Nonlinear Stabilizing Control for Ship-Mounted Cranes ... - IEEE Xplore

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Nonlinear Stabilizing Control for Ship-Mounted Cranes With Ship Roll and Heave Movements: Design, Analysis, and Experiments Ning Sun, Member, IEEE, Yongchun Fang, Senior Member, IEEE, He Chen, Yiming Fu, and Biao Lu

Abstract—Presently, ship-mounted cranes are playing more and more important roles in modern ocean transportation and logistics. Different from traditional land-fixed crane systems, ship-mounted cranes present much more complicated nonlinear dynamical characteristics and they are persistently influenced by different mismatched disturbances due to harsh sea environments, e.g., sea waves, ocean currents, sea winds, and so forth; these unfavorable factors bring about many challenges for the development of effective control schemes. This paper presents a novel nonlinear stabilizing control strategy for underactuated ship-mounted crane systems. Specifically, some novel coordinate change procedures are first introduced to tackle the disturbing terms by transforming the original dynamics into a new form, which facilitates both controller design and stability analysis. After that, a nonlinear control law is constructed to regulate the cargo position to the desired location asymptotically, in the presence of ship roll and heave movements. The boundedness and convergence of the closed-loop signals are proven with Lyapunovbased analysis. To the best of our knowledge, this is the first closed-loop scheme that can achieve asymptotic control results, without linearizing/approximating the original nonlinear dynamics when performing controller design and stability analysis, for underactuated ship-mounted cranes with ship roll and heave movements. Hardware experimental results are included to show that the proposed control method can achieve satisfactory control performance and it admits strong robustness against external perturbations. Index Terms—Mechatronics, nonlinear systems, oscillations, underactuated cranes.

I. I NTRODUCTION N RECENT years, ocean engineering-oriented automation has become a mainstream research focus in many research fields. As an important component, large scale ships

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Manuscript received January 12, 2017; revised March 16, 2017; accepted April 17, 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 61503200 and Grant 61325017, in part by the Natural Science Foundation of Tianjin under Grant 15JCQNJC03800, and in part by the China Postdoctoral Science Foundation under Grant 2016M600186. This paper was recommended by Associate Editor T. Li. (Corresponding author: Yongchun Fang.) The authors are with the Institute of Robotics and Automatic Information Systems, College of Computer and Control Engineering, Nankai University, Tianjin 300350, China, and also with the Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin 300350, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMC.2017.2700393

are playing more and more important roles in modern transportation and logistics. Ship-mounted boom cranes (or termed as offshore cranes), which serve as powerful transportation tools, are indispensable for fulfilling basic manipulation tasks of transferring heavy cargos between different ships or between ships and harbors. From the mechanical structure perspective, ship-mounted boom cranes have very complicated dynamical characteristics, and similar to many other mechatronic systems [1]–[15], they have more degrees of freedoms (DOFs) than available control inputs, or in other words, from a pure control theory viewpoint, the configuration space dimension of the ship-mounted crane system is larger than the corresponding input space dimension. It is a generally accepted fact that the control problem for underactuated systems, including ship-mounted cranes, is very challenging, which receives much attention from both mechatronics and control systems communities. In particular, for ship-mounted cranes, the basic control objective is to place the cargos precisely and efficiently to the desired locations (e.g., other ships or land), with negligible cargo sway during the operation process and no residual swing at the end. This problem itself is already very challenging owing to the complex mechanical structure and inherent underactuated nature. Moreover, different from land-fixed cranes, due to the harsh (sea) working conditions, ship-mounted cranes additionally suffer from unfavorable influences induced by various extraneous perturbations (e.g., sea waves, sea winds, etc.), which makes the control problem extremely challenging. For the above reasons, the issue of effectively controlling a shipmounted crane has both theoretical and practical importance. Over the past several decades, the control problem for land-fixed cranes, including overhead cranes, boom cranes, and tower cranes [16], has been extensively studied, and many ambitious solutions have been reported in the literature. According to the fact whether the nonlinear dynamics are approximated/linearized when designing controllers or performing closed-loop stability analysis, the currently available control methods for land-fixed crane systems can be grouped into two categories, i.e., linear control [17]–[22] and nonlinear control [23]–[34]. In comparison with linear control approaches, since nonlinear control methods are developed on the basis of the original nonlinear dynamical equations, they can maintain satisfactory control performance even when the state variables are far from the equilibrium point. Additionally, some intelligent control strategies, including fuzzy logics [35],

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neural networks [36], and genetic algorithms, are incorporated to improve the antiswing and positioning performance for land-fixed cranes. Although the control problem for land-fixed cranes has been deeply studied and many mature control strategies have already been developed, much fewer results have been reported for the stabilization problem of ship-mounted crane systems, mainly, because of their complicated dynamical characteristics, strong nonlinearity, and unexpected disturbances. After making some careful comparisons between traditional (landfixed) cranes and ship-mounted cranes, the main difference is that the supporting bases of traditional cranes are fixed and they are operated in the inertia (earth-fixed) frame, while the bases of ship-mounted cranes are equipped on ships, whose motions are influenced by sea waves and ocean currents, and hence work in noninertial frames. Aside from the fact that ship-mounted cranes have much more complicated dynamics than land-fixed cranes, the undesired sea waves and ocean currents-induced ship motions may include energy close to the natural frequency of the crane system, which will easily excite the cargo swing and even lead to unexpected resonance effects [37]. As a consequence, it is unpracticable to apply the existing control methods, which are designed for landfixed cranes, to solve the control issue for ship-mounted crane systems. Additionally, as previously mentioned, ship-mounted cranes are usually operated in harsh sea conditions where various matched and mismatched disturbances (e.g., sea winds) are present, which add further difficulties for their control problem. These facts reveal that the problem of effectively controlling ship-mounted cranes is much more challenging than that of land-fixed cranes, which remains a rather open problem and requires in-depth investigations. In the literature, there are some innovative works devoted to the control of ship-mounted cranes. In particular, for shipmounted cranes with the well-known “Maryland rigging” a feedforward control strategy is designed with gain scheduling to suppress the cargo swing caused by ship roll movements [38]. Later on, Al-Sweiti and Söffker [39] derived the mathematical model of an elastic Maryland rigged shipmounted crane and then develop a novel control scheme consisting of a variable-gain observer and a variable-gain controller, which is demonstrated to be effective. In addition, there are also some other meaningful works on both dynamics analysis [40] and advanced controller design, including predictionbased control [41], preview tracking control [42], nonlinear feedback control [43], sliding mode control (SMC) [44]–[48], composite control [49], linear matrix inequality-based control [50], delayed feedback control [51], active rate-based control [52], combination-based control [53], external modelbased control [54], and so on. In particular, in [43], a novel nonlinear feedback technique-based control approach is proposed for underactuated ship-mounted cranes subject to the effects of seawater viscoelasticity and rope flexibility. Two interesting nonlinear sliding mode controllers are developed in [44] which are demonstrated to be effective and robust. In addition, an effective robust controller is presented to deal with bounded disturbances and uncertainties for ship-mounted crane systems [45].

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After carefully summarizing the existing control approaches for ship-mounted cranes, the following aspects require further improvements. 1) Most of them are designed or their stability analyzing procedures are implemented based upon reduced/simplified crane dynamics. Due to the complicated working scenarios, the system state variables (e.g., cargo swing) may not be close enough to the equilibrium point caused by unexpected disturbances, which usually degrades the performance of these control methods or even results in system instability. 2) Some of the currently available control approaches can only achieve ultimate uniform stability results; in other words, they can merely assure that the positioning errors (between the cargos’ final actual positions and the desired locations) converge to some specific ranges instead of zero (the equilibrium point). 3) Some existing methods even require the ship roll/heave acceleration for constructing the feedback control laws, which is usually difficult to measure in practice. 4) Some methods have not considered the rope length variation, while cargos may need to be lowered or hoisted during different operations in practice. 5) The existing approaches usually separate the entire dynamics into a crane-related component and a disturbance-related component, which makes the disturbances very cumbersome to analyze and brings much difficulty for controller design and analysis. In response to the issues associated with the existing control methods, this paper presents a novel nonlinear stabilizing control scheme for underactuated ship-mounted cranes in the presence of ship roll/heave movements, which is designed based on the original nonlinear dynamics without any approximating manipulations. More precisely, the complete original nonlinear dynamical equations are first provided for underactuated ship-mounted boom cranes, which are complicated and make it difficult to directly conduct controller design/analysis. To deal with this problem, some coordinate transformation manipulations are presented to convert the nonlinear dynamics into a form more convenient for the subsequent controller design and analysis, and as a consequence, the control objective is equivalently converted on the basis of the newly defined coordinate (state) variables. After that, a Lyapunov-based nonlinear control law is proposed and the closed-loop stability analysis is carried out without linearizing or approximating the original nonlinear dynamics. In addition, a complete Lyapunov-based stability analysis is provided based upon the original nonlinear dynamical equations. To our knowledge, this paper yields the first closed-loop control method that achieves asymptotic results for ship-mounted cranes suffering from ship roll and heave movements, without the necessity of linearizing or approximating the original complicated nonlinear dynamics, when compared with the state-of-the-art methods. Finally, hardware experiments, implemented on a self-built ship-mounted boom crane prototype, are exhibited to demonstrate the practical performance of the proposed control approach in terms of its superior performance over existing methods, including the proportional derivative (PD) controller,

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. SUN et al.: NONLINEAR STABILIZING CONTROL FOR SHIP-MOUNTED CRANES WITH SHIP ROLL AND HEAVE MOVEMENTS

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and three DOFs (i.e., φ, L, and θ ), indicating its underactuated property. Moreover, the ship roll/heave movements make the ship-mounted crane a noninertial system, which brings great challenges for the system modeling, controller design, and closed-loop stability analysis. After utilizing Lagrange’s modeling method for noninertial systems and performing a lot of tedious mathematical operations, the dynamics of a ship-mounted crane system can be derived as M(q)¨q + C(q, q˙ )˙q + G(q) = u + Fd θ ]

Fig. 1.

Model of an underactuated offshore ship-mounted crane.

the SMC method in [46], and the composite controller in [49], and also its robustness against disturbances. It is worthwhile to mention that, most works on ship-mounted cranes just present simulation results, while some more convincing experimental results are provided in this paper. The paper is arranged as follows. In Section II, the dynamical equations for ship-mounted cranes are explicitly provided and some model transformation procedures are included. The main results including the controller development and closed-loop stability analysis are presented within Section III. Hardware experimental results are included in Section IV, together with some corresponding analysis, to verify the practical performance of the proposed control scheme. Some concluding remarks are provided in Section V. II. P ROBLEM F ORMULATION In this section, the dynamics of ship-mounted cranes will be described and the control objective will be claimed in details. A. Dynamics for Ship-Mounted Cranes The model of an underactuated offshore ship-mounted crane system is shown in Fig. 1. yg and zg are the axes of the earthfixed frame, which are parallel with and vertical to the ground, respectively, and ys and zs denote the axes of the ship-fixed frame, which are parallel with and vertical to the deck, respectively. For the variables, θ denotes the cargo swing with respect to the zs -axis, φ represents the boom luffing angle with respect to the ys -axis, and L is the time-varying rope length; in addition, χ1 , χ2 , and χ3 represent some new state variables, which will be defined in (5) and utilized in the subsequent controller design/analysis. As shown in Fig. 1, Mc and Fc are the actuating torque/force for controlling the boom luffing angle and the rope length, respectively. In addition, is the ship roll angle about the zg -axis, and z denotes the ship vertical heave displacement induced by sea waves. For the plant parameters, LB denotes the boom length, mp is the cargo mass, and g denotes the gravitational constant, respectively. Hence, the ship-mounted crane has two control inputs (i.e., Mc and Fc )

(1)

where q = [φ L ∈ denotes the state vector, M(q) ∈ R3×3 represents the inertia matrix, C(q, q˙ ) ∈ R3×3 is the centripetal-Coriolis matrix, G(q) ∈ R3 denotes the gravitational vector, u ∈ R3 represents the control input vector, and Fd ∈ R3 is the disturbance vector. The explicit expressions for the matrices M(q) and C(q, q˙ ) are provided as1 ⎡ ⎤ J + mp LB2 −mp LB Cθ−φ mp LLB Sθ−φ ⎦ mp 0 M = ⎣ −mp LB Cθ−φ 2 mp LLB Sθ−φ 0 mp L ⎡ ⎤ 0 c12 c13 0 c23 ⎦ C = ⎣ c21 (2) c31 −c23 mp LL˙ R3

where the elements in C(q, q˙ ) are given as ˙ θ−φ + mp LB LCθ−φ θ˙ c12 = mp LB Sθ−φ θ˙ , c13 = mp LB LS ˙ c23 = −mp Lθ˙ , c31 = −mp LB LCθ−φ φ. ˙ c21 = −mp LB Sθ−φ φ, The vectors involved in (1) are given as follows: G(q) = mp LB + md gCφ− − mp gCθ− mp gLSθ− u = [Mc Fc 0] , Fd = fd1 fd2 fd3 (3) where

˙ θ−φ fd1 = J + mp LB2 + mp LB LSθ−φ ¨ + 2mp LB LS ˙

− mp LB LCθ−φ ˙ 2 − 2θ˙ ˙ − mp LB + md Cφ− z¨

fd2 = mp L ˙ 2 − 2φ˙ ˙ 2 − 2θ˙ ˙ + mp LB Sθ−φ ˙

fd3

¨ + mp Cθ− z¨ − mp LB Cθ−φ

˙ 2 − 2φ˙ ¨ + mp LB LCθ−φ = mp L L + LB Sθ−φ ˙ ˙ − c θ˙ − ˙ − mp LSθ− z¨. (4) + 2mp LL˙

As shown from (4), fd1 , fd2 , and fd3 consist of z¨, ˙ , and ¨ , and hence they can be viewed as “disturbances” induced by the ship roll and heave movements to the entire system (1). More precisely, fd1 and fd2 are matched disturbances appearing in the same channels as the control inputs Mc and Fc , respectively, while fd3 denotes mismatched disturbances existing in a different channel from the control inputs. Existing methods usually treat the state variables and the disturbances (i.e., fd1 , fd2 , and fd3 ) separately, which makes the controller development and analysis rather cumbersome. Alternatively, after 1 In this paper, if not otherwise stated, S x−y and Cx−y are abbreviations for sin(x − y) and cos(x − y), respectively, where x and y denote any real variables.



carefully analyzing (1), it is intended to perform some coordinate transformation operations for the ship-mounted crane dynamics to facilitate the subsequent controller development and analysis. In particular, with the coordinate transformation, it is expected to decompose Fd in (1) and then integrate the ship roll and heave-related terms into the new state variables, which are closely related to the control objective, as will be shown next. To this end, some new state variables are introduced as follows (see Fig. 1 for a more intuitive relationship between them and the original state variables φ, L, θ ): χ1 := φ − , χ2 := L − z, χ3 := θ −

(5)

and the following new state vector: χ := [χ1 χ2 χ3 ] .

(6)

Then, based on the newly defined state variables and vectors in (5) and (6), (1) can be alternatively represented by the following new matrix-vector form: MN (q)χ¨ + CN (χ, χ˙ )χ˙ + GN (q) = u + FNd

(7)

where MN (q) and GN (q) are the same with M(q) and G(q) in (1), respectively, CN (χ , χ˙ ) is expressed as follows: ⎤ ⎡ 0 cn12 cn13 0 cn23 ⎦ CN (χ , χ˙ ) = ⎣ cn21 (8) cn31 −cn23 mp LL˙ where cn12 = mp LB Sχ3 −χ1 χ˙ 3 ˙ χ3 −χ1 + mp LB LCχ3 −χ1 χ˙ 3 cn13 = mp LB LS cn21 = −mp LB Sχ3 −χ1 χ˙ 1 , cn23 = −mp Lχ˙ 3 cn31 = −mp LB LCχ3 −χ1 χ˙ 1 and FNd takes the following expression: FNd = fnd1 fnd2 fnd3 where

(9)

fnd1 = − mp LB + md z¨ cos(χ1 ) + mp LB z¨Cχ3 −χ1 − mp LB Sχ3 −χ1 z˙χ˙ 3 fnd2 = mp z¨ cos(χ3 ) − mp z¨ fnd3 = −mp L¨z sin(χ3 ) − cχ˙ 3 − mp L˙zχ˙ 3 .

For MN (q) and CN (χ , χ˙ ), the following property holds. Property 1: The matrix MN (q) is positive definite. In addition, the following relationship holds for MN (q) and CN (χ, χ˙ ):

˙ N (q) M p (10) − CN (χ, χ˙ ) p ≡ 0 2 ˙ N (q)/2 − CN (χ , χ˙ ) where p ∈ R3 denotes any vector, i.e., M is an antisymmetric matrix. ˙ N (q) and Proof: By inserting the explicit expressions of M CN (χ , χ˙ ) into (10), canceling the common terms, and implementing some arrangements, the property can be proven straightforwardly, which is omitted for brevity. In addition, considering the practical working conditions of ship-mounted cranes, the cargo-swing/ship-roll angles are always within reasonable ranges, and the disturbances induced

by sea waves are amplitude-bounded. In other words, the following practical assumptions are reasonable. Assumption 1: The initial ship heave displacement is smaller than the rope length in the sense that z(0) < L(0) =⇒ χ2 (0) = L(0) − z(0) > 0.

(11)

The cargo is always beneath the boom tip, and the ship roll/swing angular velocity is amplitude-bounded, that is π |χ3 | = |θ − | < , |χ˙ 3 | ≤ βχ3 . (12) 2 Additionally, for the ship heave movement z and its derivatives (induced by sea waves), they satisfy the following conditions: ... |z| ≤ βzp , |˙z| ≤ βzv , |¨z| ≤ βza , | z | ≤ βzj z, z˙, z¨ ∈ L2 . (13) Remark 1: For ship-mounted cranes, sea waves and winds are the original disturbances which cause the ship roll and heave movements. Because cranes are equipped on ships, the ship motions (induced by sea waves and winds) will directly influence the cranes. That is, the influences of these disturbances on the cranes are reflected by ship roll and heave movements. Hence, in this paper, we focus on designing a control approach for ship-mounted cranes suffering from the effects of ship roll and heave movements. Remark 2: Lyapunov-based stability analysis for nonlinear control systems, including ship-mounted cranes, is famous for its conservatism. For the ship heave movements, although it is assumed that z, z˙, z¨ ∈ L2 , the controller (25) works well even when the disturbances do not satisfy these assumptions. As will be shown in the experiments, the ship heave movements are chosen as sine wave functions, and the experimental results will show that the proposed controller works satisfactorily. In addition, abundant simulation tests have also been implemented where the condition of z, z˙, z¨ ∈ L2 is not satisfied, and the results demonstrate the satisfactory performance as well, which are omitted for brevity. Similar phenomena are also commonly seen in control applications of many other mechatronic systems, since there is still a gap between practice and theory. In addition, although the condition of z, z˙, z¨ ∈ L2 is assumed for the ship heave movements, similar assumptions are not needed for the ship roll movements. B. Control Objective For underactuated offshore ship-mounted cranes, the control objective is to stabilize the cargo at a specific position without swinging in the earth-fixed frame (see Fig. 1), so that further control tasks can be carried out. Specifically, the desired cargo position in the earth-fixed frame is denoted as [ygd zgd ] . After performing careful geometric analysis for Fig. 1, the cargo position in the ship-fixed frame can be expressed as ys = LB cos (φ) + L sin (θ ), zs = LB sin (φ) − L cos (θ ).

(14)

In addition, its position in the earth-fixed frame reads as yg = LB Cφ− + LSθ− = LB cos (χ1 ) + L sin (χ3 ) zg = LB Sφ− − LCθ− + z = LB sin (χ1 ) − L cos(χ3 ) + z

(15)


where (5) has been utilized. According to the control objective, to eliminate the cargo swing angle in the earth-fixed frame, it is essentially required to render χ3 = θ − → 0. Then, by using (5), the desired cargo location can be described as ygd = LB Cφ− = LB cos (χ1 ) zgd = LB Sφ− − L + z = LB sin (χ1 ) − χ2 and the control objective can be formulated by yg → ygd , zg → zgd .

(17)

In order to facilitate the controller development, it is needed to derive the desired settings χ1d , χ2d , and χ3d for χ1 , χ2 , and χ3 from (16). Toward this end, solving (16) yields ygd , χ2d = LB2 − y2gd − zgd , χ3d = 0. (18) χ1d = arccos LB Then, the control objective in (17) can be alternatively achieved by designing suitable controllers such that χ1 → χ1d , χ2 → χ2d , χ3 → χ3d

To achieve the control objective in (19), the following error signals eχ1 , eχ2 , and eχ3 are constructed: eχ1 = χ1 − χ1d , eχ2 = χ2 − χ2d , eχ3 = χ3 − χ3d =⇒ e˙ χ1 = χ˙ 1 , e˙ χ2 = χ˙ 2 , e˙ χ3 = χ˙ 3

(16)

(19)

which will be used for the subsequent controller design. III. M AIN R ESULTS We will develop a novel stabilizing control scheme for underactuated ship-mounted crane systems and then provide the corresponding theoretical analysis. A. Control Law Development The mechanical energy of the ship-mounted crane system, denoted as Em , can be expressed as 1 q˙ M(q)˙q + mp gL[1 − cos(θ )]. (20) 2 Based on the form of (20), the following scalar function is constructed (a part of the final Lyapunov function): Em =

1 χ˙ MN (q)χ˙ + mp gχ2 [1 − cos(χ3 )] (21) 2 whose derivative versus time can be calculated as

1 ˙ N (q)χ˙ + mp gχ˙ 2 [1 − cos(χ3 )] E˙ = χ˙ MN (q)χ¨ + M 2 + mp gχ2 sin(χ3 )χ˙3 = χ˙ u − GN (q) + FNd + mp gχ˙ 2 [1 − cos(χ3 )] + mp gχ2 sin(χ3 )χ˙3 (22) E=

where (7)–(10) are used. By further inserting for the expressions of u, GN (q), and FNd [see (3), (9)] and canceling the common terms, it can be derived from (22) that E˙ = Mc − mp LB + md (¨z + g) cos(χ1 ) + mp LB Cχ3 −χ1 z¨ − mp LB Sχ3 −χ1 z˙χ˙ 3 χ˙ 1 + Fc + mp g + mp cos(χ3 )¨z − mp z¨ χ2 − cχ˙ 32 − mp gz sin(χ3 )χ˙ 3 − mp L sin (χ3 )¨zχ˙ 3 − mp L˙zχ˙ 32

5

(24)

where χ1d , χ2d , and χ3d are in (18). Based on (23), the control laws for luffing and hoisting/lowering are proposed as2 Mc = −kpχ1 eχ1 − kdχ1 χ˙ 1 + mp LB + md g cos (χ1 ) − kaχ1 χ˙ 32 χ˙ 1 + mp LB Sχ3 −χ1 z˙χ˙ 3 − βza × (mp LB + md )| cos(χ1 )| + mp LB |Cχ3 −χ1 | sgn(χ˙ 1 ) kr χ2d eχ2 − kaχ2 χ˙ 32 χ˙ 2 − mp g Fc = −kpχ2 eχ2 − kdχ2 χ˙ 2 − χ23 − βza mp [1 + | cos (χ3 )|]sgn(χ˙ 2 ) (25) where kpχ1 , kpχ2 , kdχ1 , kdχ2 , kaχ1 , kaχ2 , and kr ∈ R+ are positive control gains, βza ∈ R+ is defined in (13), and sgn(∗) denotes the standard sign function. The proposed scheme in (25), for boom luffing and rope length control, can place the cargo to the desired location in the earth-fixed frame [see (16)] in the presence of ship motions, which is summarized via the ensuing theorem. For a real ship-mounted crane system, the ship motion can be conveniently measured by such sensors as an inertial measurement unit equipped on the ship, which then allows the proposed control scheme to be implemented in real time. Additionally, it is worthwhile to mention that, when compared with some existing methods (e.g., the controller in [49]), the controller (25) only needs the upper bound of |¨z(t)|, i.e., βza , which has relaxed the implementation requirements to some extent, since it is traditionally more practical to get an upper bound of |¨z(t)| (i.e., βza ) than to measure z¨(t) in real time. In the future, we will further consider the problems of dead zones, state constraints, and so on, as in [55]–[58]. Remark 3: The control gains are obtained by trial and error. To facilitate the control gain tuning procedure, a few items of laws are summarized after many simulation and experimental tests. In particular, as kpχ1 (respectively, kpχ2 ) increases, the boom luffing angle φ (respectively, the rope length L) will generally vary faster but undesired overshoots will increase to some extent. For kdχ1 and kdχ2 , when the value of kdχ1 (respectively, kdχ2 ) increases, φ (respectively, L) will generally vary slower with smaller overshoots. In addition, kaχ1 and kaχ2 play the role of injecting additional damping for swing suppression, and as kaχ1 (respectively, kaχ2 ) increases, the residual swing will be damped faster, but φ (respectively, L) will generally vary slower. Remark 4: From the structure viewpoint, the dynamics of ship-mounted crane systems present some similarity with those of some other underactuated mechanical systems, e.g., ball and beam systems, translational oscillations with a rotational actuator systems, underactuated manipulators, and so forth. Hence, it is promising that the designed approach can be applicable to the control of these systems after proper modifications.

(23) which will be employed next.

2 The designed controllers (25) present no singularity, as will be shown by (33) in the proof for Theorem 1.



Remark 5: In most cases, for nonlinear underactuated systems, it is usually challenging to prove finite-time convergence conclusions with Lyapunov-based analysis due to its conservative nature, especially when the complicated nonlinear dynamics are not linearized. For most control methods, although Lyapunov-based analysis just gives a conservative theoretical conclusion (i.e., asymptotic convergence as t → ∞), the actual control performances (provided by either simulation or experimental results) are usually better than the theoretical conclusions, and this fact is true for many systems not limited to cranes. In the future, we will focus on designing finite-time control schemes for ship-mounted cranes. B. Stability Analysis Theorem 1: The proposed nonlinear stabilizing controllers in (25) can regulate the cargo’s horizontal and vertical positions to the desired location denoted by (16), in the presence of ship roll and heave movements, in the sense that lim eχ1 = 0, lim eχ2 = 0, lim eχ3 = 0.

t→∞

t→∞

(26)

t→∞

Proof: To start with, in view of E in (21), define V=

1 χ˙ MN (q)χ˙ + mp gχ2 [1 − cos(χ3 )] 2 1 1 kr eχ2 2 + kpχ1 e2χ1 + kpχ2 e2χ2 + . 2 2 2 χ2

(27)

Then, differentiating (27) with respect to time, inserting for (23) and (25), and making some tedious arrangements yields that

V˙ ≤ −kdχ1 χ˙ 12 − kdχ2 χ˙ 22 − kaχ1 χ˙ 12 + kaχ2 χ˙ 22 χ˙ 32 − cχ˙ 32 − mp L sin(χ3 )¨zχ˙ 3 − mp L˙zχ˙ 32 − mp gz sin(χ3 )χ˙ 3

c ≤ −kdχ1 χ˙ 12 − kdχ2 χ˙ 22 − kaχ1 χ˙ 12 + kaχ2 χ˙ 22 χ˙ 32 − χ˙ 32 4 m2p 2 2 L z¨ + z˙2 χ˙ 32 + g2 z2 (28) + c where the arithmetic mean geometric mean inequality has been utilized. Further, noting that |χ˙ 3 | ≤ βχ3 from (12), the result in (28) can be simplified into m2p 2 2 c L z¨ + βχ23 z˙2 + g2 z2 . V˙ ≤ −kdχ1 χ˙ 12 − kdχ2 χ˙ 22 − χ˙ 32 + 4 c (29) By using z, z˙, z¨ ∈ L2 [see (13)], integrating both sides of (29) with respect to time shows V(t) ≤ V(0) − kdχ1 +

m2p

0

t

χ˙ 12 dt

− kdχ2

t 0

χ˙ 22 dt

t

L2 z¨2 + βχ23 z˙2 + g2 z2 dt

c 0 ≤ βM +∞

c − 4

t 0

χ˙ 32 dt

(30)

where βM is a positive bounding constant. Then, considering that χ2 (0) > 0 [see (11)], if χ2 → 0 during the control process, then (kr /2)(eχ2 /χ2 )2 → +∞ [see (27)], indicating that V → +∞. This leads to a contradiction with the conclusion of V +∞ in (30); thus, χ2 0, which implies that χ2 > 0, ∀ t ≥ 0 =⇒ V ≥ 0

(31)

clearly indicating that V in (27) is always non-negative and hence is a Lyapunov function. Further, it is known from (30) that V ∈ L∞ =⇒ χ˙ 1 , χ˙ 2 , χ˙ 3 , eχ1 , eχ2 , eχ2 /χ2 ∈ L∞ .

(32)

It will be shown that 1/χ2 is bounded. To this end, two situations are considered. First, if eχ2 → 0, then 1/χ2 → 1/χ2d ∈ L∞ . Second, if eχ2 0, it follows from eχ2 ∈ L∞ and eχ2 /χ2 ∈ L∞ in (32) that 1/χ2 ∈ L∞ . Hence, as a summary 1/χ2 ∈ L∞ =⇒ Mc , Fc ∈ L∞

(33)

where the conclusions in (32) have been utilized. In addition, (30) may be alternatively re-expressed as follows: t t c t 2 2 2 kdχ1 χ˙ 1 dt + kdχ2 χ˙ 2 dt + χ˙ dt 4 0 3 0 0 m2p t 2 2 L z¨ + βχ23 z˙2 + g2 z2 dt ≤ V(0) − V(t) + c 0 (34) ∈ L∞ which indicates that χ˙ 1 , χ˙ 2 , χ˙ 3 ∈ L2 . On the other hand, it follows from (7), (31), and (32) that χ¨ 1 , χ¨ 2 , χ¨ 3 ∈ L∞ .

(35)

Then, Barbalat’s lemma [59] can be directly utilized to show that lim χ˙ 1 = 0, lim χ˙ 2 = 0, lim χ˙ 3 = 0.

t→∞

t→∞

t→∞

(36)

Subsequently, the convergence of eχ1 , eχ2 , and χ3 to zero will be proven. Expanding the matrix-vector structure of (7), the latter two dynamical equations, after inserting for (25), read as mp χ¨ 2 = mp LB Cχ3 −χ1 χ¨ 1 + mp Lχ˙ 32 + mp LB Sχ3 −χ1 χ˙ 12 kr χ2d + mp g cos(χ3 ) − kpχ2 eχ2 − kdχ2 χ˙ 2 − eχ2 χ23 − kaχ2 χ˙ 32 χ˙ 2 − mp g − mp z¨[1 − cos (χ3 )] − βza mp [1 + | cos (χ3 )|]sgn(χ˙ 2 ) mp Lχ¨3 =

(37)

−mp LB Sχ3 −χ1 χ¨ 1 + mp LB Cχ3 −χ1 χ˙ 12

− mp χ˙ 2 χ˙ 3 ˙ − mp Lχ˙ 3 − mp g sin(χ3 ) − mp z¨ sin(χ3 ) − mp z˙χ˙ 3 c − χ˙ 3 . (38) L Then, by substituting (25), (37), and (38) into the first dynamical equation expanded from (7), canceling the common terms, and rearranging the obtained equation, the following result is derived: J χ¨ 1 = g1 + g2

(39)


where g1 and g2 are auxiliary functions that, respectively, represent

cLB Sχ3 −χ1 χ˙ 3 − kaχ1 χ˙ 32 + kdχ1 χ˙ 1 g1 = mp LB Sχ3 −χ1 z˙χ˙ 3 + L − LB Cχ3 −χ1 kdχ2 χ˙ 2 + kaχ2 χ˙ 32 χ˙ 2 + βza mp × [1 + | cos (χ3 )|]sgn(χ˙ 2 ) − βza (mp LB + md )| cos(χ1 )| + mp LB |Cχ3 −χ1 | × sgn(χ˙ 1 ) (40) g2 = mp gLB cos(χ1 ) − kpχ1 eχ1 − LB Cχ3 −χ1 kr χ2d × kpχ2 eχ2 + mp g + eχ2 χ23 − LB Cχ3 −χ1 mp z¨[1 − cos(χ3 )] − mp LB + md z¨ cos (χ1 ) + mp LB z¨Cχ3 −χ1 + mp LB z¨Sχ3 −χ1 sin(χ3 ).

lim J χ¨1 = 0, lim g2 = 0 =⇒ lim χ¨ 1 = 0. t→∞

t→∞

(41)

(42)

(43)

g3 = mp LB Cχ3 −χ1 χ¨ 1 + mp Lχ˙ 32 + mp LB Sχ3 −χ1 χ˙ 12 − kdχ2 χ˙ 2 − kaχ2 χ˙ 32 χ˙ 2 − βza mp [1 + | cos (χ3 )|]sgn(χ˙ 2 ) g4 = mp g cos(χ3 ) − kpχ2 eχ2 − mp z¨[1 − cos (χ3 )] − mp g kr χ2d − eχ2 . (44) χ23 In a similar way, it is obtained from (36) and (42) that limt→∞ g3 = 0. Also, it is known from (32) that g˙ 4 ∈ L∞ . It then follows from extended Barbalat’s lemma [59] that: lim mp χ¨ 2 = 0, lim g4 = 0 =⇒ lim χ¨ 2 = 0. t→∞

t→∞

(45)

Further, similar to (37), the result in (38) can be rewritten as follows: mp Lχ¨3 = g5 + g6

t→∞

t→∞

(48)

which further implies from (12) and g6 of (47) that lim mp g sin(χ3 ) = 0 =⇒ lim χ3 = 0.

t→∞

t→∞

Inserting (45) and (49) into g4 [see (44)] yields kr χ2d eχ2 = 0 =⇒ lim eχ2 = 0. lim kpχ2 + t→∞ t→∞ χ23

lim eχ1 = 0.

where

t→∞

lim mp Lχ¨3 = 0, lim g6 = 0 =⇒ lim χ¨ 3 = 0

t→∞

t→∞

In addition, (37) can be decomposed into mp χ¨ 2 = g3 + g4

Extended Barbalat’s lemma [59] can again be used to conclude that

(49)

(50)

Finally, substituting (42), (49), and (50) into (41) and canceling the common terms can show that

It follows from (36) that limt→∞ g1 = 0. Additionally, together with (13) and (32), it is derived that g˙ 2 ∈ L∞ . Recalling (36), extended Barbalat’s lemma3 [59] can be used to show that t→∞

7

(46)

where g5 and g6 are, respectively, defined by g5 = −mp LB Sχ3 −χ1 χ¨ 1 + mp LB Cχ3 −χ1 χ˙ 12 − mp χ˙ 2 χ˙ 3 c − mp L˙ χ˙ 3 − mp z˙χ˙ 3 − χ˙ 3 L g6 = −mp z¨ sin(χ3 ) − mp g sin(χ3 ). (47) Again, it follows that limt→∞ g5 = 0 in accordance with (36) and (42), and further it is derived that g˙ 6 ∈ L∞ from (32). 3 Extended Barbalat’s Lemma [59]: If a continuous differentiable function f (t) : R≥0 → R has a finite limit as t → ∞, and its derivative with respect to time is expressed by f˙ (t) = f1 (t) + f2 (t), where f1 (t) is uniformly continuous [or alternatively, f˙1 (t) ∈ L∞ ] and limt→∞ f2 (t) = 0, then limt→∞ f1 (t) = 0, limt→∞ f˙ (t) = 0.

(51)

It is clear from (36), (42), (45), (48), and (49)–(51) that the conclusions in (26) are valid. The proof is hence completed. Remark 6: In the strict sense, the control laws in (25) are not sliding mode controllers. Although two robust sign function-related terms, sgn(χ˙ 1 ) and sgn(χ˙ 2 ), are included, sliding surfaces are not involved in (25). The main purpose of introducing these robust terms is to counteract the effects of the heave acceleration z¨, by utilizing βza . For the designed control method, although it is difficult to analyze its robustness against system uncertainties theoretically, we will try to design more advanced controllers that can theoretically take care of parametric uncertainties as an important future research direction. IV. E XPERIMENTAL R ESULTS AND A NALYSIS To verify the effectiveness of the proposed approach, some hardware experimental results and analysis will be presented. A. Self-Built Ship-Mounted Crane Hardware Prototype In this paper, the performance of the proposed control method will be verified on a self-built ship-mounted boom crane hardware prototype, as shown in Fig. 2. In this crane prototype, the boom can pitch under the actuation of a luffing servo motor (equipped with a reduction gear) through a worm and gear mechanism. The cargo can be lifted and lowered via a steel rope by a winding drum connected to a servo motor (equipped with a reduction gear). The pitching angular displacement and the rope length can be captured in real time by coaxial encoders, which are embedded within the luffing and winding drum servo motors. The cargo swing angle with respect to the boom can be measured online by angular encoders. In addition, as seen in Fig. 2, a tilter is also designed and built to mount the boom crane, which is used to imitate the ship roll and heave movements induced by sea waves under the actuation of three large power servo motors. The computer-based control system is composed of a hosting personal computer (PC) embedded with a GTS-800PV-PCI eight-axis motion control board (made by Googol Technology Limited), a GT2-800-ACC2-V I/O interface board, and the MATLAB/Simulink 2012b Real-Time Windows


Fig. 2.


Self-built ship-mounted crane hardware prototype. (Left) The mechanical body and (right) the computer-based control system.

B. Hardware Experiments and Analysis In this section, two groups of hardware experiments are carried out on the ship-mounted crane prototype shown in Fig. 2. In the experiments, the desired location for the cargo is set as √ (52) ygd = 3 3/10 m, zgd = 0.12 m. Then, it follows from (18) that the desired equilibrium point for χ1 (t), χ2 (t), and χ3 (t) is as follows: χ1d = π/6 rad, χ2d = 0.18 m, χ3d = 0 rad.

(53)

Except for scenario 2 of experiment group 2, the initial values of χ1 (t), χ2 (t), and χ3 (t) are, respectively, taken as χ1 (0) = 0 rad, χ2 (0) = 0.52 m, χ3 (0) = 0 rad. Fig. 3. Ship-mounted crane hardware prototype, marked up with state variables.

Target (RTWT) software control system. More precisely, the motion control board can receive realtime data from the encoders and deliver them to the PC. By making use of the collected data, the control algorithms are implemented in MATLAB/Simulink RTWT to derive the desired control forces/torques for the motors; these control commands are first converted and then sent to the servo actuators by the motion control board, and then the motors will follow the control orders given by the servo actuators to complete the tasks. In addition, the state variables are marked up in the built ship-mounted crane hardware prototype in Fig. 3, which shows similarity with the model plotted in Fig. 1. The hardware prototype has been carefully designed to reflect the major dynamical characteristics of a ship-mounted crane, since it has all the functions of a ship-mounted crane, including luffing, hoisting/lowering, cargo swing, roll and heave movements, and so on. Also, abundant experimental tests have been implemented to verify its performance, which indicates that the hardware prototype can satisfactorily capture the operation principle of a ship-mounted crane system.

(54)

Also, except for scenarios 3 and 4 of experiment group 2, the disturbances induced by the tilter roll and heave movements are set as follows: √ = 4 sin(t) deg, z = 3 3 sin( )/50 m. (55) In addition, the main physical parameters of the ship-mounted crane system are given as follows: J = 6.93kg·m2 , mp = 0.5 kg, LB = 0.52 m, g = 9.8 m/s2 . Experiment Group 1: Three existing control approaches, i.e., the PD controller, the SMC method4 in [46], and the composite controller in [49], are chosen as comparative control methods to fully validate the effectiveness of the proposed controller in the presence of ship (tilter) roll and heave movements. Due to space limitation, the expressions for the comparative control laws are not provided. After careful tuning, the control gains for the PD controller are selected as kp1 = 16.9, kd1 = 5.5, kp2 = 4.04, and kd2 = 7.1; the control parameters for the SMC method in [46] and the composite controller in [49] are determined, respectively, as k1 = 2.3, k2 = 3.5, μ = 1.1, 4 It is worthwhile to mention that the original SMC method in [46] does not take the rope length control into account; here, an extra hoisting controller is added by following the preliminary design idea in [46] so that the rope length can be controlled for comparison purposes.


9

TABLE I C ONTROL I NPUT E NERGY

Fig. 4.

Experiment group 1. Results of different control methods.

k1 = 1.5, k2 = 4.3, k3 = 3.6, kα = 0.1, kβ = 0.2, kL1 = 9.5, kL2 = 5.4, kx = 1.0, and σ = 0.01; as for the proposed controller, the control gains are chosen as kpχ1 = 11.4, kdχ1 = 15.1, kpχ2 = 3.3, kaχ1 = 0.31, kdχ2 = 6.5, kaχ2 = 3.0, βza = 0.1, and kr = 0.1. In addition, as widely done in the literature, sgn(∗) has been replaced with tanh(∗) for both the proposed controller and the comparative controllers to avoid chattering during implementation. After implementing the proposed controller and the three comparative ones, the obtained experimental results are shown in Fig. 4. To be more intuitive, the unit for the angular variables has been changed from radian (rad) to degree (deg). It is seen from the dashed lines in Fig. 4 that the PD controller can drive eχ1 and eχ2 to tend toward zero with significant overshoots; additionally, the cargo swing with respect to the ground, i.e., eχ3 , oscillates seriously, which is mainly caused and excited by the ship (tilter) roll and heave movements. From the dotteddashed lines in Fig. 4, it is found that the trajectory of eχ1 , under the action of the SMC method in [46], still presents significant overshoots; as an improvement compared with the PD controller, the overshoots in eχ2 are reduced to some extent, and the cargo swing also gets better suppressed. Further, it can be seen from the dotted lines in Fig. 4 that the composite control method in [49] achieves better control performance for the convergence of eχ1 than the first two comparative ones while maintaining similar transient performance in terms of eχ2 and eχ3 with the SMC method in [46]. Finally, after comparing the proposed method (solid lines) with the comparative methods discussed above, it is clear that the proposed controller performs more satisfactorily than the comparative methods in

suppressing the overshoots of eχ1 and eχ2 , which improves the overall working efficiency. As for the cargo swing eχ3 , the designed control method restricts the (swing) amplitude to be smaller. Also, all three comparative methods result in significant residual cargo swing, while there is almost no residual swing for the proposed method, even though the system is influenced by the ship (tilter) roll and heave movements. In addition, some quantitative comparisons are also provided for the control inputs of different methods. To this end, the control input energy indices EMc and EFc are intro20 duced, which are, respectively, defined as EMc 0 Mc2 dt and 20 2 EFc 0 Fc dt, to depict the control input energies required by different controllers. These two indices for the four control methods are given in Table I. It is found that, the required energies for Fc (i.e., EFc ) are similar for all the four controllers. Regarding EMc , the proposed method consumes the least energy, which is only 51.6% of that of the PD method, 49.5% of that of the SMC method [46], and 30.9% of that of the composite method [49]. Hence, based on the above analysis, the proposed controller behaves better than the three comparative control methods in terms of faster error convergence and reduced control input energy consumption, and the performance improvement over the comparative methods is significant. Define the cargo position errors as eyg = yg − ygd and ezg = zg − zgd . Then, to more intuitively see whether the cargo has been successfully placed at the desired location (corresponding to eyg = ezg = 0), a figure which records the histories of eyg and ezg has been additionally included as Fig. 5, which corresponds to the experimental results in Fig. 4. It is clearly seen from Fig. 5 that, in comparison with the comparative control approaches, the proposed controller makes the cargo positioning errors eyg and ezg converge to zero more rapidly and efficiently, in the presence of ship (tilter) roll and heave movements. Experiment Group 2: To further explore its robustness against extra external disturbances (the tilter roll and heave movements are still present), four sets of experiments are implemented by taking into account the following scenarios. 1) Scenario 1 (External Disturbances to the Cargo Swing): The cargo swing is manually perturbed at approximately 4.0 s, 6.3 s, and 10.0 s on purpose, respectively. The maximum amplitudes for these three waves of perturbations are 5.9 deg, 3.95 deg, and 2.1 deg, respectively, (see the middle plot of Fig. 6), which are 155%, 104%, and 55.2% of the maximum swing amplitude without perturbations (occurring at about 0.5 s), respectively. These perturbations are created by manually knocking



Fig. 5. Experiment group 1. The cargo position errors eyg and ezg for all the four control methods.

Fig. 7.

Fig. 6. swing.

Experiment group 2–scenario 2. Nonzero initial cargo swing.

Experiment group 2–scenario 1. External disturbances to the cargo

the cargo with a stick, as a means to imitate external disturbances, such as collisions or winds. 2) Scenario 2 (Nonzero Initial Cargo Swing): The initial cargo swing with respect to the ground is set as 5.9 deg. 3) Scenario 3 (Increased Tilter Roll and Heave Frequency): The frequency of the tilter roll and heave frequency is increased from 1/2π Hz to 1/π Hz. 4) Scenario 4 (Increased Tilter Roll and Heave Amplitude): The disturbances induced by the tilter roll and heave movements are changed √ from those in (55) to = 6 sin(t) deg and z = 5 3 sin( )/50 m, respectively.

Fig. 8. Experiment group 2–scenario 3 Increased tilter roll and heave frequency.

The experiments for the proposed method in the above four scenarios are carried out by using the same control gains as in experiment group 1, and the derived experimental results are


11

law is designed to stabilize the cargo position to the destination, which is supported by rigorous stability analysis. To illustrate the practical performance, hardware experiments are implemented to verify that the presented control method performs better than the comparative methods and shows good robustness against external disturbances. ACKNOWLEDGMENT The authors would like to thank all the reviewers and the Associate Editor for their valuable suggestions, which have greatly improved the quality of this paper. R EFERENCES

Fig. 9. Experiment group 2–scenario 4 Increased tilter roll and heave amplitude.

plotted in Figs. 6–8. It is found from Fig. 6 that the influences of the nonship-induced swing disturbances are eliminated out in an efficient manner. Fig. 7 illustrates that the proposed controller can deal with the influences of nonzero initial system states without degrading the control performance. It can be seen from Fig. 8 that, even though the frequency of the ship (tilter) roll and heave movements is increased, the proposed controller still performs satisfactorily in terms of cargo positioning and swing elimination. At last, by comparing Fig. 9 with the solid lines in Fig. 4, it is found that, in the presence of increased disturbance amplitudes (induced by ship roll and heave movements), the proposed controller still maintains satisfactory control performance by successfully making the error signals eχ1 , eχ2 , and eχ3 converge to zero rapidly. V. C ONCLUSION In this paper, a novel nonlinear stabilization control scheme has been proposed for underactuated ship-mounted cranes influenced by ship roll and heave movements. The kernel contribution lies in the fact that, to our knowledge, it is the first closed-loop control method, developed and analyzed without linearizing or approximating the original complex nonlinear dynamics, which can effectively stabilize a ship-mounted crane system in the presence of ship roll and heave movements. To do so, some coordinate transformation operations are utilized to tackle the ship roll and vertical movements, through which the original dynamics are equivalently transformed into a new form convenient for the controller design and analysis. Based on the transformed model, a Lyapunov-based nonlinear control

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Ning Sun (S’12–M’14) received the B.S. (Hons.) degree in measurement and control technology and instruments from Wuhan University, Wuhan, China, in 2009, and the Ph.D. (Hons.) degree in control theory and control engineering from Nankai University, Tianjin, China, in 2014. He is currently with the Institute of Robotics and Automatic Information Systems, Nankai University. His research interests include cranes, wheeled robots, magnetic suspension systems, and nonlinear control with applications to mechatronic systems. Dr. Sun was a recipient of the Outstanding Ph.D. Dissertation Award from the Chinese Association of Automation (CAA) in 2016, the Best Application Paper Award from the 31st Youth Academic Annual Conference of CAA in 2016, and the Nomination Award of the Guan Zhao-Zhi Best Paper Award at the 32nd Chinese Control Conference in 2013. He is an Organizing/Program Committee Member for several international conferences, including the Program Chair of 2017 International Conference on Computer Vision Systems.

Yongchun Fang (S’00–M’02–SM’08) received the B.S. degree in 1996 and M.S. degree in control theory and applications in 1999 from Zhejiang University, Hangzhou, China, and the Ph.D. degree in electrical engineering from Clemson University, Clemson, SC, USA, in 2002. From 2002 to 2003, he was a Postdoctoral Fellow with the Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA. He is currently a Professor with the Institute of Robotics and Automatic Information Systems, Nankai University, Tianjin, China. His research interests include nonlinear control, visual servoing, control of underactuated systems, and AFM-based nano-systems. Dr. Fang is an Associate Editor of the ASME Journal of Dynamic Systems, Measurement, and Control.

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He Chen received the B.S. degree in automation from Nankai University, Tianjin, China, in 2013, where he is currently pursuing the Ph.D. degree in control science and engineering with the Institute of Robotics and Automatic Information Systems. His research interests include control of mechatronics, overhead cranes, and wheeled mobile robots.

Yiming Fu received the B.S. degree in intelligent science and technology from Nankai University, Tianjin, China, in 2013, where he is currently pursuing the M.S. degree in control science and engineering with the Institute of Robotics and Automatic Information Systems. His research interests include control of underactuated overhead cranes and offshore cranes.

Biao Lu received the B.S. degree in intelligent science and technology from Nankai University, Tianjin, China, in 2015, where he is currently pursuing the Ph.D. degree in control science and engineering with the Institute of Robotics and Automatic Information Systems. His research interests include control of overhead cranes and offshore cranes.