Abstract— Several fuzzy control schemes of overhead cranes have been proposed. Most of these schemes are valid for a specific crane configuration only. Extensive experimentation is needed to apply such schemes to a different crane. This paper presents an approach for automatically creating anti-swing fuzzy logic controllers for two-dimensional overhead cranes with hoisting. Inverse dynamics and desired motion parameters of the overhead crane are used to determine the ranges of the variables of the controllers. The control action is divided into two phases. In the first phase, two fuzzy logic controllers (FLCs) drive the system toward its final destination: travel controller and hoist controller. The second phase is initiated after this point. It includes an anti-swing controller in addition to the travel and hoist controllers. The simulation example presented shows that the proposed controller can successfully drive overhead cranes under various operating conditions.

I. INTRODUCTION he area of swing control of overhead cranes has attracted considerable attention. Abdel-Rahman et al. [15] presented a thorough review of various proposed crane

models. They also presented an extensive review of various control strategies for overhead cranes. Recently, many researchers applied fuzzy logic to control overhead cranes since fuzzy logic can mimic human behavior. The following is a brief overview of recent relevant published research on the use of fuzzy logic in overhead crane control. Nowacki et al. [1]0 proposed two independent feedback loops: a proportional regulator for crane velocity, and a fuzzy proportional derivative controller to improve damping of load swinging. Sousa et al. [2] presented an experimental comparison of different cost functions, using an example of container crane control. They compared conventional quadratic criterion with a conjunctive aggregation of fuzzy goals. Moon and VanLandingham [3] derived a set of energy reducing rules for asymptotic stable feedback control using only the position and angle information, to simultaneously regulate crane load angle as well as the transport distance. Liang and Koh [4] presented a concise approach for

Mohamed B. Trabia is with the Department of Mechanical Engineering, University of Nevada, Las Vegas, Las Vegas, NV 89154-4027, USA (702-895-0957; Fax: 702-895-3936; e-mail: mbt@me.unlv.edu).

Jamil M. Renno is with the Center for Intelligent Material Systems and Structures, Virginia Polytechnic Institute and State University, Blacksburg, VA 24060-0261 USA (e-mail: renno@vt.edu).

Kamal A. F. Moustafa is with the Department of Mechanical Engineering, United Arab Emirates University, P.O.Box 17555, Al Ain, UAE (e-mail: kamalam@uaeu.ac.ae).

implementing an anti-swing algorithm for overhead cranes using fuzzy control. Their algorithm could be incorporated into an existing fuzzy crane controller through creating proper displacements on the fuzzy rule plane. Gutiérrez and Soto [6] used fuzzy logic to resemble the behavior of a pre-calculated trolley acceleration profile. They showed that load angular displacement overshoot does not depend on trolley travel distance. Nalley and Trabia [8] considered the problem of controlling three-dimensional overhead cranes with large payloads. They divided the control action between trolley travel and swing controllers for each motion to enable the tuning of the controllers. A bang-bang profile was used, based on the natural frequency of the payload, to ensure that large amplitudes are not excited. Cho and Lee [9] designed an overall controller consisting of a position servo controller, via loop shaping method, and a fuzzy logic controller to suppress the load swing. Ha et al. [10] designed a controller using an intelligent algorithm. The trolley position, velocity, and load weight were used as parameters to find proper control rules and design a controller. Cho and Lee [11] proposed a controller consisting of a position servo controller and a fuzzy logic controller for a three-dimensional crane system. The position servo controller was used to control the position of the crane and the length of the cable, while the fuzzy logic controller was used to suppress load swing. A gain tuning criteria was provided, and results were demonstrated experimentally. Akbarzadeh and Meghdadi [12] applied a look-up table scheme for fuzzy systems to learn and imitate human control strategy in tracking control of an overhead crane. They focused on fuzzy learning accounting for the random nature of human behavior. Yi et al. [13] proposed an anti-swing and positioning fuzzy controller, based on SIRM (Single Input Rule Modules), for an overhead crane. Lui et al. [14] proposed a GA based on a two-stage fuzzy controller where Two fuzzy sub-controllers were designed to respectively control the swing and trolley displacement. A real-valued GA was used twice in different stages to adjust the parameters of the controllers and ensure the trolley positioning precision, minimizing travel time, and damp swing angle at target position. Lui et al. [16] used two sliding mode controllers (SMC) to control an overhead crane with hoisting. One SMC controller is dedicated to hoist and lower payload while the other controls trolley position and payload swing angle. A fuzzy tuning algorithm is used to adjust the relationship between trolley position control and payload sway control. Sharkawy et al. [15] studied the stability of overhead cranes with fuzzy logic control using Lyapunov functions. Their control law resulted in a stable scheme that was able to exploit the dynamic variables of the

A GENERAL ANTI-SWING FUZZY CONTROLLER FOR AN OVERHEAD CRANE WITH HOISTING Mohamed B. Trabia, Jamil M. Renno, and Kamal A. F. Moustafa

T

0-7803-9489-5/06/$20.00/©2006 IEEE

2006 IEEE International Conference on Fuzzy SystemsSheraton Vancouver Wall Centre Hotel, Vancouver, BC, CanadaJuly 16-21, 2006

627

system in a linguistic way. The proposed method can enable the designer to derive the rule base of the control systematically. Renno et al. [18] proposed a one-phase fuzzy controller that was based on inverse dynamics for overhead cranes. While this controller achieved stability and robustness, it exhibited some trolley overshoot and load swinging during motion.

The authors of [15] stated that “fuzzy logic strategies are especially hard to tune.” The above literature survey confirms this observation. Most of the reviewed research assumed no hoisting while transporting the load (except for [16]). Moreover, efficient crane operation necessitates hoisting or lowering the load during crane travel. In addition, fuzzy logic controllers proposed in most of these publications are valid only for specific crane parameters and/or motion variables. This is a severe restriction on general implementation of these controllers since extensive re-tuning will be required whenever there is a change in the specifications of the crane, payload, and/or motion parameters.

The objective of this paper is to develop a two-phase-anti-swing fuzzy logic controller for an overhead crane that moves while the payload is being hoisted or lowered. The proposed controller is divided into two phases. In the first phase, two fuzzy logic controllers (FLC’s) drive the system toward its final destination: travel controller and hoist controller. The second phase starts after the system reaches its desired target. It includes an anti-swing controller in addition to the travel and hoist controllers. Simulation example shows that the proposed controller can successfully drive overhead cranes under various operating conditions. In addition to proposing fuzzy rules and formulas for spacing the fuzzy variables, the paper presents a novel method for calculating the ranges of the variables of the controllers based on the inverse dynamics of the crane and the parameters of its desired motion. The proposed control strategy can thus be easily modified to work with any modification of desired motion or system parameters.

II. DYNAMIC MODEL OF THE OVERHEAD CRANE WITH VARIABLE CABLE LENGTH

Fig. 1 shows a schematic of an overhead crane with hoisting. The crane is composed of a trolley of mass mT that moves along horizontal straight rails. The trolley has a drum that hoists a cable of length l and linear density ρC. The payload mP is attached to the end of the cable. Based on the model proposed by Lee [5], a system of three equations, is used to represent the dynamics of the overhead crane as given by Equation (1). The cable is assumed inextensible. Friction and slipping are neglected throughout this model of the system. Two forces, Fx at the trolley and Fl along the cable control the crane.

[ ] { } { }FBl

xA =+

θ , (1)

where A, B, and F matrices are defined as:

[ ]

( ) ( )

( )

( )

+

+

+

+

+

+++

=

40sin

2

03

cos2

sin2

cos2

2

lm

lm

ll

mll

m

lml

lmlmm

A

CP

CP

CP

CP

CP

CPCTP

ρθ

ρ

ρθ

ρ

θρ

θρ

ρ

,

(2) { }

( ) ( )

( )

( ) ( ) ( )( )( )( )

( ) ( )

+

−+−

+−

++++

+

+

+−

−

=

θρ

θθρθρρ

θθρθρθθρ

ρθρθ

θθρθθρ

cos

cos21

21

81

cos42sin221

23

2cos

sin21sin

21

22

2222

glm

xxlllml

llmllglml

xlml

llml

B

CP

CCPC

CPCCP

CCP

CPC

, (3)

{ }

=

l

x

F

FF 0 . (4)

Fig. 1. Schematic representation of an overhead crane with hoisting

III. DEVELOPMENT OF THE FUZZY LOGIC CONTROLLERS Study of the equations of motion of the crane system that

is presented in the previous section shows that x, θ, and l are coupled. Attempting to design a single fuzzy logic controller for an overhead crane that is traveling and hoisting simultaneously can be daunting as it may be difficult to establish fuzzy relations between all the variables based on intuition and observation of the system. Therefore, control action is distributed among three controllers. Each controller has two inputs and a single output. Errors with respect to desired motion and velocity profiles of the trolley, cable swinging, and payload hoisting are used as inputs to the respective controllers.

In this work, the desired trolley motion profile is a bang-bang acceleration profile. This profile excites two opposite load swings and ends in zero values of the swing angle and angular velocity at the end of the active motion. These concepts are extended to the case when the load is being

628

hoisted. A desired payload swing motion profile are proposed in Appendix A. Since swinging is bounded during the desired motion phase by the choice of the bang-bang trolley motion profile, two controllers are used during this phase (t ≤ tD), Fig. 2. These controllers are: 1. Travel Fuzzy Logic Controller (TFLC) whose inputs are

displacement and velocity errors of the trolley travel with respect to desired values, ex and edx. The output of the controller is a force signal to the trolley actuator, FT.

2. Hoisting Fuzzy Logic Controller (HFLC) whose inputs are displacement and velocity errors of the payload with respect to desired values, el and edl. The output of the controller is a force signal to the drum actuator, Fl. Gravity effects of the payload are canceled using feed-forward control.

After the desired motion is reached (t = tD), a second phase of the controller is implemented, Fig. 3. This phase differs from the first one in two aspects: • Output range of the TFLC is significantly reduced as the

objective of the controller at this phase is to keep the trolley at the desired position.

• A third controller is used to suppress remaining load swinging. This controller is labeled Anti-Swing Fuzzy Logic Controller (ASFLC). The inputs of this controller are angular displacement and velocity errors of the payload with respect to desired values, eθ and edθ. The output of ASFLC is a force signal to the trolley actuator, Fθ. FT and Fθ are added as Fx.

Fig. 2. Distributed fuzzy logic controller for an overhead crane with

hoisting during desired motion time (Gravity feedforward is not shown) A. Design of the Controllers

Fuzzy rules for the three controllers are derived based on observing the behavior of the crane. Gaussian membership functions are used to describe the fuzzy variables of all the controllers.

22

2)(

)( σµ

cx

ex

−−

= (5) The center of gravity method is used for defuzzification, [7].

Fig. 3. Distributed fuzzy logic controller for an overhead crane with

hoisting after desired motion time (Gravity feedforward is not shown) Design of the TFLC and HFLC

To simplify the controllers, three membership functions (negative big (NB), zero (Z), and positive big (PB)) are used to describe each of the inputs of the two controllers. Five membership functions (negative big (NB), negative small (NS), zero (Z), positive small (PS), and positive big (PB)) are used to describe the output of each of the two controllers. The rules of the two controllers are listed in Table I and Table II respectively. The rules for the TFLC and HFLC are based on observing the behavior of a PD controller for an inertial system.

Table I

RULES OF THE TROLLEY FUZZY LOGIC CONTROLLER (TFLC) ex ↓

edx ↓ NB Z PB NB NB NS Z Z NS Z PS

PB Z PS PB

Table II RULES OF THE HOISTING FUZZY LOGIC CONTROLLER (HFLC)

el ↓ edl ↓ NB Z PB NB NB NS Z Z NS Z PS PB Z PS PB

The membership functions for all variables are

symmetrical about the zero value of each variable. Membership functions for the nth input variable of a controller j are arranged according to the following equations. In the sequel, the subscripts i and o will refer to input and output, respectively.

nijPBnijnijPB ca ,,,,,,,, =σ , (6)

629

nijPBnijnijZ ,,,,,,,, σβσ = , (7) 0,,, =nijZc . (8)

where α j,i,n and β j,i,n are design parameters controlling the mean and the standard deviations of the Gaussian membership functions. These two variables in addition to cPB,j,i,n control the shape and distribution for other membership functions for a variable. Due to symmetry of membership functions,

nijPBnijNB ,,,,,, σσ = , (9)

nijPBnijNB cc ,,,,,, −= . (10) These design parameters are to be selected by the user to achieve best performance.

Similarly, the membership functions for the output variable of a controller j are arranged according to the following equations:

1,,,1,,1,,, ojPBojojPB cασ = , (11)

1,,,1,,1,,, ojPBojojPS σβσ = , (12)

1,,,1,,1,,, ojPSojojZ σβσ = , (13)

1,,,1,,1,,, ojPBojojPS cc β= , (14)

01,,, =ojZc . (15) Due to symmetry, equations similar to Equations (7), (8) and (10) can also be written for the parameters of the NB and NS membership functions.

Choosing the fuzzy rules and shapes of membership functions for TFLC and HFLC results in a control surface, Fig. 4, that is flat near the zero point of input variables as well as the extremes of the range. This surface design ensures that these two controllers do not excite the system near zero errors or when it is approaching stability.

Fig. 4. Typical control surface for TFLC/ HFLC

Design of the ASFLC

Experimenting with the model of the crane showed that a more complex controller is needed to deal with the swinging of the payload. Therefore, ASFLC uses seven membership functions (negative big (NB), negative medium (NM), negative small (NS), zero (Z), positive small (PS), positive medium (PM), and positive big (PB)) to describe each of the two inputs and the output of this controller. The rules of the controller are listed in Table III. The rules for the ASFLC

are selected to produce input whenever the payload is swinging away from the vertical direction.

Table III RULES OF THE ANTI-SWING FUZZY LOGIC CONTROLLER (ASFLC)

eθ ↓ edθ ↓

NB NM NS Z PS PM PB NB PB PB PM PM PS PS Z NM PB PM PM PS PS Z NS NS PM PM PS PS Z NS NS Z PM PS PS Z NS NS NM PS PS PS Z NS NS NM NM PM PS Z NS NS NM NM NB PB Z NS NS NM NM NB NB

The membership functions for all variables are symmetric about the zero value of each variable. Membership functions of the controller variables are arranged similar to the method used in the previous section. Choosing the fuzzy rules and shapes of membership functions for ASFLC results in a control surface, Fig. 5, that is only flat at two corners.

Fig. 5. Typical control surface for ASFLC

B. Defining the Ranges of the Controllers’ Variables Using

Inverse Dynamics The previous section shows that the proposed controllers

depend on the ranges of input and output variables. Instead of leaving these ranges static or empirically modifying them, this work proposes a method for adjusting these ranges whenever the parameters of the crane or the desired crane path change. For each of the three controllers, the ranges of input variables are chosen as a function of the desired crane motion and system parameters. The remainder of this section presents an approach to obtain the range of the desired forces for each controller.

The absolute values of the desired forces, Appendix B, correspond to the center of gravity of the PB membership function of the output variables of the controllers. For an output variable,η, the maximum value of the PB membership function, cPB,η,ο,1, can be calculated using the following equation:

630

( )

( )

∫

∫

∞−

−−∞−

−−

=1,,,

21,,,

21,,,

1,,,2

1,,,

21,,,

2

2

1,,,oPB

oPB

oPB

oPB

oPB

oPB

c cx

c cx

oPB

dxe

dxex

CGη

η

η

η

η

η

σ

σ

η . (16)

Solving the above equation symbolically shows that ( )1,,,1,,1,,, oPBooPB CGRc ηηη = , (17)

where

1,,1,,

22

2

ooR

ηη

αππ

−= . (18)

Defining the Ranges of the Parameters of TFLC

The ranges of the variables of TFLC are defined as follows:

DePBTFLCePBTFLC xcxx

max,,,, γ= , (19)

DePBTFLCePBTFLC xcdxdx

max,,,, γ= . (20)

xePBTFLC ,,γ anddxePBTFLC ,,γ are design parameters controlling

the range of the TFLC inputs. The center of gravity of the PB membership function of the output force of TFLC is calculated using

( ) ( ) ( )( )( )

( ) ( )( )DlD

DDxDFPBTFLC

Fl

TFCG

T

θ

θθ

maxsinmaxmin

maxcosmaxmax,,

+

+= . (21)

The above equation uses the inverse dynamics of Appendix B to calculate the necessary force that the TFLC should produce to generate the desired motion. In the second phase of the control (t > tD), the second term is dropped as it is to be used in the anti-swing controller, which means that the above equation reduces to,

( ) ( ) ( )( )DlDxDFPBTFLC FFCGT

θmaxsinmaxmax,, += . (22) The center of the output PB membership function is,

( )TT FPBTFLCoTFLCFPBTFLC CGRc ,,1,,,, = . (23)

Defining the Ranges of the Parameters of HFLC

The ranges of the variables of HFLC are defined as follows:

( ) ( )DePBHFLCePBHFLC tllcll

−= 0,,,, γ , (24)

DePBHFLCePBHFLC lcdldl

max,,,, γ= . (25)

lePBHFLC ,,γ and dlePBHFLC ,,γ are design parameters

controlling the range of the HFLC inputs. The center of the PB membership function of the output force of HFLC can be calculated by first identifying the center of gravity of this function using the following equation:

( )lDFPBHFLC FCGl

max,, = . (26) The center of the output PB membership function is,

( )ll FPBHFLCoHFLCFPBHFLC CGRc ,,1,,,, = . (27)

Defining the Ranges of the Parameters of ASFLC

Based on the desired swinging payload of Appendix B, the ranges of the variables of ASFLC are defined as follows:

max,,,, DePBASFLCePBASFLCc θγθθ

= , (28)

max,,,, DePBASFLCePBASFLC ddc θγ

θθ= . (29)

θγ ePBASFLC ,, and

θγ

dePBASFLC ,, are design parameters controlling the range of the ASFLC inputs. As the trolley motor is used to dampen swinging errors, it should produce a force that is necessary to counteract this swinging. The center of the PB membership function of the output force of ASFLC can be calculated using

( ) ( )( )( )l

TCG DD

FPBASFLC minmaxcosmax

,,θθ

θ= . (30)

The center of the PB membership function, θFPBASFLCc ,, , is

( )θθ FPBASFLCoASFLCFPBASFLC CGRc ,,1,,,, = . (31)

IV. CASE STUDY The controllers developed in the preceding sections are

applied to an overhead crane with the following specifications:

mT = 20 Kg, mP =10 Kg, ρC= 0.1 Kg/m. Table IV shows the values of the design parameters that achieved good overall performance.

Table IV

DESIGN PARAMETER VALUES Design Parameter Value

2,1,,,,,, === nniASFLCniHFLCniTFLC ααα 0.3

1,,1,,1,, oASFLCoHFLCoTFLC ααα == 0.3

2,1,,,,,, === nniASFLCniHFLCniTFLC βββ 0.5

1,,1,,1,, oASFLCoHFLCoTFLC βββ == 0.5

xePBTFLC ,,γ 0.01

dxePBTFLC ,,γ 0.10

lePBHFLC ,,γ 0.01

dkePBHFLC ,,γ 0.10

θγ ePBASFLC ,, 0.1

θγ

dePBASFLC ,, 1

A simulation scenario is tested. The total simulation time

is 40 seconds and the sampling frequency is 10 Hz. Table V shows the parameters of the crane motion. The duration of both the acceleration and deceleration periods, ta, of the trolley motion are six seconds. Maximum desired swing, θDmax, is equal to five degrees. The payload goes through a three-stage motion. The payload is hoisted and lowered at nine seconds. The duration of acceleration and deceleration periods for both hoisting and lowering are three seconds. The same crane and motion parameters was used by Liu et al. in [16]. The results of the proposed controller are compared to the results of that algorithm.

Fig. 6 through Fig. 9 show the motion time history of the overhead crane. The trolley experiences limited amount of overshoot; namely, 0.30%. Overshoot of the trolley motion occurs immediately after the end of the desired motion stage

631

when the anti-swing controller is activated. Similarly, overshoot of the cable motion is of the same order of the trolley motion; namely, 0.36%. The maximum actual swinging of the load is 3.4 degrees. Fig. 10 and Fig. 11 show the forces of the trolley and hoisting, respectively. Travel and anti-swing forces cancel each other largely after the end of the desired motion stage. Force-time histories show that the trolley and hoisting forces roughly correspond to the bang-bang profile used for generating the desired motion.

Table V

CHARACTERISTICS OF THE SIMULATION SCENARIO Desired

Trolley Travel (m)

Active Motion Time,

tD, (s)

Initial Cable Length (m)

Final Cable Length (m)

10 5

5 5 40 30

5 10

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

45

Time(sec)

Tra

vel(m

)

ActualDesiredLiu et al. [16]

Fig. 6. Motion of the trolley for the short trolley travel

0 5 10 15 20 25 30 35 404

5

6

7

8

9

10

11

Time(sec)

Hoi

st(m

)

AcutalDesiredLiu et al. [16]

Fig. 7. Hoisting of the payload

0 5 10 15 20 25 30 35 40-5

-4

-3

-2

-1

0

1

2

3

4

5

Time(sec)

Sw

ing

Ang

le(d

egre

es)

Actual

DesiredLiu et al. [16]

Fig. 8. Swinging of the payload

0 10 20 30 40

-5

0

5

10

15

20

25

30

35

40

45

xy

Fig. 9. Motion of the crane payload (Motion is plotted every second)

0 5 10 15 20 25 30 35 40-15

-10

-5

0

5

10

15

Time(sec)

Forc

e(N

)

Travel force

Anti-Swing ForceTotal Trolley Force

Fig. 10. Travel and anti-swing forces

0 5 10 15 20 25 30 35 40-112

-110

-108

-106

-104

-102

-100

-98

Time(sec)

Forc

e(N

)

Hoisting Force

Fig. 11. Hoisting force

632

V. CONCLUSIONS The paper presents a two-phase distributed fuzzy logic

controller for an overhead crane that moves while hoisting the payload. The control action is divided into two phases. The objective of the first phase is to drive the payload toward the target location. The control action in this phase is distributed between two controllers as follows: • Trolley Fuzzy Logic Controller (TFLC) whose inputs

are displacement and velocity errors of the trolley with respect to desired values. The output of the controller is force signal to the trolley actuator.

• Hoisting Fuzzy Logic Controller (HFLC) whose inputs are displacement and velocity errors of the payload with respect to desired values. The output of the controller is force signal to the drum actuator. Gravity effects of the payload are canceled using feed forward control.

The second phase is activated after the end of the desired motion stage. In this phase, the output range of the TFLC is significantly reduced as the objective of the controller is to maintain the trolley at its final destination. In addition to the two controllers described above, this phase has a third controller: • Anti-Swing Fuzzy Logic Controller (ASFLC) whose

inputs are angular displacement and velocity errors of the payload with respect to desired values. The output of the controller is force signal to the trolley actuator.

The outputs of TFLC and ASFLC are added to drive the trolley.

The paper presents a novel method for adjusting ranges of the variables of the inputs and outputs of the three fuzzy logic controllers based on the systems characteristics and desired motion using inverse dynamics. The relative shapes and distribution of membership functions with respect to each other are maintained fixed. The proposed method, which can be extended to other dynamic systems, has the advantage of avoiding guessing acceptable ranges of the variables.

The two-phase controller is applied to an overhead crane that travels while hoisting, and lowering a payload. The results show that the proposed controller produces stable motion with minimal overshoot and payload swing. The values of the design parameters, α, β, and γ that control the shape of the membership functions and the ranges of the input variables of the fuzzy controllers are selected in the present study to obtain good performance by simulation. The results of the paper are compared to results of a previously published work and the superiority of the present approach is made clear. The optimum choice of these design parameters to satisfy an appropriate criterion function is currently under investigation.

APPENDIX A. DESIRED SWINGING OF THE PAYLOAD The crane system is under-actuated as it does not have an

actuator that supplies torque to counteract the swinging of the payload. Observation of actual crane motion show that a typical payload swinging profile has the following features:

a. Payload swings opposite to the trolley desired motion. After reaching a maximum value, swinging goes close to zero. b. Swinging angle stays close to zero. c. Near the end of the trolley desired motion, the payload swings in the opposite direction of step (a). Swinging angle should come close to zero at the end of this stage.

In this work, the duration of each of the first and third stages is selected to be, ta, which matches the desired motion profile of the trolley. Fig. A.1 shows the stages of this motion.

ta

taTime

Swing Angle ( D)

tD0

Dmax

Dmax

Fig. A.1 Desired swing profile

This motion can be achieved by several means. In this paper a bang-bang acceleration profile as in Fig. A.2 is used to create the desired swing profile. The following maximum acceleration is used to achieve this motion:

2max

max16

a

DD

tθθ = (A.1)

ta

taTime

Swing Angular Acceleration ( )

tD0

Dθ

maxDθ

maxDθ−

Fig. A.2 Desired swing acceleration profile

APPENDIX B. INVERSE DYNAMICS OF THE OVERHEAD CRANE MODEL

Based on the equations of the crane dynamics, the forces needed to produce a desired path (xD, θD, lD) can be expressed as:

633

( )

( )

( ) ( )

( )

+

++

−

−

+

+

+

++

=

DCDCPDD

DDDCDDDPDC

D

D

D

T

DDCP

DDDCP

DCTP

xD

xlml

llml

l

x

lm

llm

lmm

F

ρθρθ

θθρθθρ

θ

θρ

θρ

ρ

232cos

sin21sin

21

sin21

cos21

2222 , (B.1)

( )

( ) ( )( )

++

++

+

+

+

=

DDDCDDP

DCDDDDCPD

D

D

D

T

DDCP

DDDCP

D

llm

llglml

l

xllm

llm

T

θθρθ

ρθθρ

θρ

θρ

θ

cos4

2sin221

031

cos21

2

, (B.2)

( )

( )( ) ( ) ( )DDCPDDDDDC

DDDCPDC

D

D

D

T

DCP

DDCP

Dl

glmxxl

llml

l

x

lm

lm

F

θρθθρ

θρρ

θρ

θρ

coscos21

21

81

410

sin21

22

+−+−

+−

+

+

+

=

. (B.3)

REFERENCES [1] Nowacki, Z., Owczarz, D., Woźaniak, P., “On

Robustness of Fuzzy Control of an Overhead Crane,” Proceedings of the IEEE International Symposium on Industrial Electronics, v: 1, 17-20 June 1996.

[2] Sousa, J.M., Babuška, R., Bruijn, P., Verbruggen H.B., “Comparison of Conventional and Fuzzy Predictive Control,” IEEE International Conference on Fuzzy Systems, v: 3, 1996, pp. 1782-1787.

[3] Moon, M.S., VanLandingham, H.F., “Fuzzy Feedback Control of Crane Using Energy Reducing Rules,” Proceedings of the 1997 Artificial Neural Networks in Engineering Conference, ANNIE’97, Nov 9-12 1997, St. Louis, MO, USA.

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