IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 10, OCTOBER 2016 6407

Slew/Translation Positioning and SwingSuppression for 4-DOF Tower CranesWith Parametric Uncertainties: Design

and Hardware ExperimentationNing Sun, Member, IEEE, Yongchun Fang, Senior Member, IEEE, He Chen, Biao Lu, and Yiming Fu

Abstract—As a powerful large-scale construction tool,a tower crane is a strongly nonlinear underactuated sys-tem presenting complicated dynamical characteristics.Existing control methods for tower cranes are developed onthe basis of simplified (i.e., linearized/approximated) cranedynamics, and most of them require exact model knowl-edge. However, practical tower cranes usually suffer fromuncertainties (e.g., unknown rope length and payloadmass); moreover, when the state variables are not closeenough to the equilibrium point due to unexpected distur-bances, simplified models might not reflect the actual dy-namics any longer, which usually badly degrades the con-trol performance. To tackle these problems, this paper pro-poses an adaptive control scheme for underactuated towercranes to achieve simultaneous slew/translation position-ing and swing suppression, which can reduce unexpectedovershoots for the jib/trolley movements. The closed-loopstability is backed up with the rigorous mathematical analy-sis. To the best of our knowledge, the proposed controller isthe first method for tower cranes with parametric uncertain-ties, which is developed without linearizing/approximatingtheir nonlinear dynamics. Finally, we introduce our self-built multifunctional hardware crane experiment testbedand present experimental studies for the proposed method.Experimental results show that the new control approach iseffective and admits satisfactory robustness.

Index Terms—Nonlinear systems, tower cranes, underac-tuated systems, vibration/swing suppression.

I. INTRODUCTION

FOR decades, cranes have been widely utilized and playdominant roles as powerful transportation tools in many

fields, which, based on their dynamical properties, can beclassified into three types, i.e., tower cranes, rotary cranes, and

Manuscript received September 24, 2015; revised January 31, 2016and March 13, 2016; accepted March 31, 2016. Date of publicationJuly 7, 2016; date of current version September 9, 2016. This workwas supported in part by the National Natural Science Foundation ofChina under Grant 61503200 and Grant 61325017 and in part by theNatural Science Foundation of Tianjin under Grant 15JCQNJC03800.(Corresponding author: Yongchun Fang.)

The authors are with the Institute of Robotics and Automatic Infor-mation Systems, College of Computer and Control Engineering and theTianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin300353, China (e-mail: sunn@nankai.edu.cn; fangyc@nankai.edu.cn;chenh@mail.nankai.edu.cn; lubiao@mail.nankai.edu.cn; fuym@mail.nankai.edu.cn).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2016.2587249

overhead cranes [1]. Though they have different mechanicalstructures and working principles, the common nature theyshare is that there are less independent actuators than thedegrees of freedom (DOFs) that need to be controlled. Hence,analogous to many mechanical systems such as nonholonomicsystems [2], [3], mobile robots [4], aerial/underwater vehicles[5]–[7], wheeled pendulum robots [8], underactuated robots[9]–[11], moving liquid containers [12], and so on, cranes areunderactuated systems whose control issues are challengingand remain open.

Among different types of cranes, tower cranes have beenplaying indispensable roles in construction sites for putting uptall buildings (skyscrapers). Like other mechanical systems [13],vibration/suppression control is a preliminary task for cranes. Atpresent, almost all tower cranes employed in practical environ-ments are still manually manipulated by human operators, whichpresents many drawbacks, e.g., long training time required,unsatisfactory control accuracy, low working efficiency, highsafety risks, and so on. With the development of mechatronicsand advanced control technology, it has turned out to be an ur-gent practical demand to solve the slew/translation positioningand swing suppression problem for 4-DOF underactuated towercranes, by means of developing effective control schemes.

During the past several decades, many researchers have laidtheir focuses on dealing with the control problem of overheadcrane systems with a lot of ambitious and meaningful worksreported in the literature. Based upon the fact whether or not thedeveloped methods require realtime state feedback to generatethe control commands, one may roughly partitioned them intotwo categories, i.e., open-loop controllers and closed-loopfeedback controllers. Traditionally, for open-loop control meth-ods, the control commands are predefined trolley accelerationreference trajectories, which are planned by utilizing the trolleytranslation/payload swing coupling; see, e.g., input shaping[14], [15] and trajectory planning [16]. As the trolley follows theplanned reference trajectories, the trolley motion-induced swingwill be reduced, and the trolley can move toward the target loca-tion at the same time. However, when uncertainties are presentor the state variables are drifted from the predefined referencetrajectories due to unexpected extraneous perturbations, the per-formance of open-loop control methods might be influenced. Inorder to surmount these unfavorable factors, some closed-loopfeedback control methods, including nested saturation control[17], sliding mode control [18]–[20], nonlinear coupling control

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6408 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 10, OCTOBER 2016

[21]–[24], partial feedback linearization control [25], [26],linear control [27]–[29], adaptive control [30]–[32], observer-based control [33], [34], fuzzy logic-based control [35]–[37],genetic algorithm-based control [38], etc., are designed andapplied to tackle the antiswing problem for overhead cranes.

For 4-DOF tower cranes, their dynamics are underactuated,complicated, and highly nonlinear with strong state coupling,and these factors make it challenging to conduct controllerdesign and stability analysis, especially when not linearizing thenonlinear dynamics around the equilibrium point or neglectingsome highly nonlinear terms. In particular, compared withoverhead cranes (where only translation motion is involved),tower cranes additionally involve slew movement whichproduces the complex centrifugal force and inertia force (theinertia force direction is always changing). Hence, tower cranesystems have more complicated dynamical characteristicsthan overhead cranes, and it is more difficult and challengingto achieve the dual control objectives of slew/translationpositioning and swing elimination for tower cranes.

So far, there are much fewer available control methodsreported for tower cranes than overhead cranes, since thecomplicated nonlinear dynamics are cumbersome to analyzefor controller development. In [39], the dynamical responseof underactuated tower cranes, which is coupled with payloadswing motion, is thoroughly analyzed based on a linearizedmodel. To minimize payload swing, Omar and Nayfeh developscheduling feedback-based control schemes for tower cranesystems [40], [41]. In [42], Lee et al. develop a laser-based pathtracking system for tower cranes and validate its effectiveness.In [43], command shaping-based control approaches aredeveloped and utilized to reduce the payload oscillations oftower cranes, by means of shaping the reference commands ac-cording to the system natural frequencies; interested readers arerecommended to [44] and[45] for more details about commandshaping and its applications. By simplifying the crane dynamicswith small angle and slow slew/translation assumptions, anoptimal velocity controller is proposed by utilizing Pontryagin’smaximum principle in [46] to achieve antiswing control fortower cranes. In [47], a novel model predictive control approachis presented, by neglecting some specific terms within thetower crane dynamics, to make the payload follow a predefinedpath so as to fulfill the control task. Besides those model-basedcontrol approaches, Duong et al. design a recurrent neuralnetwork-based control strategy for underactuated tower cranesystems, where a hybrid evolutionary algorithm is utilized tofurther improve the overall control performance [48].

After carefully summarizing the existing approaches fortower cranes, the following issues need to be solved or requirefurther improvements.

1) Existing tower crane controllers are designed and an-alyzed based upon simplified (linearized/approximated)tower crane dynamics. Once the state variables are notclose enough to the equilibrium point (e.g., due to unex-pected disturbances), there will be significant differencesbetween the simplified models and the real dynamics,which then influences the control performance and evenleads to instability.

2) Most existing controllers require exact model knowledge(EMK), that is, they involve system parameters (e.g.,rope length, trolley/payload masses, jib moment of in-ertia). However, the exact values of these parameters areusually difficult to obtain in practice. Moreover, the sys-tem suffers from the effects of uncertain frictions. Theseuncertain factors will degrade the performance of EMK-based control approaches.

3) For existing closed-loop tower crane controllers, thereis no theoretical guarantee for reducing overshoots sincemerely asymptotic stability results can be achieved. Oncethe control parameters are not well chosen, there willbe obvious overshoots not only influencing the entire effi-ciency but also causing unnecessary energy consumption.

In this paper, we present a new adaptive control scheme forunderactuated 4-DOF tower crane systems, which can achievesuperior control performance in the presence of uncertaintiesand reduce unexpected overshoots. Specifically, by carefullyanalyzing the system energy storage function, an adaptivecontrol scheme is developed by incorporating some elaboratelyconstructed terms to improve the transient control performanceas well as to reduce unexpected overshoots for the jib slewand trolley translation movements. Then, Lyapunov-basedanalysis is carried out to prove the performance of the designedcontrol scheme. Finally, we introduce a multifunctional craneexperiment testbed designed and built by our research groupon which we implement extensive experiments to examine theeffectiveness and robustness of the proposed control methodand compare it with the two existing approaches.

The originality of this paper is as follows.1) For underactuated tower cranes, this paper derives the

first control scheme which is designed without lineariz-ing the tower crane’s nonlinear dynamical equations orneglecting partial nonlinear terms. Moreover, we providea complete rigorous theoretical closed-loop stability anal-ysis without linearizing or approximating the nonlineardynamics, either. One benefit is that, even when the statevariables are not close enough to the equilibrium point,the controller can still perform satisfactorily.

2) The designed controller does not need the exact valuesof the system parameters (e.g., trolley/payload masses,rope length, jib moment of inertia), and it can increasethe transient performance (in terms of swing suppressionand elimination) and reduce unexpected overshoots.

3) We develop and build a hardware tower crane testbed andinclude experimental results to verify that the proposedcontrol scheme achieves better control performance thanthe existing comparative methods and exhibits good ro-bustness to uncertainties and extraneous disturbances.

We arrange the remaining parts as follows. In Section II,we introduce the nonlinear dynamics of underactuated towercranes and describe the control objective mathematically.Then, the detailed controller design procedures are provided inSection III. Further, Section IV presents the stability analysis.After that the proposed controller is experimentally examinedin Section V. Finally, the paper is wrapped up with conclusionsin Section VI.

SUN et al.: SLEW/TRANSLATION POSITIONING AND SWING SUPPRESSION FOR 4-DOF TOWER CRANES WITH PARAMETRIC 6409

Fig. 1. Model of an underactuated tower crane system.

II. PROBLEM FORMULATION

This section will provide the dynamical equations of motionfor tower cranes and then give the control objective.

A. Dynamics of Tower Cranes

The dynamical equations for an underactuated 4-DOF towercrane system are as follows (see Fig. 1)1:[

mp

(sin2θ1 cos2θ2 + sin2θ2

)L2 + 2mp xL cos θ2 sin θ1 + J

+ (mp + mt ) x2]φ − mp L sin θ2 x − mp L2 cos θ1

× cos θ2 sin θ2 θ1

+ mp L (cos θ2x + L sin θ1 ) θ2 + 2 (mp + mt ) xxφ

+ 2mp L cos θ1 cos θ2xφθ1 − mp L sin θ2 (2φ sin θ1 + θ2 )xθ2

+ 2mp L sin θ1 cos θ2 xφ + mp L2 sin (2θ1 ) cos2 θ2 φθ1

+ mp L2 sin θ1 sin θ2 cos θ2 θ21 + mp L2 cos2 θ1 sin(2θ2 )φθ2

+ 2mp L2 cos θ1 sin2θ2 θ1 θ2 = M − Mf (1)

−mp L sin θ2 φ + (mp + mt ) x + mp L cos θ1 cos θ2 θ1

−mp L sin θ1 sin θ2 θ2 − (mp + mt ) xφ2 − 2mp L cos θ1

× sin θ2 θ1 θ2

−mp L cos θ2

[sin θ1 (φ2 + θ2

1 + θ22 ) + 2φθ2

]= F − Ff (2)

−mp L2 cos θ1 cos θ2 sin θ2 φ + mp L cos θ1 cos θ2 x + mp L2 cos2θ2

·θ1 − mp L cos θ1 cos θ2 (x + L sin θ1 cos θ2 )φ2 − 2mp L2 cos θ2

·(φ cos θ1 cos θ2 + θ1 sin θ2 )θ2 + mp gL sin θ1 cos θ2 = 0 (3)

mp L(cos θ2x + L sin θ1 )φ − mp L sin θ1 sin θ2 x + mp L2 θ2

+ 2mp L cos θ2 xφ + mp L(x sin θ1 sin θ2 − Lcos2θ1 sin θ2 cos θ2 )

·φ2 + 2mp L2 cos θ1 cos2θ2 φθ1 + mp L2 θ21 sin θ2 cos θ2

+ mp gL cos θ1 sin θ2 = 0 (4)

1Throughout the paper, for a time-related variable [e.g., M (t)], when itappears in equations, we omit the time variable “(t)” if no confusion happens,and when it appears in the text, we keep “(t)” to distinguish with the text.

where φ(t) denotes the jib slew angle, x(t) denotes the trol-ley translation displacement, θ1(t), θ2(t) are used to depict thepayload’s swing, J denotes the moment of inertia of the jib, mt

represents the trolley mass, mp is the payload mass, L is thesuspension rope length, g is the gravitational constant, M(t)denotes the slew control toque, F (t) is the translation controlforce, and Mf (t), Ff (t) are the friction torque and force, re-spectively. After a lot of experimental measurements, we adoptthe following friction models for Mf (t), Ff (t)2:

Mf = Mf 1 tanh(ξφ φ) + Mf 2 |φ|φFf = Ff 1 tanh(ξxx) + Ff 2 |x|x (5)

where Mf 1 , Ff 1 , Mf 2 , Ff 2 , ξφ , ξx ∈ R are friction-relatedparameters. ξφ , ξx hardly vary for different payload massesand, hence, can be determined offline and regarded to be apriori known [16]. Without loss of generality, the initial trolleyhorizontal displacement and jib slew angle are taken as zero,i.e.,

φ(0) = 0, x(0) = 0. (6)

The dynamical equations (3) and (4) reflect the internal cou-pling that exists between the actuated trolley/jib motion andthe nonactuated payload swing motion, and the mere way toeliminate the payload swing is to make full use of this couplingrelationship by applying a proper feedback controller.

For crane systems working in practice, the payload is alwaysbeneath the jib. Therefore, as widely done in the crane-related lit-erature [14]–[48], the following reasonable assumption is made:

Assumption 1: The payload swing angles satisfy −π/2 <θ1(t) < π/2, −π/2 < θ2(t) < π/2.

B. Control Objective

The ultimate control objective is to transfer the payload fromits initial position toward the target location accurately whilesufficiently suppressing and eliminating the unexpected pay-load swing motion, in the presence of parametric uncertainties.Nevertheless, due to the underactuation nature, the payload isnonactuated and cannot be directly controlled.

To resolve the above-mentioned issue, we alternatively de-compose the control objective into two subtasks as follows:

1) make the jib arrive at its target angular position φd anddrive the trolley to reach its target position xd , which canbe mathematically described as follows:

limt→∞

φ(t) = φd, limt→∞

x(t) = xd

2) at the same time, suppress and eliminate the swing of thepayload in the sense that

limt→∞

θ1(t) = 0, limt→∞

θ2(t) = 0.

Then, the control task is transformed into designing a suit-able control scheme, without linearizing or approximating the

2As suggested in [49]–[51], disturbance observers can be used to effectivelyeliminate the effects of frictions. Different from that we utilize a model-basedadaptive scheme to compensate for the uncertain friction force/torque, since thefriction structure has been determined except for the unknown parameters.

6410 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 10, OCTOBER 2016

complicated nonlinear tower crane dynamics, to achieve theabove-mentioned two points in the presence of parametric un-certainties.

To be concise, the abbreviations S1 � sin θ1 , S2 � sin θ2 ,C1 � cos θ1 , C2 � cos θ2 will be used in the subsequent contextunless otherwise claimed.

III. CONTROLLER DEVELOPMENT

The total energy of the tower crane system is as follows:

E =12q�MT (q)q + mpgL(1 − C1C2) (7)

where q � [ φ x θ1 θ2 ]� and MT (q) ∈ R4×4 denotes thefollowing positive-definite symmetric matrix:

MT (q) =

⎡⎢⎢⎣

m11 m12 m13 m14m12 mp + mt mpLC1C2 −mpLS1S2m13 mpLC1C2 mpL

2C22 0

m14 −mpLS1S2 0 mpL2

⎤⎥⎥⎦

with

m11 � mp L2 (S21 C2

2 + S22 ) + 2mp LxS1C2 + (mp + mt )x2 + J

m12 � −mp LS2 , m13 � −mp L2C1S2C2

m14 � mp L(C2x + LS1 ). (8)

Then, we can take the time derivative of E(t), insert for (1)–(4),and make tedious mathematical operations to obtain

E = (M − Mf )φ + (F − Ff )x (9)

implying that the tower crane system is passive. It is also in-ferred from (9) that the energy can merely be eliminated via theactuated jib and trolley motion, i.e., φ(t) and x(t).

Further, in view of (5), one can rewrite (9) into

E = (M − η�φ ωφ)φ + (F − η�

x ωx)x (10)

where the vectors ηφ , ηx , ωφ , ωx ∈ R2 are defined as

ηφ � [ tanh(ξφ φ) |φ|φ ]�, ωφ � [Mf 1 Mf 2 ]�

ηx � [ tanh(ξxx) |x|x ]�, ωx � [Ff 1 Ff 2 ]�. (11)

Next, we will design an adaptive control scheme, which doesnot involve system parameter-related terms, for tower cranesystems.

At first, let the following error signals be introduced:

eφ = φ − φd, ex = x − xd =⇒ eφ = φ, ex = x. (12)

After that to incorporate the control objective into the controllerdesign, we further construct the following continuous differ-entiable positive definite scalar function W (t) on the basis of(7):

W � 12q�MT (q)q + mpgL(1 − C1C2) +

12kpφe2

φ +12kpxe2

x

+12ω�

φ Π−1φ ωφ +

12ω�

x Π−1x ωx (13)

where kpφ , kpx ∈ R+ are positive control gains to be intro-duced later, ωφ(t), ωx(t) ∈ R2 are estimation error vectorsdefined by

ωφ = ωφ − ωφ , ωx = ωx − ωx =⇒ ˙ωφ = ˙ωφ , ˙ωx = ˙ωx

(14)

with ωφ(t), ωx(t) ∈ R2 being the online estimated signals forωφ , ωx [see (11)], respectively, i.e.,

ωφ � [ Mf 1 Mf 2 ]�, ωx � [ Ff 1 Ff 2 ]� (15)

and Πφ = diag{πφ1 , πφ2} ∈ R2×2 , Πx = diag{πx1 , πx2} ∈R2×2 are positive definite matrices to be introduced later withπφ1 , πφ2 , πx1 , πx2 ∈ R+ being positive tunable update gains.After differentiating (13) with respect to time, inserting for (1)–(4) and (10), and implementing some tedious mathematical ma-nipulations, we can obtain the following result:

W = (M + kpφeφ − η�φ ωφ)φ + (F + kpxex − η�

x ωx)x

+ω�φ Π−1

φ˙ωφ + ω�

x Π−1x

˙ωx (16)

where (14) is used. Then, based on the form of (16), as a meansto cancel the crossing terms and render W (t) nonpositive, wefirst give the following preliminary control law:

Mbc = −kpφeφ − kdφ φ + η�φ ωφ

Fbc = −kpxex − kdxx + η�x ωx (17)

where kdφ , kdx ∈ R+ are positive control gains and the updatelaw for ωφ(t), ωx(t) is designed as follows:

˙ωφ = −Πφηφ φ, ˙ωx = −Πxηx x. (18)

Inserting (17) and (18) into (16) yields

W = −kdφ φ2 − kdxx2 ≤ 0 (19)

indicating that (13) merely vanishes along the actuated statevariables φ(t) and x(t), which is due to the underactuated nature.

Although we can prove that the closed-loop equilibrium pointis asymptotically stable by using (17), the useful nonactuatedswing-related information is not directly used. On the otherhand, one common issue associated with most feedback controlmethods [including (17)] is the probable presence of overshoots.

To overcome the above-mentioned drawbacks, we present thefinal adaptive controller as follows:

Mic = −βφ

[(φd + εφ)2 − φ2 + φeφ

][(φd + εφ)2 − φ2 ]2

eφ − khφ(θ21 + θ2

2 )φ

−kpφeφ − kdφ φ + η�φ ωφ

Fic = −βx

[(xd + εx)2 − x2 + xex

][(xd + εx)2 − x2 ]2

ex − khx(θ21 + θ2

2 )x

−kpxex − kdxx + η�x ωx (20)

where khφ , khx , βφ , βx ∈ R+ are positive control gains, andthe positive parameters εφ and εx ∈ R+ are introduced to re-strict the maximum overshoot amplitudes for φ(t) and x(t),

SUN et al.: SLEW/TRANSLATION POSITIONING AND SWING SUPPRESSION FOR 4-DOF TOWER CRANES WITH PARAMETRIC 6411

respectively. The designed controller (20) does not have thesingularity problem, as will be proven by (26)–(28).

Remark 1: The preliminary version in (17) is the basis and apart of the final controller (20). More precisely, compared with(17), the final controller (20) contains two additional nonlin-ear terms, where the first terms in (20) are used to reduce theovershoots for the jib slew and trolley translation movements,and the second terms in (20) have incorporated swing-relatedinformation as a means to enhance the swing suppression andelimination control performance.

Remark 2: There is no rigorous theoretical guarantee for an-alyzing the overshoots of (17) [a part of the final controller (20)],whose control gains can only be derived via trial and error ex-perimentation. The major drawback of such a tuning procedureis that we need to try many times before getting a suitable setof control gains, and in most cases, significant overshoots arepresent. By contrast, for the final controller (20), the overshootshave been carefully taken into consideration when we performcontroller design and analysis, and the benefit is that, as long aswe choose βφ and βx as nonzero, the maximum amplitudes ofthe overshoots will not exceed εφ and εx , respectively.

IV. CLOSED-LOOP STABILITY ANALYSIS

This section presents theoretical analysis for the stability ofthe closed-loop system’s equilibrium point.

Theorem 1: The designed controller (20), along with the up-date law in (18), can drive the actuated variables [i.e., the jibslew angle φ(t) and the trolley displacement x(t)] to reachtheir desired positions, respectively, and simultaneously sup-press/eliminate the nonactuated payload angles θ1(t) and θ2(t).In addition, during the overall process, the slew angle φ(t) andthe trolley displacement x(t) never exceed φd + εφ and xd + εx ,respectively. In other words, the overshoots for φ(t) and x(t)are less than εφ and εx , respectively. These conclusions can bemathematically described as follows:

limt→∞

[φ(t) x(t) φ(t) x(t)

]� =[

φd xd 0 0]�

limt→∞

[θ1(t) θ2(t) θ1(t) θ2(t)

]� =[

0 0 0 0]�

φ(t) < φd + εφ , x(t) < xd + εx (21)

where εφ and εx are defined below (20).Proof: To begin with let the following scalar function V (t)

be constructed:

V � 12q�MT (q)q + mpgL(1 − C1C2) +

12kpφe2

φ +12kpxe2

x

+βφe2

φ

2 [(φd + εφ)2 − φ2 ]+

βxe2x

2 [(xd + εx)2 − x2 ]

+12ω�

φ Π−1φ ωφ +

12ω�

x Π−1x ωx . (22)

Then, we can differentiate V (t) with respect to time and insertfor (10), (18), and (14) to derive the following result:

V =

{M + kpφ eφ +

βφ [(φd + εφ )2 − φ2 + φeφ ][(φd + εφ )2 − φ2 ]2

eφ − η�φ ωφ

}φ

+

{F + kpx ex +

βx [(xd + εx )2 − x2 + xex ][(xd + εx )2 − x2 ]2

ex − η�x ωx

}x

− ω�φ ηφ φ − ω�

x ηx x. (23)

Further, by substituting (20) into (23) and then taking advantageof (18), it can be obtained that

V = −kdφ φ2 − kdxx2 − (khφ φ2 + khxx2)θ21

−(khφ φ2 + khxx2)θ22 ≤ 0 (24)

which indicates that

V (t) ≤ V (0) � +∞. (25)

We can infer from (6) that |φ(0)| < φd + εφ , |x(0)| < xd +εx . Assuming that φ(t) [respectively, x(t)] tends to exceed theboundary of |φ(t)| < φd + εφ [respectively, |x(t)| < xd + εx ]from the interior, then it is clear from (22) that V (t) → +∞which invalidates V (t) � +∞ in (25). Thus, one has

|φ(t)| < φd + εφ , |x(t)| < xd + εx (26)

which indicates that the overshoots of φ(t) and x(t) are, respec-tively, less than εφ and εx . It is also indicated from (26) that(φd + εφ)2 − φ2 > 0 and (xd + εx)2 − x2 > 0, which furtherimplies that V (t) in (22) is always nonnegative, i.e., V (t) ≥ 0,and it is hence a control Lyapunov function candidate. Then,noting that V (0) is bounded, it is easy to conclude that

V ∈ L∞ =⇒ eφ , ex , φ, x, θ1 , θ2 , ωφ , ωx , ωφ , ωx ∈ L∞,

e2φ

(φd + εφ )2 − φ2 ,e2x

(xd + εx)2 − x2 ∈ L∞. (27)

Moreover, if eφ(t) → 0, then 1(φd +εφ )2 −φ2 → 1

(φd +εφ )2 −φ2d∈

L∞. In addition, if eφ(t) � 0, then it follows from eφ(t) ∈ L∞

ande2

φ

(φd +εφ )2 −φ2 ∈ L∞ that 1(φd +εφ )2 −φ2 ∈ L∞. Similar analy-

sis can be implemented for 1(xd +εx )2 −x2 . Hence, it is indicated,

along with (20) and (27), that

1(φd + εφ)2 − φ2 ,

1(xd + εx)2 − x2 , M, F ∈ L∞. (28)

To prove (21), we need to perform further analysis. To thisend, define Ξ as the set of all the trajectories such that V (t) = 0,i.e.,

Ξ � {(q, q) | V = 0}. (29)

Then, we further denote Λ as the largest invariant set includedin Ξ. Then, it follows from (5), (24), and (29) that, in Λ,

φ = 0, x = 0 =⇒ φ = 0, x = 0, Mf = 0, Ff = 0 =⇒eφ = λ1 , ex = λ2 =⇒ φ = λ1 + φd, x = λ2 + xd (30)

where λ1 , λ2 are constants to be determined. Then, in the setΛ, it is known from (30) that M(t) and F (t) in (20) reduce to

M = λ3 , F = λ4 (31)

where λ3 , λ4 are constants explicitly expressed as

λ3 � −βφ

[(φd + εφ)2 − (λ1 + φd)2 + (λ1 + φd)λ1

][(φd + εφ)2 − (λ1 + φd)2 ]2

λ1

6412 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 10, OCTOBER 2016

−kpφλ1 ,

λ4 � −βx

[(xd + εx)2 − (λ2 + xd)2 + (λ2 + xd)λ2

][(xd + εx)2 − (λ2 + xd)2 ]2

λ2

−kpxλ2 (32)

i.e., both M(t) and F (t) keep constant in Λ. We will completethe invariant-set-based analysis with two steps.

Step 1. We first prove that eφ(t) = 0, ex(t) = 0, F (t) = 0,M(t) = 0 in Λ. Dividing both sides of (1) with mpL and apply-ing the conclusions in (30), one has the following equation:

−LC1C2S2 θ1 + (λ2 + xd)C2 θ2 + LS1 θ2 − (λ2 + xd)S2 θ22

+LS1S2C2 θ21 + 2LC1S

22 θ1 θ2 =

λ3

mpL. (33)

To proceed, we intend to integrate both sides of (33) with respectto time; however, the left-hand side is not directly integrable.As a means to deal with this issue, by using the fact that S2

2 ≡1 − C2

2 , one can expand LC1S22 θ1 θ2 in the following way:

LC1S22 θ1 θ2 = LC1 θ1 θ2 − LC1C

22 θ1 θ2 . (34)

Using (34) and performing arrangements, we rewrite (33) as

−LC1C2S2 θ1 + LS1S2C2 θ21 + LC1S2

2 θ1 θ2 − LC1C22 θ1 θ2

+LS1 θ2 + LC1 θ1 θ2 + (λ2 + xd)(C2 θ2 − S2 θ22 ) =

λ3

mpL

⇐⇒ ddt

[−LC1S2C2 θ1 + LS1 θ2 + (λ2 + xd)C2 θ2

]=

λ3

mpL.

(35)

The integral of (35) in time can be calculated as follows:

− LC1S2C2 θ1 + LS1 θ2 + (λ2 + xd)C2 θ2 =λ3

mpLt + λ5 (36)

where λ5 denotes a constant yet to be determined. If the constantλ3 in (31) is nonzero, then as t → ∞,∣∣∣−LC1S2C2 θ1 + LS1 θ2 + (λ2 + xd)C2 θ2

∣∣∣ → +∞ (37)

which contradicts with the conclusions of S1 , C1 , S2 , C2 ∈ L∞,and θ1(t), θ2(t) ∈ L∞ [see (27)]. Hence, λ3 = 0, which furtherimplies from (30)–(32) and (36) that, in Λ,

− LC1S2C2 θ1 + LS1 θ2 + (λ2 + xd)C2 θ2 = λ5 , M = 0. (38)

By employing φ = λ1 + φd [see (30)] and λ3 = 0, one canrewrite the first equation in (32), after some arrangements, as{

kpφ +βφ

[(φd + εφ)2 − φφd

][(φd + εφ)2 − (λ1 + φd)2 ]2

}λ1 = 0. (39)

As φ < φd + εφ [see (26)] and φd < φd + εφ , one has (φd +εφ)2 > |φφd | ≥ φφd , indicating that the bracket term in (39) isalways positive. Thus, it is known from (39) and (30) that

λ1 = 0 =⇒ eφ = 0, φ = φd. (40)

Similar with (33), by using (30), one can reduce (2) into

C1C2 θ1 − S1C2 θ21 − 2C1S2 θ1 θ2 − S1S2 θ2 − S1C2 θ2

2 =λ4

mpL

⇐⇒ ddt

(C1C2 θ1 − S1S2 θ2

)=

λ4

mpL. (41)

Then, we can integrate (41) with respect to time to yield

C1C2 θ1 − S1S2 θ2 =λ4

mpLt + λ6 (42)

where λ6 is constant. Similar arguments with those in (36)–(40)can be carried out for (42) to conclude that

λ4 = 0, F = 0, ex = 0, λ2 = 0, x = xd (43)

C1C2 θ1 − S1S2 θ2 = λ6 . (44)

Step 2. We further illustrate that θ1(t) = 0, θ2(t) = 0, θ1(t) =0, θ2(t) = 0 in Λ. By utilizing (30), after some arrangements,one can reduce (3) and (4) into

LC22 θ1 = 2LS2C2 θ1 θ2 − gS1C2 (45)

Lθ2 = −Lθ21S2C2 − gC1S2 . (46)

In addition, multiplying both sides of (41) with LC2 and usingλ4 = 0 [see (43)] yields

LC1C22 θ1 − LS1C

22 θ2

1 − 2LC1S2C2 θ1 θ2

−LS1S2C2 θ2 − LS1C22 θ2

2 = 0. (47)

Then, by inserting (45) and (46) into (47), canceling the commonterms, and rearranging the obtained equation, we are led to

gS1C1C32 + LS1C

42 θ2

1 + LS1C22 θ2

2 = 0 (48)

where we have used the triangular identity S22 + C2

2 ≡ 1. Notingthat C2 > 0 (see Assumption 1), one can divide both sides of(48) with C2

2 to derive

S1

(gC1C2 + LC2

2 θ21 + Lθ2

2

)= 0. (49)

Again, using C1 > 0, C2 > 0 (see Assumption 1), the bracketterm in (49) is always larger than zero. Hence, it follows from(49) that

S1 = 0 =⇒ θ1 = 0 =⇒ θ1 = 0, θ1 = 0 (50)

where Assumption 1 is used. Inserting (50) into (44) yieldsλ6 = 0.

Further, we can use (43) and (50) to simplify (38) as

C2 θ2 =λ5

xd=⇒ S2 =

λ5

xdt + λ7 (51)

where λ7 is a constant number. Again, by performing somereduction to absurdity analysis similar with those in (36) and(37), it is not difficult to show, by using |S2 | ≤ 1, that

λ5 = 0, S2 = λ7 . (52)

SUN et al.: SLEW/TRANSLATION POSITIONING AND SWING SUPPRESSION FOR 4-DOF TOWER CRANES WITH PARAMETRIC 6413

It can be concluded from θ2(t) ∈ L∞ [see (27)] that θ2(t)is uniformly continuous. Thus, it follows from (52) that θ2(t)takes a constant value in Λ. Hence, in the set Λ, we always have

θ2(t) = 0 =⇒ θ2(t) = 0. (53)

Then, substituting (50) and (53) into (46) produces

gC1S2 = 0 (54)

which, together with C1 > 0 (see Assumption 1), indicates that

S2 = 0 =⇒ θ2 = 0 (55)

upon using Assumption 1.After gathering the results in (26), (30), (38), (40), (43),

(50), (53), and (55), the largest invariant set Λ contains onlythe closed-loop equilibrium point. Then, the conclusions ofTheorem 1 can be proven by applying LaSalle’s invarianceprinciple [52].

Remark 3: The first terms in (20) stem from V (t) givenin (22). It is seen that V (t) includes two nonnegative termsβφe2

φ/{2[(φd + εφ)2 − φ2 ]} and βxe2x/{2[(xd + εx)2 − x2 ]},

which will tend to infinity if the overshoots of φ(t), x(t) ap-proach εφ , εx . Thus, the first terms in (20) are motivated tocancel the crossing terms in V (t) of (23) to prevent V (t) fromescaping to infinity, so that the overshoots of φ(t) and x(t) canbe suppressed to be less than εφ and εx , respectively, as shown in(26). Additionally, by comparing (19) with (24), two extra non-positive terms −(khφ φ2 + khxx2)θ2

1 and −(khφ φ2 + khxx2)θ22

are included in (24) because of the second terms in (20). Dueto the underactuated nature, the actuated and nonactuated statevariables are always coupled with and influence each other,i.e., if the actuated variables vary with time, they will affectthe trajectories of the nonactuated ones, and vice versa. Aslong as θ1(t), θ2(t) are nonzero, they will always influenceφ(t), x(t) in turn. As a result, the nonpositive “coefficient”−(khφ φ2 + khxx2) in (24) will not be always zero until θ1(t)and θ2(t) are regulated to zero, that is, the last two terms in (24)are active before V (t) ≡ 0. It is inferred from (19) that W (t)merely vanishes along the actuated state variables; by contrast,we see from (24) that V (t) vanishes along both the actuated andnonactuated state variables, which implies that the controller(20) induces further swing-related damping to accelerate swingsuppression and elimination.

V. HARDWARE EXPERIMENTS

This section exhibits hardware experimental results to verifythe performance of the proposed control method.

A. Self-Built Multifunctional Crane Experiment Testbed

We first briefly describe the utilized multifunctional hardwarecrane experiment testbed, as shown in Fig. 2. The multifunc-tional crane testbed is made up of a mechanical body, actuatingdevices/sensors, and a computer-based control system.

1) Mechanical Body and Actuating De-vices/Sensors: As illustrated in Fig. 2, the jib canrotate around the mast fixed to the base, and the trolley can

Fig. 2. Self-built multifunctional hardware crane experiment testbed.

move along the jib. All the movements are generated byac motors (accompanied with reduction gears) which arecontrolled by servo actuators connected to the computer-basedcontrol system. The trolley translation displacement and the jibslew angle are measured by coaxial encoders embedded withinthe ac servo motors. The payload is suspended from the trolleythrough a steel rope, whose swing motion can be convenientlycaptured in real time by angular encoders equipped beneath thetrolley. With these moving devices, we are admitted to studythe dynamical characteristics of a tower crane system. On theother hand, the jib can also perform pitch motion so that thebuilt crane testbed turns to a boom crane by fixing the trolleyto some position on the jib. In addition, we have specificallydesigned the base in such a way that it can undulate, if needed,under the actuation of three ac servo motors, which can beutilized to imitate the effects of sea waves, and in this case,the crane testbed acts as an offshore crane working in the seaenvironment. In this paper, the crane experiment testbed is setto work in the mode of a tower crane. It is worthwhile to pointout that, when the self-built testbed works as a tower crane,although its size is not large, it is carefully and elaboratelydesigned in order to capture the dynamical characteristics ofa tower crane, and it has all the functions of a tower crane,including trolley translation, jib slew, payload swing, and soforth. Additionally, we have carried out a lot of experimentaltests to verify its performance, which demonstrate that it canwell reflect the working principle of tower cranes.

2) Computer-Based Control System: The controlsystem mainly consists of a hosting PC, a motion control board(embedded within the PC), and a software control system (seeFig. 2). In particular, a GTS-800-PV-PCI eight-axis motion con-trol board made by Googol Technology Limited, together witha GT2-800-ACC2-V I/O interface board, is used to collect datafrom the sensors and convey them to the hosting PC in realtime, and simultaneously, it outputs the control commands gen-erated by the hosting PC to make the ac servo motors generatethe corresponding actuating forces/torques. As for the software

6414 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 10, OCTOBER 2016

control system, the MATLAB/Simulink 2012b Real-Time Win-dows Target which runs in the environment of the Windows XPoperating system is used to implement the developed controlapproach in real time, with the control period being configuredas 5 ms.

For the utilized testbed, the physical parameters, includ-ing the trolley mass and the jib moment of inertia, are set asmt = 3.5 kg, J = 6.8 kg · m2 , g = 9.8 m/s2 . In addition, thepayload mass mp and the rope length L can be adjusted asneeded. Unless otherwise mentioned, mp and L are set as

mp = 0.5 kg, L = 0.5 m. (56)

B. Experimental Results and Analysis

In order to fully verify the actual control performance, wehave carried out two groups of hardware experiments. Moreprecisely, we compare the performance of the proposed con-troller with those of the robust extra-insensitive (EI) controlapproach [44] and the linear quadratic regulator (LQR) methodin Experiment 1. After that, in Experiment 2, the performanceof the proposed method is then experimentally examined in thepresence of different unfavorable conditions.

For all the experiments, without loss of generality, the initialjib slew and trolley translation positions are taken as φ(0) = 0◦

and x(0) = 0 m, respectively. The target jib slew and trolleytranslation positions are set as φd = π/2 rad and xd = 0.4 m,respectively. In order to guarantee soft start, as in [30], φd andxd are changed to be φd = (π/2)(1 − e−8.33t3

) rad and xd =0.4(1 − e−8.33t3

) m for implementation, respectively.1) Experiment 1: In this group of experiments, to better

show the designed controller’ performance, the well-knownrobust EI control method in [44] and the LQR method arechosen for comparisons. For the comparative EI controlapproach, the tolerance level of vibration is chosen as 5% as in[44] and the natural period is calculated as T = 1.4192 s, whichcan be used to determine the control commands. Regardingthe LQR method, by linearizing the tower crane dynamicsaround the equilibrium point, we can obtain two decoupledlinear systems in state space forms. Their performance indicesare chosen as Jlqr =

∫ ∞0

(X�QX + RF 2

)dt where X =

[φ(t) − φd, φ(t) − φd , θ2(t), θ2(t)]� for the slew subsystemand X = [x(t) − xd, x(t) − xd , θ1(t), θ1(t)]� for the trans-lation subsystem, respectively. After careful tuning, we choosetwo sets of Q and R as Q = diag {10, 14, 150, 0} , R = 0.1 andQ = diag {13, 14.5, 165, 0} , R = 0.1, respectively; the corre-sponding LQR controllers are derived by MATLAB as Mlqr =−10(φ − φd) − 18.0514(φ−φd)+24.4277θ2 +5.82θ2 , Flqr =−10(x− xd) − 16.746(x−xd)+29.6098θ1 +4.1224θ1 (whichwe call LQR 1) and Mlqr = −11.4018(φ − φd) − 19.0151(φ −φd) + 26.4348θ2 + 5.9425θ2 , Flqr = −11.4018(x − xd) −17.6068(x − xd) + 31.7093θ1 + 4.1266θ1 (which we callLQR 2), respectively. After tuning, the control parameters forthe designed controller (20) are selected as kpφ =26, kdφ =3.1, kpx = 71.4, kdx = 13, khx =3.5, khφ =4.5, βx =βφ =0.01, εx = 0.002, εφ = 0.003, πφ1 = 0.6, πφ2 =0.8, πx1 =πx2 =10, Mf 1(0)=Mf 2(0)= Ff 1(0)= Ff 2(0)=0. To better

Fig. 3. Results for Experiment 1: the robust EI method in [44] (dashedlines: φd = π/2 rad and xd = 0.4 m).

exhibit the experimental results, we introduce the followingperformance indices:

1) θ1 max , θ2 max : maximum swing amplitudes, i.e.,θ1 max � maxt{|θ1(t)|}, θ2 max � maxt{|θ2(t)|}.

2) θ1res , θ2res : residual swing defined as the max-imum swing amplitudes after 6 s, i.e., θ1res �maxt≥6{|θ1(t)|}, θ2res � maxt≥6{|θ2(t)|}.

3) Mmax , Fmax : maximum control force/torque am-plitudes, i.e., Mmax � maxt{|M(t)|}, Fmax � maxt

{|F (t)|}.The obtained experimental results are recorded by the curves

in Figs. 3–6, and the corresponding quantified performanceindices introduced above are provided in Table I. Whenimplementing the three kinds of control methods, the trolleyand the jib all arrive at the target positions within 6 s. It is seenthat the proposed adaptive control approach (see Fig. 6) ex-hibits superior antiswing performance over the comparative EIand LQR methods (see Figs. 3–5), in terms of maximum swingamplitude suppression (see θ1 max and θ2 max in Table I) andresidual swing elimination (refer to θ1res and θ2res in Table I).There is almost no residual swing for the proposed controller(20), while obvious residual swing is observed for the compar-ative approaches.

By observing the locally zoomed plots in Figs. 3–5, the jibslew and trolley translation positions exceed the target positions.By contrast, Fig. 6 verifies that the proposed controller (20) caneffectively prevent the trolley and jib motion from going be-yond the target positions (i.e., avoiding overshoots), which isbeneficial for practical applications. By comparing the resultsof LQR 1 and LQR 2 in Figs. 4 and 5 and also Table I, one findsthat although LQR 2 reduces the overshoots to some extent by

SUN et al.: SLEW/TRANSLATION POSITIONING AND SWING SUPPRESSION FOR 4-DOF TOWER CRANES WITH PARAMETRIC 6415

Fig. 4. Results for Experiment 1: LQR 1 (dashed lines: φd = π/2 radand xd = 0.4 m).

Fig. 5. Results for Experiment 1: LQR 2 (dashed lines: φd = π/2 radand xd = 0.4 m).

adjusting the values of Q and R, it results in larger swing am-plitudes and more significant residual swing as a tradeoff. Addi-tionally, it is seen from the zoomed plots for M(t) in Figs. 3–6that the control input signals are a bit noisy. This is caused by thefact that the velocity signals are obtained by using the numeri-cal difference technique together with low-pass filters, because

Fig. 6. Results for Experiment 1: the proposed control method (dashedlines: φd = π/2 rad and xd = 0.4 m).

TABLE IPERFORMANCE INDICES OF EXPERIMENT 1

θ1 m a x θ2 m a x θ1 r e s θ2 r e s Mm a x Fm a x

Methods (degree) (degree) (degree) (degree) (N · m) (N)

EI method 4.22 4.67 1.13 1.37 5.36 14.91LQR 1 6.29 4.80 2.31 1.14 6.48 12.31LQR 2 6.59 4.83 1.58 1.44 6.32 14.56Proposed method 2.21 3.03 0.26 0.56 5.39 11.88

they are not directly measurable; then, there unavoidably existunfiltered noises in the velocity signals, which are further incor-porated into the control inputs. For practical implementation, ifneeded, one can further use low-pass filters to smooth the con-trol signals, as commonly done in many mechatronic controlsystems.

2) Experiment 2: In order to further validate the robust-ness performance of the designed control method, we carry outexperiments by considering the following cases.

1) Case 1: Parametric uncertainties. We change mp andL to mp = 1 kg and L = 0.8 m, respectively, while thenominal values are still regarded the same as in (56).

2) Case 2: External perturbations and time-varying ropelength. The rope length is online changed from 0.4 mat t = 0 s to 0.6 m at t = 5 s with an average rate of0.04 m/s. At the same time, external perturbations areexerted on the swing on purpose for three times at about6.1, 8.6, and 13.1 s, respectively; see the marked parts inFig. 8.

3) Case 3: Initial swing angle disturbances. We set the initialswing angles as θ1(0) ≈ −6.0◦, θ2(0) ≈ −3.0◦.

6416 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 10, OCTOBER 2016

Fig. 7. Results for Experiment 2–Case 1: parametric uncertainties(dashed lines: φd = π/2 rad and xd = 0.4 m).

Fig. 8. Results for Experiment 2–Case 2: external perturbations andtime-varying rope length (dashed lines: φd = π/2 rad and xd = 0.4 m).

For all the three cases, the proposed method’s control param-eters are selected the same with those used in Experiment 1. Thecurves in Figs. 7–9 depict the experimental results correspond-ing to the above three cases. It is seen, by comparing Figs. 6with 7, that the overall performances of the presented controlscheme in different system parameter settings are quite similar,

Fig. 9. Results for Experiment 2–Case 3: initial swing angle distur-bances (dashed lines: φd = π/2 rad and xd = 0.4 m).

implying its satisfactory adaptability to parametric changes (un-certainties). Further, as shown in Fig. 8, one can find that thepresented control method still performs satisfactorily even whenthe rope length is time-varying and unexpected extraneous per-turbations are incorporated. Finally, we can observe from Fig. 9that the designed method is insensitive to nonzero initial swingconditions. From the above analysis, it is evident that the pre-sented method is effective for controlling underactuated towercranes and shows satisfactory robustness.

VI. CONCLUSION

We have proposed a new adaptive control approach for towercrane systems, where parametric uncertainties are present. Com-pared with existing tower crane control methods, the proposedmethod has both theoretical and practical significance. In thetheoretical aspect, the proposed approach does not need to lin-earize the tower crane dynamical equations around the equilib-rium point or to neglect nonlinear terms, and rigorous theoreticalanalysis has been carried out based upon the full nonlinear dy-namics. On the other hand, from the practical viewpoint, the pro-posed scheme achieves superior control performance over thecomparative EI and LQR methods and shows good robustness,which has been verified by hardware experiments implementedon a self-built multifunctional crane testbed.

ACKNOWLEDGMENT

The authors would like to thank all the reviewers and the As-sociate Editor for their constructive suggestions and comments,which have greatly improved the quality of this paper.

SUN et al.: SLEW/TRANSLATION POSITIONING AND SWING SUPPRESSION FOR 4-DOF TOWER CRANES WITH PARAMETRIC 6417

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Ning Sun (S’12–M’14) received the B.S. de-gree in measurement and control technologyand instruments (Hons.) from Wuhan University,Wuhan, China, in 2009, and the Ph.D. degree incontrol theory and control engineering (Hons.)from Nankai University, Tianjin, China, in 2014.

He is currently with the Institute of Roboticsand Automatic Information Systems, NankaiUniversity. His research interests include cranes,wheeled robots, and nonlinear control with ap-plications to mechatronic systems.

Dr. Sun is a Member of the IEEE Industrial Electronics Society Tech-nical Committee on Control, Robotics, and Mechatronics and a ProgramCommittee Member for several international conferences. He receivedthe Nomination Award of the Guan Zhao-Zhi Best Paper Award at the32nd Chinese Control Conference.

Yongchun Fang (S’00–M’02–SM’08) receivedthe B.S. and M.S. degrees in control the-ory and applications from Zhejiang Univer-sity, Hangzhou, China, in 1996 and 1999,respectively, and the Ph.D. degree in electricalengineering from Clemson University, Clemson,SC, USA, in 2002.

From 2002 to 2003, he was a PostdoctoralFellow with the Sibley School of Mechanicaland Aerospace Engineering, Cornell University,Ithaca, NY, USA. He is currently a Professor with

the Institute of Robotics and Automatic Information Systems, Nankai Uni-versity, Tianjin, China. His research interests include nonlinear control,visual servoing, control of underactuated systems, and AFM-basednano-systems.

Prof. Fang is an Associate Editor of the ASME Journal of DynamicSystems, Measurement, and Control and Control Theory and Applica-tions.

He Chen received the B.S. degree in automationfrom Nankai University, Tianjin, China, in 2013,where he is currently working toward the Ph.D.degree in control theory and control engineeringin the Institute of Robotics and Automatic Infor-mation Systems.

His research interests include control ofmechatronics, overhead cranes, and wheeledmobile robots.

Biao Lu received the B.S. degree in intelligencescience and technology from Nankai University,Tianjin, China, in 2015, where he is currentlyworking toward the Ph.D. degree in control the-ory and control engineering in the Institute ofRobotics and Automatic Information Systems.

His research interests include control of over-head cranes and offshore cranes.

Yiming Fu received the B.S. degree in intel-ligence science and technology from NankaiUniversity, Tianjin, China, in 2013, where he iscurrently working toward the M.S. degree in con-trol theory and control engineering in the Insti-tute of Robotics and Automatic Information Sys-tems.

His research interests include control ofunderactuated overhead cranes and offshorecranes.