#### 1. Introduction

Casting cranes are the main equipment used in the steelmaking process, and they are used more and more frequently due to the increasing demand for steel. They are widely used in the casting industry to transport molten liquids in hot and dusty workshops. Since the molten metal is lifted, the casting crane must be safe, reliable, and efficient when operating. Even in the presence of various disturbances, the controller can effectively restrain the swing of the ladle and avoid the accident of ladle dumping and the threat to the safety of workers [1]. At present, few scholars at home and abroad consider the deviation of the liquid centroid when conducting research on the control of casting cranes. Therefore, the establishment of a casting crane model and controller design is difficult and challenging [24].

The other parts of this paper are organized as follows: In Section 2, the dynamic model of the casting crane with a double-pendulum casting crane based on viscous damping and radial spring damper is provided. After that, we design an enhanced coupling adaptive sliding mode controller and conduct rigorous theoretical analysis of it in Section 3. Then, simulation experiments are carried out with Simulink to verify the control performance and robustness of the proposed control method in Section 4. At last, some summarizing remarks of this article are given in Section 5.

#### 2. Dynamic Model of Casting Crane System

##### 2.1. Analysis of Liquid Centroid

During ladle transportation by casting crane, the molten metal liquid in the ladle will shake at different degrees, which will cause the position of the load centroid to change. The microdeformation of the ladle under the impact of molten steel is ignored. We only consider the two-dimensional XOY plane, as shown in Figure 1. The expression for the centroid of molten liquid is as follows:

In (1), the abscissa of the liquid centroid is expressed as xp, and its ordinate is expressed as yp. Note that this type of calculation method mainly uses the infinitesimal method, assuming that the molten liquid is divided into n parts and that each part is the same. The quality of one part is mi, .

In actual industrial production, it is difficult to accurately obtain the state of liquid sloshing and the fluctuation of the free liquid surface, let alone the mass and coordinates of each liquid unit. Therefore, it is difficult to calculate the liquid mass point by (1) coordinates. (1) is not practical for the calculation of liquid centroid.

To find a more practical way to calculate the centroid, some reasonable assumptions have been made to treat the surface of the sloshing liquid as a plane when the free surface of the sloshing liquid does not fluctuate. This particular case is shown in Figure 1.

A rectangular parallelepiped tank is selected as the load in the experiment, and its two-dimensional XOY plan is shown in Figure 1. The length of the tank is 2d, and the height is a. To facilitate observation, the blue part is represented as liquid, and the height of the liquid at rest is H. In the process of liquid sloshing, the horizontal liquid level inclines to varying degrees. The inclination angle is indicated by . Combining Figure 1 and through geometric analysis, the coordinates of the liquid centroid are as follows:

The trajectory of the liquid centroid in Figure 2 is obtained by using MATLAB software when the conditions that the length of the ladle is 160 mm, the height is 120 mm, and the sloshing angle is .

The liquid position when the liquid is at rest is indicated by a blue dashed line, and the centroid locus during the swing process is indicated by a blue arc. The centroid of the ladle load is represented by a red dot; the green dotted line is a circle with whose centroid of the ladle as the center and whose the distance from the centroid of the liquid at rest to the centroid of the ladle as radius. According to Figure 2, the blue arc and the green dotted line basically coincide, indicating that under the condition of small sloshing of the liquid surface, the motion trajectory of the liquid centroid is a calculable arc. It is worth noting that the movement of the liquid centroid exhibits a pendulum effect. Therefore, in the process of ladle transportation, the swing of the ladle is regarded as a first pendulum, and the swing of the liquid centroid is regarded as a second pendulum, which effectively transforms the complex casting crane system into a double-pendulum model.

A casting crane system is large-scale transportation equipment that mainly transports high-temperature molten liquid. To prevent the molten liquid from dumping due to the violent swing of the ladle, a radial spring-damper antisway structure is designed. During the swings of the ladle, radial tension along the rope direction is generated, which causes the spring vibrator to deform. The spring shape variable is r, and the rope length from the trolley to the ladle load is l1. In the process of liquid sloshing, there is viscous damping in the contact part between the liquid and the inner wall of the ladle, which is recorded as C2. Overall, this paper establishes a mechanical model of a double-pendulum casting crane based on viscous damping and radial spring damper, as shown in Figure 3.

u is the resultant force imposed on the trolley in the X direction, the friction between the trolley and the rail is f, and the mass of the trolley is m. The cable length is l1, the spring shape variable is r, and the ladle swing angle is . The equivalent cable length is l2, and the liquid sloshing swing angle is φ. The ladle mass is m1, and the liquid mass is m2. In addition, the radial spring damper consists of a spring oscillator Kr and a damping coefficient C1. The viscous damping between the liquid and the inner wall of the ladle is C2. Assuming that the ladle is a rigid rectangular tank, ignoring its deformation, the length of the tank is 2d, the height of the tank is a, and the height of the liquid at rest is H.

##### 2.2. System Dynamics Equation

In the casting crane system, the control input is only the trolley driving force u, and the state variables that need to be controlled are the trolley displacement x, the ladle swing angle θ, the liquid sloshing swing angle φ, and the deformation variable r. It can be seen that the input quantity is less than the state vectors, which obviously shows that the double-pendulum casting crane based on viscous damping and radial spring damper is a typical underdriven system. For this kind of multivariable and strongly coupled nonlinear system, not only is the trolley required to reach the target position smoothly but also the swing angles are required to be fully suppressed. This greatly increases the control difficulty of the system. To improve controller performance, the Lagrange’s equation is used to establish the dynamic equation of the casting crane system. The general expression is as follows:

In (3), L denotes the Lagrange operator, qk denotes the state variable, denotes the first derivative of the state variable with respect to time, k = 1, …, n (n denotes the number of state quantities), Qc denotes the consuming energy of liquid damping, and Fk denotes the generalized force.

Establishing a two-dimensional XOY rectangular coordinate system, as shown in Figure 3, the position coordinates of the trolley are (x, 0), the position coordinates of the ladle centroid m1 are (x1, y1), and the position coordinates of the liquid centroid m2 are (x2, y2). Then, according to the geometric relationship:

Then, we calculate the first derivative of (4) with respect to time to obtain the velocity of movement of the ladle centroid m1 and the liquid centroid m2:

The force analysis of the system is further carried out, and the total kinetic energy of the casting crane system is expressed as follows:

When the casting crane is not in operation, the horizontal plane where the centroid m2 is located in a zero potential energy surface. We analyze the potential energy of the double-pendulum crane system, which can be written as follows:

The consuming energy of the system is mainly the consumption of the damper, so the consuming energy of the casting crane system is as follows:

According to (6) and (7), the Lagrange operator is as follows:

The state vector of the casting crane system is q = [x, θ, φ, r]T. According to (3) and (4), the following dynamic equations describing the underactuated double-pendulum casting crane system can be obtained by using the Lagrange equation under four generalized coordinates:where , represents the resistance of the spring damper, , are the viscous damping force of the liquid and the air resistance of the ladle, with denoting the corresponding coefficients.

Considering that the trolley is simultaneously affected by Coulomb friction and viscous friction, the nonlinear friction model is expressed as follows:where are the corresponding friction factors.

For the convenience of subsequent analysis, the dynamic equation of the system is transformed into the matrix form of the system state:

The expressions of each item are as follows:

, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .

For the underactuated double-pendulum casting crane system shown in equation (15), the two important properties are as follows:

Property 1. is a positive definite symmetric matrix.

Property 2. is an oblique symmetric matrix. We can obtain thatBefore the follow-up analysis, we make the following assumptions.

Assumption 1. The sloshing angle of the ladle and the liquid is always within the following range:

Assumption 2. The initial cable length l1 and the deformation variable r are in a reasonable range:

#### 3. Controller Design and Stability Analysis

In this section, an enhanced coupled adaptive sliding mode control (ECASMC) is designed for the double-pendulum casting crane model based on viscous damping and radial spring damper. In addition, using the Lyapunov theorem and LaSalle's invariance principle, the asymptotic stability of the system at the desired equilibrium point is strictly proven. The controller design flow chart is shown in Figure 4.

It can be clearly seen from Figure 4 that the entire control framework is mainly composed of four parts: the structure of the composite signal, the design of the enhanced coupling adaptive sliding mode controller, the establishment of the double-pendulum casting crane system, and various disturbances. Specifically, a new coupling variable ε is obtained through the nonlinear coupling between the target value and the output value firstly. From this, a composite signal eε containing each state vector is obtained. Then, the composite signal and a novel adaptive quantity are used to construct the sliding mode surface. Next, the driving force u of the system is obtained by combining the coupling variable and the adaptive sliding surface, which will drive the trolley to move reasonably to achieve the control goal of antiswing positioning. Finally, three disturbances factors (external wind disturbance system, coefficient disturbance, and nonzero initial swing angle disturbance) are designed to test the robustness of the casting crane system.

##### 3.1. Controller Design

As is well known, the control objective of casting cranes is to achieve precise trolley positioning, fast payload, and liquid swing angles elimination. Due to the underactuated nature of the casting crane system, ladle swing and liquid sloshing can only be eliminated by the control of the trolley movement. Unfortunately, it is actually difficult to ensure the stability of the system and, at the same time, introduce enough swing information into the closed-loop controller. Therefore, many existing control methods are mainly based on displacement. However, if there is little (or no) swing feedback, sloshing feedback, and spring deformation feedback in the driving force, the trolley cannot effectively react to ladle swinging and liquid sloshing. In other words, the ladle swinging and liquid sloshing can only be attenuated between x, r, θ, and φ through the internal couplings of the system, which leads to undesirable transient control performance and robustness.

To address the previously mentioned problems, the composite signals are constructed as follows:where is the undetermined parameter, xd denotes the desired trolley position, and are parameters to be determined. The merit of the signal is that it can reflect the four objectives of the double-pendulum casting crane. To facilitate the description, we define as follows:where rd is the desired spring deformation and er is the deviation of spring deformation. Subsequently, we intend to design a controller to drive eε toward 0. After numerous experiments and massive analyses by predecessors, we design the following:where k1 > 0 is a control parameter to be determined. For the convenience of subsequent calculations, we define the following:

The controller will be introduced in the subsequent analysis that, based on the composite deviation signal and sliding mode control in (19). Different from the traditional sliding mode control, this method can enhance the coupling of each state vector and the sliding surface and improve the control output's constraint performance for each state vector. Finally, the underactuated characteristics of the casting crane are weakened. Moreover, adaptive technology is used to adaptively adjust the parameters of the sliding surface to accomplish the dynamic adjustment of the sliding surface. Based on the enhanced coupling signal , a new type of sliding surface is designed as follows:where λ is the sliding mode control parameter, and S is derived with regard to time as follows:

(22) shows that to obtain good control performance (especially the suppression of swing angle and spring deformation), the sliding surface needs to be dynamically adjusted. In practice, manual adjustment may deteriorate the transient performance and result in unpredictable difficulties. To deal with this drawback, we introduce an adaptive equation to dynamically adjust the sliding surface parameter λ, so that only its approximate value of the sliding surface parameter λ is needed for controller design. We define the following deviation signal:where is the preliminary estimation for λ, and is the deviation between the set value λ and the estimated value . Correspondingly, one further defines the following:

Based on the proposed sliding surface, the following controller and adaptive equation are designed:where are positive control gains. To overcome the chattering during the switching process of the sliding mode surface caused by the discontinuity of the sign function, a continuous switching function is designed as follows:

After a series of sliding surface designs and related algebraic calculations, the driving force u of the casting crane system is obtained. The controller contains the feedback information of the four-state quantities of x, θ, φ, and r, which can improve transient performance. In addition, by dynamically adjusting the parameters of the sliding mode controller through adaptive technology, the controller has strong dynamic performance, and the system can quickly and accurately locate and effectively suppress the swing angle. The stability of the ECASMC will be rigorously proofed in the subsequent analysis by using LaSalle's invariance principle and Lyapunov theorem.

##### 3.2. Stability Analysis

Theorem 1. For the double-pendulum casting crane system (10)–(13), the designed control inputs (24) ensure that the closed-loop system is asymptotically stable at the equilibrium point, which can be mathematically expressed as follows:

Proof of Theorem 1. The proof process is shown in Figure 5. Specifically, we select an appropriate positive definite Lyapunov function and ensure that its first derivative is seminegative definite. Then, we prove that the closed-loop signal is bounded and analyze the final trend of the set based on the invariant set. Finally, LaSalle's invariance principle is used to complete the proof of asymptotic stability.
The Lyapunov candidate function is designed as follows [4]:Differentiating V(t) with regard to time yield and substituting (26) into it produces the following result:To simplify (31), the second derivative of w in (20) is obtained:In addition, comparing (32) with (10) on the basis of considering the uncertain system parameters, we can conclude that:where is the amount of influence of parameter perturbation on auxiliary coupling signal .
Combining with equation (30) yields:Then, we can substitute into (31) to yield that:Substituting the system driving force u of (27) into (35) implies thatAfter analysis, , is negative semidefinite. Hence, is always satisfied. It follows from equations (16), (17), (19), (20), and (36) thatThen, based on (37), each state vector is bounded. It is necessary to further use invariant-set-based analysis to complete the proof.
To this end, this paper uses LaSalle's invariance principle to prove that each state vector eventually converges to the largest invariant set over time. The set is defined as follows:It can be inferred that the largest invariant set satisfies the following:Using equations (20), (21), and (36) can be further organized as follows:Then, we can conclude from equations (17), (24), (36), and (40) thatSubstituting into the friction (14), we have the following:Then, we will prove the fact that is also asymptotically stable. Substituting into equations (11), (12), and (13), we can derive thatAccording to Assumption 2 combined with (43), we can obtain thatSubstitute into to get the following:Based on , we combine equations (37), (42), and (43) to obtain the following:In summary, gathering the results in (39), (43), and (44) show that the state quantities of the casting crane system eventually converge to the largest invariant set as time changes. Hence, according to the LaSalle’s invariance theorem, Theorem 1 is successfully proven.

#### 4. System Simulation and Analysis

In this section, some simulation tests are carried out to verify the performance of the ECASMC method.

The system parameters for the testbed are configured as m = 6.5 kg, m1 = 2 kg, m2 = 0.6 kg, l1 = 0.53 m, l2 = 0.2 m  = 9.8 m/s2.

Unless otherwise specified, each state variable at the initial time are set to x(0) = 0 m, θ(0) =0°, φ(0) =.

The friction coefficients between the trolley and the rail are as follows [33]:

fr = 4.4, εx = 0.01, krx = −0.5.

The physical parameters of the spring and damping coefficient are shown in Table 1. Then according to reference [11], we obtain ξ = 0.3296, so the parameters of the spring damper can be further obtained: elastic coefficient Kr = 11.21 and damping coefficient C1 = 1.788. For the air resistance model and the sloshing viscous model, the coefficient of air damping d1 = 0.03 [4] and coefficient of viscous damping C2 = 0.75 [9].

To comprehensively verify the control performance of the ECASMC controller, several experimental tests are elaborately performed. In detail, we first compare the proposed controller with two existing controllers. Then, we further test the robust performance of the proposed controller. Specifically, the robustness of the proposed controller with respect to external disturbances, nonzero initial conditions, and parameters uncertainties is verified.

##### 4.1. Comparison with Existing Methods

Experiment 1. For literature completeness, the expression of the CSMC controller [32] and VP-TVSMC controller [3] is provided as follows:

###### 4.1.1. CSMC Controller

where γ, α, and β are the sliding mode surface control gains. To overcome the chattering during the switching process of the sliding mode surface, a saturation function sat(s) is designed. Then, the definition of a sliding surface can be expressed as follows:

###### 4.1.2. VP-TVSMC Method

where μ1, μ2, and μ3∈ℝ+ are the control gains for the sliding mode controller. λ1, λ2, and λ3∈ℝ+ are positive time-varying function coefficients. Therefore, the driving force of the system can be obtained as follows:where and ξ∈ℝ+ are the sliding mode switching gains.

Without loss of generality, the control gains for the three controllers are elaborately tuned by quantum particle swarm optimization [34]. After a lot of off-line optimization experiments, the parameters of three controllers are shown in Table 2:

The first group of experimental results is recorded in Figure 6. One can see that the ECASMC method achieves precise trolley positioning and fast swing elimination. Specifically, it can be seen from Figure 6(a) that the designed adaptive law can accurately predict the sliding mode controller parameter and dynamically adjust around the target parameter λ = 1.1. In addition, it can be seen from Figure 6(b) that three methods can achieve accurate trolley positioning and reaches the desired location within about 6 s. Further observation of Figures 6(c)6(e) shows that the proposed ECASMC method has better performance than the comparative methods in the sense of swing suppression and spring deformation. Specifically, in the above three types of controls, the lower swing angles θ and φ were suppressed in about 7 s, but the maximum amplitude of θ of ECASMC reached 0.97°, and the maximum amplitude of φ reached 0.71°; the maximum amplitude of θ of CSMC and VP-TVSMC is 2.22° and 1.99°, and the maximum amplitude of φ is 1.22° and 1.19°, respectively. In addition, the control effect of ECASMC is significant in terms of the spring deformation r; the maximum deformation is 0.355 mm, while the maximum deformations of CSMC and VP-TVSMC are 0.999 mm and 0.905 mm, respectively. It can be seen from Figure 6(f) that the ECASMC method can not only effectively reduce the swing angle and spring deformation but also effectively reduce the driving force output.

##### 4.2. Robustness Verification

To further verify the antidisturbance ability of the proposed method against three kinds of external disturbances, the following experiments are considered in this simulation.

Experiment 2. External disturbances. We intentionally exert the following three kinds of disturbances on the payload:(1)Between 12 and 14 s, a sine wave perturbation is induced, with a relative amplitude of 30% and a frequency of 0.5 Hz.(2)Between 19 and 19.5 s, a pulse disturbances is added, with a relative amplitude of 30% and a pulse width of 0.5 s.(3)Between 24 and 26 s, random disturbances are added, with a relative amplitude of 30%.The relative amplitude is defined as the percentage of the swing angle disturbances amplitude to the maximum swing amplitude of the load. The ECASMC system control parameters are chosen the same as those in Table 2.
The results of Experiment 2 are shown in Figure 7. It can be clearly found that the method proposed in this paper can quickly attenuate external disturbances. In other words, the casting crane system can still reach a convergent state during the continuous action of external disturbances. The system can quickly return to the target position, effectively attenuating the residual swing angle of the load and the residual deformation of the spring. The results show that the ECASMC method has good control performance even in the condition of various external disturbances.

Experiment 3. Nonzero initial conditions. Initial swing disturbance is intentionally exerted on the ladle and liquid to disturb the casting crane system.
The simulation result of Experiment 3 is shown in Figure 8. When the swing angles θ and φ both have initial angles, the trolley can still achieve rapid and precise positioning, and its convergence time is 4.7 s. In addition, the swing angle θ can still quickly converge to 0° within 6 s, and the swing angle remains within 2.45° when the initial swing angle is 2°. Similarly, the swing angle φ can still quickly converge to 0° within 6.9 s, and the swing angle remains within 1.51° when the initial swing angle is 1°. For the spring deformation, the initial angle disturbance has a greater impact on the deformation r, and the maximum deformation length is 1.5 mm, which is 1.145 mm longer than that without disturbance. Under the inertia of the initial swing angle, the driving force of the trolley is significantly reduced to 7.92 N. Moreover, one can see that the initial ladle swing and liquid swing hardly influence the overall control performance of the casting crane systems.

Experiment 4. Parameter uncertainties. The payload mass is changed from 6.5 kg to 7 kg, and the cable length is changed from 0.53 m to 0.6 m abruptly at t = 3s.
It can be concluded from Figure 9 that even when some system parameters are uncertain, the proposed ECASMC method can still achieve precise trolley positioning, suppress the ladle swing, and effectively reduce spring deformation. Therefore, the proposed control method is not sensitive to these parameter uncertainties.

#### 5. Conclusions

Aiming at the antiswing and positioning problem of a casting crane system, we propose a double-pendulum casting crane model based on viscous damping and radial spring damper and an enhanced coupled adaptive sliding mode control method. To improve the control performance and enhance the robustness, more feedback information is introduced into the controller. To reduce the chattering of the system, a continuous switching function is designed to ensure the continuity of the switching process, and an adaptive parameter is designed for the sliding mode surface to realize dynamic adjustment. Under the strict proof of Lyapunov theorem and LaSalle's invariance principle, the system is asymptotically stable at the desired equilibrium point. Compared with the existing CSMC method and VP-TVSMC method, the ECASMC method proposed in this paper has better control performance and stronger robustness. Since it is difficult to measure the liquid pendulum angle in real time at this stage, in the future, we will focus on studying the state observation technology to estimate the state vector and improve the controller to improve the robustness against parameter uncertainty.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 51505154 and 51437005) in part, the Science and Technology Project of Jiangmen City (Grant nos. 2020JC01035 and 2019JC01005) in part, and the Teaching Quality Engineering and Teaching Reform Project in Guangdong Province (Grant no. GDJX2019012).