Fuzzy Sliding Mode Control on Positioning and Anti-swing for Overhead Crane

This paper proposes a novel positioning and anti-swing controller based on fuzzy sliding mode for an overhead crane. For the underactuated characteristics of the overhead crane, the trolley displacement and load swing angle are integrated into the same sliding mode surface. The boundness and convergence of each state on the sliding mode surface are analyzed. Appropriate fuzzy rules are introduced to adjust the control quantity to ensure that the system state is always on this surface. The sliding mode function is used to replace the tracking error signal to reduce the dependence of the control method on the system model, so that the control system has better robustness to parameter changes and external disturbances. The experiment and results analysis show that this proposed method can effectively deal with the dynamic deviation of the system model and external random disturbances. The load swing has a good suppression effect and the positioning of the trolley can be accomplished quickly and accurately. The proposed positioning and anti-swing controller based on fuzzy sliding mode is able to provide the high-quality and fast transportation of overhead cranes.


Introduction
Overhead cranes are often affected by adverse climate and environmental factors in practical applications, such as strong wind, heavy fog, sand and dust, etc. Similarly, the control systems of overhead cranes are largely subject to these unpredictable external disturbances [1][2][3][4]. Moreover, the control systems contain some unmodeled dynamics that are hard to be accurately described by mathematical model, which will also make it difficult for the conventional control methods on overhead cranes to have good performance [5][6][7][8]. As the special equipment, the overhead crane must properly deal with the above problems for its safe operation, otherwise it may lead to the instability of the control system and serious accidents.
The conventional controllers design for overhead cranes, whether open-loop methods or closed-loop methods, don't comprehensively consider the influence of unknown parameters and external interference on the control systems. At present, many researchers begin to study the model parameter independent control method of bridge crane, and make it have strong robustness and good resistance to unknown variables and external disturbances. Such methods mainly include sliding mode control, fuzzy control, genetic algorithm, neural network and so on.
For the different dimension modelling on overhead cranes, Yang et al. [9] propose the novel adaptive method applied to overcome the influence of unknown parameters and external disturbs. Moreover, this method can realize the positioning and an-swing control for overhead cranes. Ngo et al. [10] adopt the idea of the slide mode dimension reduction to propose a time-varying adaptive an-swing controller, which can alleviate the chattering in conventional sliding mode control strategy to some extend. Tuan et at. [11] introduce the adaptive function into the fuzzy sliding mode control to design an adaptive controller for overhead cranes based on the fuzzy sliding mode. They provide the experiments to verify this method's effectiveness. Lee [12] takes the horizontal coordinate of the load as the fuzzy 1 3 variable, and realizes the anti swing positioning control of the crane by reformulating the fuzzy rules. Sato, et al. [13] adopt a container mass estimation method when a crane system performs rolling up control. The observer parameter can be selected using the estimated mass value. A robust antisway control is applied in crane parallel shift control even when there is a wind disturbance. A wind disturbance observer is designed and a wind disturbance estimator is used to separate the friction observer output from the wind disturbance observer output. For two-dimension underactuated overhead crane systems, Zhang et al. [14] provide an adaptive proportional-derivative sliding mode control(APD-SMC) method. The new idea involves that the PD control part is used to stabilize the controlled system, the SMC part is used to compensate the external disturbances and system uncertainties, and the adaptive control part is utilized to estimate the unknown system parameters. Hamdy et al. [15] provide an experimental verification of a hybrid partial feedback linearization (PFL) and deadbeat (DB) control scheme with chaotic whale optimization algorithm (CWOA) for a nonlinear gantry crane (GC) system. The CWOA is used to tune the controller parameters. A sliding-mode observer (SMO) is utilized to estimate the unmeasured states. Using this hybrid scheme, a better payload sway elimination can be obtained. Shen et al. [16] presents a robust passivity-based adaptive control method for payload trajectory tracking of a three degree-of-freedom overhead crane with a varyinglength rigid hoist cable. The Passivity Theorem is used to prove that the closed-loop system is input Coutput stable when an OSP negative feedback controller is implemented.
Artificial intelligence and machine learning are also used in the studies on non-model control strategy for overhead cranes. As a classical evolutionary algorithm, genetic algorithm has a good ability of global optimization. At the same time, genetic algorithm also has the advantages of simple principle and operation, strong universality and unrestricted constraints. In [17], the optimal control problem of crane is solved by genetic algorithm, which effectively enhances the robustness of the system. Because neural network control has the characteristics of high operation speed, strong adaptability, strong fault tolerance and self-organization ability, neural network control has been widely used in the antiswing control of overhead cranes. In [18], NASA provide and emphasis on the compensation for the uncertainties and unknown disturbances of the system model by the omnidirectional automatic optimization ability of neural network. In contrast, the sliding mode control method itself has strong anti-interference ability and can effectively deal with unmodeled dynamics and disturbances. For the full drive system, it is not difficult to design the method to stabilize each state of the system to a given sliding surface. However, for the underactuated overhead crane systems, how to construct an appropriate sliding surface has become the main difficulty of this method. When designing the sliding mode surface, the trolley displacement and load swing angle must be integrated into the same sliding mode surface. Because there is more than one variable on the sliding mode surface, it is difficult to analyze the boundedness and convergence of each state on the sliding mode surface. Several scholars have conducted their study on the sliding mode control for overhead crane systems, and designed different control laws for different types of sliding mode surfaces. Xu et al. [19] propose a multi-sliding mode controller to solve the problem of antiswing and positioning controller for an overhead crane with rope length variations. This controller can provide a simultaneous trolley-position regulation, sway suppression, and load hoisting control. Tuan et al. [20] proposes the control method combining partial feedback linearization and sliding mode control, in which the anti-sway control based on PFL reduces the vibration of the load in two directions, and the sliding mode control accurately tracks the crane and trolley and completes the lifting of the load at the same time. Chwa [21] designs a robust limited time anti sway tracking control method based on sliding mode control by introducing the finite time stability method of sliding surface and deflection rate dependent on position tracking error, and verifies these methods effectiveness including position tracking control, anti-sway control and anti-deflection control respectively. Wang et al. [22] present a novel dynamic sliding mode variable structure control algorithm; this proposed controller can ensure global stability through improved design of sliding surface for the drive control force, and obtain the continuous driving control force in time domain.
The positioning and anti-swing control method based on fuzzy sliding mode control for overhead cranes is proposed in this paper. The sliding mode surface of the nonlinear and underactuated system for the overhead crane is constructed hierarchically. Combined with the variable universe fuzzy control method, a new nonlinear controller is designed by using new fuzzy rules to ensure that the system state is always on the manifold surface. Then a series of simulation experiments are carried out to verify the effectiveness of the proposed method. The results show that it can achieve better control effect than the existing methods.

Dynamic Model of Bridge Crane and Load generalized Motion Analysis
The transport load of the bridge crane mainly depends on the action of the crane, trolley and lifting rope, so the threedimensional mathematical model of the bridge crane with five degrees of freedom is established [23]. The two degree of freedom swing angle in the model is determined by the increase (decrease) speed and rope length of the crane and trolley, and the motion of the crane and trolley is in decoupling state, so it is only necessary to study the motion in one direction, and the control law in the other direction is the same. In the twodimensional coordinate system, establish a simplified bridge crane model, as shown in Fig. 1, in which, m and M represent the mass of trolley and load respectively; F represents driving force of trolley; l is the length of lifting rope and is the load swing angle. In order to ensure the safety of transportation, the length of the lifting rope is not changed in the process of load transportation. Based on the simplified model in Fig. 1 and the Lagrangian dynamic equation [24][25][26] in generalized coordinates, the following two-dimensional bridge crane mathematical model with fixed rope length is established as follow: where, x represents the horizontal displacement of the trolley; ̇ and ̈ represent the angular velocity and acceleration of the load swing respectively; g represents the acceleration of gravity.
By sorting out the second equation in (1), it is obtained as follow: For the convenience of controller design, the dynamic model (1) of crane is expressed as follows: T is state vector; (1) According to Fig. 1, the horizontal displacement of the load is as follow: where, x m is horizontal displacement of load. Equation (4) can be seen as the generalized displacement, which includes the displacement of the driven trolley, x(t), and can also reflect the load swing without driving, (t) . Derivation (4), it is obtained as follow: where, the load swing angle (t) ∈ (− ∕2, ∕2) . The transformation model (3) can express the energy of the system directly and provide convenience for the construction of new energy function. The horizontal displacement of the load includes the coupling information of the trolley position and load swing angle, which provides conditions for the design of the controller.

Fundamentals of Sliding Mode Variable Structure Control
As a special type of nonlinear control, the nonlinearity of variable structure control is a discontinuous switching characteristic, which forces the control process to be discontinuous. The formation of the sliding mode of the system is closely related to this characteristic, which makes the system move back and forth along the path of a specific state with small amplitude and high frequency. Therefore, variable structure control is also called sliding mode variable structure control. Because the sliding mode can be designed and is not affected by system parameters and external disturbances, sliding mode variable structure control has the advantages of fast response, insensitive to disturbances and parameter changes, strong robustness and easy implementation.

The Principle Sliding Mode
In general, the state space of the system is as follow: ⋯ , x n ) = 0 is defined. This switching surface divides the above state space into two In sliding mode control, the above first two items (1) and (2) have no value, while the third item (3) has special value. If all points in a certain range on the switching surface operate according to the item (3), as long as other moving points are close to the range, they will slide towards the range. Therefore, it is known that all points on the switching surface s = 0 are in C point area, which are "sliding mode" area. The motion of the system in the sliding mode region is called "sliding mode motion". According to the requirement that all points on the sliding mode must be the termination point, when the moving point is near the switching surface s = 0 , it is obtained as follow: and transforming the form of (6), it can reformed as follow: The inequalities (7) presents the necessary condition for the system in the form of Lyapunov function as follow.
Equation (8) is positive definite in the range of switching surface. It can be seen from (7) that the derivative of s 2 is semi-negative definite. Therefore, if the system meets the requirements of (7), it is stable under the condition s = 0 , and (8) is a conditional Lyapunov function of the system.

S u p p o s e d t h a t t h e r e i s a c o n t r o l s yst e m ,
The control function is obtained by determining the switching function s(x), as follow: where, u + ≠ u − . Making the system control as sliding mode variable structure control, the following three basic conditions must be satisfied.
(1) The function equation (9) is ensured to hold, that is, the sliding mode of the system exists. In this paper, A novel control scheme based on sliding mode variable structure is proposed for the overhead cranes. The sliding mode functions of position control and swing angle control are defined respectively. The synthetic sliding mode surface is designed to reduce the chattering phenomenon in the conventional variable structure control methods as much as possible. When the condition of sliding mode control is ensured, variable universe fuzzy control is introduced. The appropriate fuzzy rules are designed according with analysis on the overhead crane model. The proposed method in this paper can compensate the control force and weaken the chattering phenomenon of sliding mode control.

The Adaptive Fuzzy Sliding Mode Control
The adaptive fuzzy sliding mode control is able to closely integrate the advantages of conventional sliding mode variable structure control and fuzzy control. The design of adaptive fuzzy sliding mode control system has the lowest dependence on the system model, and has strong robustness to external disturbances and changes of system parameters [27][28][29]. Fuzzy control can optimize and improve the sliding mode control signal and reduce the occurrence of chattering. At the same time, sliding mode control can optimize the fuzzy structure and simplify the problems with many fuzzy rules. In order to solve the disturbance to the system caused by external unknown disturbance and uncertain parameters, meet the requirements for the existence of sliding mode, and greatly reduce the vibration of sliding mode control, the fuzzy control rules is employed to adjust the magnitude of the control quantity u to ensure the condition ṡs < 0 . At the same time, sliding mode control has two effects on the design of fuzzy control: the control quantity of fuzzy control is changed from tracking error to sliding mode function, and the tracking error can achieve the same effect by controlling the sliding mode function s; for high-order systems, compared with traditional fuzzy control, the input [s,̇s] of fuzzy sliding mode control is always two-dimensional.
Definition: r is the reference input of the control system, y is the output of the control system and e = r − y is the deviation value. The sliding mode function can be defined as follow: The two-dimensional fuzzy controller with the input s,̇s and output u is established. The domains are defined as The fuzzy subset on the domain Z is C 1 , C 2 , ..., C 7 . These subsets correspond to the language values of "negative large, negative medium, negative small, zero, positive small, positive medium, positive large", or recorded as "NB, NM, NS, ZO, PS, PM, PB". Equidistant triangular membership function is adopted. The fuzzy reasoning rules are as follows: In order to make the sliding mode control satisfy the inequality ṡs < 0 , the fuzzy control rule table is designed, shown in Table 1.

Design of Controller on Positioning and Anti-swing for Overhead Cranes
The structure diagram of the fuzzy sliding mode control system designed in this paper in the x-axis direction is shown in Fig. 3, in which, xd ∈ R and dx ∈ R are the desired trolley position and swing angle respectively; x ∈ R and x ∈ R are the current trolley position and swing angle respectively. The trolley position tracking error is defined as e px = x d − x ∈ R and the trolley swing angle error is defined as e The sliding mode functions for position control and angle control are defined as: Combining equations (12), the synthetic slip surfaces are defined as: When the adaptive fuzzy controller based on variable universe is introduced [30,31], the input of the controller is obtained as:  In order to meet the necessary condition, ṡs < 0 , and ensure to reach the sliding surface, the symbols on both sides of the sliding surface should be controlled to be opposite. At the same time, the distance between the state quantity and the sliding mode surface should be proportional to the amplitude of the control quantity. The domains X and Y of the slide mode functions s and ̇s are initiated be X = [−1, 1] and Y = [−1, 1] . The output domain is initiated as Z = [−1, 1] . According to the conditions that meet the sliding modal characteristics, the fuzzy rules are designed, shown in Table 2.
The values within Table 2 are the peaks values of the fuzzy set on the output domain. Each of them corresponds to uij. The membership function of "triangle type" is selected as fuzzy set, and equidistant division is made on the domain, as shown in Fig. 4. Table 2 is designed on the premise that the fuzzy rules satisfy the sufficient necessary condition for sliding mode control. Therefore, we designed a fuzzy sliding mode control system that is stable. According to the fuzzy rule table and analysis on the fuzzy rules, it can be seen that the control output is zero (if sisPBandisNB, thenuisZO) when ṡs < 0 , which conforms to the expectation of sliding mode control. If s > 0 and ̇s > 0 , ṡs > 0 . In order to reduce ṡs rapidly, it is necessary to input a large positive control quantity (if sisPBandisPB, thenuisPB); When s < 0 and ̇s < 0 , ṡs > 0 . Therefore, it is necessary to input a large negative control quantity to reduce ṡs rapidly (if sisNBandisNB, thenuisNB).
Only when the necessary and sufficient conditions of sliding mode control are met, can the designed fuzzy sliding mode control system always be stable. No matter what conditions, the system state can be quickly reached the sliding mode surface. And the states finally reach the stable point of the system along the sliding surface. All fuzzy rules in Table 2 are designed to meet the condition ṡs < 0.

Experimental and Results Analysis
The experimental platform of the bridge crane, shown in Fig. 5, is built according to the practical application of the bridge crane. The hardware system includes: bridge mechanical structure, trolley, cart, guide rail, DC servo motor, servo driver, cable, load, motion controller, computer, coder and other devices.  The components in the platform are connected with the cables, power lines and signal lines to ensure the normal data transmission and realize the coordinated operation. The motion controller and servo driver are integrated in the control box. As the core central equipment of the platform system, the control box is powered by 220V AC, communicates with the computer with the cable, and completes the three degrees of freedom driving execution and displacement, deflection angle and rope length information feedback with the bridge crane motion by the signal lines, as shown in Fig. 6. The experimental parameters are set in Table 2. The proposed method is compared with the conventional sliding mode control method. The results is shown in Fig. 7.
In Fig. 7a, both the conventional sliding mode control method and the proposed fuzzy sliding mode control method can achieve accurate positioning function. Compared with the conventional sliding mode control method, the proposed fuzzy sliding mode control can be completed about one second in advance. In Fig. 7b, the conventional sliding mode control method and the proposed fuzzy sliding mode control method can control the swing angle to zero degree almost at the same time. However, compared with the conventional sliding mode control method, the proposed fuzzy sliding    (Table 3). In order to verify the anti-disturbance ability and robustness of the proposed method, under the condition of constant control parameters, when t = 1s , the external disturbance of swing angle is added, shown in Fig. 8. The curves of displacement and swing angle with disturbed and undisturbed are shown in Fig. 9. It is obviously that, the proposed fuzzy sliding mode control method has a small change in displacement (shown in Fig. 9b) when the external disturbance of swing angle is added, and can quickly respond to the specified position. In Fig. 9b, when the disturbance occurs, the load swing angle suddenly increases but the the proposed fuzzy sliding mode control method can quickly suppress the swing angle and quickly approach the balance state. The above experimental results shows that the proposed fuzzy sliding mode control method in this paper has the better performance of positioning and anti-swing as well as the good robustness.

Conclusion
The traditional controller used to solve the positioning and anti swing of bridge crane does not comprehensively consider the influence of unknown parameters and external disturbances on the system. The sliding mode control method has strong anti-disturbance ability and can effectively deal with the dynamic offset of system model and external random disturbances. For the full drive system, it is not difficult to design the method to stabilize each state of the system on the given sliding mode surface. However, for the underactuated overhead crane system, the trolley displacement and load swing angle must be considered as an integration for the same sliding mode surface. Since there is more than one variable on its sliding surface, the problems on the boundness and convergence of states on the sliding mode surface must be resolved. The chattering phenomenon of sliding mode control is also a problem that must be confronted.
In this paper, the fuzzy control method is introduced to the traditional sliding mode controller on positioning and anti-swing for overhead cranes. The fuzzy control rules are designed to adjust the control quantity so that the control system of the overhead crane can not only weaken the chattering phenomenon coming from the conventional sliding mode control, but also make the control system more robustness to the changes of parameters and external disturbances. The comparative analysis of experiments' results shows that this method has better control effect on the control of positioning and anti swing for the overhead crane. The proposed method not only improves the response speed of the system, but also ensures the high-quality and high-speed transportation of the overhead crane.
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Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Qianqian Zhang received the B.S. degree from Henan University of Science and Technology, Luoyang, China, the M.S. degree from Henan University of Science and Technology, Luoyang, China, all in electrical engineering and Computer Science and Technology. Her current research interests include Stability Analysis on Complex system, underactuated modeling, detection and control.