Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.1683 AN ENHANCED COUPLING NONLINEAR TRACKING CONTROLLER FOR UNDERACTUATED 3D OVERHEAD CRANE SYSTEMS Menghua Zhang, Xin Ma, Xuewen Rong, Rui Song, Xincheng Tian, and Yibin Li ABSTRACT An enhanced coupling nonlinear tracking control method for an underactuated 3D overhead crane systems is set forth in the present paper. The proposed tracking controller guarantees a smooth start for the trolley and solves the problem of the payload swing angle amplitude increasing as the transferring distance gets longer for the regulation control methods. Different from existing tracking control methods, the presented control approach has an improved transient performance. More speciﬁcally, by taking the operation experience, mathematical analysis of the overhead crane system, physical constraints, and operational efﬁciency into consideration, we ﬁrst select two desired trajectories for the trolley. Then, a new storage function is constructed by the introduction of two new composite signals, which increases the coupling behaviour between the trolley movement and payload swing. Next, a novel tracking control strategy is designed according to the derivation form of the aforementioned storage function. Lyapunov techniques and Barbalat’s Lemma are used to demonstrate the stability of the closed-loop system without any approximation manipulations to the original nonlinear dynamics. Finally, some simulation and experiments are used to demonstrate the superior transient performance and strong robustness with respect to different cable lengths, payload masses, destinations, and external disturbances of the enhanced coupling nonlinear tracking control scheme. Key Words: Tracking control, Overhead crane systems, Underactuated systems, Lyapunov techniques, Barbalat’s lemma, Coupling behavior. I. INTRODUCTION Due to high payload capacity, less energy consumption, and wide ﬂexibility of operation, overhead cranes have been widely applied in construction sites, harbors, workshops, and so on. Overhead crane systems are underactuated systems, which have fewer independent control inputs than the number of to-be-dominated degrees of freedoms (DOFs) [1–6]. Underactuated systems have some advantages, such as simple structure, low power consumption, low cost, and low weight due to the omission of partial actuators. However, unexpected payload swings occur for reasons like inertia or external disturbances during the transportation process, which makes it challenging to design the crane controllers due to the underactuated nature Manuscript received March 3, 2016; revised June 23, 2017; accepted August 19, 2017. The authors are with School of Control Science and Engineering, Shandong University, Jinan 250061, China. Menghua Zhang, Xin Ma (corresponding author, e-mail: maxin@sdu.edu.ch), Xuewen Rong, Rui Song, Xincheng Tian, and Yibin Li are with the School of Control Science and Engineering, Shandong University, Jinan 250061, China. This work is supported by Shandong Province Science & Technology Development Foundation, Shandong Province Independent Innovation & Achievement Transformation Special Fund, the Fundamental Research Funds of Shandong University, China, and the National High-tech Research and Development (863 Program) of China. of cranes. At present, payload swing suppression and elimination are usually achieved by skilled workers through reducing crane speeds, leading to a loss of transportation efﬁciency. Moreover, accidents may occur due to workers’ fatigue and negligence. Therefore, it is of great practical signiﬁcance and engineering value to design crane controllers. In order to improve the reliability and transportation efﬁciency of overhead cranes, a considerable number of studies have been undertaken. Fang et al. [7,8] proposed several nonlinear coupling regulation control laws for underactuated overhead crane systems by analyzing system energy, and demonstrated that the more enhanced coupling of the controllers the better transient responses performance for crane systems. Based on this, Sun et al. [9,10] designed some enhanced coupling regulation controllers for 2D overhead crane systems. However, for the above coupling-based regulation control method, the payload swing amplitude increases as the transferring distance gets longer. Furthermore, when analyzing state convergence, either approximation operations are required or speciﬁc terms are neglected. In [11], an adaptive controller based on dissipation theory was designed, achieving rapid stabilization control. In [12,13], a hierarchical sliding-mode controller and an adaptive sliding-mode controller were proposed, respectively, © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018 realizing precise positioning of the trolley and effective swing suppression control. An input shaping technique proved to be a simple and effective anti-swing control method through a series of experiments [14–19]. In [20–23], reference trajectories subject to trolley displacement, velocity, and acceleration limitation were deﬁned, and then combined with anti-swing components, guaranteeing precise trolley positioning and anti-swing control. Radac et al. [24] presented an optimal behavior prediction mechanism for multi-input–multi-output control systems in a hierarchical control system structure, using previously learned solutions to simple tasks called primitives. Zhang et al. [25] achieve simultaneous motion regulation and payload swing suppression and elimination within a ﬁnite time. Hashim et al. [26] proposed an adaptive synchronization control method with prescribed performance to control transient as well as steady state behavior. In addition to the aforementioned model-based control methods, intelligent control methods such as fuzzy logic control method [27,28], which was introduced into the sliding-mode control method in order to eliminate the swing angle induced by the sliding-mode control method, has been applied in crane systems. The aforementioned studies were presented for 2D overhead crane systems. Because 3D overhead crane systems have more states, stronger state coupling and stronger non-linearity, it is more challenging to propose appropriate control methods for them. Passivity and energy-based frameworks [29], second-order sliding-modes controller [30], fuzzy controller [31], and motion planning control methods [32] were employed to drive the trolley to the desired positions and suppress the payload swing effectively. Sun et al. [33] proposed an energy coupling output feedback (OFB) controller of a 4-DOF underactuated crane with inputs constraints, achieving both accurate trolley positioning and efﬁcient payload swing elimination. Zhang et al. [34] proposed an ofﬂine minimum-time trajectory planning approach for underactuated overhead cranes. Khatamianfar and Savkin [35] presented a novel tracking control method based on model predictive control, leading to high control performance with small payload swing. It should be pointed out that most of the above control methods are aimed at stabilization control. From the perspective of practical application, workers prefer to operate the trolley along a positioning reference trajectory to improve transportation efﬁciency and offset adverse effects, such as the large initial control moment brought by stabilization control. For this reason, Fang et al. [36] planned an Sshaped smooth trajectory for the trolley based on operation experience and mathematical analysis, and then designed an adaptive control law to make the trolley track the planned trajectory. By taking system friction and air resistance into account, an adaptive trajectory tracking controller was designed for a 3D overhead crane systems in [37]. In [38], by introducing the potential function, a nonlinear tracking controller for 2D overhead crane systems guarantees the tracking error always within a priori set bounds. However, the existing tracking control schemes can only guarantee the trolley displacement and the payload swing angels converge to the desired value (trajectory) and zero, but cannot guarantee the transient performance of the cranes. In response to the above-mentioned practical issues, a nonlinear coupling tracking control scheme for 3D overhead crane systems is proposed in this paper. Speciﬁcally, we take operation experience, the mathematical analysis of the system, physical constraints, and operational efﬁciency into consideration. Two desired trajectories are selected for the trolley. Then, inspired by [7–10], a novel positive deﬁnite function is constructed by introducing two new composite signals integrating the trolley velocity and payload swing, increasing the transient performance of the controller. Lyapunov techniques and Barbalat’s Lemma are used to demonstrate the asymptotic stability of the closed-loop system. Simulation and experimental results are included to prove the superior robustness and transient performance of the nonlinear coupling tracking control method set forth in the present paper. The merits of the proposed nonlinear tracking controller are as follows. 1. As validated by numerical results, the proposed controller admits strong robustness against different cable lengths, payload masses, destinations, and external disturbances. 2. By comparing the proposed controller with PD controller and OFB controller, an increased transient performance of the proposed controller is achieved. Furthermore, the proposed control method guarantees “soft trolley start”, addressing the problem of existing crane regulation control methods. 3. A strict mathematical analysis of the control method without any approximation to the original nonlinear dynamic equations is presented in this paper, providing theoretical support for the superior performance of the controller. 4. As will be seen from simulation and hardware experiment results, the transient performance of the controller is improved. The rest of the paper is organized as follows: In Section II, the model of 3D overhead crane systems is illustrated. Section III provides the nonlinear coupling tracking controller. In Section IV, we exhibit some numerical simulation and hardware experiment results to demonstrate the superior performance of the proposed controller. Section V summarizes the paper. © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd M. Zhang et al.: Enhanced coupling nonlinear tracking controller Z II. 3D OVERHEAD CRANE SYSTEMS trolley In this paper, the control problem for a 3D overhead crane system is addressed. The dynamic equations of 3D overhead cranes are described as follows [30–38]: M ðqÞ€ q þ C ðq; q_ Þq_ þ G ðqÞ ¼ F (1) where q ∈ R represents the state vector, M(q) ∈ R denote the inertia matrix, C ðq; ; q_ Þ∈R44 is the Centripetal-Coriolis matrix, G(q) ∈ R4 denote the gravity vector, F ∈ R4 is the resultant force imposed on the trolley. The vectors/ matrixes are detailed deﬁned as T q ¼ x y θx θy 4 2 4×4 mp þ M x 0 6 0 mp þ M y 6 M ðq Þ ¼ 6 6 mp lC x C y 0 4 mp lS x S y mp lC y 2 mp lC x C y 0 mp lS x C y θ_ x mp lC x S y θ_ y 0 0 0 mp l2 S y C y θ_ y mp l 2 S y C y θ_ x G ðqÞ ¼ 0 0 mp glS x C y mp lC y mp l 2 C 2y 0 0 6 60 0 C ðq; q_ Þ ¼ 6 6 40 0 mp lS x S y 0 3 7 7 7; 7 5 mp l 2 0 3 mp lC x S y θ_ x mp lS x C y θ_ y 7 _ 7 mp lS y θy 7; 7 2 _ mp l S y C y θ x 5 0 mp glC x S y h F ¼ F x f rx F y f ry d θx C 2y θ_ x T d θy θ_ y Y x y payload Fig. 1. Illustration for 3D overhead crane. [Color ﬁgure can be viewed at wileyonlinelibrary.com] 1 _ M ðqÞ C ðq; ; q_ Þ δ ¼ 0; ∀δ∈R4 (4) 2 For ease of the forthcoming controller design and stability analysis, 1 can be rewritten as δT M x þ mp €x þ mp lC x C y θ€ x mp lS x S y θ€y mp lS x C y θ_ 2x 2mp lC x S y θ_ x θ_ y mp lS x C y θ_ 2y ¼ F x f rx M y þ mp €y þ mp lC y θ€y mp lS y θ_ 2y ¼ F y f ry mp lC x C y €x þ mp l 2 C 2y θ€x 2mp l 2 S y C y θ_ x θ_ y (5) (6) (7) þmp glS x C y ¼ d θx C 2y θ_ x iT mp lS x S y €x mp lC y €y mp l 2 θ€y mp l 2 S y C y θ_ 2x where mp, Mx and My denote the payload mass, the trolley mass, the sum mass of the trolley and the bridge, respectively, l is the cable length, g represents the gravitational constant, Sx, Cx, Sy and Cy are the abbreviations of sinθx, cosθx, sinθy and cosθy, respectively, C1 and C2 are two auxiliary variables, Fx and Fy represent the control inputs along the X and Y axis, respectively, frx and fry denote the frictions in direction X and Y, respectively, dθx and dθy are air resistance coefﬁcients, x(t) and y(t) are the trolley displacements along the X and Y axis, respectively, θx(t) and θy(t) are projected swing angles (as shown in Fig. 1). According to the nature of friction, the following friction models are selected [3,9,33]: f rx ¼ f r0x tanhðx_ =ηx Þ þ k rx jx_ j_x f ry ¼ f r0y tanh y_ =ηy þ k ry jy_ j_y bridge X (2) (3) where fr0i, ηi∈R+, kri∈R1, i = x, y, are friction-related parameters. The Centripetal-Coriolis matrix C ðq; q_ Þ and the inertia matrix M(q) have the following relationship: (8) mp glC x S y ¼ d θy θ_ y In the process of crane transportation, for safety’s sake, the cable length remains the same and the payload swing angles are within permitted ranges. Based on this, we made the following assumptions. 1. The cable connecting the trolley and the payload should be mass-less and rigid. 2. During the whole transportation, the swing angles should be kept within the following bounds: π π < θx ; θy < 2 2 III. NONLINEAR COUPLING TRACKING CONTROLLER DESIGN In this section, we propose a nonlinear coupling tracking controller for 3D overhead crane systems and present the stability analysis of the closed-loop system. © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018 3.1 Selection of target trajectory It is well known that to keep the trolley running smoothly, the target trajectory should be an S-curve or a series of shaped impulses [36]. Therefore, we use the S-curve trajectories in [36] as the trolley positioning reference trajectories, which are given by " # pdx k 2vx coshð2k ax t=k vx εx Þ þ ln xd ¼ 2 4k ax cosh 2k ax t=k vx εx 2pdx k ax =k 2vx (9) 3 2 k 2vy pdy cosh 2k ay t=k vy εy 4 5 þ ln yd ¼ 2 2 4k ay cosh 2k ay t=k vy εy 2pdy k ay =k vy (10) where kvx and kax represent the maximum velocity and the maximum acceleration of the trolley in direction X, respectively. The maximum velocity and the maximum acceleration of the trolley along the Y axis are denoted by kvy and kay, respectively. Pdx and pdy are the destinations in direction X and Y. The purpose of introducing εx and εy is to regulate the initial accelerations of the trolley. The desired trajectories of 9 and 10 have the following properties [36]: 3.2 Nonlinear coupling tracking control law development To facilitate the design of the controller, the tracking errors are deﬁned as T e ¼ x xd y yd θx θy T ¼ ex ey θx θy (16) where ex and ey denote the tracking errors of the trolley along the X and Y axis, respectively. Based on the energy form of the crane systems, a positive deﬁnite function is constructed as 1 (17) E ðt Þ ¼ e_ T M ðqÞ_e þ mp gl 1 C x C y 2 By taking the derivative of 17 with respect to time, along the trajectories of 4, we are led to the following result: 1_ e_ þ M€e þ mp glS x C y θ_ x þ mp glC x S y θ_ y E_ ðt Þ ¼ e_ T M 2 ¼ e_ x F x f rx mp þ M x €x d h i þ_ey F y f ry mp þ M y €y d þmp l S x S y θ_ y C x C y θ_ x €x d mp lC y θ_ y €y d d θx C 2y θ_ 2x d θy θ_ 2y (18) lim xd ðt Þ ¼ pdx ; lim yd ðt Þ ¼ pdy ; lim x_ d ðt Þ t→∞ t→∞ t→∞ ¼ 0; lim y_ d ðt Þ ¼ 0; lim €x d ðt Þ t→∞ ¼ 0; lim €y d ðt Þ ¼ 0 t→∞ t→∞ (11) 0≤_xd ðt Þ≤k vx ; 0≤_yd ðt Þ≤k vy (12) j€x d ðt Þj≤k ax ; j€y d ðt Þj≤k ay (13) x_ d ∈L2 ; €x d ∈L2 ; y_ d ∈L2 ; €y d ∈L2 (14) During the transportation process, we cannot control the payload swing angles directly due to the underactuated nature of the crane systems. The only way for achieving swing damping is to utilize the coupling relationship between the trolley motion and the payload swing. Hence, we can’t plan the trajectory of payload swing. In this paper, the desired swing angles are selected as ξ y ¼ e_ y þ γΨ θy Then, the desired state variable of a 3D overhead crane system is written as (15) (20) where λ, γ∈R+ are positive control parameters, g(θx)denotes one θx-related function to be determined, f (θy) and Ψ(θy) represent θy-related functions to be determined. By taking the time derivative of 19 and 20, it can be obtained that 0 0 ξ_ x ¼ €e x þ λg ðθx Þf θy θ_ x þ λgðθx Þf θy θ_ y (21) 0 ξ_ y ¼ €e y þ γΨ θy θ_ y θ x ¼ θy ¼ 0 qd ¼ ½xd yd 0 0T We notice that, for the ﬁrst two terms of E_ ðt Þ related to control inputs, no terms are associated with the payload swing motion θx or θ_ x andθy or θ_ y . To solve this problem, two generalized signals are introduced as ξ x ¼ e_ x þ λg ðθx Þf θy (19) (22) where g0 (θx) denotes the derivative of g(θx) with respect to θx, f 0 (θy) represents the derivative of f (θy) with respect to θy , and Ψ0 (θy)is the derivative of Ψ(θy) with respect to θy. © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd M. Zhang et al.: Enhanced coupling nonlinear tracking controller 0 λmp lC x C y g ðθx Þf θy θ_ 2x 0 þλmp lC x C y gðθx Þf θy θ_ x θ_ y 0 λmp lS x S y g ðθx Þf θy θ_ x θ_ y 0 λmp lS x S y gðθx Þf θy θ_ 2y 2 ¼ λmp l C x C y θ_ x S x S y θ_ y ⇒gðθx Þ ¼ sin θx ; f θy ¼ cos θy By integrating 19 and 20 with respect to time, one has t t ∫0 ξ x dt ¼ ∫0 e_ x þ λgðθx Þf θy dt t (23) ¼ ex þ λ∫0 g ðθx Þf θy dt t t t ∫0 ξ y dt ¼ ∫0 e_ y þ γΨ θy dt ¼ ey þ γ∫0 Ψ θy dt (24) Then, the new state vector is accordingly constructed as T ξ ¼ ξ x ξ y θ_ x θ_ y T ¼ e_ x þ λg ðθx Þf θy e_ y þ γΨ θy θ_ x θ_ y Further, according to the structure of 27 and the conclusions of 29 and 30, the controller is designed as (25) t F x ¼ k d1 ξ x k p1 ∫0 ξ x dt þ f rx þ mp þ M x €x d þ λ mp þ M x C x C y θ_ x λ mp þ M x S x S y θ_ y Inspired by the form of 17, we deﬁne a new positive definite function as 1 E L ðt Þ ¼ ξ T M ðqÞξ þ mp gl 1 C x C y 2 the time derivative of which is calculated as (26) 0 C y Ψ θy ≤0 (28) Considering the practical conditions where the payload swing is always beneath the trolley, that is, π2 < θy < π2 ⇒ Cy > 0. Therefore, one natural and convenient choice for Ψ (θy) is obtained as 0 Ψ θy ¼ 1⇒Ψ θy ¼ θy (29) (31) t F y ¼ k d2 ξ y k p2 ∫0 ξ y dt þ f ry þ mp þ M y €y d þ λ mp þ M y θ_ y h 0 i 0 E_ L ðt Þ ¼ ξ x F x f rx mp þ M x €x d þ λ mp þ M x g ðθx Þf θy θ_ x þ gðθx Þf θy θ_ y h 0 i þ ξ y F y f ry mp þ M y €y d þ γ mp þ M y Ψ θy θ_ y þ mp l S x S y θ_ y C x C y θ_ x €x d mp lC y θ_ y €y d d θx C 2y θ_ 2x d θy θ_ 2y 0 0 þ λmp lC x C y g ðθx Þf θy θ_ 2x þ λmp lC x C y gðθx Þf θy θ_ x θ_ y 0 0 0 λmp lS x S y g ðθx Þf θy θ_ x θ_ y λmp lS x S y g ðθx Þf θy θ_ 2y þ γmp lC y Ψ θy θ_ 2y Hence, to guarantee the last term of 27 non-negative, the following formula should be satisﬁed: (30) (32) (27) where kp1, kp2, kd1, and kp2 ∈ R+ represent positive control gains. 3.3 Stability analyses of the closed-loop system Theorem 1. The nonlinear coupling tracking controller 31 and 32 can guarantee the displacement, velocity, and acceleration of the trolley converges to the desired trajectories and the payload swing angles, angular velocity, and angular acceleration converge to zero, in the sense that T lim x y x_ y_ θx θy θ_ x θ_ y ¼ ½xd yd x_ d y_ d 0 0 0 0 T t→∞ Further, to guarantee the following terms of 27 nonnegative, that is, (33) 0 0 λmp lC x C y g ðθx Þf θy θ_ 2x þ λmp lC x C y gðθx Þf θy θ_ x θ_ y ; 0 0 λmp lS x S y g ðθx Þf θy θ_ x θ_ y λmp lS x S y gðθx Þf θy θ_ 2y ≤0 Proof. The following positive deﬁnite function is chosen as the Lyapunov function candidate based on complete square formula, one choice for g(θx) and f (θy) are obtained as (34) 1 V ðt Þ ¼ ξ T Mξ þ mp gl 1 C x C y 2 2 1 t 2 1 t þ k p1 ∫0 ξ x dt þ k p2 ∫0 ξ y dt 2 2 © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018 We can easily conclude from 36 that By taking the time derivative of 34, and substituting the conclusions of 27, 31, and 32, it can be derived that t t t 3d θ t 3d θ k d1 ∫0 ξ 2x dτ þ k d2 ∫0 ξ 2y dτ þ ∫0 x C 2y θ_ 2x dτ þ ∫0 y θ_ 2y dτ 4 4 2 t _ _ ≤V ð0Þ V ðt Þ λmp l∫0 C x C y θx S x S y θy dτ 2 2 2 x d mp lC y €y d t mp l 2 t mp lS x S y € €x dτ þ ∫0 þ∫0 dτ∈L∞ d θx d d θy 2 V_ ðt Þ ¼ k d1 ξ 2x k d2 ξ 2y λmp l C x C y θ_ x S x S y θ_ y þmp l S x S y θ_ y C x C y θ_ x €x d mp lC y θ_ y €y d 3d θx 2 _ 2 C θ d θx C 2y θ_ 2x d θy θ_ 2y ≤ 4 y x 2 mp lS x S y €x d mp lC y €y d m2 l 2 2 3d θy _ 2 €x θ þ þ d θx d 4 y d θy k d1 ξ 2x k d2 ξ 2y λmp l C x C y θ_ x S x S y θ_ y (40) It follows from 41 that ξ x ; ξ y ; θ_ x ; θ_ y ∈L2 2 (35) The following formula can be obtained from 37–39, 41 and the Barbalat’s Lemma [1] as lim ξ x ¼ 0; lim ξ y ¼ 0; lim θ_ x ¼ 0; lim θ_ y ¼ 0 Integrating 35 with respect to time and performing some arrangements yields 2 2 3d θx 2 _ 2 t mp l €x 2 dτ C y θx dτ þ ∫0 4 d θx d 2 x d mp lC y €y d t 3d θy 2 t mp lS x S y € _ ∫0 θ dτ þ ∫0 dτ 4 y d θy t V ðt Þ≤V ð0Þ ∫0 (36) It follows from 13 and 14 that t t V ðt Þ∈L∞ ⇒ξ x ; ξ y ; θ_ x ; θ_ y ; ∫0 ξ x dt; ∫0 ξ y dt∈L∞ t→∞ (37) (38) t→∞ t→∞ M y þ mp €y þ mp lC y θ€y mp lS y θ_ 2y t ¼ k d2 ξ y k p2 ∫0 ξ y dt þ mp þ M y €y d þ λ mp þ M y θ_ y €x ¼ g 1 þ g2 (39) (42) (44) After performing some mathematical operations like addition, subtraction, multiplication, and division on 5–6 and 43–44, one has It is clear from 13, 21–22, and 37–38 that ξ_ x ; ξ_ y ∈L∞ t→∞ Substituting 31 and 32 into 5 and 6, respectively, 5 and 6 are rewritten as M x þ mp €x þ mp lC x C y θ€x mp lS x S y θ€ y mp lS x C y θ_ 2x 2mp lC x S y θ_ x θ_ y mp lS x C y θ_ 2y t ¼ k d1 ξ x k p1 ∫0 ξ x dt þ mp þ M x €x d þ λ mp þ M x C x C y θ_ x λ mp þ M x S x S y θ_ y ; (43) Then, according to 5–6, 19–20, 31–32, and 36 we have F x ; F y ; €x ; €y ; θ€ x ; θ€y ; x_ ; y_ ∈L∞ (41) (45) where the expressions of g1 and g2 are 1 mp lM y C 2y S x C y θ_ 2x mp lM y S x C y θ_ 2y þ M y þ mp S 2y k d2 ξ y mp þ M y €y d λ mp þ M y θ_ y @ A _ _ þmp C y S x S y mp þ M x €x d þ λ mp þ M x C x C y θx λ mp þ M x S x S y θy k d1 ξ x 0 g1 ¼ g2 ¼ M x M y þ M x mp S 2y þ M y mp C 2y t t mp gM y C 2y S x C x M y þ mp S 2y mp k p1 ∫0 ξ x dtC y S x S y k p2 ∫0 ξ y dt M y þ mp S 2y M x M y þ M x mp S 2y þ M y mp C 2y © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd M. Zhang et al.: Enhanced coupling nonlinear tracking controller For 54, owing to lim ω1 ¼ 0; lim ω_ 2 ∈L∞ ; lim ξ y ¼ 0, we It follows from 37–39, and 43 that t→∞ lim g 1 ¼ 0; g_ 2 ∈L∞ (46) t→∞ It can be obtained from 21 that ξ_ x ¼ €x €x d λC x C y θ_ x þ λS x S y θ_ y (47) Substituting 47 into 42, the following result is obtained ξ_ x ¼ g1 €x d λC x C y θ_ x þ λS x S y θ_ y þ g 2 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |{z} (48) ϑ2 ϑ1 t→∞ lim θ€y ¼ 0; lim S y ¼ 0⇒θy ¼ 0 t→∞ t→∞ t→∞ t→∞ (49) By invoking the extended Barbalat’s Lemma [1], we have lim ξ_ x ¼ 0⇒ lim €x €x d λC x C y θ_ x þ λS x S y θ_ y t→∞ t→∞ ¼ 0⇒ lim €x ¼ 0 t→∞ (50) With some similar analysis, it can be obtained from 8 and 44 that €y ¼ β1 þ β2 (51) where k d2 ξ y þ λ mp þ M y θ_ y mp S x S y C y €x þ mp þ M y €y d β1 ¼ ; M y þ mp S 2y β2 ¼ mp lS y C 2y θ_ 2x t k p2 ∫0 ξ y dt (57) t→∞ It can be obtained from 7 that, t→∞ lim ϑ 1 ¼ 0; ϑ_ 2 ∈L∞ ; lim ξ x ¼ 0 t→∞ mp lS x S y €x mp lC y €y mp l 2 S y C y θ_ 2x þ d θy θ_ y gC x S y þ θ€ y ¼ l mp l 2 (56) In a similar way, it follows from 9, 37, 42, 50, and 55 and the extended Barbalat’s Lemma [1] that From the conclusions of lim €x d ¼ 0; lim θ_ x ¼ 0; lim θ_ y ¼ 0;and 46, 42, it is concluded that t→∞ can utilize again extended Barbalat’s Lemma [1] to obtain lim ξ_ y ¼ 0⇒ lim €y €y d γθ_ y ¼ 0⇒ lim €y ¼ 0 (55) t→∞ t→∞ t→∞ The following formula can be obtained from 8, dθ þ mp lS y θ_ 2y þ mp gC x S y C 2y y C y θ_ y l : M y þ mp S 2y θ€ x ¼ mp lC x €x þ 2mp l 2 S y C y θ_ x θ_ y d θx C 2y θ_ x mp l 2 C y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} α1 gS x ; lC y |{z} α2 where lim α1 ¼ 0; α_ 2 ∈ L∞ (59) t→∞ Then, by using 42 and the extended Barbalat’s Lemma [1], the following conclusions can be derived lim θ€x ¼ 0; lim α2 ¼ 0⇒θx ¼ 0 t→∞ t→∞ lim x_ ¼ lim x_ d ¼ 0 t→∞ (61) t→∞ By substituting 11, 42, and 57 into 20, one has lim y_ ¼ lim y_ d ¼ 0 t→∞ (62) t→∞ It can be obtained from 11, 42, 50, 60, and 43 that t lim ∫0 ξ x dt ¼ 0 (63) It holds from 38–39, and 42 that Rearranging 23 results in the conclusion of lim β1 ¼ 0; β_ 2 ∈L∞ ex ¼ ∫0 ξ x dt þ λ∫0 S x C y dt : |ﬄﬄ{zﬄﬄ} |ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄ} t t→∞ t ℑ1 (64) ℑ2 From 37, 61, and 63, it is derived that It can be derived from 22 that ξ_ y ¼ €y €y d γθ_ y (60) Substituting 11, 42, and 60 into 20 yields t→∞ (52) (58) (53) lim ℑ 1 ¼ 0; lim ℑ_ 2 ∈L∞ ; lim e_ x ¼ 0 t→∞ t→∞ t→∞ (65) By invoking Barbalat’s Lemma [1], we have By substituting 53 into 51, one has ξ_ y ¼ β1 €y d γθ_ y þ β2 : |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |{z} ω1 ω2 lim ex ¼ 0⇒ lim x ¼ lim xd (54) t→∞ t→∞ t→∞ We can conclude from 11, 42, 55, 57, and 44 that © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd (66) Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018 t lim ∫0 ξ y dt ¼ 0 (67) t→∞ It can be obtained from 24 that t t ey ¼ ∫0 ξ y dt þ γ∫0 θy dt : |ﬄﬄ{zﬄﬄ} |ﬄﬄﬄ{zﬄﬄﬄ} (68) Then, by using 42, 62, and 67, the following conclusions can be derived lim σ 1 ¼ 0; lim σ_ 2 ¼ 0; lim e_ y ¼ 0 t→∞ t→∞ (69) t→∞ We can utilize the extended Barbalat’s Lemma [1] again to derive lim ey ¼ 0⇒ lim y ¼ lim yd t→∞ t→∞ κy = μy + kd2υy, μ_ y ¼ k d2 μy þ k d2 υy ,where μx and μy denote auxiliary functions. σ2 σ1 m þM υy = y pdy + λySy,where λx < 0, λy ¼ mpp þM xy λx . κx and κy are dynamically generated as follows: κx = μx + kd1υx, μ_ x ¼ k d1 ðμx þ k d1 υx Þ, (70) t→∞ With the conclusions of 42, 50, 55, 57, 60–62, 66, and 70 we conclude that Theorem 1 is proved. IV. SIMULATION AND EXPERIMENTS In this section, some numerical simulation and experiments are performed to verify the superior performance of the proposed nonlinear coupling tracking control method. 4.1 Simulation To illustrate the control performance, some simulation tests are implemented for the proposed nonlinear coupling tracking control method. First, to verify the improved transient performance of the proposed controller, we compare our method with the OFB controller [33] and the motion planning-based adaptive (MPA) controller [36]. Second, the robustness against different payloads, cable lengths, desired positions, and external disturbances is further tested, and we demonstrate the proposed controller can guarantee “soft trolley start” with respect to different destinations. For literature completeness, the expressions for the OFB controller and the MPA controller are provided as follows. 2. MPA controller It should be pointed out that the MPA controller is designed for the 2D overhead crane system. To be fair, we apply it to the control of the 3D overhead crane system and obtain that ^x F x ¼ k p1 ex k d1 e_ x þ YTx ω (73) ^y F y ¼ k p2 ey k d2 e_ y þ YTy ω (74) where kp1, kp2, kd1, and kp2 ∈ R+ represent positive control gains, ex = x xd and ey = y yd denote the trolley tracking errors along the X and Y axis, respectively, xd and yd are chosen the same as those in equations 9–10, respectively, ^ y are the online estimation of ωx and ωy, respec^ x and ω ω tively, which are obtained by the following update laws: ^_ x ¼ Τx Yx e_ x , ω ^_ y ¼ Τy Yy e_ y,where Τx, Τy ∈ R3 × 3 stand for diagonal, ω positive deﬁnite, update gain matrixes. The detailed expressions for Yx, ωx, Yy, and ωy are given as follows: x_ tanh Yx ¼ €x d ηx ¼ mp þ M x f r0x " Yy ¼ €y d y_ tanh ηy h ¼ mp þ M y T jx_ j_x ; ωx T k rx ! f r0y #T jy_ j_y ; ωy iT k ry 1. OFB controller: F x ¼ k p1 tanhðυx Þ k d1 tanhðκx Þ (71) F y ¼ k p2 tanh υy k d2 tanh κy (72) where kp1, kp2, kd1, and kp2 ∈ R+ stand for positive control gains, υx and υy have the following forms: υx = x pdx + λxSxCy, As can be seen from equations (80)–(83), the MPA control laws do not take the payload swing angles into consideration. Therefore, the payload swing angles can only be damped out by air resistance. The numerical simulation is implemented by Matlab/Simulink. The control gains for the proposed controller, the OFB controller, and the MPA controller are shown in Table I. For the MPA controller, the initial online estimates of ωx and ωy are set as zero. © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd M. Zhang et al.: Enhanced coupling nonlinear tracking controller Table I. Control gains for simulation 1. Controllers kp1 kd1 kp2 kd2 λ γ λx λy Tx MPA controller OFB controller proposed controller 25 18 50 25 23 50 18 16 100 35 26 30 NA NA 0.4 NA NA 0.5 NA 1 NA NA 0.54 NA 20I3 NA NA Ty 20 I NA NA 3 M x ¼ 7kg; M y ¼ 14kg; mp ¼ 1:025kg; g ¼ 9:8m=s2 ; k ax ¼ 0:3m=s2 ; k vx ¼ 0:3m=s; k ay ¼ 0:2m=s2 ; k vy ¼ 0:2m=s; l ¼ 0:7m; d θx ¼ d θy ¼ 0:3; f rox ¼ 4:4; k rx ¼ 0:5; ηx ¼ 0:01; f roy ¼ 8; k ry ¼ 1:2; ηy ¼ 0:01; within 5 s, yet the payload swing of the proposed controller is much better suppressed and eliminated. Moreover, there is almost no residual swing as the trolley stops, and it consumes the fewest control efforts and time with the proposed controller. These results have evidently proven the superior transient performance of the proposed control method. The system parameters are deﬁned as and the desired position is set as pdx ¼ 0:7m; pdy ¼ 0:5m: 4.1.1 Comparative test As a means to validate the superior performance of the proposed controller, we compare it with the OFB controller and the MPA controller. The simulation results are recorded in Figs 2–4 and the quantiﬁed results are detailed in Table II, which include the following performance indices: 1 ﬁnal position of the trolley pﬁ; 2 maximum swing amplitude θimax; 3 payload swing residual swing θires; 4 maximum actuating force Fimax; 5 time consumption tsi, which refers to the time when positioning error is within the range of (4 mm, 4 mm); 6 4.1.2 Robustness veriﬁcation simulation In this subsetion, four parts are included. In the ﬁrst part, we consider l = 0.7 m, 0.4 m, 2 m three cable lengths. Next, three cases mp = 1.025 kg, 2 kg, 4 kg for payload masses are considered. In the third part, the destinations are set as pdx = 0.7 m, 1.2 m, 2 m, pdy = 0.5 m, 1 m, 1.7 m, respectively. In the concluding part, to emulate external disturbances, three kinds of disturbances included impulsive disturbances, sinusoid disturbances (the period is 1 s and the initial phase angle is 0°), and a uniformly distributed random disturbance to the load swing are added between 7 s and 8 s, 10 s and 11 s, 15 s and 16 s, respectively. All above 10 energy consumption ∫0 F 2i dt, where i = x, y. From Figs 2–4 and Table II, it can be seen that all of the three controllers can drive the trolley to the destination 0.5 0.7 y yd x xd 0 0 0 1 2 3 4 5 6 7 8 9 0 10 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 1.5 1 0 0 -1.5 0 1 2 3 4 5 6 7 8 9 10 10 -1 20 10 5 0 0 0 1 2 3 4 5 6 7 8 9 10 -10 Time(sec) Time(sec) Fig. 2. Comparative test: Results for the proposed controller. [Color ﬁgure can be viewed at wileyonlinelibrary.com] © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018 0.5 0.7 y yd x xd 0 0 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 2 2 0 0 -2 0 10 1 2 3 4 5 6 7 8 9 10 -2 0 20 10 5 0 0 -5 -10 0 0 1 2 3 4 5 6 7 8 9 10 Time(sec) Time(sec) Fig. 3. Comparative test: Results for the MPA controller. [Color ﬁgure can be viewed at wileyonlinelibrary.com] 1 0.7 0.7 0.5 0 0 2 1 2 3 4 5 6 7 8 9 10 0 0 1 2 3 4 5 6 7 8 9 10 5 6 7 8 9 10 5 6 7 8 9 10 1 0 0 -2 -4 0 30 -1 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 10 15 10 5 0 -5 0 1 2 3 4 15 0 -5 0 1 2 3 4 5 6 7 8 9 Time(sec) Time(sec) Fig. 4. Comparative test: Results for the OFB controller. [Color ﬁgure can be viewed at wileyonlinelibrary.com] Table II. Control performance comparison for simulation 1. Controller pfx pfy θxmax θymax θxres θyres Fxmax Fymax tsx tsy unit MPA controller OFB controller proposed controller m 0.698 0.696 0.7 m 0.409 0.502 0.501 ° 2.23 2.98 1.4 ° 1.825 1.34 0.94 ° 0.58 0.12 0.01 ° 0.64 0.08 0.01 N 7.8 14.1 6.15 N 12.6 14.75 11.4 s 4.7 4.4 4.2 s 4.9 5.0 4.4 disturbances have an amplitude of 2°. The control gains for the four parts are set as those in Table I. As can be seen from Figs 5–8, the proposed controller can accurately position the trolley as well as suppress and eliminate the payload swing even in the different cable lengths, payload masses, desired positions, external disturbances, which demonstrate strong robustness of the presented method. Furthermore, we can conclude from Fig. 7 that the payload swing amplitude stays the same as the 10 10 ∫0 F 2x dt ∫0 F 2y dt N2·s 99.5 96.1 95.3 N2·s 318.4 330.2 292.3 transferring distance gets longer, which makes up for the shortage of regulation control methods. 4.2 Experiments To further illustrate the practical control performance of the proposed nonlinear coupling tracking control method, two sets of experiments are implemented on a self-built © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd M. Zhang et al.: Enhanced coupling nonlinear tracking controller 0.7 1 l=0.7m l=0.4m l=2m xd 0.7 0 l=0.7m l=0.4m l=2m yd 0.5 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 0 -5 -10 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 2 3 2 0 0 -2 0 2 4 6 8 10 12 14 16 18 20 -2 10 20 5 10 0 2 4 6 8 10 12 14 16 18 20 Time(sec) Time(sec) Fig. 5. Robustness veriﬁcation simulation: Results for the proposed controller with respect to different cable lengths. [Color ﬁgure can be viewed at wileyonlinelibrary.com] 1 0.7 mp=1.025kg 0.7 mp=2kg 0.5 mp=1.025kg mp=2kg mp=4kg mp=4kg xd 0 0 2 4 6 8 10 12 14 16 yd 18 20 0 2 2 0 0 -2 0 2 4 6 8 10 12 14 16 18 20 -2 10 20 5 10 0 0 -5 -10 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Time(sec) Time(sec) Fig. 6. Robustness veriﬁcation simulation: Results for the proposed controller with respect to different payload masses. [Color ﬁgure can be viewed at wileyonlinelibrary.com] 2 1.7 pdx=0.7m pdx=1.2m pdx=2m 1.2 0.7 0 0 2 4 6 8 10 12 14 16 18 pdy=0.5m pdy=1m pdy=1.7m 1 0.5 20 0 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 1 1 0 0 -1 0 2 4 6 8 10 12 14 16 18 20 10 -1 0 20 10 5 0 0 0 2 4 6 8 10 12 Time(sec) 14 16 18 20 -10 0 Time(sec) Fig. 7. Robustness veriﬁcation simulation: Results for the proposed controller with respect to different destinations. [Color ﬁgure can be viewed at wileyonlinelibrary.com] © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018 1 0.7 0.5 0.7 x xd 0 0 2 4 6 8 10 12 14 16 18 y yd 0 20 0 2.5 2.5 0 0 -2.5 0 2 4 6 8 10 12 14 16 18 20 -2.5 0 10 20 5 10 0 0 -5 -10 0 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Time(sec) Time(sec) Fig. 8. Robustness veriﬁcation simulation: Results for the proposed controller with respect to different external disturbances. [Color ﬁgure can be viewed at wileyonlinelibrary.com] prototype overhead crane (as shown in Fig. 9) at the College of Engineering, Qufu Normal University. The physical parameters for the crane testbed are set as tuned for the two controllers until the best performance is derived as shown in Table III. The corresponding experiment results are shown in Figs 10, 11. M x ¼ 6:157kg; M y ¼ 15:594kg; mp ¼ 1kg; g ¼ 9:8m=s2 ; l ¼ 0:6m k ax ¼ 0:1m=s2 ; k vx ¼ 0:1m=s; k ay ¼ 0:1m=s2 ; k vy ¼ 0:1m=s; f rox ¼ 20:371; k rx ¼ 0:5; ηx ¼ 0:01; f roy ¼ 23:652; k ry ¼ 1:2; ηy ¼ 0:01 and the target location is pdx ¼ 0:3m; pdy ¼ 0:4m 4.2.1 Comparison experiments We compare the proposed control method with the OFB controller in this subsection. The control gains are As can be seen from Figs 10, 11, the proposed nonlinear coupling tracking control method is superior to the OFB control method in terms of both swing amplitude and residual swing. Moreover, the control input amplitude of the proposed controller is the smaller. Therefore, we can conclude from the direct comparisons that the transient performance of the proposed controller is improved by the introduction of two generalized signals. 4.2.2 Robustness veriﬁcation experiments The following two cases are taken into consideration. Case 1. Different cable lengths. The cable length is set as l = 0.4m, whereas the control gains remain the same as those in experiment 1. Table III. Control gains for experiment 1. Controllers Fig. 9. Overhead crane experimental testbed. [Color ﬁgure can be viewed at wileyonlinelibrary.com] kp1 kd1 kp2 kd2 OFB controller 4.5 10.018 4 Proposed controller 9 10 9 λ γ λx λy 8.5 NA NA 1 0.43 8.5 0.2 1 NA NA © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd M. Zhang et al.: Enhanced coupling nonlinear tracking controller 0.4 0.3 0.2 0 0 2 4 6 8 10 12 14 16 18 0 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 2 1 0 0 -2 -1 -4 0 2 4 6 8 10 12 14 16 18 20 20 20 10 10 0 0 -10 0 2 4 6 8 10 12 14 16 18 20 Time(sec) Time(sec) Fig. 10. Experiment 1: Results for OFB controller. [Color ﬁgure can be viewed at wileyonlinelibrary.com] 0.4 0.3 0 x xd 0 2 4 6 8 10 12 14 16 y yd 0.2 18 20 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0.5 1 0 0 -0.5 -1 0 2 4 6 8 10 12 14 16 18 20 -1 10 4 5 2 0 0 -2 0 -5 2 4 6 8 10 12 14 16 18 20 Time(sec) Time(sec) Fig. 11. Experiment 1: Results for the proposed controller. [Color ﬁgure can be viewed at wileyonlinelibrary.com] 0.4 0.3 x xd 0 0 2 4 6 8 10 12 14 16 18 y yd 0.2 20 1 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 1 0 0 -1 0 5 -1 2 4 6 8 10 12 14 16 18 20 10 5 0 0 -5 0 0.01 2 4 6 8 10 12 14 16 18 20 -5 0.01 0 0 0 2 4 6 8 10 12 14 16 18 20 Time(sec) Time(sec) Fig. 12. Experiment 2: Results for the proposed controller with respect to different cable lengths. [Color ﬁgure can be viewed at wileyonlinelibrary.com] © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018 0.4 0.3 x xd 0 0 2 4 6 8 10 12 14 16 18 20 1 0 2 4 6 8 10 12 14 16 18 20 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 0.01 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 -1 10 5 0 0 0 2 4 6 8 10 12 14 16 18 20 0.01 -5 0 0 -0.01 2 -0.5 5 -5 0 0 0.5 0 0 -1 y yd 0.2 0 2 4 6 8 10 12 14 16 18 Time(sec) 20 -0.01 0 Time(sec) Fig. 13. Experiment 2: Results for the proposed controller with respect to different payload masses. [Color ﬁgure can be viewed at wileyonlinelibrary.com] Case 2. Different payload masses. The payload mass is modiﬁed to mp = 2kg, whereas the control gains remain the same as those in experiment 1. Figs 12, 13 show the experiment results. As shown in Figs 12, 13 the performance of the overhead crane system is not inﬂuenced by the variations of cable length and payload mass, which indicates strong robustness of the proposed nonlinear coupling tracking control method and implies signiﬁcant beneﬁt for its practical use. V. CONCLUSIONS In this paper, we propose a novel nonlinear coupling tracking controller for 3D overhead crane systems, which can drive the trolley to the desired positions accurately and eliminate the payload swing quickly. Two S-shape trajectories with physical constraints are selected as the desired trajectories for the trolley based on the operation experience, mathematical analysis, and operational efﬁciency. Then, on the basis of strict mathematical analysis, the controller is proposed to track the desired trajectories, providing theoretical support for the superior performance of the controller. By invoking Lyapunov techniques and Barbalat’s Lemma, we conclude that the equilibrium point of the closed-loop system is asymptotically stable. The superior transient performance and strong robustness with respect to different cable lengths, payload masses, destinations, and external disturbances of the tracking controller are demonstrated by some simulation and hardware experiment results. In future studies, we will try to apply the method to a 3D crane system with time-variable cable length. REFERENCES 1. Fantonin, I., and R. Lozano, Non-Linear Control for Underactuated Mech. Systems, Springer-Verlag, London (2002). 2. Wu, Y., F. Gao, and Z. Zhang, “Saturated ﬁnite-time stabilization of uncertain nonholonomic systems,” Nonlinear Dyn., Vol. 84, No. 3, pp. 1609–1622 (2016). 3. Zhang, M., X. Ma, X. Rong, X. Tian, and Y. Li, “Adaptive tracking control for double-pendulum overhead cranes subject to tracking error limitation, parametric uncertainties and external disturbances,” Mech. Syst. Signal Proc., Vol. 76-77, pp. 15–32 (2016). 4. Bartolini, G., A. Pisano, and E. Usai, “Output-feedback control of container cranes: A comparative analysis,” Asian J. Control, Vol. 5, No. 4, pp. 578–593 (2003). 5. Wei, X., Z. Wu, and H. R. Karimi, “Disturbance observer-based disturbance attenuation control for a class of stochastic systems,” Automatica, Vol. 63, pp. 21–25 (2016). 6. Zhang, H., Y. Jing, and X. Wei, “Composite antidisturbance control for a class of uncertain nonlinear systems via a disturbance observer,” Trans. Inst. Meas. Control, Vol. 38, No. 6, pp. 2907–2912 (2016). 7. Fang, Y., W. Dixon, D. Dawson, and E. Zergeroglu, “Nonlinear coupling control laws for an underactuated overhead crane system,” IEEE/ASME Trans. Mechatron., Vol. 130, No. 3, pp. 1–7 (2008). 8. Fang, Y., “Lyapunov-based control for mechanical and vision –based systems,” PhD Dissertation Clemson University (2002). 9. Sun, N., Y. Fang, H. Chen, and B. Lu, “Amplitudesaturated nonlinear output feedback antiswing control © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd M. Zhang et al.: Enhanced coupling nonlinear tracking controller 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. for underactuated cranes with double-pendulum cargo dynamics,” IEEE Trans. Ind. Electron., Vol. 64, No. 3, pp. 2135–2146 (2017). Sun, N., Y. Fang, H. Chen, Y. Fu, and B. Lu, “Nonlinear stabilizing control for ship-mounted cranes with disturbances induced by ship roll and heave movements: Design, analysis, and experiments,” IEEE Trans. Syst. Man Cybern. Syst., https://doi.org/ 10.1109/TSMC.2017.2700393, (2017). Ma, B., Y. Fang, Y. Wang, and Z. Jiang, “Adaptive control for an underactuated overhead crane system,” IET Contr. Theory Appl., Vol. 25, No. 6, pp. 1105–1109 (2008). Ngo, Q. H., and K. S. Hong, “Adaptive sliding mode control of container cranes,” IET Contr. Theory Appl., Vol. 6, No. 5, pp. 662–668 (2012). Vazquez, C., L. Fridman, and J. Collado, “Second-order sliding mode control of a perturbedcrane,” Trans. ASME J. Dynamics Syst. Meas. and Control, Vol. 137, No. 8, pp. 081010, (2015). Pan, H., X. Jing, and W. Sun, “Robust ﬁnite-time tracking control for nonlinear suspension systems via disturbance compensation,” Mech. Syst. Signal Proc., Vol. 88, pp. 49–61 (2017). Matusko, J., S. Iles, F. Kolonic, and V. Lesic, “Control of 3D tower crane based on tensor product model transformation with neural friction compensation,” Asian J. Control, Vol. 17, No. 2, pp. 443–458 (2015). Maghsoudi, M. J., Z. Mohamed, and A. R. Husain, “An optimal performance control scheme for a 3D crane,” Mech. Syst. Signal Proc., Vol. 66-67, pp. 756–768 (2016). Sorensen, K., and W. Singhose, “Command-induced vibration analysis using input shaping principles,” Automatica, Vol. 44, No. 9, pp. 2392–2397 (2008). Garrido, S., M. Abderrahim, A. Gimenez, R. Diez, and C. Balaguer, “Anti-swing input shaping control of an automatic construction crane,” IEEE Trans. on Autom. Sci. Eng., Vol. 5, No. 3, pp. 549–557 (2008). Blackburn, D., W. Singhose, and J. Kitchen, “Command shaping for nonlinear crane dynamics,” J. Vib. Control, Vol. 16, No. 4, pp. 477–501 (2010). Sun, N., Y. Fang, X. Zhang, and Y. Yuan, “Transportation task-oriented trajectory planning for underactuated overhead cranes using geometric analysis,” IET Control Theory Appl., Vol. 6, No. 10, pp. 1410–1423 (2012). Zhang, M., X. Ma, H. Chai, X. Rong, X. Tian, and Y. Li, “A novel online motion planning method for double-pendulum overhead cranes,” Nonlinear Dyn., Vol. 85, No. 2, pp. 1079–1090 (2016). Uchiyama, N., H. Ouyang, and S. Sano, “Simple rotary crane dynamics modeling and open-loop control for 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. residual load sway suppression by only horizontal boom motion,” IEEE /ASME Trans. Mechatron., Vol. 23, No. 8, pp. 1223–1236 (2013). Lee, H. H., “A new design approach for the anti-swing trajectory control of overhead cranes with high-speed hoisting,” Int. J. Control, Vol. 77, No. 10, pp. 931–940 (2004). Radac, M. B., and R. E. Precup, “Optimal behavior prediction using a primitive-based data-driven model-free iterative learning control approach,” Comput. Ind., Vol. 74, pp. 95–109 (2015). Zhang, Z., Y. Wu, and J. Huang, “Differential-ﬂatnessbased ﬁnite-time anti-swing control of underactuated crane systems,” Nonlinear Dyn., Vol. 87, No. 3, pp. 1749–1761 (2017). Hashim, H. A., S. El-Ferik, and F. L. Lewis, “Adaptive synchronisation of unknown nonlinear networked systems with prescribed performance,” Int. J. Syst. Sci., Vol. 48, No. 4, pp. 885–898 (2017). Ranjbari, L., A. H. Shirdel, and M. Aslahi-Shahri, “Designing precision fuzzy controller for load swing of an overhead crane,” Neural Comput. Appl., Vol. 26, No. 7, pp. 1555–1560 (2015). Chen, W., and M. Saif, “Output feedback controller design for a class of MIMO nonlinear systems using high-order sliding-mode differentiators with application to a laboratory 3-D crane,” IEEE Trans. Ind. Electron., Vol. 55, No. 11, pp. 3985–3997 (2008). Chwa, D., “Nonlinear tracking control of 3D overhead cranes against the initial swing angle and the variation of payload weight,” IEEE Trans. Control Syst. Technol., Vol. 17, No. 4, pp. 876–883 (2009). Pisano, A., S. Scodina, and E. Usai, “Load swing suppression in the 3-dimensional overhead crane via second-order sliding-modes,” In Proc. 11 th IEEE Int. Workshop on Variable Structure Syst. (VSS), Mexico City, MX, pp. 452–457 (2010). Smoczek, J., “Experimental veriﬁcation of a GPC-LPV method with RLS and P1-TS fuzzy-based estimation for limiting the transient and residual vibration of a crane system,” Mech. Syst. Signal Proc., Vol. 62–63, pp. 324–340 (2015). Han, S., Z. Lei, and A. Bouferguene, “3D visualization-based motion planning of mobile crane operations in heavy industrial projects,” J. Comput. Civil Eng., Vol. 30, No. 1, pp. 04014127, (2016). Sun, N., Y. Fang, and X. Zhang, “Energy coupling output feedback control of 4-D of underactuated cranes with saturated inputs,” Automatica, Vol. 49, No. 5, pp. 1318–1325 (2013). © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd Asian Journal of Control, Vol. 20, No. 6, pp. 1–16, November 2018 34. Zhang, X., Y. Fang, and N. Sun, “Minimum-time trajectory planning for underactuated overhead crane systems with state and control constraints,” IEEE Trans. Ind. Electron., Vol. 61, No. 12, pp. 6915–6925 (2014). 35. Khatamianfar, A., and A. V. Savkin, “A new tracking control approach for 3D overhead crane systems using model predictive control,” Proc. of European Control Conf., pp. 796–801 (2014). 36. Fang, Y., B. Ma, P. Wang, and X. Zhang, “A motion planning-based adaptive control method for an underactuated crane system,” IEEE Trans. Control Syst. Technol., Vol. 20, No. 1, pp. 241–248 (2014). 37. Sun, N., Y. Fang, P. Wang, and X. Zhang, “Adaptive trajectory tracking control of underactuated 3dimensional overhead crane systems,” Acta Automatica Sinica, Vol. 36, No. 9, pp. 1287–1294 (2010). 38. Sun, N., Y. Wu, Y. Fang, H. Chen, and B. Lu, “Nonlinear continuous global stabilization control for underactuated RTAC systems: Design, anasianalysis, and experimentation,” IEEE/ASME Trans. Mechatron., Vol. 22, No. 2, pp. 1104–1115 (2017). Xuewen Rong received his bachelor’s and master’s degrees from Shandong University of Science and Technology, China, in 1996 and 1999, respectively. He received his Ph.D. from Shandong University, China, in 2013. He is currently a senior engineer at the School of Control Science and Engineering, Shandong University, China. His research interests include robotics, mechatronics, and hydraulic servo driving technology. Rui Song received his B.E. degree in industrial automation in 1998, M.S. degree in control theory and control engineering in 2001 from Shandong University of Science and Technology, and Ph.D. in control theory and control engineering from Shandong University in 2011. He is engaged in research on intelligent sensor networks, intelligent robot technology, and intelligent control systems. He is currently an Associate Professor at the School of Control Science and Engineering of Shandong University in Jinan, China. Menghua Zhang received her M.S. degree from the School of Electrical Engineering and Automation, University of Jinan, Jinan, China, in 2014. She is currently working toward a Ph.D. with the School of Control Science and Engineering, Shandong University, Jinan, China. Her research interests include control of mechatronics, underacutated overhead crane systems, and bionic eye systems. Xincheng Tian received his B.S. degree in industrial automation and M.S. degree in automation from Shandong Polytech University (now Shandong University), Shandong, China, in 1988 and 1993, respectively. He received his Ph.D. in aircraft control, guidance, and simulation from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2000. He is currently a Professor at Shandong University. His current research interests include robotics, mechatrotics, and CNC techniques. Xin Ma received her B.S. degree in industrial automation and M.S. degree in automation from Shandong Polytech University (now Shandong University), Shandong, China, in 1991 and 1994, respectively. She received her Ph.D in aircraft control, guidance, and simulation from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 1998. She is currently a Professor at Shandong University. Her current research interests include artiﬁcial intelligence, machine vision, human-robot interaction, and mobile robots. Yibin Li received his B.S. degree in automation from Tianjin University, Tianjin, China, in 1982, M.S. degree in electrical automation from Shandong University of Science and Technology, Shandong, China, in 1990, and Ph.D. in automation from Tianjin University, China, in 2008. From 1982 to 2003, he worked with Shandong University of Science and Technology, China. Since 2003, he has been the Director of the Center for Robotics, Shandong University. His research interests include robotics, intelligent control theories, and computer control system. © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd

1/--страниц