Nonlinear Dyn DOI 10.1007/s11071-015-2313-9
ORIGINAL PAPER
Dynamics and trajectory tracking control of cooperative multiple mobile cranes Sen Qian · Bin Zi · Huafeng Ding
Received: 21 January 2015 / Accepted: 2 August 2015 © Springer Science+Business Media Dordrecht 2015
Abstract This paper addresses the dynamics and trajectory tracking control of cooperative multiple mobile cranes. Compared with a single mobile crane, cooperative cable parallel manipulators for multiple mobile cranes (CPMMC) are more complex in configuration, which have the characters of both series and parallel manipulators. Therefore, for the CPMMC, the forward as well as the inverse kinematics and dynamics include the difficulties of both series and parallel manipulators. However, the closed kinematic chain brings about potential benefits, including sufficient accuracy, higher cost performance, better lifting capacity and security. Firstly, the forward and inverse kinematics of the CPMMC with point mass are derived with elimination method, and the complete dynamic model of the CPMMC is established based on Lagrange equation and the complete kinematics. Secondly, considering the repetitive tasks and high security and precision requirement, a robust iterative learning controller is designed for trajectory tracking on the basis of the linearization S. Qian · B. Zi (B) School of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, China e-mail:
[email protected] B. Zi School of Mechanical and Automotive Engineering, Hefei University of Technology, Hefei 230009, China H. Ding School of Mechanical Engineering and Electronic Information, China University of Geosciences (Wuhan), Wuhan 430074, China
of the dynamics. Thirdly, taking the engineering practice into consideration, two case studies are simulated with the same expected trajectory but with different weights of the loads. Finally, the designed controller is compared with traditional PD control algorithm via numerical simulation. The results demonstrate the feasibility and superiority of the CPMMC and designed controller, and provide a theoretical basis for the cooperation of multiple mobile cranes. Keywords Cooperative multiple mobile cranes · Cable parallel manipulator · Dynamics · Robust iterative learning control
List of symbols Li f s h lsi φi li lci q
The length of boom The horizontal distance between lower pivot point of boom and the slewing axis The horizontal distance between lower pivot point of cylinder and the slewing center The vertical distance between lower pivot point of cylinder and the slewing center The distance between upper pivot point of cylinder and lower pivot point of boom The slewing angle The length of cable The length of hydraulic cylinder Generalized coordinates of the CPMMC
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D
The distance between the slewing centers of each two cranes d The distance between the top point of boom and hoisting point of load m The mass of the load The kinetic energy of the load Km The gravitational potential energy of the load Pm The general force on the actuators with respect τm to the payload Inertia matrix with respect to the payload Mm Coriolis matrix with respect to the payload Cm The angle of the boom around the correspondθi ing down pivot The kinetic energy of the boom KL The rotary inertia of the boom around the corJ¯Li responding lower pivot The rotary inertia of the boom around the corJ˜Li responding slewing center The mass of the boom mL The gravitational potential energy of the boom PL The kinetic energy of the boom KL The general force on the actuators with respect τL to the boom Inertia matrix with respect to the boom ML Coriolis matrix with respect to the boom CL The rotary inertia of the turrets around the corJr responding slewing center The gravitational potential energy of the turret Pr The kinetic energy of the turret Kr The general force on the actuators with respect τr to the turret Inertia matrix with respect to the turret Mr Coriolis matrix with respect to the turret Cr τ The general force on the actuators of the CPMMC M Inertia matrix of the CPMMC C Coriolis matrix of the CPMMC j The iteration times j τ a (t) The unknown disturbance τ j (t) The input torque e The positional tracking error matrix of the actuators j The proportional gain Kp j The differential gain Kd β( j) The regulatory factor of the control gains which acts in the j-th iteration E The robust gain j The proportional gain of the j-th iteration Kp
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j
Kd K 0p K 0d
The differential gain of the j-th iteration The initial proportional gain The initial differential gain
1 Introduction Different types of cranes are widely used for material transportation in many industrial fields, such as steel enterprises, electric power construction, container terminals and many manufacturing segments. A single crane with a cable-suspended load has been widely studied in the literature. Owing to the under-actuated nature, the crane dynamics are not fully input-tooutput linearizable [1]. The severely nonlinear properties together with extraneous disturbances cause undesired swings, especially at launch and break, which badly reduce the efficiency and security [2]. Thus, fast and accurate suspending of loads with minimum swing is the objective of various crane controllers. Many recent researches have discussed the control of a single crane using different approaches. A nontimebased control strategy named delayed reference control was adopted in [3] for the swing control of the industrial robotic cranes. Schaper et al. [4] proposed a nonlinear model for the skew dynamics of a container rotator of a boom crane. A feedback control is implemented, and the effectiveness is validated. A simple vibration control method is provided in [5] for fast crane operation using an open-loop control approach, which is safe and fast without addition of any sensors. Tomczyk et al. [6] discussed a crane control system with a state simulator under different wind disturbances. In order to achieve both positioning and anti-oscillation control, a new anti-swing control method was presented in [7] and proved to be of robustness to unmodeled uncertainties and external disturbances. The last decades have seen the rapid development of world economy. Due to the ever-increasing quantity of cargo, hazardous terrain and weather, some complex tasks are impossible to handle with a single traditional crane, inevitably [8]. Hence, a new generation of cranes, cable parallel crane-type robots are proposed. The closed kinematic chain in the cable parallel cranetype robots makes the configuration more complex; however, it brings about potential benefits including better load control, sufficient accuracy, larger capacity and workspace [9–12]. For instance, Korayem et
Dynamics and trajectory tracking control
al. [13,14] presented an Iran University of Science and Technology cable-suspended robot (ICaSbot) with six DOFs. Two closed-loop nonlinear optimal controllers are proposed, based on state-dependent Riccati equation and optimal feedback linearization, respectively. In addition, the dynamic load-carrying capacity are analyzed to investigate the efficiency and optimality of the proposed methods. A robust point-topoint position control method was proposed in [15] for completely restrained cable parallel crane-type robot, and the robustness based on Lyapunov stability is discussed. Capua et al. [16] designed a novel concept of a mobile cable parallel robot called SpiderBot, and the kinematic, statics, motion planning and experiments are conducted using a SpiderBot prototype. An underconstrained cable-driven parallel robot in crane configuration was studied in [17], which has intrinsically coupled kinematics and statics. And the stability of static equilibrium is assessed based on the geometrico-static model. Obviously, the cable parallel crane-type robots have many advantages compared with traditional cranes. However, it should be noted that the cable parallel crane-type robots are still not widely used in engineering practice. One of the reasons is that most of the cable parallel crane-type robots are redesigned for specific tasks instead of general ones. Meanwhile, the cooperative multiple cranes attracts more attention recently for the redesigning cost is saved [18]. Thus, as more reconfigurable and cost-effective cable parallel cranetype robots, the cooperative multiple cranes have broad application prospects, as shown in Fig. 1. For instance, Michael et al. [19] presented a novel approach to cooperation of multiple aerial robots, which was demonstrated via simulation and experimentation. And then, Jiang and Kumar [20] addressed the kinematics of cooperative cable parallel robots for multiple quadrotors and developed an approach for its stability analysis. Ku and Ha [21] performed the dynamic response analysis of the multi-cranes for block lifting operation, and the modules for determining hydrostatic and hydrodynamic forces are developed based on the dynamic analysis. It is well known that the inverse kinematics of parallel manipulators is simple in comparison with series ones. Inversely, the forward kinematics of parallel manipulators is more difficult. In the CPMMC, each single mobile crane can be considered as a series manipulator with three degrees of freedom (DOFs).
Fig. 1 The CPMMC in engineering applications
Moreover, the cables together with the load can be treated as a cable parallel robot, of which the vertexes are moved by the corresponding mobile crane. Hence, for the CPMMC, the forward as well as the inverse kinematics includes the difficulties of both series and parallel manipulators [22]. As a version of cable parallel crane-type robot, cooperative multiple mobile cranes has also been studied in previous researches, in particular in [8,23,24]. A cost-effective, safe and compliant solution for the operation of the CPMMC is the urgent need to reach potential advantages of both cranes and the cable parallel robot [23]. And there are still some important unresolved problems, especially complete kinematics and dynamics. Specifically, quasistatic models of the CPMMC is studied in [23,24], with the assumption that the load together with the mobile cranes have motions that give rise to negligible inertial forces. In addition, the dynamics of the platform in the CPMMC was analyzed based on D’Alembert principle, regardless of the dynamics of cranes [8]. Indeed, it is impossible to make exact assertions about the results without explicitly modeling and analyzing the complete dynamics of the system. Different approaches can be used to solve the dynamics of manipulators, such as Kane equation [25], Newton–Euler equations [26], principle of virtual work [27] and Lagrange equation [9]. In addition, control of cooperative multiple mobile cranes is another key problem to be solved. Many recent researches have discussed the control of multiple robots. For instance, a multiple unmanned aerial vehicles control scheme was developed in [28], considering the full nonlinear position/orientation model of a multi-quadrotors system. Zi et al. [29] designed
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S. Qian et al. Fig. 2 CAD model of the CPMMC
an adaptive iterative learning tracking controller for a hybrid-driven cable parallel manipulator. An inverse kinematic control strategy for cooperative dual cranes was presented in [30] regardless of the ship motion on which the cranes are attached. Li et al. [31] proposed a new version of adaptive robust control for multiple mobile manipulators carrying a common object in a cooperative manner. The major contribution of this paper lies in the solution of the complete kinematics and dynamics of the cooperative cable parallel manipulators for multiple mobile cranes, which is the foundation for the further research of this new promising and significant engineering equipment. Moreover, an effective and suitable controller is designed considering the engineering practice. The rest of the paper is structured as follows. In Sect. 2, the kinematics of the cooperative three mobile cranes with point mass is established with elimination method, including both forward and inverse kinematics. Based on Lagrange equation and the kinematics, the complete dynamic model of the CPMMC is derived in Sect. 3. In Sect. 4, a robust iterative learning controller (ILC) of the CPMMC is designed on the basis of the linearization of the dynamics. Numerical simulation is conducted in Sect. 5, in order to investigate the dynamics and the designed robust ILC of the CPMMC. Finally, Sect. 6 provides concluding remarks and thoughts on future work.
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2 Kinematics 2.1 Description of mechanism Figure 2 shows the general three-dimensional CAD model of the CPMMC, which is established based on the engineering applications in Fig. 1. The simple schematic sketches of the whole CPMMC and single mobile crane are provided in Figs. 3 and 4, respectively. As shown in Fig. 3, the global frame OXYZ is attached to the ground at the slewing center O1 of the first mobile crane in order to define the position of the payload, as well as the orientation of three cranes. The three cables of cranes are connected together at the hoisting point in the load. The CPMMC can be considered as a nine-input and three-output system, in which the three cables are capable of controlling the three Cartesian DOFs of the load but not including the rotation. However, taking the engineering application into consideration, we are only interested in the Cartesian DOFs in this study and the mentioned configuration can meet most engineering tasks. Obviously, the CPMMC is a redundant actuated manipulator since the DOFs of the load are less than the inputs of the system. It is important to note that although the load is fully constrained, its maximum acceleration in the vertical direction will be limited by the acceleration due to the gravity.
Dynamics and trajectory tracking control
Referring to Fig. 4, the local frame of each single mobile crane oi xi yi is attached to the corresponding turrets at the slewing center Oi (i = 1, 2, 3). Links Ai Bi and Ci Di denote the telescopic booms and the luffing hydraulic cylinders, respectively. The length of boom is L i (i = 1, 2, 3). The lower pivot point Bi of the boom and the lower pivot point Ci of luffing hydraulic cylinder are fixed in the crane turret of respective mobile crane. Oi E i represents the slewing axis of crane turret. In addition, the horizontal distance between down pivot point of boom Bi and the slewing axis is f . The length of links Bi Di are lsi (i = 1, 2, 3). The horizontal and vertical distance between down pivot point of cylinder Ci and the slewing center Oi are s and h, respectively.
(a)
(b) 2.2 Forward kinematics of cable parallel manipulator with movable vertexes A single mobile crane can be treated as a 3-DOF rigid serial manipulator with three inputs and three outputs, of which the end effecter can reach any point in designed workspace. Hence, the parameters of nine inputs of driving devices are chosen as the generalized global coordinates of the whole system for kinematics and dynamics analysis, which are the slewing angles φi (i = 1, 2, 3) of three groups of turrets, lengths of three groups of cables li (i = 1, 2, 3) and hydraulic cylinders lci (i = 1, 2, 3). q = [φ1 , φ2 , φ3 , l1 , l2 , l3 , lc1 , lc2 , lc3 ] .
(1)
Fig. 3 Schematic sketch (a) and a top view (b) of the CPMMC
According to the structure of the CPMMC in Fig. 3, the global coordinates of each mobile crane are obtained as
Ai
Li(AiBi)
li
Di
[X O1 , Y O1 , Z O1 ] = [0 0 0]T
(2)
[X O2 , Y O2 , Z O2 ] = [D 0 0]T
(3)
[X O3 , Y O3 , Z O3 ] = [D sin(π/6) D cos(π/6) 0]T .
lsi (BiDi)
(4) Y Oi
f
Bi
lci X
h
s Ei
Ci
Fig. 4 Schematic sketch of a single mobile crane
P
The constraint equations of the pivot point Di (x Di , y Di ) (i = 1, 2, 3) on the plane Ai Bi Ci consisted by the boom and hydraulic cylinder can be expressed as (x Di li + s)2 + y Di l12 = ls2 (5) (x Di li + s − f )2 + (y Di li + h)2 − lci2 . By solving the above equation, one can obtain
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S. Qian et al. 2 + l2 −2s f + f 2 + h 2 − lci s 2f 1 2 h f 2 h + h 3 − hlci − + hls2 2 f f 2 + h2 2 +2l l −l 2 −f 2 −h 2 +l 2 +2l l +l 2 −f f 2 +h 2 −lci ci s c1 s s s ci
x Di =
(6) 1 2 − f 2 h − h 3 + hlci − hls2 2 2 +h 2 +2l l −l 2 −f 2 −h 2 +l 2 +2l l +l 2 f 2 +h 2 −lci . +f ci s ci s s s ci
y Di =
f2
(7)
Let [X Y Z ] be the coordinate of the load. Then, the constraint equation of the load and three cables in the CPM can be generated as follows based on the forward kinematics of single mobile crane in Eqs. (12)–(14) li2 = (X − X Ai )2 + (Y − Y Ai )2 + (Z − Z Ai )2
where the cable length li is measured from the hoisting point P to the pulley center Ai . Elimination method is used to solve the equations with respect to the global coordinates of three top points of booms; iterative substitution is defined as
Combining with the schematic sketch of the CPMMC, the global coordinates of three upper pivots Di (i=1, 2, 3) can be generated as
[X, Y, Z ]T = [X − X A1 , Y − Y A1 , Z − Z A1 ]T [X 1 , Y1 , Z 1 ] = [X A2 − X A1 , Y A2
Y O1 + x D1 sin (φ1 + π/6) , y D1 ] (8)
− Y A1 , Z A2 − Z A1 ]T
Y O2 + x D2 sin (π/6 − φ2 ) , y D2 ] (9) [X D3 , Y D3 , Z D3 ] = [X O3 + x D3 cos φ3 , (10)
Hence, the global coordinates of three top points of booms both on the plane Ai Bi Ci and in space can be derived as +s) x Ai = L(x Di −s ls (11) L y Di y Ai = ls [X A1 , Y A1 , Z A1 ] = [X O1 + x A1 cos (φ1 + π/6) , Y O1 + x A1 sin (φ1 + π/6) , y A1 ] (12) [X A2 , Y A2 , Z A2 ] = [X O2 + x A2 cos (π/6 − φ2 ) , Y O2 (13)
[X A3 , Y A3 , Z A3 ] = [X O3 + x A3 sin φ2 , Y O3 + x A3 cos φ2 , y A3 ].
(14)
It is well known that the inverse kinematics of parallel manipulators is simple compared with series ones. However, the forward kinematics of parallel manipulators is more difficult relatively. Therefore, as mentioned in Sect. 1, the vertexes of cable towers in traditional cable parallel manipulators are mainly fixed related to the ground. In addition, some assumptions are made for simplification of modeling, such as the equal of the height of the cable towers. Meanwhile, the workspace and flexibility of the CPM are reduced.
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(17)
[X 2 , Y2 , Z 2 ] = [X A3 − X A1 , Y A3 T
−Y A1 , Z A3 − Z A1 ]T .
[X D2 , Y D2 , Z D2 ] = [X O2 + x D2 cos (π/6 − φ2 ) ,
+ x A2 sin(π/6 − φ2 ), y A2 ]
(16)
T
[X D1 , Y D1 , Z D1 ] = [X O1 + x D1 cos (φ1 + π/6) ,
Y O3 + x D3 sin φ3 , y D3 ].
(15)
(18)
Then, the length of the cables can be expressed as ⎡
l12
⎤
⎢ 2⎥ ⎢l ⎥ = ⎣ 2⎦ l32 ⎤ ⎡ X 2 + Y 2 + Z 2 2 2 2 ⎣ (X − X 1 ) + (Y − Y1 ) + (Z − Z 1 ) ⎦ . (X − X 2 )2 + (Y − Y2 )2 + (Z − Z 2 )2
(19) Via expansion and subtraction, the following equation can be derived as l12 −l22 +X 12 +Y12 +Z 12 = 2(X 1 X +Y1 Y (20) + Z 1 Z ) 2 2 2 2 2 l1 −l3 +X 2 +Y2 +Z 2 = 2(X 2 X +Y2 Y + Z 2 Z ).
(21)
Eqs. (20) and (21) can be rewritten through substitution of the right part as follows: uY2 −vY1 +Z (Z 2 Y1 − Z 1 Y2 ) X 1 Y2 − X 1 Y2 = u 1 + u 2 Z (22) uX 2 −vX 1 +Z (Z 2 X 1 −Z 1 X 2 ) Y = X 1 Y2 − X 1 Y2 = u 3 + u 4 Z (23)
X =
Dynamics and trajectory tracking control
u= v= u1 = u2 = u3 = u4 =
⎤ X +d (X Oi − X ) (X − X Oi )2 +(Y −Y Oi )2 ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ X Ai ⎥ ⎢ 2 2 ⎥ − X Y +d −Y +(Y −Y (X ) ) (Y ) Oi Oi Oi ⎣ YA ⎦ = ⎢ ⎥. ⎢ i ⎥ ⎢ 2 ⎥ ⎢ Z Ai ⎦ ⎣ 2 2 2 L − (X − X Oi ) +(Y −Y Oi ) −d +s ⎡
where 1 2 (l − l22 + X 12 + Y12 + Z 12 ) 2 1 1 2 (l − l32 + X 22 + Y22 + Z 22 ) 2 1 uY2 − vY1 X 1 Y2 − X 1 Y2 Z 2 Y1 − Z 1 Y2 X 1 Y2 − X 1 Y2 uX 2 − vX 1 X 1 Y2 − X 1 Y2 Z 2 X 1 − Z 1 X 2 X 1 Y2 − X 1 Y2 .
(24) (25)
(27) (28) (29)
Thus, substitution variable ΔZ can be generated with respect to the global coordinates of the vertexes and the lengths of three cables l12 = (u 1 + u 2 Z )2 + (u 3 + u 4 Z )2 + Z 2 .
(31)
(26)
(30)
Through solving Eq. (30) and ignoring the unreasonable one of the two solutions, one can obtain substitution variable ΔZ , and then, the global coordinates of the load [X, Y, Z ] can be derived by combining with Eqs. (19)–(23)
On the basis of the schematic sketch of a mobile crane in Fig. 4, the coordinates of the pivot points Di relative to the global frame O can be achieved as ⎡
⎡
⎤
X Ai +
X Di ⎢ ⎣ Y Di ⎦ = ⎢ ⎢ Y Ai + ⎣ Z Di
L−Ls L−Ls
(X −X Oi
)2 +(Y −Y
Oi
)2 +s−d
√(L i −ls )(Y Oi −Y Ai )
(X −X Oi )2 +(Y −Y Oi )2 +s−d
⎤ ⎥ ⎥ ⎥. ⎦
Z Ai ls L
(32) Thus, the following equation is concluded as lci =
2 (X Di − X Di )2 + (Y Di − Y Oi )2 − f
li = (X − X Ai )2 + (Y − Y Ai )2 + (Z − Z Ai )2
+ (Z Di + h)2
(33) (34)
⎤ Y A1 /X A1 −tan(π/6) arctan 1+(Y /X tan(π/6) ) A1 A1 φ1 ⎢ ⎥ ⎥ (Y A2 −Y O2 )/(X A2 −X O2 )+tan(π/6) ⎣ φ2 ⎦ = ⎢ ⎢ arctan 1−((Y ⎥. −Y −X tan(π/6) )/(X )) A2 O2 A2 O2 ⎣ ⎦ φ3 A3 −X O3 ) arctan (X (Y −Y ) ⎡
⎤
⎡
O3
2.3 Inverse kinematics of cable parallel manipulator with movable vertexes As mentioned, the CPMMC is a redundant actuated system. Furthermore, there are infinite groups of inputs that can meet the given movement of the load. Generally, limited solutions can be obtained through adding certain constraints, such as shortest stroke, minimum power and force. Taking the engineering application into consideration, two certain constraints are proposed, as depicted in Fig. 3b. In detail, in order to avoid collision during the cooperation of mobile cranes, suppose that the horizontal distance between the top points of the boom Ai (i = 1, 2, 3) and hoisting point of load P is d. Besides, the boom and the corresponding cable are constrained in the same direction from the top view of the CPMMC, in order to minimize the lateral forces in cables and avoid side tumbling of cranes. Hence, the coordinates of the top points of the booms Ai can be obtained by
√(L−ls )(X Oi −X Ai )
A3
(35)
3 Dynamics There are different approaches for solving the dynamics of manipulators, including Newton–Euler equation, Kane equation, and Lagrange equation. Based on D’Alembert principle, Lagrange equation takes the whole mechanical system as the object, which is simple in form. Moreover, the dynamic model established based on Lagrange equation is standard in form, which can be directly used for further study of the mechanism, such as control and load distribution. In this article, Lagrange equation is adopted to solve the dynamics of the CPMMC. The three main parts of the CPMMC are analyzed, including the load, the booms and the turrets. Based on the forward kinematics of cable parallel manipulator with activities vertexes, the velocity of the load can be written in differential matrix form as
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S. Qian et al.
⎡
⎡
3 i=1
⎤
∂X ∂li li
+
⎢ vX ⎢ 3 ∂Y ⎣ vY ⎦ = ⎢ i=1 ⎢ ∂li li + ⎣ vZ 3 ∂Z i=1 ∂li li +
∂X ∂lci lci
+
∂X ∂φi
φi
⎤
⎥ ⎥ ∂Y ∂Y ⎥ ∂lci lci + ∂φi φi ⎥ . (36) ⎦ ∂Z ∂Z l + φ ∂lci ci ∂φi i
Hence, the kinetic energy of the load mass m can be expressed as 9 qi Si 1 2 2 2 K m = m v X + vY + v Z = 2 2
(37)
Si = m
∂X ∂Y ∂Z vX + vY + v Z , i = 1, 2, . . . , 9. ∂qi ∂qi ∂qi (38)
The gravitational potential energy of the load is as follows Pm = mg Z .
(39)
The Lagrange equation of the load is established as ∂ ∂ Km ∂ Km ∂ Pm − = τm + (40) ∂t ∂q ∂q ∂q where τ m stands for the general force on the nine actuators of the CPMMC with respect to the mass of the payload m, τ m = [τm1 , τm2 , τm3 , τm4 , τm5 , τm6 , τm7 , τm8 , τm9 ]·T (41) Referring to Eqs. (36)–(39), we obtain 9 ∂ Si ∂ Si ∂ ∂ Km ∂ Si = = q¨i + q˙i . ∂t ∂ q˙i ∂t ∂ q˙i ∂qi
(42)
i=1
1 ¯ 2 JLi θ˙i + J˜Li φ˙ i2 , i = 1, 2, 3 2 3
KL =
i=1
where
One should notice that the movement of each boom includes two parts, which are rotation around the lower pivot with the angle θi (i = 1, 2, 3) and rotation around the slewing center of the turret with the angle φi (i = 1, 2, 3). Therefore, the kinetic energy of the boom contains two parts, which can be represented as
i=1
θ˙i =
1 L 2 − Z 2Ai
9 ∂ Z Ai j=1
∂q j
q j
, i = 1, 2, 3
(47)
(48)
1 (49) J¯Li = m L L 2 , i = 1, 2, 3 3 where J¯Li is the rotary inertia of the booms around the corresponding lower pivots and θ˙i represents the angle velocity of the boom around the lower pivot During the rotation of booms around the slewing center of the turrets, the rotary inertia of the booms around the slewing center are variable with the angle θi . Therefore, the rotary inertia around the slewing center can be derived as 1 J˜Li = m L L 2 1 − sin2 θi 3 1 (50) = m L L 2 − Z 2Ai . 3 From Fig. 4, the coordinates of the lower boom pivots can be represented as [X B1 , Y B1 , Z B1 ] = [X O1 − s cos (φ1 + π/6) , Y O1 − s sin (φ1 + π/6) , 0]
(51)
By substituting Eq. (42) into Eq. (40), the left part of the Lagrange equation of the load can be expressed in the standard form as ∂ Km ∂ ∂ Km − = M m q¨ + C m q˙ (43) ∂t ∂q ∂q
[X B2 , Y B2 , Z B2 ] = [X O2 − s cos (π/6 − φ2 ) , Y O2
where
The assumption is made that the telescopic booms of mobile cranes are homogeneous, thus, the center of gravity of each boom in global coordinate system can be expressed as
M mi j = C mi j =
∂ Si , i = 1, 2, . . . , 9; j = 1, 2, . . . , 9 ∂q j
(44)
∂ Si ∂ S j + , i = 1, 2, . . . , 9; j = 1, 2, . . . , 9. ∂q j 2∂qi (45)
In addition, the angles of the booms around the down pivots are Z Ai , i = 1, 2, 3. (46) θi = arcsin L
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− s sin (π/6 − φ2 ) , 0]
(52)
[X B3 , Y B3 , Z B3 ] = [X O3 − s sin φ3 , Y O3 − s cos φ3 , 0].
(53)
[X Li , Y Li , Z Li ] (X Ai + X Bi ) (Y Ai + Y Bi ) (Z Ai + Z Bi ) , = , , 2 2 2 i = 1, 2, 3. (54) Thus, the gravitational potential energy of the booms are
Dynamics and trajectory tracking control
PL =
3
m L g Z Li , i = 1, 2, 3.
(55)
i=1
where M r = d i ag (0, 0, 0, 0, 0, 0, Jr , Jr , Jr )
(66)
C r = 0.
(67)
Therefore, the Lagrange equation of the booms can be established in the standard form as ∂ K L ∂ PL ∂ ∂ KL − + τL = ∂t ∂q ∂q ∂q = M L q¨ + C L q˙ + G L (56)
Hence, the Lagrange equation of the CPMMC is established in the standard form as ∂K ∂ ∂K ∂P − τ = + = M q¨ + C q˙ + G (68) ∂t ∂q ∂q ∂q
where
where
M Li j =
∂ ∂q j
∂ KL ∂qi
τ = [τ1 , τ2 , τ3 , τ4 , τ5 , τ6 , τ7 , τ8 , τ9 ]T
, i = 1, 2, 3 · · · 9; j = 1, 2, 3
(57) ∂ KL ∂ ∂ + η j +δi j , i = 1, 2, 3 · · · 9; C Li j = ∂q j ∂qi 2∂qi j = 1, 2, 3 (58) ηi =
3 n=1
L2
∂ Z An J¯L θ˙n , i = 1, 2, 3 2 − Z An ∂qi
(59)
δ is a 9 × 9 matrix generated in the process of deriving the Coriolis matrix of the nine-input and three-output system. For simplicity in form, δ is written in partitioned matrix in Eq. (60) including two block matrixes ˜ which is represented by Eqs. (61) and (62), δ¯ and δ, respectively. . δ = δ¯ ..δ˜ (60) δ¯ = 09,6 δ˜ i j
⎤ ⎡ 2 − Z2 ∂ L aj mL ⎣ φ˙ j ⎦ , i = 7, 8, 9; = 6 ∂qi j = 1, 2, 3.
(61)
(62)
Referring to Figs. 3 and 4, the kinetic energy of the turrets are K r and Pr , respectively. 1 Jr φ˙ i2 , i = 1, 2, 3 2 3
Kr =
(63)
i=1
Pr = 0
(64)
where Jr is the rotary inertia of the turrets around the corresponding slewing center. Therefore, the Lagrange equation of the turrets can be established in the standard form as ∂ Kr ∂ Pr ∂ ∂ Kr − + τr = ∂t ∂q ∂q ∂q (65) = M r q¨ + C r q˙ + G r
(69)
M = Mm + M L + Mr
(70)
C = Cm + C L + Cr
(71)
G = G m + G L + Gr .
(72)
where τ is the generalized force of the CPMMC, M is the inertia matrix of the system, which is a positivedefinite matrix, C is the vector of Coriolis and centripetal terms, and G is the vector of gravity terms.
4 Robust ILC design ILC is based on the notion that the performance of a system that executes repetitive tasks can be improved by learning from previous iteration. It is promising control scheme for rehabilitation robots due to the repetitiveness of the therapy mode. ILC has been successfully applied to industrial robots control [29]. As is known, mobile cranes are widely used for loading, mounting and carrying large heavy-duty loads in engineering practice. It should be noted that most of the tasks are repetitive and require high security and precision. In this section, an robust ILC method is designed for trajectory tracking control of the CPMMC. Take the nonrepetitive interference and uncertain dynamic terms into consideration, the dynamic equation of the CPMMC can be written as M(q j (t))q¨ j (t) + C(q j (t), q˙ j (t))q˙ j (t) + G(q j (t), q˙ j (t)) + d j (t) = τ j (t)
(73)
d (t) = M(q (t))q¨ (t) + C(q (t), q˙ (t))q˙ (t) j
j
j
j
j
+ G(q j (t), q˙ j (t)) + τ a (t)
j
j
(74)
where t ∈ [0, T ] is the time, j ∈ N denotes the j iteration number, τ a (t) denotes the unknown disturj bance, and τ (t) is the input torque. The dynamics of the CPMMC is of the following two characteristics, specifically the inertia matrix M, a symmetric, bounded
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˙ − 2C, and positive-definite matrix, and the matrix M a skew symmetric. Hence, along the desired trajectory ˙ ¨ qd(t), qd(t), qd(t) , Eq. (73) can be linearized with Taylor formula as ˙ e, t) M(t)¨e + (C − C 1 ) e˙ + Fe + n (¨e, e, = H − (M q¨ + C q˙ + G)
(75)
where e stands for the positional tracking error matrix of the actuators in the CPMMC, including the cable length, slewing angle and hydraulic cylinder length. For the j-th iteration, e j (t) = q d (t) − q j (t), ∂ C ∂ G ∂ M e¨ e − e¨ e − e˙ e˙ n (¨e, e˙ , e, t) = − ∂q q d ∂q q d ,q˙ d ∂q qd ,q˙ d + O M (·)q¨ + OC (·)q˙ − O G (·).
(76)
Substituting Eq. (73) into Eq. (75), one can obtain M(t)¨e j (t) + (C(t) + C 1 (t)) e˙ j (t) + F(t)e (t) − j
j d 1 (t)
= H(t) − T (t) j
(77)
where
j d 1 (t) = −n e¨ j , e˙ j , e j , t + d j (t) (78) ∂ M ∂ C F(t) = q¨ d (t) − q˙ (t) ∂q qd (t) ∂q qd (t),q˙ d (t) d ∂ G − (79) ∂q qd (t) H(t) = M q˙ d (t) q¨ d (t) + C q d (t), q˙ d (t) q˙ d (t) (80) + G q d (t) .
Hence, M(t)¨e j+1 (t) + (C(t) + C 1 (t)) e˙ j+1 (t) j+1
+ F(t)e j+1 (t) − d 1
j
T j (t) = K P e j (t) + K d e˙ j (t) + T j−1 (t) + E sgn ε y j−1 , j = 0, 1, · · ·, N j j+1 j E ≥ εd 1 (t) = d 1 (t) − d 1 (t)
(82)
y j = e˙ j + e j
(84)
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(83)
j
Theorem Provided that the adaptive iterative learning control law is selected so that the following conditions are satisfied ⎧ ⎨ l p = λmin K 0d + 2C 1 − 2 M > 0 lr = λmin K 0d + 2C + 2F/ − 2 C˙ 1 / > 0 (85) ⎩ l p lr ≥ F/ − (C + C 1 − M)2max . Then, for t ∈ [0, T ], the resulting CPMMC system guarantees q j (t) → qd (t) and q˙ j (t) → q˙d (t) as j → ∞, and it is asymptotically stable. Proof A Lyapunov-like function is constructed as t T j V = e−ρυ y j K 0d y j dτ ≥ 0 (86) 0
where K 0d > 0, and ρ is a positive constant. Define δ y j = y j+1 − y j and δe j = e j+1 − e j ; then, δ y j = δ e˙ j + δe j .
(87)
From Eqs. (77)–(82) and (87), one can obtain j+1 j+1 δyj K d yn = −Mδ ˙y j − C + C 1 − M + K d − [F − (C + C 1 − M)] δe j .
(t) = H(t) − T j+1 (t). (81)
The assumptions are made that the desired trajectory is of the third-order continuity with respect to time. In addition, the same initial conditions are input for each iteration. In the iterative operational domain, all information from the current and previous operations is utilized as feedforward. For system described by (73), the control law in the j-th iterative operation is defined as follows: j
j
where K p = β( j)K 0p , K d = β( j)K 0d , β( j + 1) > β( j) > 1, β( j) is the regulatory factor of the control j j gains which acts in the j-th iteration; K p and K d are the proportional and the differential gains of the j-th iteration, respectively; and K 0p and K 0d are the initial PD control gains obtained by traditional experience tuning. The control gain matrixes are adjusted from iteration to iteration. E sgn ε y j−1 is the robust term of the ( j-1)-th iteration.
(88)
Define V j = V j+1 − V j , combining Eqs. (86)–(88), one can obtain t T T j e−ρυ δ yn K 0d δ y j + 2δ y j K 0d y j dτ V = 0 t 1 T j+1 = e−ρυ δ y j K d δ y j dτ β( j + 1) 0 t T −2 e−ρυ δ yn Mδ ˙yn dτ 0 t T j+1 −2 δ y j dτ e−ρυ δ y j C + C 1 − M + K d 0 t −ρυ jT j −2 e δ y (F − (C +C 1 − M)) δe dτ . 0
(89)
Dynamics and trajectory tracking control
Considering δ y j (0) = 0 and applying the partial integration t T e−ρυ δ y j Mδ ˙y j dτ 0 t T −ρυ jT j =e δ y Mδ y + ρ e−ρυ δ y j Mδ y j dτ 0 t t T T −ρυ j j ˙ y j dτ. − e δ y Mδ ˙y dτ − e−ρυ δ y j Mδ 0
0
(90) Substituting (90) into (89), one can obtain 1 T V j = −e−ρυ δ y j Mδ y j β( j + 1) t T −ρ e−ρυ δ y j Mδ y j dτ 0 t T −2 e−ρυ δ y j (F − (C + C 1 − M)) δe j dτ 0 t T j+1 − δ y j dτ . e−ρυ δ y j 2C 1 − 2 M + K d
Using the Cauchy–Schwartz inequality gives δ e˙ T Qδe ≥ − δ e˙ Qmax δe According to Eqs. (85) and (95) 2 ω ≥ l p δ e˙ − Qmax δe lp 1 2 2 + l p − Qmax δe2 ≥ 0 lr
(95)
(96)
Since M(θ) is the symmetric positive-definite inertia matrix, and from (85) and (93), it can be concluded that V j ≤ 0. Hence, V j+1 − V j ≤ 0. In addition, K 0d is a positive-definite matrix and V j > 0 is bounded. As a result, y j (t) → 0 when j → ∞. Then, e˙ j (t) → 0 and e j (t) → 0. This completes the proof of Theorem.
The block diagram of robust ILC for the CPMMC is shown in Fig. 5.
0
(91) From Eq. (82), the following equation can be derived t T j+1 e−ρυ δ y j K d δ y j dτ 0 t T = β ( j + 1) e−ρυ δ y j K 0d δ y j dτ 0 t T ≥ e−ρυ δ y j K 0d δ y j dτ. (92) 0
Combing (87) and (91), and consider (92) gives 1 T −e−ρυ δ y j Mδ y j V j ≤ β( j + 1) t T −ρ e−ρυ δ y j Mδ y j dτ 0 t T − ρ e−ρυ δe j l p δe j dτ 0 t T −ρυ (93) − e δe j l p δe j − ωe−ρυ dτ 0
where ω = δ e˙ j
T
2C 1 − 2 M + K 0d δ e˙ j T
+ 2 δ e˙ j [F/ − (C + C 1 − M)] δe j T + 2 δe j K 0d + 2C + 2F/ − 2 C˙ 1 / δe j . Define Q = F/ − (C + C 1 − M); then, one can obtain from (85) ω ≥ l p δ e˙ 2 + 2 δ e˙ T Qδe + 2 lr δe2
(94)
5 Results and discussion In order to investigate the dynamics and the designed robust ILC controller of the CPMMC, numerical simulation is conducted with MATLAB software. First of all, by referring to the engineering practice, a certain case study is simulated. Specifically, all of the luffing angle in the multi-crane system are constrained (i.e., θi = [θ ]); thus, the CPMMC is simplified as a six-input and three-output system, so that the computational cost and simulation time can be reduced obviously. Then, the designed robust ILC is implemented on the basis of the CPMMC dynamics from iteration to iteration until the total error along the trajectory and smoothness are acceptable according to the requirements of specific engineering tasks. The main parameters of the CPMMC are listed in Table 1. During the simulation, the load is suspended in the workspace generated in the preceding part along the given spatial trajectory formulated as ⎧ πt ⎨ x = sin 2 + 10 . (97) y = cos π2t + 5 sin(π/3) 2 ⎩ z = 2t + 5 The related trajectories are obtained online through simulation of the specific engineering case represented in Eq. (97). The actual process of the operation is similar to that of the widely used industrial robots. The
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S. Qian et al. Fig. 5 Robust ILC structure of the CPMMC
Table 1 Parameters of the CPMMC System parameters
Value
Mass of the payload (m)
3 × 103 kg
Acceleration due to gravity (g)
9.81 m/s2
Mass of boom (m l )
2 × 103 kg
the rotary inertia of the turrets ( Jr )
4 × 103 kg m2
Distance between each mobile crane and the reference point ( D)
20 m
Length of boom (L)
20 m
Vertical distance between the lower hinge joints of boom and cylinder (h)
1.5 m
Horizontal distance between the lower hinge joints of boom and cylinder ( f )
2.45 m
Horizontal distance between the lower hinge joint of boom and the slewing axle (s)
0.25 m
Distance between the lower hinge joints and the center of gravity of boom (l g )
10 m
Distance between the lower hinge joint of boom and the upper hinge joint of cylinder (l s )
7m
first step is online, aimed to obtain the torque trajectories of all actuators according to the concrete target and task, specifically the mass and desired trajectory of the payload for the CPMMC. The designed robust ILC is implemented on the basis of the CPMMC dynamics from iteration to iteration until the torque trajectories meet the tracking performance requirement. The second step is offline. The nine actuators of the CPMMC are actuated according to the obtained torque trajectories in the online step. The repetitive disturbances on the six joints are expressed as τ (t) = d i ag (τ 1 , τ 1 , τ 1 , τ 2 , τ 2 , τ 2 ) (98) where τ 1 = 200 sin (5t) 1 − e−t andτ 2 = 100 sin (20t). The initial proportional and the differential gain of the designed robust ILC are obtained by traditional experience tuning K 0p = 2K 0d = d i ag (3000, 3000, 3000, 1500, 1500, 1500) . (99)
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The gain of the robust term is the updated according to the linearized dynamics of the CPMMC j+1 j (100) E = d 1 (t) − d 1 (t) . Figures 6, 7, 8, 9, 10 and 11 show the tracking performance of the six active joints, including the three cable lengths l1 , l2 and l3 and the slewing angle φ1 , φ2 and φ3 . The smooth curves indicate that the designed controller can avoid impact stress effectively, which is important for the stability of the system, especially prolonging service life the hydraulic driving system. Figures 12 and 13 represent the maximum tracking errors of cable length and slewing angle in different iterations with iteration times on the horizontal axis and maximum tracking errors on the vertical axis. As shown, the tracking performance was considerably improved with the increase in the iteration. Only after the fifth iteration, the maximum tracking errors of the cable lengths l1 , l2 and l3 are reduced to 0.027, 0.025 and 0.024 (m), respectively. And the maximum tracking errors of slewing angle φ1 , φ2 and φ3 are reduced to 0.0046, 0.0038 and 0.0015 (rad), respectively. The velocity tracking performance of the six active joints are shown in Figs. 14, 15, 16, 17, 18 and 19.
Dynamics and trajectory tracking control
Fig. 6 Tracking performance of cable length l1
Fig. 9 Tracking performance of slewing angle ϕ1
Fig. 7 Tracking performance of cable length l2
Fig. 10 Tracking performance of slewing angle ϕ2
Fig. 8 Tracking performance of cable length l3
Fig. 11 Tracking performance of slewing angle ϕ3
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Fig. 12 Maximum tracking errors of cable length in different iterations
Fig. 13 Maximum tracking errors of slewing angle in different iterations
Fig. 15 Velocity tracking performance of cable length l2
Fig. 16 Velocity tracking performance of cable length l3
Fig. 17 Velocity tracking performance of slewing angle ϕ1 Fig. 14 Velocity tracking performance of cable length l1
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Fig. 18 Velocity tracking performance of slewing angle ϕ2
Fig. 19 Velocity tracking performance of slewing angle ϕ3
Fig. 20 Maximum velocity tracking errors of cable length in different iterations
Fig. 21 Maximum velocity tracking errors of slewing angle in different iterations
Figures 20 and 21 show the maximum velocity tracking errors of the cable length and slewing angle in different iterations with iteration times on the horizontal axis and maximum tracking errors on the vertical axis, respectively. As is shown in Figs. 14, 15, 16, 17, 18 and 19, the velocity of the six active joints changes smoothly in different iterations. Figures 20 and 21 indicate that the maximum velocity tracking errors of cable lengths are reduced from 0.9534, 0.8023 and 0.7885 to 0.08912, 0.1233 and 0.08507 (m/s), respectively. The similar decreasing trend of the velocity tracking errors can be found for the slewing angles. The maximum velocity tracking errors of the slewing angles are reduced from 0.0926, 0.1039 and 0.0883 to 0.0092, 0.0131 and 0.0122 (rad/s), respectively. Figure 22 shows the desired trajectory and tracking trajectories of the load in different iterations. The tracking trajectory closes to the desired trajectory with iteration. After the fifth iteration, the actual trajectory tracks the desired trajectory with high accuracy and the tracking error of the load converges close to zero. The results indicate that tracking performance of the robust ILC is acceptable for cooperation of multiple mobile cranes in engineering practice. The traditional PID controller as well as PD controller are widely accepted for automated machines in industry and engineering for the reliable and robust performance under poor working conditions [32]. In order to investigate the effect, the robust ILC is compared with the traditional PD controller. Specifically, robust ILC and traditional PD controller are implemented in the CPMMC, respectively, in the light-duty case (i.e.,
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Fig. 22 Tracking performance of the load under robust ILC Fig. 25 The lateral forces on the cranes under light-duty condition with robust ILC for the sixth iteration
Fig. 23 Tracking performance of the load under robust ILC and PD controller
Fig. 26 Required force in the first cable under light-duty and heavy-duty conditions for the fifth iteration
Fig. 24 The lateral forces on the cranes under light-duty conditions with traditional PD control
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the mass of the load is 3 tons) under the same initial proportional gain K 0p and differential gain K 0d . The resultant trajectories of the fifth iteration of the robust ILC and traditional PD controller are shown in Fig. 23. After the fifth iteration, the actual trajectory tracks the desired trajectory with much higher accuracy than adopting traditional PD controller. Figures 24 and 25 display the lateral forces on the cranes with robust ILC and traditional PD controller for the fifth iteration. As shown, the lateral forces on the three cranes with traditional PD controller are 250.39, 348.76 and 307.84 N. However, the corresponding lateral forces reduced to 78.08,
Dynamics and trajectory tracking control
Fig. 27 Required force in the second cable under light-duty and heavy-duty conditions for the fifth iteration
Fig. 29 Required force on the first slewing motor under lightduty and heavy-duty conditions for the fifth iteration
Fig. 30 Required force on the second slewing motor under lightduty and heavy-duty conditions for the fifth iteration Fig. 28 Required force in the third cable under light-duty and heavy-duty conditions for the fifth iteration
47.56 and 43.27 N at the fifth iteration with robust ILC, which is significant for the stability of the system, especially prolonging service life of the supporting devices in cranes. The results shown in Figs. 23, 24 and 25 indicate that the designed control is significantly useful compared with traditional PD control, and acceptable for the cooperation of multi-cranes. In addition, two case studies are simulated with the same expected trajectory but different load weights of 10 and 3 tons, which are heavy-duty and lightduty conditions, respectively. The simulation results at the fifth iteration with the robust ILC are shown in Figs. 26, 27, 28, 29, 30 and 31. The max absolute value
Fig. 31 Required force on the third slewing motor under lightduty and heavy-duty conditions for the fifth iteration
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Fig. 32 The driving power of the fifth iteration of the robust ILC
Fig. 33 The driving power of the traditional PD controller
of the forces on the three cables are 7.301×104 , 7.451× 104 , 7.058×104 N under the heavy-duty condition, and reduced to 3.744 × 104 , 3.171 × 104 , 3.056 × 104 N under the light-duty condition. One can obtain that the max absolute value of the dynamic forces on the three cables are much smaller than even the corre-
Fig. 34 The maximum tracking position error of different payloads
sponding static forces on a single crane. Obviously, the CPMMC have better carrying capability than a single crane, which are more safe and stable during engineering practice. The similar decreasing trend can be found for the max absolute value of the torques on the slewing motors. In detail, under the heavy-duty condition, the max absolute value of the torques on the slewing motors are 3536, 2991 and 3272 N M, and measured 2411, 2909 and 3220 N M under the light-duty condition. However, the reduced proportion is much more less than under the heavy-duty condition. Figures 32 and 33 illustrate the corresponding driving power of the fifth iteration of the robust ILC and traditional PD controller, respectively, including the power of the three winches and three slewing motors. The corresponding max absolute values are listed in Table. 2. Compared with the max absolute values with the traditional PD control method, the values of each the driving power is smaller when the designed robust ILC is implemented. The results indicate that the three cable winches undertake the most of power for the CPMMC system. Effect of varying mass should be considered since the CPMMC is designed to lift loads whose masses are
Table 2 The driving power of the fifth iteration of the robust ILC and traditional PD controller Power
The first winch
The second winch
The third winch
The first slewing motor
The second slewing motor
Robust ILC
28.88
26.74
30.76
4.71
6.10
6.39
kW
PD
45.85
56.39
38.71
7.99
6.65
12.81
kW
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The third slewing motor
unit
Dynamics and trajectory tracking control
different, such as containers in the terminals and wind power equipment. The simulations of the engineering case presented in Eq. (97) with different masses of payloads are carried out with the proposed ILC controller. The maximum tracking position error of different payloads are shown in Fig. 34. One can obtain that the maximum tracking position error of the payload increases with the payload mass, which are 0.0550, 0.1336 and 0.2024 m when the payload mass are 3, 7 and 10 t, respectively. The tracking position error in heavy-duty working condition can be reduced by increasing the iterative times according to the accuracy requirement of the task.
6 Conclusions The kinematics of the CPMMC is derived via iterative substitution. The complete dynamic model of the nineinput and three-output system is established based on Lagrange equation and the kinematics. On the basis of the linearization of the dynamics, a robust iterative learning control strategy is presented for trajectory tracking control of cooperative multiple mobile cranes. A case study is analyzed that the CPMMC is actuated by slewing motors and cable winches with the luffing angles constrained. Numerical simulations of tracking the same desired path are carried out with different loads and control schemes. The results indicate the better lifting capacity of the CPMMC, compared with an under-actuated single crane. The robust ILC shows a better position and velocity tracking performance than the traditional PD control scheme under the same proportional and differential gains. In addition, the lateral forces on the cranes and corresponding driving power are reduced considerably, which validate the stability and energy efficiency of the proposed controller. In our future work, we will extend the proposed approach to more common engineering cases so as to take the algorithm into practice. In addition, the experimental prototype is being conducted for further research, including the optimal path planning and dynamic load-carrying capacity of cooperative multiple mobile cranes. Acknowledgments This work was supported by the National Natural Science Foundation of China (51275515) and the Fundamental Research Funds for the Central Universities (2014HGC H0015). The authors appreciate the comments and valuable suggestions of anonymous referees and editors for improving the quality of the paper.
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