## 2 Mathematical Model of Bridge Crane  
Many studies [30; 31] have indicated that the coordinated motion of the trolley and crane can be decomposed and understood as independent two-dimensional motions. Focusing on the trolley's motion allows the crane payload system to be simplified. As illustrated in Figure 1, the control system regulates the horizontal motion of the trolley on the bridge, thereby indirectly controlling the movement and sway of the payload by manipulating the trolley's motion. In practical operations, the crane is influenced by various factors, including the shape and mass of the payload, the length of the wire rope, external disturbances, and so on. This makes it a multivariable nonlinear system. Considering the complexity of mathematically modeling all these factors, in order to better establish motion models for studying sway suppression, it is necessary to simplify the system while ensuring that it conforms to engineering reality and sufficiently describes the system performance. By neglecting some relatively minor factors, the following assumptions are made about the system:  
* Assumption 1: The motion of the trolley and the load is considered to be that of point masses. * Assumption 2: During operation, the friction between the track and the wheels is proportional to the running speed. The effects of disturbances such as air resistance and wind force are neglected. * Assumption 3: The driving force of the trolley is controllable, and the attenuation of transmission mechanisms such as motors is ignored. The direction of the driving force acting on the trolley is defined as the positive horizontal direction (to the right) in the system, with the downward direction from the trolley's center of mass designated as the positive vertical direction. The general form of the Lagrangian dynamics equations can be described as follows:  
\[\left\{ \begin{array}{l}L\left(q_{i},\dot{q}_{i}\right)=T\left(q_{i},\dot{q} _{i}\right)-V\left(q_{i},\dot{q}_{i}\right)\\ \frac{d}{dt}\left(\frac{q_{L}}{\partial q_{i}}\right)-\frac{\partial L}{ \partial q_{i}}=F_{i}\end{array} \right. \tag{1}\]  
where \(L\) is the Lagrangian operator, \(T\) represents the kinetic energy, \(V\) represents the potential energy, time \(t\) is the independent variable, and the generalized coordinates  
Figure 1: Two-dimensional model of a bridge crane. (where \(i=1,2,3\)) are the dependent variables. Here, \(q_{1},q_{2}\), and \(q_{3}\), respectively, represent trtoley position \(x\), payload swing angle \(\theta\), and length of the payload's cable \(l\) in Figure 1. \(F_{i}\) represents the generalized external force acting on the generalized coordinate \(q_{i}\). Using generalized coordinates to establish the differential equations for a system consisting of a three-degrees-of-freedom trolley and load, the mathematical model of the bridge crane hoisting system can be described as follows [32; 33; 34]:  
\[L=\frac{1}{2}(m_{c}+m_{l})\dot{x}^{2}+\frac{1}{2}\left(\dot{l}^{2}+l^{2}\dot{ \theta}^{2}+2\dot{x}\dot{l}\sin\theta+2\dot{x}\dot{l}\dot{\theta}\cos\theta \right)-m_{l}gl(1-\cos\theta) \tag{2}\]  
where \(g\) is the gravitational acceleration. Simplifying the system of Equations (1) and (2), we obtain the following:  
\[\left\{ \begin{array}{l}(m_{c}+m_{l})\ddot{x}+m_{l}\ddot{l}\sin\theta+2m_{l} \dot{l}\dot{\theta}\cos\theta-m_{l}\dot{l}\dot{\theta}^{2}\sin\theta+m\dot{l} \dot{\theta}\cos\theta=F-\mu\dot{x}\\ m_{l}\ddot{l}+m_{l}\ddot{x}\sin\theta+m_{l}\dot{x}\dot{l}\dot{\theta}\cos \theta=F_{l}\\ \dot{l}\dot{\theta}+2\dot{l}\dot{\theta}+g\sin\theta+\ddot{x}\cos\theta=0\end{array} \right. \tag{3}\]  
Considering practical scenarios, bridge cranes often commence lifting operations only after the load has been raised to a predetermined height, and lowering of the load occurs once both the trolley and the load have reached their designated positions. Thus, the deformation of the wire rope can be disregarded, denoted as \(\ddot{l}=\dot{l}=0\).