Cranes are normally utilized to lift or move cargo in the field of construction, logistics, the manufacturing industry, etc. Cranes come in a variety of forms, and most cranes work by utilizing wire ropes to suspend cargo. With crane motion or external disturbances, cargo will sway owing to the flexibility of the wire ropes, and active and passive methods are taken to carry the cargo to the desired position. The swaying of the cargo is a kind of simple harmonic vibration. When the cargo carried by the cranes begins to sway, it needs several periods to be suppressed, which is time-consuming and even lead to working efficiency problems. Therefore, the swaying problem has become an urgent subject, especially for the port industry. More than that, the swaying problem will cause safety problems when carrying cargo of great mass, and it may even influence the design of the crane structure greatly.

Anti-swaying technology for cranes has been studied in the last 20 years. Attention was paid to both single-pendulum and double-pendulum effect cranes by using mechanical and automatic anti-swaying technology. When the mass of the spreader for the crane is big enough, the double-pendulum effect of the cranes cannot be neglected. In order to eliminate this double-pendulum effect, which brings stronger nonlinearity caused by interactions between the spreader and the payload into the system, controllers are designed to handle the swaying. A lot of research has been done to design such controllers for solving the double-pendulum effect. Kamal (Moustafa and Ebeid, 1988) et al. derived the nonlinear dynamical model of cranes, and, in 1988, a feedback control method was also presented to eliminate the swaying of the cargo. M. Guitierrez et al. (Gutierrez and Solo, 1998) proposed a fuzzy logic controller for an overhead crane prototype model that had obtained excellent control performances in simulations and experiments. Z Nowacki et al. (Nowacki et al., 1996) presented a PD controller with two feedback loops for an overhead crane and robustness analysis of the system was also proposed. Chwa et al. (Dongkyoung, 2017) presented a tracking control method based on a sliding mode control method for a three-dimensional overhead crane system to suppress the sway angels of the trolley, and the control performance was proved to be excellent. H.H Lee et al. (Cho and Lee, 2002) presented a novel fuzzy anti-swing control scheme combined with a position servo control and fuzzy logic control for an overhead crane prototype and the effectiveness of the proposed control strategy is proved by experiments.

Owing to the excellent approximation capability of the neural network method, researchers have presented several studies to solve the swaying problem of the cranes. Chunshien Li and Chun-yi Lee et al. (Li and Lee, 2001a; Li and Lee, 2001b) proposed a composite method combined with a neural network adaptive control method with fuzzy control for an overhead crane with high capacity to resist disturbance. Park et al. (Park and Le, 2012) had proposed a virtual prototype co-simulation by MATLAB and ADAMS to study the floating container crane. Zhang et al. (Zhang et al., 2016) proposed an adaptive tracking controller based on double-pendulum overhead cranes with uncertainties and disturbances by building a new sliding function as the desired trajectory. Ouyang et al. (Ouyang et al., 2019) presented a novel adaptive hierarchical sliding mode controller for overhead cranes with a double-pendulum effect, which can make the states of the system enter the desired sliding surface faster and even improve the precision of cart tracking. Ning Sun et al. (Sun et al., 2020a) established a dynamic model of a cooperative dual rotary crane system based on Lagrange’s method. Considering the actuator constraints, an output feedback control method that helps to increase the dual-boom positioning accuracy was also presented. Also, the control method mentioned in (Sun et al., 2020a) was extended to the positioning and tracking control for a pneumatic artificial muscle system (Sun et al., 2020b). Chen H et al. presented a new control strategy for a kind of under-actuated system by treating different constraints, and the proposed method was applied to a double-pendulum crane system with superior performances (Chen et al., 2019; Chen and Sun, 2020). Yang T et al. (Yang et al., 2020) designed an adaptive control method based on the neural network to solve the positioning problems for a ship-mounted crane system.

Based on the previous analysis, the elimination of the double-pendulum effect of cranes plays an important role to ensure the safety and effectiveness of cranes during operation. The main contribution of this work is as follows:

1. The proposed controller can realize fast positioning and anti-swing function for the crane system with a double swing effect.

In this paper, an overhead crane with a double-pendulum effect is chosen as the research object. Firstly, the mathematical model of the double-pendulum effect crane is established. Base on the mathematical model, an adaptive controller based on a radial basis function neural network compensation method is designed where the core of the controller is the robust controller, and the neural network is worked as an additional system to output compensation force. Then, the Lyapunov stability analysis is performed to testify the stability of the neural network controller crane system. Finally, hardware in a loop experimental platform is utilized to verify the proposed controller. A comparison between the sliding mode controller and the neural network controller is also presented to verify the effectiveness and feasibility of the control strategy.

The structure of the overhead crane with a double-pendulum effect is composed of a trolley, a spreader with a large mass (first-order payload), and a payload (second-order payload). Connections between the spreader, trolley, and payload are made of wire ropes, which are shown in Figure 1.

This assumes that the trolley, spreader, and payload of the crane are working on a two-dimensional plane, and the track of the trolley is horizontal. It considers the spreader and payload as particles, and the connection point of the wire rope is on the centroid of the spreader and payload. The transient wind disturbances and friction among the trolley, spreader, and payload connection point are ignored. The simplified model is shown in Figure 2.

In Figure 2, the positive direction for the x axis is pointed to the right; F denotes the driving force of the trolley; f represents the friction force between the trolley and the track; M denotes the mass of the trolley; m ₁ and m ₂ represent the mass of the spreader and the payload, respectively; l ₁ and l ₂ denote the wire rope length among the centroid of the spreader, trolley, and the payload, respectively; x represents the displacement of the trolley; θ ₁ and θ ₂ denote the swing angle of the spreader and payload, respectively.

According to Lagrange equation of the second kind, the dynamic equation set of the crane with double-pendulum can be obtained: { ( M + m 1 + m 2 ) x ¨ + ( m 1 + m 2 ) l 1 ( θ ¨ 1 ⁡ cos ⁡ θ 1 − θ ˙ 1 2 ⁡ sin ⁡ θ 1 ) + m 2 l 2 ( θ ¨ 2 ⁡ cos ⁡ θ 2 − θ ˙ 2 2 ⁡ sin ⁡ θ 2 ) + b x ˙ = F m 1 l 1 ( x ¨ cos ⁡ θ 1 + g ⁡ sin ⁡ θ 1 + l 1 θ ¨ 1 ) + m 2 l 1 [ l 1 θ ¨ 1 + l 2 θ ¨ 2 ⁡ cos ( θ 1 − θ 2 ) + l 2 θ ˙ 2 2 ⁡ sin ( θ 1 − θ 2 ) + x ¨ cos ⁡ θ 1 + g ⁡ sin ⁡ θ 1 ] = 0 m 2 l 2 2 θ ¨ 2 + m 2 l 1 l 2 θ ¨ 1 ⁡ cos ( θ 1 − θ 2 ) − m 2 l 1 l 2 θ ˙ 1 2 ⁡ sin ( θ 1 − θ 2 ) + m 2 l 2 x ¨ cos ⁡ θ 2 + m 2 l 2 g ⁡ sin ⁡ θ 2 = 0 (1) where b denotes the damping coefficient; x ˙ represents the velocity of the trolley; x ¨ denotes the acceleration of the trolley; θ ˙ 1 and θ ¨ 1 represent the angular velocity and acceleration of the swing angle 1 respectively; θ ˙ 2 and θ ¨ 2 denote the angular velocity and acceleration of the swing angle 2, respectively.

Owing to the development of neural network technology (Hunt et al., 1992; Feng, 1995), a great process had been made in the field of automatic control, signal processing, and pattern recognition. In this case, the neural network method had been introduced into the crane system, and the model of the double-pendulum crane follows Eq. 2: M ( q ) q ¨ + C ( q , q ˙ ) q ˙ + G ( q ) = τ + d (2) where q = [ x θ 1 θ 2 ] T ; q ˙ = [ x ˙ θ ˙ 1 θ ˙ 2 ] T ; q ¨ = [ x ¨ θ ¨ 1 θ ¨ 2 ] T ; τ = [ F 0 0 ] T ;

The control purpose of the double-pendulum crane is to help the trolley stop at the desired position while the first and second payloads which is set below the trolley keeps still. The tracking signal is defined as q d = [ x d θ 1 d θ 2 d ] ; q ˙ d = [ x ˙ d θ ˙ 1 d θ ˙ 2 d ] ; q ¨ d = [ x ¨ d θ ¨ 1 d θ ¨ 2 d ] .

The calculated traction is utilized to control the double-pendulum crane system. The controller is composed of a traction calculation method and a neural network compensation method. An RBF neural network correction controller is defined as an additional to recognize the online model error of the crane. As shown in Figure 3.

The hardware in the loop (HIL) simulation is introduced into the experiment, which is known as an advanced method in scientific research. In this experiment, the HIL is mainly composed of PC, the controller, and the mechanical structure. The PC had already been installed with the MATLAB/Simulink software and was used as a host computer. The motion control card is driven and communicated with the PC. The mechanical structure is driven by the motion card, and the motion parameters of the mechanical structure are collected by the motion control card and passed to the PC. The experimental physical prototype is shown in Figure 6.

In order to testify the performance of the presented neural network-based controller, the well-performed sliding mode controller in (Tuan and Lee, 2013) is also presented as a comparison group. The input of the sliding mode controller is defined as follows: u = − ( m 1 + m 2 ) l 1 θ ¨ 1 ⁡ cos ⁡ θ 1 − m 2 l 2 θ ¨ 2 ⁡ cos ⁡ θ 2 + b x ˙ + ( m 1 + m 2 ) l 1 θ ˙ 1 2 ⁡ sin ⁡ θ 1 + m 2 l 2 θ ˙ 2 2 ⁡ sin ⁡ θ 2 − ( M + m 1 + m 2 ) ( λ x ˙ + α θ ˙ 1 + β θ ˙ 2 ) − K sgn ( s ) (16) where K denotes the amplitude of the control switch gain; λ, α, β represent the designed parameters; sgn(·) denotes the sign function.

The overhead crane with the double-pendulum effect is chosen as the research object in this paper. A dynamic model of the crane has been established first. Then, an RBF-based neural network compensation adaptive controller is proposed, and simulations with MATLAB are also performed to verify the stability of the system. Finally, experiments with HIL are presented to test the presented controller where a comparison with a sliding mode based controller is also presented. The experimental results show that compares with the traditional sliding mode based controller, the neural network-based controller has a superior performance and better ability to anti-disturbances and without oscillation. Moreover, the proposed controller is also proved to be adequate for real-time control by experimental results.