Tong Yang, Ning Sun,, He Chen,, and Yongchun Fang,

Abstract—Offshore cranes are widely applied to transfer largescale cargoes and it is challenging to develop effective control for them with sea wave disturbances. However, most existing controllers can only yield ultimate uniform boundedness or asymptotical stability results for the system’s equilibrium point, and the state variables’ convergence time cannot be theoretically guaranteed. To address these problems, a nonlinear sliding mode-based controller is suggested to accurately drive the boom/rope to their desired positions. Simultaneously, payload swing can be eliminated rapidly with sea waves. As we know, this paper firstly presents a controller by introducing error-related bounded functions into a sliding surface, which can realize boom/rope positioning within a finite time, and both controller design and analysis based on the nonlinear dynamics are implemented without any linearization manipulations. Moreover, the stability analysis is theoretically ensured with the Lyapunov method. Finally, we employ some experiments to validate the effectiveness of the proposed controller.

I. Introduction

NOWADAYS, the import and export trade of large-scale cargoes mainly depends on marine transportation. Therefore, the effective control for offshore crane systems, as a class of typical nonlinear systems [1]–[13], becomes particularly important in practical applications. Different from traditional human operations, automatic control can effectively reduce manpower costs and improve transportation efficiency.However, for nonlinear offshore cranes, their complicated mechanical structures and harsh working environments may bring many challenges for the development and analysis of automatic control strategies. More precisely, the major task of crane systems is to stabilize/regulate the payload in a prescribed position with zero or negligible residual swing by only actuating the boom and changing the rope length simultaneously, which means that one must control all independent state variables with fewer control inputs. Consequently, offshore crane systems can be considered as a kind of underactuated systems [14]–[18] and the lack of controllable variables complicates controller development to a great extent. At present, many researchers show great interests in underactuated crane systems. Especially, the research results reported in the literature about land-fixed crane systems (e.g., overhead cranes and tower cranes) are much maturer than those about offshore crane systems. The existing control methods for landfixed ones can be roughly divided into two parts: open-loop control [19]–[21] and closed-loop control [22]–[31]. In real applications, open-loop controllers can reduce the hardware(e.g., sensors) costs and avoid the impacts of inaccurate feedback information, while the closed-loop control is more effective to resist uncertain disturbances and has stronger robustness.

In most cases, it is difficult to directly apply the abovementioned control methods to offshore crane systems owing to the more complicated nonlinear coupling characteristics.Moreover, unless properly controlled, boom pitch motions are usually prone to triggering large amplitude payload swing and unexpected residual swings. In addition, offshore cranes are usually adopted on ships in the sea environments, and as a result,external disturbances resulting from sea waves must be carefully taken into account in practice. Hence, it can be seen that automatic control for offshore crane systems [32]–[34] is quite challenging and attracts much more attention of researchers in recent years.

For eliminating sea wave disturbances, Küchleret al.present a method to compensate ship motions in [35] and achieve payload positioning successfully. Similarly, [36] designs a closed-loop control law to compensate external disturbances on the basis of an adaptive observer. In addition, Liuet al.focus on the heave compensation in [37] and the control gains can be adjusted by fuzzy logics. Apart from that, some effective strategies based on the sliding mode concept are presented to accomplish accurate boom/trolley trajectory tracking and payload swing suppression. In particular, [38] proposes a control approach to achieve robust trajectory tracking and resist external disturbances. By combining fuzzy control with a sliding mode-based scheme, the performance of transportation and positioning is further improved in [39]. On the basis of feedback control, [40], [41] can tackle anti-swing and positioning tasks for offshore crane systems effectively. In addition to existing closed-loop controllers, an open-loop input shaping-based control strategy [42] is derived to solve control issues for offshore cranes by considering sea wave harmonic disturbances. Recently, some intelligent algorithms, e.g., [43],are introduced into parameter adjustments to realize payload positioning.

Although there exist many control approaches to deal with boom/rope positioning and payload swing elimination for offshore crane systems, it is easily found that some important problems deserve in-depth investigations, which are summarized as follows. 1) Additional model operations for offshore crane systems, e.g., model linearization and cascade form transformation, are widely employed to reduce the difficulty of controller design and stability analysis. However,the linearized dynamic models may ignore some important nonlinear characteristics of offshore cranes in many cases.Once the payload swing angle is far away from the equilibrium point, e.g., owing to unknown disturbances, simplified linear models will become inaccurate. Additionally, in most cases,cascade form transformation is very complicated and not suitable for offshore crane models. 2) The equilibrium point of underactuated offshore crane systems is only ultimately uniformly bounded by using some existing controllers. 3) In some existing control methods for complex nonlinear systems including offshore cranes, some additional gain constraints are needed to guarantee the closed-loop system to be stabilized at the equilibrium point, which may complicate the gain selections in real applications.

By fully considering the aforementioned issues, we design a nonlinear controller based on the original dynamics with no linearization manipulations and model transformation, to accomplish the boom/rope positioning and payload swing suppression. The proposed control method can guarantee the boom and rope to arrive at their desired positions within a finite time. Additionally, it is unnecessary to introduce complicated gain constraints for stability analysis, which greatly facilitates the application of the proposed controller to practical offshore cranes. Furthermore, by using the Lyapunov method, the convergence of the payload swing angle is strictly proven in this paper. To further improve the anti-swing performance, an elaborately designed nonlinear coupling term related to the payload swing information is introduced into the suggested control law. Finally, some experimental results validate the effectiveness and robustness of the presented sliding modebased control approach.

The remainder of this paper includes the following sections.A dynamic model of offshore crane systems is constructed in Section II. Focusing on the nonlinear dynamics, Section III provides the major control task and then presents an effective sliding mode-based control scheme. Further, the stability of the system equilibrium point is rigorously proven in Section IV.Apart from that, hardware experiments are shown in Section V. In the end, Section VI provides the summary and conclusion.

A model schematic diagram for offshore crane systems is shown in Fig. 1, whereXe-o-YeandXs-o-Ysrepresent the earth coordinate and the ship coordinate, respectively. By employing the Lagrange’s modeling technique, we obtain the corresponding dynamic equations, which can be arranged as the following matrix-vector form1In the entire paper unless otherwise claimed, and are utilized to stand for s in(m?n) and c os(m?n), respectively.:

II. Dynamics Analysis

whereq(t)=[ψ(t)l(t) ?(t)]T∈R3is the system state vector,M(q) andare the inertia matrix and the centripetal-Coriolis matrix, respectively,G(q),F(t), andf(t)∈R3denote the gravity vector, the force/torque vector,and the perturbation vector, respectively. Moreover, the corresponding specific expressions are described as follows:

withandAdditionally, some plant parameters and variables are listed in Table I,MDdenotes the product of the boom mass and the distance between the boom barycenter and the axis centero, andf1(t),f2(t),f3(t)are inevitable perturbations due to the ship roll motion,which can be expressed as

with the air resistance coefficientdf.

In practice, the payload position is usually defined in the earth-fixed frameXe-o-Ye. Hence, in the presence of ship roll movements, the two-dimensional coordinate of the payload can be expressed as

Fig. 1. Schematic diagram for offshore crane systems.

TABLE I Parameters/Variables of Offshore Cranes

It is clear that the payload positioning will be achieved when ?(t) varies synchronously with the ship roll motion, i.e.,the desired value of ?(t) is δ(t). Hence, it can be easily derived that

where (xd,yd) is the desired payload position and ψd,ld, and ?dare the desired boom pitch angle, rope length, and payload swing angle relative to the ship coordinateXs-o-Ys, respectively. To facilitate controller development and analysis, after a series of strict calculations, we find that the aforementioned problem can be solved by introducing the following variables ?1(t),?2(t), and ?3(t):

Together with (2) and (3), it is obvious that

where ?1d,?2d,?3dare constant desired values for ?1,?2,?3,respectively. Furthermore, the nonlinear dynamic model (1)can be rewritten as the following form:

where

For convenience of the following analysis, the transformed matrix-vector form (4) can be further rearranged as

Noting that the payload is usually beneath the boom tip in practice, we introduce the following assumption, as is widely done in the crane literature [19]–[43]:

Assumption 1:The payload swing angle is within the scope of ( ?π/2,π/2).

In order to facilitate controller design in next section, the dynamics about the boom pitch angle and the rope length (two actuated state variables in the proposed model (4)) can be described as

where

It is not difficult to find thatnosingularities exist in the positive definite matrixP, and the invertibility ofPcan be guaranteed for the controller design.

Remark 1:It is noted that for practical crane systems, the payload mass is usually heavier than the rope mass, and the payload always swings beneath the boom tip during the entire transportation process, which ensures the validity of Assumption 1 from the practical perspective. Meanwhile, in many crane-related literatures, e.g., [19]–[43], Assumption 1 is widely employed for controller design and stability analysis.

III. Controller Development

The main control task for offshore crane systems can be summarized as follows:

1) Realize accurate boom/rope positioning within a finite time.

2) Eliminate the residual payload swing angle rapidly.

Moreover, we can illustrate the aforementioned objectives as the following equations:

whereTris the prescribed convergence time.

In order to achieve the control task (8), we firstly introduce the following auxiliary vectors:

the vectorEabout error signals as follows:

where

Hence, a 2-order sliding surface can be designed as

whereis a constant matrix,Ci=are positive control gains,andr1(t),r2(t) are bounded functions related with positioning errorse1(t),e2(t).Then, the sliding surfacess1ands2can be described as

For the sake of guaranteeing error signals to converge to zero within the prescribed time,r1(t) andr2(t) need to satisfy the following constraints:

wherei=1,2 . Hence, we can constructr1(t) andr2(t) as the following trigonometric trajectories, respectively:

The trigonometric functions are utilized to constructr1(t)andr2(t) mainly due to their smoothness, derivability, etc.;meanwhile, it is convenient to introduce some coefficients to meet the constraints in (10). In fact, many other nonlinear trajectories, e.g., polynomial trajectories, also exhibit similar advantages and are suitable for the proposed controller, as long as the constraints in (10) are satisfied.

Next, we can take the derivative ofswith respect to time and obtains˙ as follows:

Hence, based on the presented sliding surface in (9), we can design the following control law:

whereKandk?are positive control gains. Meanwhile, the last nonlinear term (related to the swing information) is introduced into the suggested control law (14) to further enhance the anti-swing performance. Additionally, we employ a block diagram in Fig. 2 to show the entire closed-loop control system. In the subsequent stability/convergence analysis, it is proven that the closed-loop system can be stabilized at the equilibrium point as long as the control gains in (14) are positive, with no needing additional constraints for gain selections,which reduces the difficulty of the proposed method’s applications to a great extent.

Fig. 2. Block diagram of the closed-loop control system.

Remark 2:Compared with other common underactuated systems, such as inertia-wheel pendulum (IWP) systems,surface vessels, and so on, offshore cranes exhibit many similar dynamic properties. As a result, after some proper extensions, the suggested method of controller design and stability analysis may also be applied to other underactuated systems.

IV. Stability Analysis

This section will provide theoretical analysis for the convergence/stability of the suggested closed-loop system’s equilibrium point. To do so, we give the following theorem.

Theorem 1:Accurate boom/rope positioning and payload swing elimination can be achieved by utilizing the proposed controller (14). Especially, the boom and the rope can reach their desired positions within a finite timeTr, respectively,which can be expressed as the following sense:

Proof:We select the following Lyapunov function candidate:

and the derivative of (15) can be calculated as

Then, by substituting (14) into (16), it can be derived that

Sincer1(t) andr2(t) are composed of the sum of boundedr1,r2,r˙1,r˙2∈L∞, which means that trigonometric functions, it is not difficult to conclude

Then, together with the definitions ofs1ands2in (9), one can further obtain the following results:

On the basis of the elaborately constructeds(see (9)) andr(see (11)–(13)), it is obvious thats(0)=0 at the beginning of the entire control process, that is, the initial state variables of offshore crane systems are kept on the proposed sliding manifolds, i.e.,V(0)=0. Consequently, in light of the conclusion ofV(t)≥0,V˙(t)≤0, ?t≥0, we can derive the following results:

Since the matrixCis composed of positive parameters, it is easy to obtainE(t)≡R(t). Moreover, due toR(t)=0,?t≥Tr(see (11) and (12)), it can be found thatE(t)=0 after the preset timeTr, in the sense that

It is found that the position and velocity errors of the boom/rope are simultaneously restricted to a sliding surface vectors. Moreover, by elaborately designing the error-related functionsr1(t) andr2(t), both the boom pitch angle and rope length can converge to their desired values within a finite time, and the difficulty of stability analysis is reduced in this paper. Apart from that, considering the fact ofs≡0 and (17),we can conclude that

By combining the conclusions of (18) and (19), it is clear from (7) that

Then, we can take the integral of (20) in a finite time ρ to infer that

In the following analysis, we will prove the asymptotical convergence of the payload swing angle by utilizing the LaSalle’s invariance principle. From (20) and (21), we can find that ?¨3(ρ)∈L∞. Therefore, when the boom does not stop moving, the angle, angular velocity, and angular acceleration of the state variable ?3are all bounded before the preset time.

Now, we study the stability of the payload swing angle after error signalse1,e2have been completely eliminated. Based on the conclusions of, we can simplify (7) as the following form:

where ?2d∈R+represents the desired rope length. Obviously,Vp≥0 . Then, we can differentiateVp(t) and utilize (22) to derive that

It can be easily seen that

In order to accomplish the proof of Theorem 1, a set Ξ is defined as

Furthermore, Π is introduced as the largest invariant set in Ξ. In terms of (22), (23) and (24), we can find that, in the set Π,

The above-mentioned conclusion (25) indicates that the largest invariant set Π merely includes the equilibrium point of the state variable ?3. Hence, the asymptotical stability of the equilibrium point can be concluded by the LaSalle’s invariance principle. Together with (18), the conclusion of this theorem is proven.

Remark 3:It is worthwhile to point out that in this paper,both controller design and stability analysis are carried out based on the original nonlinear dynamics of offshore cranes without any linearizing operations, which is of great significance in practice. Meanwhile, compared with some existing studies, this paper does not need to transform the system dynamic model into specific forms. However, in order to make the control inputs not appear in unactuated subsystems, additional cascade transformation is implemented before controller design in, for instance, [44]–[46]. However,for the considered offshore crane systems with unexpected disturbances induced by the ship roll motion, the transform process is very complicated, if not completely impossible.

V. Experimental Verification

In order to verify the feasibility and robustness of the proposed controller (14), some hardware experiments are carried out in this section.

A. Experimental Testbed

The self-built offshore crane mechanical testbed (see Fig. 3)mainly consists of a boom, a payload, several servo motors,and some angular/displacement sensors. In particular, there are two servo motors to drive the boom and the rope,respectively, and the corresponding angular/length signals can be measured by sensors embedded within the motors.Moreover, we can use an encoder fixed above the rope to detect the payload swing angle in real time. Additionally,considering the influence of external sea waves in real applications, we build a disturbance device to imitate the ship roll motion during the control process, which is set to move along with a sinusoidal signal: δ(t)=8sin(0.2t+0.4). In order to realize real-time communications between the host computer and the hardware testbed, a control board produced by Googol Co., Ltd. is employed to transmit the measured signals of the state variables and the control inputs with the sampling period of 5 ms. Some physical parameters of the mechanical testbed are provided as follows:

Fig. 3. The self-built offshore crane mechanical testbed.

B. Experimental Results and Analysis

There are two groups of experiments to be implemented in this section. Firstly, the comparative experiments in Group 1 indicate the effectiveness of the presented control approach(14), and secondly, we introduce external disturbances to verify the robustness in Group 2. In practical applications, the initial and desired values of the boom pitch angle and the rope length can be selected arbitrarily. Without loss of generality,we choose the following values in the experiments:

Group 1:For the proposed method, the convergence timeTris chosen as 1 s and the other control parameters/gains can be set as follows:

Apart from that, we utilize the nonlinear control scheme[40] as a comparison. Hence, by careful adjustments, the involved control parameters for the method in [40] are chosen askL1= 34,kL2= 10,k1= 17,k2= 2.5,k3= 2.6,kα= 0.2,kβ=0.25,kx= 0.6, and σ =0.01.

From Fig. 4, we can see that the boom and the rope reach their desired positions within the prescribed time and there are almost no overshoots and positioning errors. Moreover, the payload swing angle can be eliminated rapidly. The experimental results above are in accordance with Theorem 1 and verify the effectiveness of the sliding mode-based control method (14). By contrast, the comparative nonlinear controller[40] exhibits inferior control performance, as shown in Fig. 5.In particular, for the comparative method, the boom and the rope move around the desired positions constantly, which may badly attenuate the operating efficiency and lead to extra energy consumption. Moreover, the convergence time of the state variables (i.e., ?1and ?2)cannotbe guaranteed by means of the comparative approach, and the residual payload swing also impacts the entire control performance of offshore crane systems.

Fig. 4. Group 1: The suggested control method (red dotted lines: the desired positions; blue solid lines: experimental results).

Group 2:In this group, we choose two types of external perturbations to validate the robustness of the proposed controller (14). Specifically, we knock the payload at the beginning and during the transportation process, respectively,to introduce unknown disturbances into the system, which are described as follows:

1) Initial Disturbances:The initial swing angle of the payload is artificially set as ? 5.1 deg.

2) Process Disturbances:The external disturbance makes the payload maximum swing amplitude reach as much as 39.4 degat 11.7 s.

The corresponding experimental results are illustrated in Figs. 6 and 7. It can be seen from Fig. 6 that the boom/rope positioning performance is not appreciably influenced by the initial perturbation. Moreover, the payload swing can be quickly suppressed. In addition, it is clear that the influences induced by external perturbations can be eliminated rapidly by means of the presented control approach, which exhibits the effective anti-swing ability when the system is disturbed, as depicted in Fig. 7.

From the above-mentioned hardware experiments, we can conclude that the suggested controller (14) performs satisfactorily in terms of positioning, swing damping, and perturbation elimination in the presence of sea waves.

VI. Conclusions

Fig. 5. Group 1: The comparative control approach in [40] (red dotted lines:the desired positions; blue solid lines: experimental results).

Fig. 6. Group 2: Initial disturbances added to the payload (red dotted lines:the desired positions; blue solid lines: experimental results).

Fig. 7. Group 2: Process disturbances added to the payload (red dotted lines: the desired positions; blue solid lines: experimental results).

For underactuated offshore crane systems, an effective sliding mode-based control method is derived to achieve precise boom/rope positioning and payload swing suppression, and simultaneously, effectively resist external disturbances induced by sea waves. Furthermore, the boom pitch angle and the rope length can converge to their desired values within a finite time based on the elaborately designed sliding surface. In addition, the stability of the equilibrium point for offshore crane systems can be proven with the Lyapunov-based stability analysis. It is worthwhile to mention that the entire process of controller design and theoretical analysis is carried out on the basis of the nonlinear offshore crane dynamics. In the end, the feasibility and robustness of the suggested controller are verified by a series of hardware experiments. For the future study, we will try our best to extend the suggested control method and take the time-variant trajectory tracking problem into account.