# Research on Variable Universe Fuzzy Multi-Parameter Self-Tuning PID Control of Bridge Crane
## 2 Bridge Crane Mechanical Dynamics Modeling
### System Dynamics Modeling  
Assuming the coordinate of the crane's center of mass is (x, 0), according to the mechanical relationships in Figure 1, the displacement of the bridge crane anti-sway system's trolley and load can be obtained as follows.  
\[\left\{\begin{array}{c}x_{m1}=x\\ y_{m1}=0\\ x_{m2}=x+lsin\theta\\ y_{m2}=lcos\theta\end{array}\right. \tag{1}\]  
In Equation (1), \(x_{m1}\), and \(y_{m1}\) are the displacement components of the bridge crane trolley in the X and Y directions, respectively, while \(x_{m2}\) and \(y_{m2}\) are the displacement components of the load in the X and Y directions. Taking the derivative of Equation (1), the velocity components of the crane trolley and load in the X and Y directions can be obtained as follows [12]:  
\[\left\{\begin{array}{c}\dot{x_{m1}}=\dot{x}\\ y_{m1}=0\\ x_{m2}=\dot{x}+lsin\theta+l\dot{\theta}cos\theta\\ y_{m2}=lcos\theta-l\theta sin\theta\end{array}\right. \tag{2}\]  
where \(\dot{x_{m1}}\) and \(\dot{y_{m1}}\) refer to the speed components of the trolley in the X and Y directions, respectively; \(x_{m2}\) and \(y_{m2}\) are the velocity components of the load in the X and Y directions. The kinetic energy \(T\) of the bridge crane system is the following:  
\[T=\frac{1}{2}m_{1}v_{m1}^{2}+\frac{1}{2}m_{2}v_{m2}^{2} \tag{3}\]  
where \(m_{1}\) and \(m_{2}\) are the masses of the crane and the load respectively, and \(v_{m1}\) and \(v_{m2}\) are the velocities of the crane and the load. Choosing the reference point for zero potential energy at the position of the crane, the potential energy \(V\) of the bridge crane anti-sway system is the following:  
\[V=-m_{2}glcos\theta \tag{4}\]  
Figure 1: Motion diagram of bridge crane.  
where \(g\) is the acceleration due to gravity, \(g=9.8\) m/s\({}^{2}\). Therefore, the Lagrange equation for the operation system of the bridge crane is the following:  
\[\left\{\begin{array}{c}L=T-V\\ \frac{d}{dx}\left(\frac{\partial L}{\partial\dot{q}_{i}}\right)-\frac{\partial L }{\partial\dot{q}_{i}}=f_{i}\ \ \ \ i=1,2\end{array}\right. \tag{5}\]  
where \(L\) is the Lagrangian, \(q_{i}\) is the generalized coordinates with \(q_{1}\) and \(q_{2}\) being the generalized coordinates for displacement and angle, respectively, and \(f_{i}\) represents the generalized external forces on the \(i\)-th coordinate. The frictional force between the trolley wheels and the track is denoted as f and \(\mu\) is the friction coefficient. By combining Equations (1) to (5), the nonlinear dynamic differential equations for the bridge crane operating system can be obtained as follows:  
\[\left\{\begin{array}{c}(m_{1}+m_{2})\ddot{x}+m_{2}\ddot{l}sin\theta+2m_{2} \dot{\mathfrak{l}}\partial cos\theta+\\ m_{2}\ddot{l}\partial cos\theta-m_{2}\dot{l}^{2}\ddot{sin\theta}+\mu\dot{x}=F \\ 2\dot{l}\dot{\theta}+\ddot{l}\ddot{\phi}+\ddot{x}cos\theta+gsin\theta=0\end{array}\right. \tag{6}\]  
Considering the actual situation, in the process of the bridge crane's trolley movement, the swing angle of the load should be as small as possible, generally not exceeding \(10^{\circ}\), so \(sin\theta=\theta\) and \(cos\theta=1\), at the same time, due to the small swing angle of the load, the length of the rope can be approximately regarded as constant, so \(\dot{l}=\dot{l}=0\). Substituting the approximate values into Equation (6), the differential equation of the bridge crane operating system can be simplified as the following.  
\[\left\{\begin{array}{c}(m_{1}+m_{2})\ddot{x}+m_{2}\ddot{l}\ddot{\theta}+\mu \dot{x}=F\\ l\ddot{\theta}+\ddot{x}+g\dot{\theta}=0\end{array}\right. \tag{7}\]  
Taking the Laplace transform of the above equation yields the transfer functions for the positioning of the trolley and the swing angle of the load.  
\[\left\{\begin{array}{c}\frac{X(s)}{F(s)}=\frac{l^{2}+\xi}{m_{1}ls^{4}+\mu ls ^{3}+(m_{1}+m_{2})gs^{2}+lgs}\\ \frac{\theta(s)}{F(s)}=\frac{g^{2}}{m_{1}ls^{4}+\mu ls^{3}+(m_{1}+m_{2})gs^{2 }+lgs}\end{array}\right. \tag{8}\]