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2012 18: 1081 originally published online 21 September 2011Journal of Vibration and ControlNan-Chyuan Tsai, Din-Chang Chen, Li-Wen Shih and Chao-Wen Chiang

techniqueodel reduction and composite control for overhead hoist transport system by singular perturbatio

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Article

Model reduction and composite controlfor overhead hoist transport systemby singular perturbation technique

Nan-Chyuan Tsai, Din-Chang Chen, Li-Wen Shih and

Chao-Wen Chiang

Abstract

An innovative Overhead Hoist Transport (OHT) system is proposed and analyzed to transport fragile semi-finishedproducts in factories. A triplet of double-link arm is used to carry the load, in replacement of the cables used conven-tionally. Unlike conventional OHT, the proposed OHTexhibits superior capability for high-speed transportation, flexible

stiffness and is able to account for the inherent auto-sway characteristics and parameters uncertainties of the OHTsystem. The three-time-scale plant model of the OHT system, including the drive motors, flexible links and rigid links, isdeveloped. By singular perturbation order-reduction technique, the highly nonlinear high-order dynamics of the OHTsystem can be modeled as a low-order linearized plant so that the synthesis of the feedback controller becomes simpler.The composite control, composed of sliding mode control and input shaping technique, is proposed. The sliding modecontrol is, as usual, employed to account for the system parameters uncertainties. On the other hand, to suppressthe residual vibration, i.e., auto-swaying, the input shaping technique is utilized by implementation of a finite-lengthsequence of impulses in the appropriate amplitude and time epoch. Finally, the efficacy of the proposal composite controlstrategy is examined and verified by intensive computer simulations.

Keywords

Composite control, multiple-time-scale system, overhead hoist transportReceived: 13 September 2009; accepted: 9 April 2011

1. Introduction

The semiconductor fabrication is one of the most signif-

icant industries. Highly complicated production facilities

with various processes are involved. The semi-finished

products have to be transported backwards/forwards

among stations in the factory. For example, a stack of

300 mm wafers is transported approximately from 8 to

10 miles during the processing and typically about 250

fabrication procedures have to be undertaken before

finished goods are completed (Agrawal and Heragu,

2006). It is evident that the transport system in semicon-

ductor industries is one of the crucial factors for the

quality of the products. Besides, due to the high cost of

wafers, manual transport is not practicable at all such

that a highly reliable wafer transport system is intensively

required. However, the commercially-available Overhead

Hoist Transport (OHT) unit, which is currently employed

in industries, is not equipped with an active suspension

controller so that fragile material (e.g., wafers) suffers

from potential damage due to collision or collapse

(Chung and Jang, 2007; Kuo, 2002; Liao and Wang,

2006).

In general, the conventional OHT is a type of pendu-

lum so that it is inherently less eligible for high speed

transportation with auto-swaying (Jerman and Kramar,

2008; Jerman et al., 2004). That is, though it is easy to

use, there are a few inherent drawbacks, such as swaying

Department of Mechanical Engineering, National Cheng Kung University,

Taiwan

Corresponding author:

Nan-Chyuan Tsai, Department of Mechanical Engineering, National

Cheng Kung University, Tainan City 70101, Taiwan

Email: [email protected]

Journal of Vibration and Control

18(8) 10811095

! The Author(s) 2011

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and the collision and collapse of piled-up carried load. In

order to prevent those threats, an innovative triplet of

double-link arm is proposed to be equipped with

the OHT unit in our work. Compared with the conven-

tional OHT, the proposed OHT unit possesses a certain

degree of superior properties, such as eligibility for high

traveling speed, adjustable stiffness and capability ofanti-sway and anti-uncertainty.

The singular perturbation theory is usually employed

to deal with the plant with the presence of parasitic

parameters. By a singular perturbation approach, the

system model can be split into two lower-order subsys-

tems in two time scales (Kokotovic et al., 1987). For

example, the quarter-car suspension system is usually

analyzed by a two-time-scale model (Salman et al.,

1988). In addition, the electric power system can be some-

how converted to a generator voltage regulator by a sin-

gular perturbation method (Vournas et al., 1995). As to

automation, flexible robot links have often been studied

by a perturbation technique (Ge and Cheng, 2005; Khalil

and Kokotovic 1979; Ladde and Siljak, 1983; Prljaca and

Gajic, 2008; Rasmussen and Alleyne, 2004; Shtessel and

Shkolnikov, 2003; Shtessel et al., 2002; Spong, 1989; Tsai

et al., 2007).

Although the system dynamics of the OHT system is

highly nonlinear and of a high-order, by a singular per-

turbation technique, the system model can be split into

three subsystems in three time scales, i.e., slow-mode

subsystem, intermediate-mode subsystem and fast-

mode subsystem. The order-reduced models are simpli-

fied but the characteristics of the fast-mode subsystem

are preserved to some extent so that the controllersynthesis becomes simpler. A composite controller, com-

posed of the Sliding Mode Control (SMC) loop and

input shaping technique, is proposed in this paper. The

SMC is synthesized, on the basis of a slow-time-scale

model, to account for the system parameters uncertain-

ties and to regulate the position of carried load. On the

other hand, the input shaping technique is developed, on

the basis of the intermediate-time-scale model, to suppress

the residual vibration. That is, the anti-sway controller is

implemented by the input shaping technique which pro-

vides a sequence of impulse in appropriateamplitudes and

time epoch. Finally,the efficacy of the proposalcomposite

control strategy is examined and verified by computer

simulations.

Innovative design and dynamic

analysis of OHT

The proposed OHT mainly consists of a trolley, a set of

DC (Direct Current) motors, three rigid links and three

flexible links, shown in Figure 1. The OHT is equipped

with a triplet of double-link arms to carry the load. The

motions of double-link arms are controlled by the set of

DC motors. Evidently, the OHT is a mechatronic

system composed by an electric subsystem and a

mechanical subsystem. Both the dynamics of electric

and mechanical subsystems are modeled to describe

precisely the dynamics of the OHT. At last, the

Multiple-time-scale (MTS) property is applied for

order-reduction and controller synthesis.

2.1. Description of OHT

The schematic diagram of the proposed overhead crane

system, hereafter named as OHT, is depicted in

Figure 1. The mass of cargo to be carried forward/back-

ward between two stations in a factory is denoted as m c.

Instead of passive cables equipped conventionally, a

triplet of double-link arms is facilitated with the OHT

to carry the load. Each double-link arm is composed

of two links, which are connected by pin-joint, and

named as a manipulator in this paper. The upper link

of each arm, Ri, is assumed rigid and controlled by a

DC motor so that each rigid link can actively move in

planar fashion. It is noted that the three rigid links

together can dominate any 3-dimensional movement

of the carried load within the specified boundary con-

structed by the three manipulators. The lower link of

each arm, Fi, is flexible and controlled by the associated

rigid link so that the load, m c, can be prevented from

any potential shock, unexpected tilting or high-fre-

quency excitation, especially if it is considerably fragile.

The ith drive motor is attached and fixed to the OHT

at location A i, specified by the actuator coordinate,

Figure 1. Schematic diagram of the OHT system.

1082 Journal of Vibration and Control 18(8)


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Xa i xa i ya i za i T

, i 1, 2, 3, with respect to theinertial reference frame fX, Y, Zg, shown in Figure 2.The center of the triangle constituted by A 1, A 2,A 3,O, is the origin of the actuator frame. Assumethe pin joints for the rigid links and flexible links, Ji,

are frictionless. The displacement vector of the load is

Xc xc yc zc T

with respect to inertial reference

frame. The constrained planar displacement of manip-

ulators are defined and shown in Figure 3. The

coordinates X0, Y0, Z0f g and X00, Y00, Z00f g are the inter-mediate coordinates whose origins are located on A iand Ji (i.e., the top ends of i

th rigid link and ith flexible

link) respectively. X, X0 and X00 are along the track ofthe OHT while Z, Z0 and Z00 are always the verticalaxes. Both of the rigid links R 1 and R 2 are constrained

to retain a constant angle, , with respect to the coor-dinate plane X0, Y0f g, shown in Figure 3(a).On the otherhand, i is the component of angular displacement ofR i i 1, 2 about Y0-axis. Similarly, i i 1, 2 is thecomponent of angular displacement of the undeformed

flexible links about Y00-axis. The motion of rigid link,R 3, is constrained on the plane, X

0, Z0f g, shown inFigure 3(b). 3 and 3 are the angular displacementsof rigid link and undeformed flexible link of the

Manipulator #3 about X0-axis and X00-axis respectively.The elastic deformations of the flexible links of manip-

ulators are expressed with respect to the coordinate,

x0, y0, z0 , whose origin is Ji i 1, 2 , shown inFigure 4. In finite-mode sense, the deformations of theflexible links can be described as follows:

vy0 is, t Xnj1

y0 jsi qy0 i jt, i 1, 2, 3: 1a

vz0 i s, t Xnj1

z0 jsi qz0 i jt, i 1, 2, 3: 1b

where y0 j si and z0 j si are the mode shape functionsabout y0-axis and z0-axis respectively. qy0 i j

t

and

qz0 i jt are the generalized coordinates in time vari-able. s i 2 0, l2 is the axial position along the flexiblelinks while n is the number of modes and t denotes

the time variable. With the boundary conditions sat-

isfied, the normalized shape functions can be found:

jsi sinj

l2si

, i 1, 2, 3, j 1, . . . , n, 2

where l2 is the length of individual flexible link before

any deformation or elongation occurs.

2.2. Dynamics of Mechanical SubsystemFrom the geometric relation among the proposed

manipulators, the linear displacement of the carried

load, which is assumed as a lumped rigid-body mass,

can be described from the geometric relation of

Manipulator #3 as follows:

xcyczc

24

35 cos 3 cos30 0

sin 3 sin 3

24

35 l1

l2

!

xa 3ya 3za 3

24

35 3

Figure 3. Schematic diagram of the manipulators (a) links R1and link R2 (b) link R3..

Figure 2. Allocation of drive motors (a) top view (b) side view.

Figure 4. Deformations of manipulators #1 and #2.

Tsai et al. 1083


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where l1 is the length of individual rigid link. Then the

absolute velocity of carried load, V*

mc , can be evaluated

as follows:

V*

mc V*

mc=T !* r* V*

M

_xc i _yc ^j _zc k

_x i _y ^j

xc i yc ^j zc k

_x i _xc zc _y _x

i

_yc

zc _x ^j

_zc

yc _x

xc _y k

4

where V

*mc=T is the relative velocity of carried load

with respect to the trolley, !*

the angular velocity of

carried load, V*

M the velocity of trolley and x the dis-

placement of trolley in X-axis. x and y are the swayangles of carried load about X-axis and Y-axis respec-

tively, shown in Figure 5. In fact, x, y

can be

described by the location of carried load as follows:

x sin1zc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2c z2c

p

5a

y sin1 zcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2c z2c

p 5b

Taking advantage of the three flexible links sepa-

rated 120 degree orderly against the carried load, the

net moment on the carried load about Z-axis is almost

zero by adjustment of the flexible links positions. That

is, the carried load does not rotate about Z-axis.

Therefore, the total kinetic energy of manipulators

and trolley can be evaluated by:

T 12

X2i1

l10

1 _rTo i _ro i d so i

1

2

X2i1

l20

2 _rT

f i _rf i d sf i

1

2

l1

0

1 _rTo3 _ro3 d so3

1

2

l2

0

2 _rT

f3 _rf3 dsf3

12

M 31 l1 32l2 _x2 12

mc ~V2mc

6

where 1 and 2 are the mass density (in length) of rigidlinks and flexible links respectively. ro i is the displace-

ment vector of the rigid link, R i, i 1, 2: That is:

roi sinicos cosicos sin

cosi sini 0

sinisin cosisin cos

264

375

so i

0

0

264

375

x

0

0

264

375,

so i

20, l1

i

1, 2:

7

ro 3 is the displacement vector of rigid link R 3:

r o 3 cos 3 sin 3 0

0 0 1

sin 3 cos 3 0

264

375

so 3

0

0

264

375

x

0

0

264

375,

so 3 2 0, l1 : 8

rf i is the displacement vector of the flexible links,

Fi, i 1, 2: That is:

rf i sin icos cos icos sin

cos i sin i 0

sin isin cos isin cos

264375 l10

0

264375 x0

0

264375

sinicos cosicos sin

cosi sini 0

sinisin cosisin cos

264

375

sf i

vy0 i

vz0 i

264

375,

sf i 2 0, l2 , i 1, 2: 9

rf3 is the displacement vector of flexible link F3:

rf3 cos 3

sin 3 0

0 0 1

sin 3 cos 3 0

264 375l1

0

0

264 375x

0

0

264 375

cos 3 sin 3 0

0 0 1

sin 3 cos 3 0

264

375

sf i

vy0 i

vz0 i

264

375, sf i 2 0, l2 :

10

On the other hand, the overall potential energy of

manipulators, with respect to the origin of the actuator

Figure 5. Schematic diagram of swaying load.



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frame, O, can be expressed as:

VX2i1

1

2

l20

E I v00y0 i 2 v00z0 i 2

d s2 i mc g zc

X2

i1

l1

0

1 g rzo i d so i

l1

0

1 g rzo3 d so3

X2i1

l20

2 g rz

f i d sf i l2

0

2 g rz

f3 d sf3 11

where Eand Iare the Youngs modulus and moment of

inertia of flexible links respectively and g denotes

the gravitation constant. rzo i, rzo3, r

zf i and r

zf3 represent

the Z-axis components of ro i, ro 3, rf i and rf3 respec-

tively i 1, 2 and can be described as follows:

rzo i sf i sin i sin 12a

rzo 3 so 3 sin 3 12b

rzf i l1 sin i sin sf i sin i sin vy0 i cos isin vz0 i cos 12c

rzf3 l1 sin 3 sf3 sin 3 vy0 i sin 3 12d

Assume HX 0 is the constraint equation invector form:

H

X

h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 h 10 h 11

T

0 13

where the scalar constraint equations, h i 0, i 1,2, , 11, can be referred to Appendix A for more

details. Finally, by Lagrangian multiplier approach, the

integrated equations of motion in matrix-vector form for

manipulators and trolley can be obtained as follows:

M qr, qf qr

qf

! 0 0

0 K

!qr

qf

! fr qr, _qr

ff qr, _qr

!

grqr, _qr, qf, _qf0 !

U

0 !ATr

0 ! l 14where M is symmetric inertia matrix, qr 2 R12 the statevector of rigid links, qf 2 R6 n the state vector of flexiblelinks, U2 R12 the applied torque by motors, l theLagrangian multiplier vector, fr, ff and gr are the terms

due to gravity, Coriolis and centripetal forces. It is noted

that bothfr andff are independent of _qf. Kis the stiffness

matrix and A r the constraint matrix.Definitions ofM,fr,

ff, gr, K, U and A r can be referred to Appendix B for

more details. The inertia matrix, M, is to incorporate all

the mass properties of the whole OHT. The quadratic

form associated with the inertia matrix of the OHT rep-

resents the kinetic energy. Kinetic energy is always

strictly positive unless the system is at rest (Asada and

Slotine, 1986).Therefore, the inertia matrix is assumed to

be positive definite. It is noted that the entries of inertia

matrix, i.e., equation (B.7), are time-varying. In other

words, the inertia matrix is structure-configuration-dependent and reflects the instantaneous overall mass

properties of OHT.

2.3. Dynamics of electric subsystem

The armature-controlled DC motors are employed to

regulate the angular displacements of rigid links in our

work. The effects of magnetic flux leakage, hysteresis

and fringing effect are all ignored.In addition, assume

the inductance and resistances of all the three armature

circuits of drive motors are identical. Therefore, the

dynamics of the individual armature circuit for any

drive motor can be described as:

Lad ia

d t Ra ia KB _qr ea 15

where La diag L k

, Ra diag R k

and KB diag Kkb

039

. The superscript, k 1, 2, 3,

denotes the k t h drive motor. L k is the inductance of

the armature, R k the resistance of the armature, Kkbthe electric constant, ia i1 i2 i3 T the armaturecurrent and ea

e 1 e 2 e 3

T the applied armature

voltage. The torque generated by the drive motor canbe described as:

KT ia 16where KT diag Kkt

, k 1, 2, 3, and Kkt is the torque

constant.

2.4. Dynamics of OHT

Since the rigid links are controlled by the drive motors,

the equations of motion of the OHT can be established

as follows:

M qr, qf qr

qf

! fr qr, _qr

ff qr, _qr

! gr qr, _qr, qf, _qf

0

!

0 00 K

!qr

qf

! U

a

0

! A

Tr

0

!l 17a

Lad ia

d t Ra ia KB _qr ea 17b

8>>>>>>>>>>>>>:where Ua KT ia 019

T. In general, the free

response of the electrical systems is much faster than

that of the mechanical systems. On the other hand, the

Tsai et al. 1085


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major time constant of the electrical subsystems is much

smaller than any of the mechanical subsystems. In addi-

tion, for the rigid-flexible pin-joined mechanical system,

the natural frequency of the rigid link is usually much

smaller than that of the first mode of the flexible link.

In other words, the rigid mode and flexible modes exhi-

bit in fairly distinct time scales. That is, the proposalOHT possesses the MTS property for it is composed of

electric (fast mode), rigid (slow mode) and flexible

(intermediate mode) mechanical subsystems. For the

sake of controller synthesis, a simple linearized mathe-

matic model is generally required. Since the OHT is an

MTS system, the coupled and nonlinear dynamic

model in equation (17) can be further simplified by a

singular perturbation technique addressed in next

section.

3. Reduced model of OHT

Since the physical value of the inductance of the arma-

ture at drive motors is much smaller than that of any

other system parameters, the first singular perturbation

parameter can be defined as "1 La.Therefore, equation(17) can be rewritten as follows:

M qr, qf qr

qf

! frqr, _qr

ffqr, _qr

! gr qr, _qr, qf, _qf

0

!

0 00 K

!qr

qf

! U

a

0

! A

Tr

0

!l 18a

"1d ia

d t Ra ia

KB _qr

ea

18b

8>>>>>>>>>>>>>:By setting "1 0 and substituting equation (18b)

into equation (18a), then the mechanical subsystem

with synergistic electrical subsystem can be obtained

as follows:

M ~qr, ~qf ~qr

~qf

" #

fr ~qr, _~qr

ff ~qr, _~qr

264

375 gr ~qr, _~qr, ~qf, _~qf

0

" #

0 0

0 K !

~qr

~qf

! ~Ua

0" #

~ATr

0" # l 19

where ~qr qr qr f and ~qf qf qf f represent the newstate vectors for rigid-mode and flexible-mode links

respectively. For order-reduction method to be under-

taken, qr f and qf f are the fast-time-scale components of

the original state vectors (i.e., qr and qf) for rigid-mode

and flexible-mode links respectively. ~Ua Ua Uf isthe primary control input by excluding the fast-time-

scale component, Uf, of the control input vector Ua.

Since the electrical subsystem (i.e., R-L circuit) is inher-

ently a stable system, in fact the fast component of the

control (to be synthesized later) is not required any

more. That is Uf 0. It is noted that the 5th-ordermathematic model in equation (18) is reduced to 4th-

order in equation (19). By separation of slow and inter-

mediate modes, equation (19) can be rewritten as

follows:

~q r G r r fr ~q r, _~q r

g r ~q r, _~q r, ~qf, _~qf

~ATr lh i

G r f ff ~q r, _~q r

G r f K ~qf G r r ~Ua 20a

~qf Gf r fr ~q r, _~q r

g r ~q r, _~q r, ~qf, _~qf

~ATr lh i

Gf f ff ~q r, _~q r

Gf f K ~qf Gf r ~Ua 20b

8>>>>>>>>>>>>>>>>>>>>>:where G is the inverse of inertia matrix, M, such that its

entry Gi, j, i, j

1, 2, is the submatrix corresponding to

the state vectors ~q r and ~qf respectively, i.e.,

G G r r G r fGf r Gf f

!21

Define K E I "K "K=, the second singular pertur-bation parameter "2 ffiffiffiffip and ~qf "K1&, then equa-tion set (20) can be rewritten as:

~qr G r r fr ~qr, _~qr

g r ~qr, _~qr, "K1&, "K1 _& h

~

A

T

rli G r fff ~qr, _~q r G r f& G r r ~Ua 22a

"K1 & Gf r fr ~qr, _~qr h

g r ~qr, _~qr, "K1&, "K1 _&

~ATr li

Gf fff ~qr, _~qr

Gf f& Gf r ~Ua 22b

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

By setting " 22 0 in equation (22b), the displace-ment vector, "&, for slow mode contributed by flexiblelinks can be obtained as follows:

"& G1f f Gf r fr "q r, _"q r g r "q r, _"q r, 0, 0 "ATr l

Gf f ff "q r, _"q r Gf r "U 23

By substituting equation (23) into equation (22a),

the dynamic of the slow-mode subsystem can be

obtained as:

Mr "qr, 0 "q r fr "qr, _"qr g r "qr, _"qr, 0, 0 "ATr l Us

24



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where Mr is the inertia matrix of rigid links, whose

dimension is corresponding to the state vector q r. "q rand Us are the slow-mode components of state vector

q r and control vector Ua respectively. It is noted that

the 4th-order mathematic model in equation (21) is

further reduced to 2nd-order in equation (23). These

two order reductions are using the same approach:singular perturbation technique.

Define # 1 "K1 &and # 2 "2 "K1 _&, then equation(22b) can be further split into two subsystems:

"2 _#1 #2 25a"2 _#2 Gf r fr g r ~ATr l

Gf fff

Gf f "K#1 Gf r ~Ua

8>>>: 25b

To find the intermediate-mode subsystem, the slow-

mode time scale is stretched by "2, i.e., t="2 and twonew state vectors are introduced as follows:

1 # 1 "K1 "& 26a 2 # 2 26b

By substituting equation (23) and equation set (26)

into equation (25) and setting "2 0, the mathematicmodel of intermediate subsystem leads to:

d1d

2 27ad2d

Gf f "K1 Gf r Um 27b

8>:where Um ~Ua Us.

Since the time constant of the drive motor is much

smaller than that of the manipulators, by singular

perturbation technique (by setting "1 0), the dynam-ics of the manipulators with synergistic drive motors

can be order-reduced. Nevertheless, the lower-order

model is constructed without ignoring the dynamics

of the drive motors so that the armature current ia is

included in the control vector, Ua.On the other hand,

due to the presence of the flexible modes, the dynamics

of the manipulators possess a certain degree of nonli-

nearity and additional order in the mathematic model.

Once again, the nonlinear dynamic model of manipu-

lators can be further simplified and order-reduced by

singular perturbation technique (by setting "2 0) sothat the controller synthesis (to be addressed in the next

section) becomes simpler.

The eigenvalues of the nominal open-loop system are

inspected and shown in Figure 6. It is observed from

Figure 6 that the eigenvalues of the nominal open-loop

system can be clustered into three groups, i.e., the

eigenvalues of rigid-mode mechanical subsystem, flex-

ible-mode mechanical subsystem and electric subsys-

tem. Since there are six marginal poles along the

imaginary axis of the complex plane in addition to

the other stable poles, the system is marginally stable.

This implies that the OHT system is most likely to oscil-

late back and forth (i.e., in harmonic motion) if no any

anti-sway control loop is employed. This is why the

anti-sway control component (to be addressed and

synthesized in the next section) is absolutely required.

4. Composite control

Due to a variety of operation conditions, such as accel-

eration or brake of the trolley, loading/unloading and

mass eccentricity, a few system parameters of the OHT

are uncertain and might be nonlinear. From the view-

point of vibration, the OHT has marginal poles so thatthe carried load suffers from periodic swaying.

Therefore, the composite controller is basically synthe-

sized by integration of the SMC and input shaping

technique. The SMC is inherently, to some extent,

robust to account for system uncertainties while the

input shaping technique is used for anti-swaying. By

the two reduced-order subsystems in equation (24)

and equation (27), a composite control law can be syn-

thesized for the studied OHT in equation set (17).

4.1. Smc for slow-mode dynamics

Define two new variables, 1 "qr and 2 _"qr. Then theslow-mode model, equation (24), can be converted into

a compact form as follows:

_C N 1, 2 Q Us 28where

CT 1 2

29a

Figure 6. Clusters for eigenvalues of nominal OHT unit.

Tsai et al. 1087


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N 2M1r fr gr "ATr l

29bQ 0

M1r

29c

For SMC design (Tsai and Wu, 2008; Tsai, Huangand Chiang, 2009), the sliding hyperplane is defined as

follows:

S P C 0 30

where P is a positive definite matrix. At sliding mode in

which sliding hyperplane is unchanged with respect to

time, i.e.,

_S @P@C

@C

@t 0 31

so that the equivalent control (first component of

SMC), to bring the OHT back to the equilibrium, can

be easily obtained as:

Us e @P

@CQ

!1@P@C

N 32

Mainly to account for the system parameters uncer-

tainties, the switch control (second component of

SMC), Us n, is designed to meet the Reaching

Condition: _S S5 0.

Us n @P@C

Q !1

NU B Sat S= 33

where is positive definite and named as the reachingfactor. The condition N "N

NU B is established todefine the maximum estimated uncertainty of the

system parameters with respect to the nominal system

matrix of the slow-mode subsystem, "N. Sat is thesaturation function, which is introduced to prevent

severe chattering as the SMC is engaged. denotesthe boundary layer thickness. That is,

Sat S

S, S 1Sgn S , otherwise

( 34

where Sgn is Signum function. Therefore, the anti-

uncertainty component of composite control (i.e.,

SMC) can be obtained:

Us Us e Us n 35

4.2. Input Shaping Technique for

Intermediate-mode Dynamics

In general, the deformation of flexible link is a rela-

tively small and the amplitude of the first flexible

mode can be regarded to approximate the vibration

of the flexible links. Hence equation set (27) can berewritten as follows:

d21d2

Gf f "K1 Gf r Um 36

where Um is the intermediate-mode component of con-

trol vector. The intermediate-mode system response

based on equation (36) due to an impulse input can

be described as follows (Singer and Seering, 1990):

ym

Am

!0ffiffiffiffiffiffiffiffiffiffiffiffiffi1 2p e

!0 0 " # sin !0 ffiffiffiffiffiffiffiffiffiffiffiffiffi1 2p 0

37

where Am is the amplitude of the impulse, !0 theundamped natural frequency of the intermediate-

mode subsystem and the damping ratio of theintermediate-mode subsystem. and 0 denote thetime variable and the time instant of the impulse

exerted. The amplitude of vibration of the flexible

links for a multi-impulse input sequence, which is

applied to account for inherent swaying of OHT, is

given by (Singer and Seering, 1990):

Aamp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNij1

Bj cosj

2

XNij1

Bj sin j

2vuut 38awhere

Bj Aj!0ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2p e !0 Nij 38b

j !0ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2p

j 38cB

jis the coefficient of the sine term in equation (37),

corresponding to the Ni th impulse input, Aj theamplitude of the Ni th impulse, j the time instantat which the impulse is applied and Ni the time instantat which the Ni th sequence has just emigrated. Tosuppress the vibration of OHT, Aamp has to be zero

after the time instant at which the input sequence has

emigrated.That is, both squared terms in equation (38a)

have to be vanished and hence referred to as the Zero

Vibration (ZV) constraints:



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XNij1

Aje !0 Nij sin !0

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 2

pj

0 39a

XNi

j1Aje

!0 Nij cos !0ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2

pj

0 39b

For a single-mode system, two impulses can be

employed to completely suppress the system vibration

in a very short period, as illustrated in Figure 7.

However, the natural frequencies of the OHT studied

in our work are position-dependent and multiple. In

order to improve the robustness of the anti-sway con-

trol, an additional set of constraint equations is

included. By differentiating equation (39) with respect

to natural frequency, !0, and setting it to be zero, itleads to:

XNij1

Ajj e !0 Nij cos !0


pj

0 40a

XNij1

Ajj e !0 Nij sin !0


pj

0 40b

equation (40) is referred to Zero Vibration

Derivative (ZVD) constraint. Therefore, the impulse

inputs for drive motors, i.e., the anti-sway control,

can be obtained as:

Um PNi

j1A 1j

PNij1

A 2jPNi

j1A 3j 019

" # T

41

where A i j, i 1, 2, 3, is the amplitude of impulse attime instant j and it has to meet the constraintequations, i.e., equation set (39) and equation set (40).

Because the OHT is inherent marginally stable and

potentially suffered from external disturbance during

the operation mode, the undesired oscillation (for

Figure 8. Schematic diagram of input shaping technique.

Figure 7. System response to (A) two individual impulses (B)

two consecutive impulses. Figure 9. Schematic diagram of composite control.

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example, response to the first impulse in Figure 7) of

the OHT is most likely to be present. If the second

impulse (see Figure 7) with an appreciate amplitude is

applied to the system at the right timing, the vibration

induced by the first impulse can be completely sup-

pressed. In order to describe how the anti-swaying con-

trol is synthesized, the schematic diagram of input

shaping technique is shown in Figure 8. A tilt sensor

is used to provide the actual swaying angle of the car-ried load. The required applied amplitudes and timing

of the input shaper are calculated by equation set (38)

and equation set (40) respectively.

Briefly speaking, the input shaping technique is imple-

mented by convolving the command (to counterbalance

the swaying of the carried load) with a sequence of

impulses. The resultant control (see Figure 8(c)) is

named as the intermediate-mode control component to

discriminate against the slow-mode control component

stated in section 4.1.

4.3. Composite control lawCombining the slow-mode control component, i.e.,

equation (35), and the intermediate-mode control com-

ponent, i.e., equation (41), the composite control for

the OHT system can be obtained as follows:

Ua Us Um 42

The schematic diagram of the control strategy is

illustrated in Figure 9. The errors between the com-

mands and the displacements of rigid links are provided

Figure 12. Position deviation regulation under composite

control.

Figure 11. Position deviations of carried load under SMC.

Figure 10. Acceleration and velocity profiles for trolley.

Table 1. Physical parameter values for overhead crane system

Mass of trolley 20 Kg

Mass of payload 10 Kg

Length of rigid links 0.3 m

Length of flexible links 0.3 m

Mass per length of rigid links 10 Kg/m

Mass per length of flexible links 5 Kg/m

Stiffness of flexible links 0.001 N-m2



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to the slow-mode controller to generate the driving

torque at the DC motors to follow the desired motion

trajectory (i.e., the low-frequency portion). Once the

vibration of the carried load is induced by the margin-

ally stable property of the OHT itself or an external

disturbance, the control sequences by the intermedi-

ate-mode controller are generated for vibration attenu-

ation (i.e., high-frequency portion).Finally, two control

inputs are integrated to constitute the composite

control. It is noted that once the swaying of the carried

load is completely suppressed, the intermediate-mode

control component vanishes. By singular perturbation

technologies, the complicated dynamic model of the

OHT can be simplified and its order is reduced. Based

on the reduced linear dynamic model, the composite

controller can be easily synthesized for dual purposes:anti-sway and anti-uncertainty.

5. Simulations and discussions

The physical parameter values of the proposed OHT

are listed in Table 1. The acceleration and velocity

profiles for trolley are illustrated in Figure 10. The

acceleration of trolley is increased to 10 m/s2 from rest

within 0.1 sec. From 0.1 sec to 0.2 sec, the acceleration

10 m/s2 is retained. At the time instant, 0.2 sec, the

acceleration of trolley starts to be decreased to step

within 0.1 sec. Based on the acceleration profile, the

velocity of trolley is increased to 2 m/s within 0.3 sec.

The position deviation of the carried load under SMC

is shown in Figure 11. The position of the load can be

regulated to the vicinity of the equilibrium in 0.1 sec but

the residual vibration is still present. On the other hand,

the composite control, which is composed by SMC and

ASC (Anti-sway Control), is employed to be compared.

The position deviation regulation on the carried load

under composite control is shown in Figure 12. The

residual vibration is successfully suppressed in 0.6 sec.

On the other hand, it is noted that the aforesaid model

is valid only under the assumption that the parameters

of the OHT models are known beforehand, such as themass of the carried load, the moments of inertia of links

and the positions of joints etc. However, some param-

eters of the OHT changes during operation mode and

are not always known prior to controller synthesis. For

example, some of the motors parameters and the coeffi-

cients of viscous and the static frictions are slowly-vary-

ing. These parameters are altered very slowly so that they

are usually considered as constants by most traditional

approaches. Therefore, the robustness against the uncer-

tainties of the system parameters is hereby examined.

Assume the system stiffness varies up by 20%, the posi-

tion deviation regulation under the proposed control law

is shown in Figure 13. It is obviously observed that the

proposed control law exhibits its robustness, to some

extent, with respect to the parameters uncertainties. In

addition, since the semi-finished products have to be

transported backwards/forwards between stations in

the factory, it has to avoid unexpected collision. As an

illustrative example, the position deviation regulation

under an abrupt impulsive disturbance to the OHT is

shown in Figure 14. The OHT is disturbed at the time

instant, t 10 sec. The position deviations of the carriedload are regulated to the equilibrium point after 0.05 sec.

Figure 14. Position deviation regulation of OHT under

disturbance.

Figure 13. Position deviation regulation of OHT under para-metric uncertainties.

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By computer simulations, it is verified that the composite

control exhibits superior anti-sway capability and

robustness to account for system parameters uncertain-

ties and external disturbances.

As to the hardware implementation, in addition to

DC motors, servo-motors are relatively straightforward

to be controlled by digital computers. The angular posi-tions of the links are measured by using rotary encoders

attached to the motors. The encoder signals are fed

back to the controller through a digital I/O (Input/

Output) board. The angular velocities can be obtained

numerically from the position signals (Burden and

Faires, 2005). The desired torque is estimated from

the desired trajectories and the feedback signals, i.e.,

the angular positions of the links. The torque of the

motors is controlled by the applied currents through

D/A (Digital to Analog) converter.

The trolley and the carried load both move along the

track. The errors between the desired trajectories of

links and the feedback signals are fed to the controller

to regulate the position deviation of the carried load.

Once the swaying is caused by the inherent marginal

poles or the external disturbances, the required impulse

sequence in order to attenuate the undesired swaying

can be real-time synthesized by the proposed controller.

That is, the anti-sway and anti-uncertainty are expected

to be both accomplished by the proposed composite

controller.

6. Conclusions

The composite control, composed of the SMC andinput shaping technique, is synthesized for anti-sway

and anti-uncertainties of the parameters of the OHT

system. Since the OHT system inherently possesses

the marginal poles at origin of the complex plane, the

auto-swaying property, like a pendulum, is fairly evi-

dent in the transportation system in factories. The input

shaping technique is thus employed to suppress auto-

swaying by exerting a sequence of impulses in appro-

priate amplitude and time epoch. In addition, the

system matrix of the OHT system is position-depen-

dent, instead of being constant, so that a certain

degree of uncertainty for the system parameters is pre-

sent. This is the reason why SMC is included in the

composite control strategy.

Unlike the traditional method to equip two or three

passive cables with the OHT unit, the proposed

OHT system is facilitated by a triplet of double-link

arm and a set of DC motors so that the carried load

can be protected from shock, tilt or high-frequency

excitation. However, the novel OHT module then

becomes a three-time-scale system which consists of

slow-mode (rigid links), intermediate mode (flexible

links) and fast-mode (DC motors) subsystems. By

singular perturbation technique, the reduce-order

model of the OHT system is established for the synthe-

sis of composite controller.At last, the efficacy of com-

posite control, to suppress auto-swaying and account

for system parameters, is verified by intensive computer

simulations.

Funding

This research was supported by National Science Council

(Taiwan) under Grant NSC99-2622-E-006-008-CC2. The

authors would like to express their appreciation.

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Appendix A

The holonomic constraint equations in equation (13) can be

described as follows:

h 1 xa1 l1 sin 1 cos l2 sin 1 cos xc 0 A:1

h 2 ya1 l1 cos 1 l2 cos 1 yc 0 A:2

h 3 l1 sin 1 sin l2 sin 1 sin zc 0 A:3

h 4 xa2 l1 sin 2 cos l2 sin 2 cos xc 0A:4

h 5 ya2 l1 cos 2 l2 cos 2 yc 0 A:5

h 6 l1 sin 2 cos l2 sin 2 cos zc 0 A:6h 7 xa3 l1 cos 3 l2 sin3 xc 0 A:7

h 8 ya3 yc 0 A:8

h 9 l1 sin 3 l2 sin 3 zc 0 A:9

h 10 zc=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

y2c z2cq

sin x 0 A:10

h 11 zc=ffiffiffiffiffiffiffiffiffiffiffi

x2c2cq

sin y 0 A:11

Appendix B

The generalized coordinate of the manipulators is composed

of the state vector of rigid links, qr, and the state vector of

flexible links, qf, i.e.,

X

qr qf T2 R126n B:1where

qr Xc T

, 1 2 3 T

,

1 2 3 T

, qf q1f q2f q3f T

qi f qy00 i1 qy00 i n qz0 i1 qz0 i n T2 R2n, i 1, 2, 3:

The generalized force vector is described as follows:

U 019 T2 R12 B:2

where, 1 2 3 T

is the torque vector.

The constraint matrix is described as follows:

Ar A r1 A r2 A r3 2 R1112 B:3

where

A r1 l1

cos 1 cos 0 0 sin 1 0 0

cos 1 sin 0 00 cos 2 cos 00 sin 2 00 cos 2 sin 00 0 sin 30 0 cos 3

266666666664

377777777775

B:4a

A r2 l2

cos 1 cos 0 0 sin 1 0 0

cos 1 sin 0 00 cos 2 cos 00 sin 2 00 cos 2 sin 00 0 sin 30 0

cos 3

266666666664

377777777775

B:4b

A r3

1 0 0 0 00 1 0 0 00 0 1 0 0

1 0 0 0 00 1 0 0 00 0 1 0 0

1 0 0 0 00 0 1 0 0

266666666664

377777777775

B:4c

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Define a vector as follows:

n n 1 n 1 T2 R2n1 B:5

where

n1 1 0 1=3 . . . n=n

2 Rn1 B:6a

i 1, if i is odd0, if i is even&

B:6b

The inertia matrix in equation (14) can be described as

follows:

M

M11 M12 0 M14 M15 0

MT12 M22 0 M24 M25 0

0 0 M33 M34 0 0

MT14 MT24 M

T34 M44 M45 M46

MT15 MT25 0 M

T45 M55 0

0 0 0 MT46 0 M66

2666666664

37777777752 R 126n 126n

B:7

where

M11 1

31l

31 2l21l2

1 0 00 1 0

0 0 1

264

375,

M12 122l1l2

M121 0 0

0 M122 0

0 0 M123

264

375

B:8a; b

M14 l10 0 M141

0 0 M142

0 0 M143

264 375,

M15 2l1l2M151 0 0

0 M152 0

0 0 M153

264

375

B:8c; d

M22 2l2M221 0 0

0 M222 0

0 0 M223

264

375,

M33 mc 0 0

0 mc 00 0 mc

264 375B:8e; f

M24 2l20 0 M241

0 0 M242

0 0 M243

264

375,

M25 2

2l

22

Tn 0 00 Tn 00 0 Tn

264

375

B:8g; h

M34 0 mczc mc

mczc 0 0mcyc mcxc 0

264

375,

M44 mc z

2c y2c

mcycxc 0mcycxc mc z2c x2c mczc

0 mczc "M

264

375

B:8i;j

M45 22L2

0 0 0

0 0 0

M451 M452 M453

264

375,

M46 0 0 0

0 0 02 2l2

Tn sin

2 2l2

Tn sin 0

264

375

B:8k; l

M55 M66 1

22l2I3n, I3n 2 R3n3n B:8m; n

M121 l2 cos 1 1 4

sin 1 1 Tn qy01 B:9a

M122 l2 cos 2 2 4

sin 2 2 Tn qy02 B:9b

M123 l2 cos 3 3 4

sin 3 3 Tn qy03 B:9c

M141 1

21l1 cos 1 cos 2l2 cos 1 cos B:10a

M142 1

21l1 cos 2 cos 2l2 cos 2 cos B:10b

M143 1

21l1 sin 3 2l2 sin 3 B:10c

M151 2

Tn cos 1 1 , M152

2

Tn cos 2 2

B:11a; b

M153 2

Tn cos 3 3 B:11c

M221 1

3 l22

1

2 qTy01qy01, M222

1

3 l22

1

2 qTy02qy02

B:12a; b

M223 1

3l22

1

2qT3fq3f B:12c

M241 12

l2 cos1 cos 2

sin1 cos Tn qy01 B:13a

M242 12

l2 cos2 cos 2

sin2 cosTn qy02 B:13b



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16/16

M243 1

2l2 sin 3

2

cos3

Tn qy03 B:13c

"M M mc 3 1l1 2l2 B:14

M451 cos1 cos Tn , M452 cos2 cos Tn ,M453 sin 3 Tn B:15a; b; c

The stiffness matrix of flexible links is:

K 5

2l42

Kf 0 0

0 Kf 0

0 0 Kf

24

35, Kf diag 124 . . . n4 2 R1n,

B:16

fr fr1 fr2 fr3 fr4 T B:17

where

fr1 l1 fr11 fr12 fr13

T B:18

fr12 1

22l

22

_22 sin 2 2 1

21l1g cos 2 sin

2l2g cos 2 sin 2

2l2 _

22

Tn qy02 cos 2 2

B:19a

fr13 1

22l

22

_23 sin 3 3 1

2 1l1g cos 3

2l2g cos 3 2

2l2 _

23

Tn qy03 cos 3 3

B:19b

fr2 l2 fr21 fr22 fr23 T B:20

fr21 1

22l1l2 _

21 sin 1 1

1

22l2g cos 1 sin

22l1 _

21

Tn qy01 cos 1 1 B:21a

fr22 1

22l1l2 _

22 sin 2 2

1

22l2g cos 2 sin

2

2l1 _22

Tn qy02 cos 2

2

B:21b

fr23

1

22l1l2 _

23 sin 3 3

1

22l2g cos 3

22l1 _

23

Tn qy03 cos 3 3 B:21c

fr3 mc2 _zc _y _x _yyc xc _2y_x _yxc 2 _x _zc yc _2x

2 _xyc 2 _xc _y zc _2x _y _x g

2664

3775 B:22

fr4 mc2 _xzc _zc 2 _xyc _yc _yxc _yc _yyc _xc

_zc _x 2 _yzc _zc 2 _yxc _xc _xxc _yc _xyc _xcfr41

24 35B:23

fr41 2l2 _21 cos1

2l2 sin 1

2

cos1

Tn qy01

2l2 _22 cos1

2l2 sin 2

2

cos2

Tn qy02

l11

21l1 2l2

_21 sin 1 cos

_22 sin 2 cos _23 cos 3 2l2 _23 12 l2 cos3

2

sin 3Tn qy03

B:24

ff

2 2l1l2

_21Tn sin 1 1 12 2l2 _21qy01

2 2l1l2

_21Tn sin 2 2 12 2l2 _22qy02

2 2l1l2 _21Tn sin 3 3 12 2l2 _23qy03031

266664

377775

B:25

gr

gr1 gr2 gr3 gr4

T

B:26

gr1 2l1l2 4 _ 1Tn _qy01 sin 1 1 4 _ 2Tn _qy02 sin 2 2 4 _ 1Tn _qy03 sin 3 3

264

375,

gr2 2l2qTy01 _qy01

_ 1

qTy02 _qy02_ 2

qTy03 _qy03_ 3

2664

3775 B:27a; b

gr3 015 gr31 T, gr4 0 0 0 0 0 0

T

B:27c; d

gr31

4

2l2 _ 1 _qy01 sin 1 cos

Tn

4

2l2 _ 2 _qy02 sin 2

cos Tn 4

2l2 _ 3 _qy03 cos 3

Tn B:28

Tsai et al. 1095

Documents

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