Dynamic Modeling of the 3-D Ship-mounted Overhead Crane

Nội dung text: Dynamic Modeling of the 3-D Ship-mounted Overhead Crane

1. Tuyển tập Hội nghị khoa học toàn quốc lần thứ nhất về Động lực học và Điều khiển Đà Nẵng, ngày 19-20/7/2019, tr. 132-139, DOI 10.15625/vap.2019000269 Dynamic Modeling of the 3-D Ship-mounted Overhead Crane Nguyen Van Quyen*, Le Xuan Hai , Pham Viet Hung*, Nguyen Quoc Hieu*, Pham Hai Quan*, and Pham Van Trieu * Hanoi University of Science and Technology, Hanoi, Vietnam College of Urban Works Construction, Hanoi, Vietnam Vietnam Maritime University, Hai Phong, Vietnam E-mail: xhaicuwc.edu.vn@gmail.com Abstract system parameters and developed a tracking control This work proposes the dynamics of a 3-D overhead-type crane method using linear quadratic regulation [9]. Another operated in offshore condition that considers the effects of sea type of the 2-D crane mounted on a ship was studied in wave excitation as well as wind-induced force. The crane [10], in which the 6-DOF model was constructed with trolley movement, hoisting drum rotation, axial container mounted on a ship with actuator saturations is taken into oscillation, container swing, roll, and heave of the ship consideration and its modeling is derived from the Lagrange and nonlinear feedback control was utilized for control formulation. The simulation studies, which showing system design. Pursuing control development for that model of performance in two case, without feedback control and floating container crane, in [11] the authors presented controlled by nonlinear sliding-mode method, are given to two robust controllers include conventional sliding-mode evaluate the properties of constructed modeling. and backstepping sliding-mode scheme and the Keywords: Dynamic Modeling, Marine System, 3-D Offshore simulation results compared the performance of these Crane controllers. Our previous research [2] investigated the modeling of a 2D offshore crane with hoisting motion, and the control problem was addressed by integral 1. Introduction hierarchical sliding-mode control. Nevertheless, most of the previous studies focused Container crane is the machine which plays a pivotal on the operation of the offshore crane in 2-D motion, role in hoisting and transferring goods at the harbors. whereas the modeling and control of the 3-D floating One of the big challenges in automatically controlling the crane, which is more common in the actual offshore crane systems is the underactuated feature. That means terminal, have not been concerned. That fact can be an ideal controller must not only tracks the trolley and attributed to some possible reasons. Firstly, the motions cargo to the desired destination but also reduces the of sea wave as well as the wind force exert a significant swing of the payload as small as possible. For this influence on the system dynamics, and adequately reason, analyzing the dynamics of cranes and improving integrating those factor into the description is highly the control quality of the crane are always considered and complicated. Besides, the control approaches for inland developed in numerous studies. Specifically, several crane system are not always suitable for the offshore one, works of our research group pertaining to modeling and because in practice, the excitation of wave and the force applying modern control theory on control of various generated by wind are difficult to measure and feedback crane systems has been published [1]–[6]. to the controller. Nowadays, due to the fact that numerous ports have a Due to the aforementioned analysis, this paper narrow and shallow channel, the oversize ships cannot investigates the modeling of a 3-D overhead-type reach that kind of harbors. Therefore, the process has to offshore crane, in which three wave-stimulated motion be conducted at the offshore terminal, where the that consists of roll rotation, pitch rotation, and heave container on a large ship is lifted and transferred to the displacement are taken into account. The 8-DOF model smaller ships which also called as the “mobile harbor” of crane mounted on the ship is formulated using the [7] using a crane mounted on the ship. Afterward, the Lagrange method. Next, the order-reduced modeling is mobile harbor delivers the cargo to the inland terminal. obtained by considering the system states which While there are a huge number of studies which pay represent for kinematic stimulation of sea waves as attention to control the conventional overhead crane non-controlled variables. Two simulation exmaples system, the studies related to the modeling and control of conducted in MATLAB/Simulink show the responses of crane system that operated in offshore condition has not system dynamics in both controlled and uncontrolled been investigated extensively. The dynamics and control cases. for ship-mounted overhead crane are mentioned in some The rest of this paper is organized as follows. Section following articles. In [7], Hong et.al. presented the 2 formulates the dynamical model of a 3D offshore dynamics of a 3-DOF container crane mounted on the container crane when considering the effect of seawave mobile harbor which considers roll, pitch motions and excitation, wind-generated force and saturation heave displacement of sea waves and the control problem nonlinearity. The simulation studies are conducted in of that system was solved by using the sliding-mode Section 3 to assess the properties of the derived technique in [8]. Kim and Park linearized the offshore modeling. Finally, the conclusions and future works are crane in [7] then proposed an algorithm to identify the given in Section 4.
2. Nguyen Van Quyen, Le Xuan Hai, Pham Viet Hung, Nguyen Quoc Hieu, Pham Hai Quan, Pham Van Trieu 2. System modelling 0012AAAA= 3123 (1) = Trans(Z,z)Rot(X ,)Rot(Y  ,) 2.1. Motion Equations 012 The physical model of a ship-mounted crane is where shown in Fig. 1. The system contains a bridge ( m1 ) with 1000 h F the height , is pushed by the force x , and a trolley 0100 (m ), is driven by the force F , move correspondingly Trans(Z, z), = 2 y 0 001 z along X -axis and Y -axis of the ship coordinate frame. 0001 3 3 The positions of bridge and trolley with respect to the origin are defined as s and s , respectively. Defining 1000 x y 0cossin0 − the angle between the cable and its projection on the Rot(X,), = 1 0sincos0 XZ00− plane of the inertial coordinate frame is  , 0001 and the angle between the projection of cable on the XZ00− plane and Z0 -axis is . The cable length is l which can be adjusted by lifting force F to hoist the cos0sin0 l container ( m ). 0100 c Rot(,).Y  = 2 − sin0cos0 0001 Fig. 1: Physical model of a 3-D offshore crane. In order to facilitate the control design, we assume that the friction is ignored, the cable is massless and Fig. 2: The motion of ship hull and coordinate frames rigid, the cargo and the trolley are considered as material particles and the bridge is considered as a homogeneous Consequently, the positions of the bridge and the straight rod with the length d1 . trolley in the fixed coordinate frame are correspondingly The motion of ship hull as a result of sea wave calculated as follows: excitations is illustrated in Fig. 2. In this paper, three rA003= r motion of ship body consists two translational motion 13 1 and one rotational motion, are taken into account, that is s chs+ x pitch, roll, and heave. The ship coordinate frame s s shc s x − (2) XYZ = 3 3 3 is constructed by the above mentioned motions zhc+− cs c  s  x as follows: Firstly, translating the fixed coordinate frame 1 XYZ0 0 0 along Z0 -axis by the displacement z ,the coordinate frame XYZ is obtained. Next, rotating the 0 0 3 111 r2= A 3 r 2 coordinate frame XYZ around X -axis by the angle , c 0ss 0 111 1 x s s  c − c  s 0 s we obtain the coordinate frame XYZ ,in which y 2 2 2 = (3) −c s  s c c  z h XX .Finally, rotating the coordinate frame XYZ 21 2 2 2 0 0 0 1 1 around Z -axis by an angle , the ship coordinate frame 0 sx c+ hs XYZ YY 3 3 3 is constituted with 32. s c −+ hc  s s s s  = yx The homogeneous transformation matrix from the z+ s s + hc c  − s c s  ship coordinate frame to the fixed coordinate frame is yx 1 determined as follows:
3. Dynamic Modeling of the 3-D Ship-mounted Overhead Crane where c(.) and s(.) briefly indicate cos(.) and b skew symmetric matrix ω1 as follows: functions, respectively. ωωωωbbbb= (9) The container position in the fixed coordinate is 11,321,131,21 derived as follows: b The skew symmetric matrix ω1 is calculated as lcs  follows: ls 00  rrc =+2 hlcc−  ωbbTb= RR (10) 100 1 (4) s chslcs ++ b x where R is the rotational transformation matrix from 0 s chcss  −++ ssls = yx zs++−+− shccs    cshlcc the bridge coordinate frame to the fixed coordinate frame yx and is given by: 1 cs0 The dynamic model of the container crane mounted b 3 RR==− ssccs   (11) on the ship is constructed using Lagrange formulation: 00 −csscc   dLL  After some calculation, the model of the system can −= , i=1, ,5 (5) dtqq i be rewritten in matrix form as follows: ii M(q)q + C(q, q)q + G(q) =Wτ + (12) where, LT=− , in which T and  is kinetic energy and potential energy of the system, q is the here, M(q) = mR55 is the inertial matrix, i ij i th element of the vector of generalized coordinates C(q, q) = cR55 is the centrifugal damping matrix, ij q = xyl  , which correspond to the G(q) = gR51 is the vector of gravitational force, i generalized forces τ = FFF 00 . The xyl W = wR51 denotes the matrix of external kinetic energy and potential energy are calculated as: i disturbances, which include sea wave excitation and wind 111 000000TTT Kmmm=++ 1rrrrrr 112 22 c cc force. 222 (6) 1 + ωΘbTbb ω The components of the Mq()are given as follows: 2 111 m11= m 1 + m 2 + mc ; mm1221==0;  =++m1122 gzm gzm gz cc (7) mmm1331==++ c s csc ( s sc c c s    ); b where ω1 is the projections of angular velocity 1 in mmm1441== lcc cc sc  s (  -;  ) b the bridge coordinate frame, Θ1 is the matrix of inertia mmmlssc1551==+ ; c ( cssccss       ) tensor of the bridge which can be determined as follows: mmm=+; 222 c J1x 00 b Θ1 = 0 0 0 (8) m23== m 32 mc ( c s -; c c  s ) 00J 1z m24== m 42 mc lc s s ; b The components of vector ω1 are obtained via the
4. Nguyen Van Quyen, Le Xuan Hai, Pham Viet Hung, Nguyen Quoc Hieu, Pham Hai Quan, Pham Van Trieu Gq() mmm2552==+ lcccss c (   ); are presented as: mm33 = c ; mm3443==0; mm3553==0; ggcsmmm112=++-();  c 22 2 m44 = mc l cos ; mm45== 54 0; m55 = mc l g22=+ gs ( m mc ); g3 = -; gmc c  c C( q ,q ) The elements on the matrix are expressed as: g4 = gmc lc s ; g5 = gmc lc  s . The elements of the vector of wave-wind disturbance c = 0; cmms=+-();  11 122 c W is determined as follows:    (ssccssccss−+ ) wmmmzcccz= −−−−−−   ; cm= -; 1161718161718 13 c +−   ccssccc ( ) wmmmzcccz2262728262728= −−−−−−   ; lccsscc   − ( ) cmlcscccs-; 14 =++ c   ( ) wmmmzcccz3363738363738= −−−−−−   ; +−  lscccss ( ) w4= fwx − m 46 − m 47  − m 48 z − c 46 − c 47  − c 48 z; lssccssccss  −+   ( ) cmlscccss15 =+−-;c   ( ) wfmmmzcccz5565758565758=−−−−−−wy   . +++  lcscssscccs    ( ) where mhmmm(); cmms212 =+ (); c c22 = 0; mssmm162 =+-();yc 1712=++ c cmc cc(); ssc s s 23 =++c (    ) mcsmmm1812=++-();  c cm24 =+ slcslcc  clss(   -;) mmmhcs262 =+ ;( s cx)( ) lcccss(  + ) m = 0; msmm=+ (); 27 282 c cmlccscs25 =+−c  (  ) ; sssccc +  - lsss y ( ) mmh36 =+ c ; sccccc ( s     ) (c  s  sc − c s  s ) +−s sccsc  s  cm= ; x ( ) 31 c ++− s c sc    c c cc s s ( ) hcscssscccs(  ++    ) mm= ; cmssccc=+-();   37 c ++ss cscc  ccc  s s -  32 c x ( ) 2 c33 = 0; c34 = − mc lc ; cml35 = -; c mm38 = -coscos;c  cm=+ lcc ; sc s cs   s s  m=+ mlcs ( sc - hcs  sss  ); 41 c ( ( ) ) 46 c y x 2 cm= lcc  s ; cm= lc ; sx ( c s + c s c  ) 42 c 43 c m= -; m lc 47 c +h c s s - c c  ( ) c m l2 s c ; cm44 = lclclsc ( -;) 45 = − c   mm48 = l c cossin; (c  c s + c s  s s  ) c= m l ; 51 c s c s - c c s  ++c  c s s  s s- c  c s c y ( ) ( ) m56 = − mlc + hccc(   + cscs   ) ; c= - m l ( c  s - c c s  ); c=  m l; -s c c s + c s  s s  52 c 53 c x ( ) m= m l2 s  c ; c= lm l. hssc(  - css  + ccss   ) 54 c 55 c m= -; m l 57 c +sccsc   ccs   sss   The components of the gravitational force vector x ( )
5. Dynamic Modeling of the 3-D Ship-mounted Overhead Crane The wind force that directly acts on the container m m58 l= c s c ; can be divided into two components along X0 -axis and c = s () m+ m + m hc  + s s  16 12cx( ) Y -axis as follows: −+syc(); m2 m s 0 fqCCA cos csmmm1712=++-(); xc wxwwgx=  (13) cmms=+++()-; sshss c   fqCCA= sin (14) 262 cxyx( ( )) wywwgy C cmmhssc272 =++ ( cx)( ); where w is the wind force coefficient,C g is the gust effect factor, A and A are the projections of surface x y ssscccsyx(      sccscs++) ( - ) area exposed to wind on ZY− plane and XZ− scsccs   - 00 00 y ( ) cms= sssccc −+++   ; 36 cx ( ) plane, respectively,  is the angle between the direction -hccccss   + ( ) hsccscs  ( - ) of wind force and X -axis. Finally, q = 0 .5 2 + 0 wa +s cccscs   - x ( ) denotes the wind velocity pressure, where a indicates the air density, and v is the design wind speed calculated +scx ( s ss  cscc  cc   ) as follows: hsc ( c  sc s - ) cm= −+ ; vvKKK= (15) 37 c +s cc c  sc s - 0 wdhr x ( ) hcssscscccc(      ) where v is the basic wind speed of each geographic + 0 +++sc scs  sscc  cs   x ( ) location, Kwd denotes the wind directionality factor sc sc  s+ cs  shss c  xx( ) ( ) that reflects the directional characteristics of extreme cm47 = lc-;c  h( c sc  + s c  ) + wind, and K indicates the profile factor of the wind +sc cc  s- s  h x ( ) speed at a reference height. 2.2. Control law sccsyx( - csss  ) ++  cc(  css ) In practice, the majority of used actuators has several -s c c + c s  s nonlinearity or constraint such as backlash, dead-zone y ( ) cm=+ lhc c +  s c  c s -;  band, or saturation feature. In this paper, the saturation 56 c ( ) nonlinearity of actuators is taken into account, and a +s s c c s - c  s x ( ) compensator based on neural network is constructed. The hs( c  c + c s  s ) definition of saturating actuator is given mathematically + by [30]: ++sx c( c  c c s  s ) maxmax: u = uu: minmax (16) sccs  + sss  - ccsc   : u x ( ) minmin hs( c  c + c s  s ) c=+ m l ; where  is the actuator output and u is the control 57 c ++s c c  c c s  s x ( ) input of the actuator,  is the upper bound and  hccs(   + sss  - ccsc   ) max min + ++sssc  - css   ccss   is the lower bound of actuator characteristic. If the x ( ) inputu is outside the linear range of actuator, saturation c18= c 28 = c 38 = c 48 = c 58 = 0; nonlinearity appears and the calculated control signal
6. Nguyen Van Quyen, Le Xuan Hai, Pham Viet Hung, Nguyen Quoc Hieu, Pham Hai Quan, Pham Van Trieu cannot act sufficiently to the plant. The eliminated term of control signal can be described as follows: =−u − uu: maxmax (17) = 0 : minmax u − uu: minmin Integrating the saturation nonlinearity of actuator, the dynamic model of an offshore crane can be rewritten as: M(q)q + C(q, q)q + G(q) = UW++δ (18) Fig. 3: Roll motion of ship hull where U UUU 00denotes the control xyl input of actuator and δτ=−U represents the influence of saturation feature. 3. Simulation In this section, to evaluate comprehensively the properties of the presented modeling, two different cases of simulation are conducted. The system specifications which are obtained approximately from a real model in the laboratory is utilized for simulation and depicted as follows. Note that all quantities are expressed in the Fig. 4: Pitch motion motion of ship hull International System of Units (SI). m1 = 24.3; m2 = 11.2(kg); mc = 2.7(kg); 2 g = 9.81(m/ s); h = 1.2(m); d1 = 1(m); JJ11xy==0.93(Nm); q(010.50.400;) =−− T q0(0;) = τ = 503030; T τ =−−− 503030. T max min The wind load parameters can be listed as follows: Fig. 5: Heave motion of ship hull 3 a =1.22(kg/ m ); Cw = 1.2; Kr = 0.85; Kh = 1.15; In the first one, we consider the operation of model without controller. The offshore crane is assumed to be 2 Kwd = 0.9; Cg = 1.05; Ax = 0.0064(m ); driven by the following inputs: 10 if t 2 2  = / 6; v= 18( m / s ). Ay = 0.01(m ); 0 Ftx ()= , 0 if t 2 The sea wave excitation in simulation is obtained by using the MSS Toolbox [12]. Three motions of ship 5 if t2 Fty ()= , hull including roll, pitch and heave are illustrated in Figs. 0 if t2 3-5, respectively.
7. Dynamic Modeling of the 3-D Ship-mounted Overhead Crane −+mgc 0.1 if t2 and Ftl ()= −mgc if t2 Fig. 10: Swing angle  in open-loop case In the second simulation case, we use the hierarchical Fig. 6: Bridge motion in open-loop case slidng-mode control (HSMC) to design the feedback controller of the closed-loop system. The HSMC is detailed in [1]. The system responses are given in Figs. 6-10, for open-loop, and in Figs. 11-15, for closed-loop. As can be seen from the figure, without control, the system will be unstable, whereas using the feedback control scheme, the system states will oscillate around the equilibrium stable points as a result of wind-wave disturbances. Fig. 7: Trolley motion in open-loop case Fig. 11: Bridge motion in closed-loop case Fig. 8: Lifting motion in open-loop case Fig. 12: Trolley motion in closed-loop case Fig. 9: Swing angle in open-loop case
8. Nguyen Van Quyen, Le Xuan Hai, Pham Viet Hung, Nguyen Quoc Hieu, Pham Hai Quan, Pham Van Trieu Dung, V. H. Thuat, and P. Q. Truong, “Modeling and Integral Hierarchical Sliding-Mode Control for 2D Ship-mounted Crane,” in 2019 First International Symposium on Instrumentation, Control, Artificial Intelligence, and Robotics (ICA-SYMP), 2019, pp. 82–85. [3] L. Viet-Anh, L. X. Hai, and N. Linh, “An Efficient Adaptive Hierarchical Sliding Mode Control Strategy Using Neural Networks for 3D Overhead Cranes,” 2007. Fig. 13: Lifting motion in closed-loop case [4] P. Van Trieu, D. D. Luu, H. M. Cuong, and L. A. Tuan, “Neural network integrated sliding mode control of floating container cranes,” Proceeding 11th Asian Control Conf., pp. 847–852, 2017. [5] P. Van Trieu and L. A. Tuan, “Combined Controls of Floating Container Cranes,” in 2015 International Conference on Control, Automation and Information Sciences (ICCAIS), 2015, pp. 442–447. [6] L. A. Tuan, H. M. Cuong, P. Van Trieu, L. C. Nho, V. D. Thuan, and L. V. Anh, “Adaptive neural network sliding mode control of shipboard container cranes considering actuator backlash,” Mech. Syst. Signal Fig. 14: Swing angle in open-loop case Process., vol. 112, pp. 233–250, Nov. 2018. [7] K. S. Hong and Q. H. Ngo, “Dynamics of the container crane on a mobile harbor,” Ocean Eng., vol. 53, pp. 16–24, 2012. [8] Q. H. Ngo and K. S. Hong, “Sliding-mode antisway control of an offshore container crane,” IEEE/ASME Trans. Mechatronics, vol. 17, no. 2, pp. 201–209, 2012. [9] D. Kim and Y. Park, “Tracking control in x-y plane of an offshore container crane,” JVC/Journal Vib. Control, vol. 23, no. 3, pp. 469–483, 2017. [10] L. A. Tuan, H. M. Cuong, S. G. Lee, L. C. Nho, and K. Moon, “Nonlinear feedback control of container crane Fig. 15: Swing angle  in closed-loop case mounted on elastic foundation with the flexibility of suspended cable,” JVC/Journal Vib. Control, vol. 22, 4. Conclusion no. 13, pp. 3067–3078, 2016. This paper presented the dynamics of the 3-D [11] L. A. Tuan, S. G. Lee, L. C. Nho, and H. M. Cuong, overhead crane mounted on a ship, while the influence of “Robust controls for ship-mounted container cranes wave excitation, wind-induced force and saturating with viscoelastic foundation and flexible hoisting actuator were fully taken into consideration. Our future cable,” Proc. Inst. Mech. Eng. Part I J. Syst. Control work will focus on developing the control law for this Eng., vol. 229, no. 7, pp. 662–674, 2015. system and applying it on the prototype in the laboratory. [12] T. I. Fossen and T. Perez, “Marine Systems Simulator (MSS),” 2004. References [1] L. V. Anh, L. X. Hai, V. D. Thuan, P. Van Trieu, L. A. Tuan, and H. M. Cuong, “Designing an Adaptive Controller for 3D Overhead Cranes using Hierarchical Sliding Mode and Neural Network,” 2018 Int. Conf. Syst. Sci. Eng., pp. 1–6. [2] N. Van Thai, H. T. K. Duyen, L. Viet Anh, P. Tien