative assessment of the hybrid

control schemes has shown that the LQR controller with positive ZVDD shaper provides higher levelof vibration reduction of the flexible joint as compared to the LQR controller with SNA-ZVDD shaper.

However, by using the LQR controller with modified SNA-ZVDD, the robustness of the controller can

be improved with slightly increased in the speed of response.


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References[1] Yim, W., 2001. Adaptive Control of a Flexible Joint Manipulator. Proc. 2001 IEEE,

International Robotics & Automation, Seoul, Korea, pp. 34413446.

[2] Oh, J. H. and Lee, J. S., 1997. Control of Flexible Joint Robot System by Backstepping DesignApproach.Proc. of IEEE International Conference on Robotics & Automation, pp. 34353440.

[3] Ghorbel, F., Hung, J.Y. and Spong, M.W., 1989. Adaptive Control of Flexible JointManipulators. Control Systems Magazine, Vol. 9, pp. 9-13.

[4] Lin, L.C. and Yuan, K., 2007. Control of Flexible Joint Robots via External LinearizationApproach.Journal of Robotic Systems, Vol. 1 (1), pp. 1-22.

[5] Spong, M. W., Khorasani, K. and Kokotovic, P. V., 1987. An Integral Manifold Approach tothe Feedback Control of Flexible Joint Robots.IEEE Journal of Robotics and Automation, Vol.

RA-3, No.4, pp. 291-300.

[6] Tomei, P., 1991. A Simple PD Controller for Robots with Elastic Joints. IEEE Trans. onAutomatic Control, Vol. 36, No. 10, pp. 1208-1213.

[7] Yeon, J. S. and Park, J. H., 2008. Practical Robust Control for Flexible Joint RobotManipulators. Proc. 2008 IEEE International Conference on Robotic and Automation,Pasadena, CA, USA, pp. 33773382.

[8] Ogata, K., 2002. Modern Control Engineering, Prentice Hall.[9] Singhose, W., Singer, N. C. and Seering, W. P., 1994. Design and Implementation of Time-

optimal Negative Input Shapers.Proc. of International Mechanical Engineering Congress and

Exposition, Chicago, pp. 151-157.

[10] Singhose, W. and Mills, B. W., 1999. Command Generation Using Specified-negative-amplitude Input Shapers.Proc. of the American Control Conference, San Diego, California, pp.61-65.

[11] Mohamed, Z., Chee, A.K., Mohd Hashim, A. W. I., Tokhi, M. O., Amin, S. H. M. and Mamat,R., 2006. Techniques for Vibration Control of a Flexible Manipulator. Robotica 24, pp. 499-511.

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Optimal Control with Input Shaping for Input Tracking and Vibration Suppression of a Flexible Joint Manipulator