296 Quoc Toan Truong, Anh Huy Vo and Quoc Chi Nguyen VCM2012 Nonlinear Adaptive Control of a 3D Overhead Crane Quoc Toan Truong, Anh Huy Vo and Quoc Chi Nguyen Ho Chi Minh City University of Technology e-Mail: tqtoan@hcmut.edu.vn ; vahuy@yahoo.com; nqchi@hcmut.edu.vn Tóm tắt Trong bài báo này, vấn đề phát triển một bộ điều khiển phi tuyến của hệ thống cầu trục trong không gian 3D được trình bày. Mô hình phi tuyến của cầu trục được xây dựng với giả thuyết hệ thống có khối lượng tập trung (lumped mass model). Giải pháp điều khiển phi tuyến sẽ nhằm đạt được ba mục tiêu: (i) điều khiển hệ thống crane đến vị trí mong muốn, (ii) triệt tiêu dao động lắc của tải trong khi di chuyển, (iii) điều khiển bám cho vận tốc nâng hạ tải. Trong bộ điều khiển phi tuyến, luật ước lượng các thông số chưa xác định rõ của hệ thống (lực ma sát và khối lượng tải) được sử dụng. Với luật điều khiển được thiết kế, ổn định tiệm cận của hệ thống cầu trục được chứng minh bằng phương pháp Lyapunov. Hiệu quả của luật điều khiển được kiểm chứng thông qua mô phỏng số. Abstract: In this paper, a nonlinear adaptive control of a 3D overhead crane is investigated. A dynamic model of the overhead crane is developed, where the crane system is assumed as a lumped mass model. Under the mutual effects of the sway motions of the payload and the hoisting motion, the nonlinear behavior of the crane system is considered. A nonlinear control model-based scheme is designed to achieve the three objectives: (i) drive the crane system to the desired positions, (ii) suppresses the vibrations of the payload, and (iii) velocity tracking of hoisting motion. The nonlinear control scheme employs adaptation laws that estimate unknown system parameters, friction forces and the mass of the payload. The estimated values are used to compute control forces applied to the trolley of the crane. The asymptotic stability of the crane system is investigated by using the Lyapunov method. The effectiveness of the proposed control scheme is verified by numerical simulation results. 1. Introduction Overhead crane systems are widely used to move heavy cargo from one place to another place in factories and harbors. A crane is naturally an underactuated mechanical system, in which the number of actuators is less than the degree of freedom of the system. For an overhead crane, the degree of freedom is five (i.e., trolley and rail positions, rope length, and two sway angles), but the number of actuators is three (i.e., trolley, rails, and hoisting motors). To improve the transferring efficiency, the trolley and rails should travel as fast as possible. However, fast trolley/rail movement will cause sway of the load, which is dangerous for the operator and the crane system. Therefore, the research field of crane control (i.e., sway vibration control, trolley motion control, and hoisting motion control) is focused by many researchers. 3D models of overhead crane systems were developed in a number of researches. The development of modeling and control method for a 3D crane has been reported in Lee (1998). Nonlinear dynamic modeling and analysis of a 3D overhead gantry crane system with system parameters variation was introduced Ismail et al. (2009). These researches developed 3D models of overhead cranes with four degrees of freedom (DOF), i.e., trolley and rail positions and two sway angles. However, the two factors, hoisting motion and effects of friction were not considered. In practice, it should be noted that the variation of the length of cable affects significantly to the crane dynamic. Moreover, it is well known that the consideration of the friction forces is very important in many mechanical systems, especially, in crane systems, the effects of friction forces are considerable under heavy payload. In this paper, we introduce a 3D model of an overhead crane system having five DOF (trolley and rail positions, rope length, and two sway angles) under two friction forces (appearing between trolley and horizontal rails and between horizontal rails and vertical rails). In crane control, there are two approaches, semi- automated and automated approaches. In the first approach, operator is kept in the loop and Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 297 Mã bài: 64 dynamics of the load are modified to make his job easier. A number of researchers developed damping controllers employing feedback signals of the load sway angles and their rate (Henry et al., 2001; Masoud et al., 2002). Robinett et al. (1999) developed a filter to remove noise of the input to avoid exciting the load near its natural frequency. Balachandran et al., 1999 used a mechanical absorber to suppress the vibration of the payload. Implementation of these methods requires considerable energy consumption, which is not cost effective. In the second approach, the operator is removed from the loop and the operation is completely automated. Many various techniques can be applied for this. The first technique is based on generating trajectories to transfer the load to its destination with minimum swing. Sakawa and Shindo (1982), in which optimal-velocity profiles of the trolley that minimize the sway angle and its derivative were proposed. In their work, the trolley motion was split into five different sections. Another important method of generating trajectories is input shaping, which consists of a sequence of acceleration and deceleration pulses. These sequences are generated such that there is no residual swing at the end of the transfer operation (Karnopp et al., 1992; Teo et al., 1998; Singhose et al., 1997; Singhose, Porter, Kenison, & Kriikku, 2000; Hong, Huh, & Hong, 2003; Sorensen, Singhose, & Dickerson, 2006; Ngo & Hong, 2009; Kim & Singhose, 2010). Unfortunately, it often meets great difficulty when trying to obtain these system coefficients which vary with the changes of the payload or the rope length, and coefficient friction. As mentioned before, the objective of the crane control is to move load from point to point and at the same time minimize the load swing. Two tasks will be done by designing two feedback controllers: controller for prompt sway suppression by a proper feedback of the swing angle and its rate and controller designed to make trolley follow a reference trajectory with trolley position and velocity are used for tracking feedback. The tracking controller can be either a classical Proportional-Derivative (PD) controller (Henry, 1999; Masoud 2000) or a Fuzzy Logic Controller (FLC) (Yang et al., 1996; Nalley and Trabia, 1994; Lee et al., 1997; Ito et al., 1994; Al-Moussa, 2000). Similarly, the anti-swing controller is designed by different methods. Henry (1999) and Masoud (2002) used delayed-position feedback, whereas Nalley and Trabia (1994), Yang et al. (1996), and Al-Moussa (2000) used FLC. Generally, the cable length is considered in the design of the anti-swing controller. Lee, Liang, & Segura (2006) proposed a sliding-mode anti-swing control for overhead cranes to realize an anti- swing trajectory control with high-speed load hoisting. A feedback linearization control of container cranes with varying rope length proposed by Park, Chwa, & Hong (2007); these studies were used for 2-D modeling overhead crane. The friction between the trolley and the rails, the varying payload weight (from 12 Ton to 72 Ton) are uncertainties in crane dynamics. Adaptive controller for 2D modeling overhead crane with the friction force model (Aschemann, 2000) was proposed by Ma, Fang & Zhang (2008). An adaptive tracking control of 3-D overhead crane systems was investigated by Yang et al. (2006) but they didn’t consider influence of hoisting mechanism to the system. We extend this research by developing a nonlinear adaptive control for 3D overhead crane under friction forces, where the hosting motion is considered. In this paper, a modeling of 3D overhead crane is derived based on using Lagrange-Euler equations, where a five degrees of freedom-crane (motions of trolley on rail, motion of rail, motion rolling the rod by hosting mechanism, two sways of payload) system is considered. Effects of friction forces are also included. A nonlinear adaptive controller is proposed to drive the crane to its desired position and to suppress the sway motion of payload. Adaptation laws are used to estimate the unknown parameters, i.e., coefficients friction and the payload mass. Lyapunov method is employed to investigate stability of the crane system under the proposed controller. The effectiveness of the proposed control method is illustrated by numerical simulation results. 2. Dynamics of a 3D overhead crane This section provides detail description on the modeling of the 3D overhead crane system, as a basis of a simulation environment for the study on the effects of parameters variation to the system. The Euler-Lagrange formulation is considered in characterizing the dynamic behavior of the crane system incorporate payload and length of the rope. Fig. 1 shows the swing motion of the load caused by trolley movement in XYZ is the inertial coordinate system. 298 Quoc Toan Truong, Anh Huy Vo and Quoc Chi Nguyen VCM2012 Table 1 Parameters of 3D overhead crane system. Symbol Description x(t) Trolley position in X-direction y(t) Trolley position in X-direction l(t) rope length c m Trolley mass r m Rail mass p m Mass of the payload ( ) t g Swing angle of the payload in OXYZ coordinate. Angle ( ) t g has two components ( ) t q and ( ) t f , which are angle  1 0 3 A A A and  2 0 3 A A A , respectively c F Control force vector includes two components X F and Y F applied to the trolley in the X- and Y-directions, respectively l F Control force applied to the hoisting cable and X Y f f Friction forces in the X- and Y- direction, respectively Fig. 1 Three-dimension overhead crane. The following assumptions are made: (i) The payload and the trolley are connected by a massless, rigid link. (ii) The position of the trolley and the swing angle of the payload are measureable. (iii) The trolley mass is unknown. (iv) The unknown friction forces, which occur in the contact surfaces between the wheels of the trolley and the rails Y, the wheels of the rail Y and the rails X are unknown (v) The time-varying length of the rope. As shown in Fig. 1, the rail, the trolley, and the payload position vectors are given as follows: Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 299 Mã bài: 64 [ ,0,0], [ , ,0], [ sin cos , sin , cos cos ], r c p x x y x l y l l q f f q f       r r r (1) where x and y are the trolley position in X- and Y- directions, respectively. 5-DOF-crane model yielding the generalized coordinate vector 5 ( ) t R q is defined as follows:   T ( ) ( ) ( ) ( ) ( ) ( ) t x t y t l t t t q fq . (2) The forces applied to the system are given by [( ) ( ) 0 0]. x cx y cy l F f F f F    F (3) The friction forces in the X- and Y-directions respective are given as follows. ( ) ( ), ( ) ( ), cx x cy y f t c x t f t c y t     (4) where and x y c c are the viscous friction coefficients in X- and Y-directions, respectively. The total kinetic energy K and the potential energy P of the crane system are derived as , , trolley rail payload payload K K K K P P     (5) where 1 2 car c c c K m r r  , 1 2 rail r r r K m r r  , 1 2 payload p p p K m r r  , (1 cos cos ) payload p P m gl q f   . The equations of motion using the Lagrange Euler equations are derived as follows: ( 1,2,3,4,5), i i i d L L T i dt q q                     (6) with L K P   . The dynamic equations (6) can be rewritten as: ( ) ( , ) ( ) , m M q q C q q q G q u       (7) where 11 13 14 15 22 23 25 31 32 33 41 44 51 52 55 0 0 0 0 0 0 0 0 0 0 m m m m m m m m m m m m m m m                        M 13 14 15 23 24 25 34 35 43 44 45 53 54 55 0 0 0 0 0 0 0 0 0 0 0 C C C C C C C C C C C C C C                        C 0 cos cos sin cos cos sin 0 p p p p m g m g m gL m gL q f q f q f                         G     T ( ) ( ) 0 0 f fc X x Y y l F c x t F c y t F            u u u   11 13 14 15 sin cos cos cos sin sin c r p p p p m m m m m m m m l m m l q f q f q f        22 23 25 sin cos c p p p m m m m m m m l f f     31 32 33 sin cos sin P P P m m m m m m q f f    41 2 2 44 cos cos cos P P m m l m m l q f f   13 14 15 cos cos sin sin cos cos sin cos cos sin sin sin sin cos cos sin p p p p p p p p C m m C m l m l m l C m l m l m l q fq q ff q f q fq q ff q f q ff q fq                  23 25 cos cos sin p p p C m C m l m l ff f ff       2 34 35 cos P P C m l C m l fq f       2 43 2 2 44 2 45 cos cos sin cos sin cos P P P P C m l C m l l m l C m l fq f f ff f fq          53 2 54 55 sin cos P P P C m l C m l C m ll f f fq       300 Quoc Toan Truong, Anh Huy Vo and Quoc Chi Nguyen VCM2012 where 5 5 ( ) R  M q is inertia matrix of the crane system, 5 5 ( ) m R  C q represent the centripetal Coriolis, and 5 ( ) R G q is the gravity term. Based on the structure of ( ) M q and ( , ) m C q q  given by (7), it should be noted that the following skew- symmetric relationship is satisfied: 5 1 ( ) ( , ) 0, 2 T m R                ξ M q C q q ξ ξ   , (8) where ( ) M q  represent the time derivative of ( ) M q and ( ) M q can be upper and lower bounded by the following inequality: 2 2 T 5 1 2 ( ) , n n R    ξ ξ M q ξ ξ ξ , (9) where 1 2 and n n R  are positive bounded constants. 3. Control design For convenience, we define a generalized coordinate as follows:   T , m a  q q q (10) where   T ( ) y( ) ( ) m x t t l t q ,   T ( ) ( ) a t t q fq . The equations of motion of the overhead crane (7) are partitioned in the following: ( ) ( ) , (11) 0 0 mm ma m mm ma m m am aa a am aa a a mf mcf                                                             M M q C C q G q M M q C C q G q u u     where 11 13 22 23 31 32 33 0 0 mm m m m m m m m              M , 14 15 25 0 0 0 ma m m m              M 41 51 52 0 0 0 am m m m          M , 44 55 0 0 aa m m          M 13 23 0 0 0 0 0 0 0 mm C C              C , 14 15 25 34 35 0 ma C C C C C              C 43 53 0 0 0 0 ma C C          C , 44 45 54 55 aa C C C C          C sin cos ( ) cos sin p a p m gl q m gl q f q f            G , x mf y l F F F              u , ( ) ( ) 0 x mcf y c x t c y t              u   . To achieve the control objective, with given desired signals , , d d d q q q   (which are assumed to be bounded), it needs to determine a control law mf u that guarantees the asymptotically convergence of q to d q . The error signals are defined as: T T T , d m a          e q q e e (12) where   T m m dm d d d T x y l x x y y l l e e e              e q q   T T a a da d d e e q f q q f f             e q q where , , , , d d d d d x y l q f are defined trajectories of , , , , x y L q f , respectively. We define r q  as following: rx ry dm m m rm rl r d da a a ra r r q q q q q q f                                              q K e q q q Ke q K e q            (13) where 0 0 0 0 0 , 0 0 m a              K K K 1 2 3 0 0 0 0 0 0 m k k k              K , 4 5 0 0 a k k          K , and m a K K are arbitrary positive definite matrices. We define a combined error: m rm m r a ra a                       q q s s q q q q s       (14) Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 301 Mã bài: 64 Then, using (11), the dynamics in terms of the newly defined signals m s and a s can be derived as: , mm ma m mm ma m m mf am aa a am aa a a                                            M M s C C s τ u M M s C C s τ   (15) where                 ( ) ( ). m mm rm ma ra mm rm ma ra m mcf a am rm aa ra am rm aa ra a                    τ M q M q C q C q G q u τ M q M q C q C q G q         We can be expressed m τ and a τ as in term of a known matrix m ω , a ω and unknown parameter vectors, m ψ and a ψ . 11 13 22 23 32 0 0 0 0 , 0 0 0 0 c r p c p m m p m m m m m x m y m m m m m x m y c c t t t t t                                        τ ω ψ   (16) where     11 13 sin cos cos cos sin sin (cos cos sin sin ) cos cos sin cos cos sin sin sin cos sin sin cos m rx m rl r r rl r r q q l q l q q l l l q l l l q q f q f t t q f q f q f q fq q ff q f q fq q ff q f q fq q ff                                22 23 32 2 sin cos cos cos sin sin cos sin cos cos cos . m ry m rl r rl r m rx ry rl r r q q l q q l l q q q q l q l q g g f f q f t t f f ff f ff t q f f fq f q f                                11 22 0 , 0 p a a a a p a m m t t                     τ ω ψ (17) where   2 2 2 2 11 2 2 2 2 22 2 cos cos cos cos cos sin cos cos sin sin cos sin sin cos cos sin a rx ry rl r r a rx ry r rl r r l q l q l q l l l q l q gl l q l q l q l q l q llq q f f q t q f f ff f f ff f fq q f t q f f f f fq                                 cos sin . gl f q f As a majority of the adaptive controller, the following signal is defined: 2( ( ) ( ), ( ) 0 2 ( ), ( ) 0, ( ) 0 , ( ) 0, ( ) 0 x x x x x x x x x x x Z a t b t Z t Z b t Z t b t Z t b td                      (18) where x d is some small positive constant and     2 T T 2 2 2 T T 2 2 2 ˆ ( ) ˆ ( ) m x a a av a m x a a av a m a t b t e e e         s s ω ψ s K s s s ω ψ s K s s (19) Note that (18) is convenience to define a differential equation, where its variable ( ) x Z t remains positive. Define a positive function ( ) x h t Z  . It can be shown that:   2 T T 2 2 2 1 ˆ ( ) 0 ( ) ( 20) m a a av a m h h t h h t e e                      s s ω ψ s K s s  Next, we assume that there exists a measure zero set of time sequences   1 i i t   such that ( ) 0 i Z t  (i.e., ( ) 0, 1,2,3, , i h t i    ), and then, the existence assumption is verified. Let the adaptive control law be designed as: ˆ mf 1 1 v mv m     u ω ψ τ K s , (21) where   T T 2 ( 1) ˆ m v a 2 2 a av a m h e                s τ s ω ψ s K s s , where ˆ ˆ and 1 2 ψ ψ are the estimates of and 1 2 ψ ψ , respectively. The adaption laws are given as T 1 1 1 T 2 1 2 ˆ ˆ m a            ψ s ω λ ψ s ω λ   (22) 302 Quoc Toan Truong, Anh Huy Vo and Quoc Chi Nguyen VCM2012 Then the error dynamics can be obtained as: 0 0 ˆ , mm ma m mm ma m mv m am aa a am aa a av a 1 1 v 2 2 av a                                                               M M s C C s K s M M s C C s K s ω ψ τ ω ψ K s   which can be rewritten as ˆ ( ) ( , ) 1 1 v m 2 2 av a              ω ψ τ M q s C q q s Ks ω ψ K s   , (23) where ˆ ˆ 1 1 1 2 2 2                    ψ ψ ψ ψ ψ ψ   Since 1 2 , y y are constant parameters, we obtain ˆ ˆ . ˆ ˆ 1 1 1 1 2 2 2 2                                   ψ ψ ψ ψ ψ ψ ψ ψ           (24) Theorem: Consider the system (7) or (23) with the parameters systems unknown. The proposed control law (21) employing the adaption laws (22) guarantees the asymptotic stability of the systems, i.e., 0  s , 0 e  , and 0 e   as t   . Proof: Lyapunov function candidate can be defined as T T 1 T 1 1 1 1 2 2 2 1 1 1 1 ( ) ( ) 2 2 2 2 x V t Z      s M q s ψ λ ψ ψ λ ψ     (25) In the previous, due to the quadratic form of system states as well as the definition of ( ) x Z t and ( ) V t is always positive-definite and indeed a Lyapunov function candidate. By taking the time derivative of ( ) V t , we have   T T T 1 T 1 1 1 1 2 2 2 T T 1 T 1 1 1 1 2 2 2 1 1 T T 1 T 1 1 1 1 2 2 2 2 2 1 ( ) ( ) ( ) 2 ( ) ( ) v av a V t hh hh hh                                              s M q s s M q s ψ λ ψ ψ λ ψ s M q s C q s ψ λ ψ ψ λ ψ ω ψ τ s Ks ψ λ ψ ψ λ ψ ω ψ K s s                             T T 1 1 2 2 T 2 2 2 2 T 1 T 1 T T 1 1 1 2 2 2 2 2 2 T 1 1 ( 1) ˆ ˆ ( m a a av a mv m m m a a av a m m h h h e e e                                                                                    s ω ψ s ω ψ s K s K s s ω ψ s ψ λ ψ ψ λ ψ s ω ψ s K s s s ω ψ             2 T T T 2 2 2 2 T T 1 T 1 T T 2 2 1 1 1 2 2 2 2 2 2 T T T T T 1 T 1 1 2 2 1 1 1 2 2 1) ˆ ˆ m a a av a m mv m m m a a a av a m m mv m a av a m a h e e e                                                 s s ω ψ s K s s K s s s s ω ψ ψ λ ψ ψ λ ψ s ω ψ s K s s s K s s K s s ω ψ s ω ψ ψ λ ψ ψ λ                     1 T 2 2 2 T T T 1 T 1 T 1 1 1 1 2 2 2 2 ˆ a m a            ψ s ω ψ s Ks s ω ψ λ ψ ψ λ s ω ψ        (26) Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 303 Mã bài: 64 The substitution of (22) into (26) yields and due to the positive-definiteness of K , we have: T ( ) 0 V t    s Ks  , when ( ) 0 h t  (27) The solution of ( ) Z t from equation (18) is defined and continuous for all 0 t  , so ( ) h t is continuous for all i t . Because ( ) V t is a continuous function of ( ) h t so it keep remains to be continuous at time i t , i.e. ( ) ( ) i i V t V t    . From the hypothesis, ( ) 0 i V t    and ( ) 0 i V t    then we can conclude that ( ) V t  is non-increasing at time i t , which then readily implies that , s h L   . Therefore, , , m a e L t t   directly from (14) and definitions of , m a t t , then following (23) that s L    . On the other hand, it is clear that the set of time instants is   1 i i t   measure zero thus 0 ( ) (0)Vdt V V         , or equivalent. Therefore by invoking the Barbalat’s lemma, we readily obtain that 0  s asymptotically as t   , therefore implies and 0  e e  as t   . Finally, to complete the proof in theory, we need to show that the above hypothesis that the set of time instants   1 i i t   is indeed measure zero. However, if is quite straightforward from (18) simply using the fact that all signals are uniformly bounded after the proposed control is employed. 4. Simulation results To illustrate the controller performance, we simulate the proposed nonlinear adaptive controller of (21) employing the adaption laws (22) in a crane with the following parameters: 2 5 kg, 5 kg, 0.5 kg, 9.81 m/s c r p m m m g    . The friction parameters are given as 0.01, c 0.01 x y c   . The trolley moves to a desired position selected as follows: 3 m, y 1 m, 0.5 m, 0, y 0, L 0. d d d d d d x l x          The initial state of the system is chosen as: (0) 0 , (0) 0, y(0) 0 , (0) 0, (0) 1 m, (0) 0, (0) 0, (0) 0, (0) 0, (0) 0. x x y l l q q f f                The control law is turned until a best performance is achieved, which yields the following control gains: 1 0 0 8.05 0 0 0.5 0 , , 0 8.05 0 0 0.45 m a                       K K 4.9 0 0 6.9 0 0 4.9 0 , , 0 6.9 0 0 4.6 vm va                       K K 1 2 3.9, 3.2, 1. l l e    Figs. 2, 3 and 4 plot the tracking of trolley. Fig. 5 and 6 are the swing angles. It can be seen that when the tracking of trolley positions reaches desired positions after 8 seconds and the swing angles go to zero asymptotically at seconds 10. The swing angles are about 2 degrees in the transferring process. Selection of controller parameters can affect the system performance. Unfortunately, there is no systematic approach for the selection of these values. They must be chosen using iterative simulation and a tradeoff between system response and control gains should be made. Fig. 2 Position of the trolley in X-direction 304 Quoc Toan Truong, Anh Huy Vo and Quoc Chi Nguyen VCM2012 Fig. 3 Position of the trolley in Y-direction Fig. 4 Rope length Fig. 5 Sway angle ( ) t q Fig. 6 Sway angle ( ) t f Fig. 7 Estimated parameters ˆ ( ) m t ψ Fig. 8 Estimated Parameters ˆ ( ) a t ψ Figs. 7 and 8 are the parameters estimation results. The estimate values will converge to constant values, if the plant is stable. As shown in these figures, the values may not get the true values. However, getting true values of the parameters was not the purpose of this paper. 5. Conclusion In this paper, a 5-DOF dynamic model of the 3D overhead crane was developed under the effects of friction forces and the unknown parameters. A nonlinear adaptive controller was proposed for the overhead crane to drive it to its desired point and to suppress the swing of payload. Under the proposed controller, asymptotic stability of the overhead crane system is proved by using Lyapunov method. Simulation results illustrate the effectiveness of the proposed controller. An experiment system is under construction at Mechatronics Lab (HCMUT) to verify the effectiveness of the controller. Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 305 Mã bài: 64 References [1] Sakawa, Y., & Shindo, Y. (1982). Optimal control of container cranes. Automatica, 18(3), 257-266 [2] Auernig, J. W., & Troger, H. (1987). Time optimal control of overhead cranes with hoisting of the load. Automatica, 23(4), 437- 447. [3] Chun-Yi Su and Yury Stepanenko, Adaptive Variable Structure Set-Point Control of Underactuated Robots, IEEE transactions on automatic control, vol. 44, no. 11, november 1999 [4] H. Aschemann, O. Sawodny , S. Lahres and E. P .Hofer, Disturbance estimation and compensation for trajectory control of an overhead crane, Proceedings of the 2000 IEEE American Control Conference, Illinois, pp.1027-1031, 2000 [5] Hong, K S., Park, B. J., & Lee, M. H. (2000). Two-stage control for container cranes. JSME International Journal-Series C, 43(2), 273-282. [6] D. C. D. Oguamanam andj. S. Hansen, Dynamics of A Three-dimensional Overhead Crane System, Journal of Sound and Vibration (2001) [7] Y. Fang, W. E. Dixon, D. M. Dawson, and E. Zergeroglu, Nonlinear Coupling Control Laws for an Underactuated Overhead Crane System, IEEE/ASME Transactions on Mechatronics, Vol. 8, No.3, 2003 [8] Omar, H. M., & Nayfeh, A. H. (2005). Gantry cranes gain scheduling feedback control with friction compensation. Journal Sound and Vibration, 281(1-2), 1-20. [9] Yang Jung Hua, Yang Kuang Shine, Adaptive coupling control for overhead crane systems, ScienceDriect, Mechatronics 17 (2007) page 143–152. [10] Hua, Y. J., & Yang, K. S. (2007). Adaptive coupling control for overhead crane systems. Mechatronics, 17(2-3), 143-152. [11] Park, H., Chwa, D., & Hong, K S. (2007). A feedback linearization control of container cranes: varying rope length. International Journal of Control, Automation, and System, 5(4), 379-387 [12] Bojun Ma, Yongchun Fang, Xuebo Zhang, Adaptive Tracking Control for an Overhead Crane System, Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 [13] Dongkyoung Chwa, Nonlinear Tracking Control of 3-D Overhead Cranes Against the Initial Swing Angle and the Variation of Payload Weight, IEEE Transactions on Control Systems Technology, Vol. 17, No. 4, July 2009 [14] R.M.T. Raja Ismail, M.A. Ahmad, M.S. Ramli, F.R.M. Rashidi, Nonlinear Dynamic Modeling and Analysis of a 3-D Overhead Gantry Crane System with Payload Variation, ems, pp.350- 354, 2009 Third UKSim European Symposium on Computer Modeling and Simulation, 2009. [15] Ngo, Q. H. & Hong, K S. (2009). Skew control of a quay container crane. Journal of Mechanical Science and Technology, 23(12), 3332-3339. [16] Neupert, J., Arnold, E., Schneider, K., & Sawodny, O. (2010). Tracking and anti-sway control for boom cranes. Control Engineering Practice, 18(1), 31-44. [17] Jung Hua Yang·Shih Hung Shen, Novel Approach for Adaptive Tracking Control of a 3- D Overhead Crane System, Journal of Intelligent & Robotic Systems, Vol 62, Number 1 (2011), 59-80 [18] Ngo, Q. H., & Hong, K S. (2012). Sliding- mode antisway control of an offshore container crane. IEEE/ASME Transactions on Mechatronics, 17(3), 201-209. [19] Nguyen, Q. C., & Hong, K S. (2012). Adaptive control of container cranes with friction compensation. Manuscript Draft submitted to Control Engineering. Quoc Toan Truong received B.C degree in mechanical engineering at the Chi Minh City University of Technology (HCMUT) in 2010. He has been pursuing a M.E program at the HCMUT since 2011. He has been a faculty member at Department of Mechatronics Engineering, Faculty of Mechanical Engineering (HCMUT) since 2010. His research interests include nonlinear control of dynamical systems, robotics, and industrial applications of control engineering. Anh Huy Vo received the B.Eng. degree and M.Eng. degree both in mechanical engineering at the Ho Chi Minh University of Technology (Vietnam), in 1998 and 2003, respectively. He has been a faculty member at the Department of Mechatronics Engineering, Ho Chi Minh University of Technology since 1998. His research interests include control of offshore cranes, [...]... robotics, fast automation system, and industrial applications of control engineering Quoc Chi Nguyen received the B.Eng degree in mechanical engineering, the M.Eng degree in automatic control engineering at the Ho Chi Minh University of Technology (Vietnam), in 2002 and 2006, respectively, and the Ph.D degree in intelligent control & automation at Pusan National University (Republic of Korea) in 2012 He has... National University (Republic of Korea) in 2012 He has been a faculty member at the Department of Mechatronics Engineering, Ho Chi Minh University of Technology since 2002 His research interests include control of infinite dimensional systems, nonlinear control of dynamical systems, and control of MEMs VCM2 012 Quoc Toan Truong, Anh Huy Vo and Quoc Chi Nguyen . July 2009 [14] R.M.T. Raja Ismail, M .A. Ahmad, M.S. Ramli, F.R.M. Rashidi, Nonlinear Dynamic Modeling and Analysis of a 3-D Overhead Gantry Crane System with Payload Variation, ems, pp.350- 354,. modeling overhead crane with the friction force model (Aschemann, 2000) was proposed by Ma, Fang & Zhang (2008). An adaptive tracking control of 3-D overhead crane systems was investigated. (1998). Nonlinear dynamic modeling and analysis of a 3D overhead gantry crane system with system parameters variation was introduced Ismail et al. (2009). These researches developed 3D models of