# An Anti-Swing Closed-Loop Control Strategy for Overhead Cranes
## 2 Mathematical Model  
Usually, an overhead crane consists of a hoist, load, and transport trolley, as shown in Figure 1. Its corresponding two-dimensional simplified physical model is shown in Figure 2.  
Figure 1: A real crane diagram with hoist and load.  
Assuming that the length of hoisting rope \(l\) does not change during transportation and the friction between the trolley and the platform \(\mu\) is negligible, that is \(\dot{l}=\bar{l}=0,\ \mu=0\), the dynamic equation of the 2-D overhead crane system is obtained as follows:  
\[\left[\begin{array}{cc}ml^{2}&ml\cos\theta\\ ml\cos\theta&M+m\end{array}\right]\left[\begin{array}{c}\ddot{\theta}\\ \ddot{x}\end{array}\right]+\left[\begin{array}{cc}0&0\\ -ml\dot{\theta}\sin\theta&0\end{array}\right]\left[\begin{array}{c}\dot{ \theta}\\ \dot{x}\end{array}\right]+\left[\begin{array}{c}mgl\sin\theta\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ F(t)\end{array}\right] \tag{1}\]  
where \(M\), \(m\) is the mass of the trolley and the hoist, respectively, \(x(t)\) is the horizontal displacement, \(\theta(t)\) is the vertical direction angle of the upload, and \(F(t)\) is the driving force. \(M(\textbf{x},\ \theta)=\left[\begin{array}{cc}\textbf{m}_{11}&\textbf{m}_{12}\\ \textbf{m}_{21}&\textbf{m}_{22}\end{array}\right]=\left[\begin{array}{cc}ml ^{2}&ml\cos\theta\\ ml\cos\theta&M+m\end{array}\right]\) is the inertial matrix of the system (1).  
Known from [31], although a 2-D underactuated system cannot be fully state-feedback linearized, \(\ddot{x}\) can be implemented as a system input, and the driven state quantity part \([x,\dot{x}]\) can be linearized. Inspired by this, the model of the 2-D overhead crane can be transformed by homeomorphism coordinate transformation:  
\[\Gamma\left\{\begin{array}{l}z_{1}=x+\alpha(\theta)\\ z_{2}=\dfrac{\partial L(x,\ \theta)}{\partial\dot{x}}=m_{11}\dot{x}+m_{12}\dot{x} \\ z_{3}=\theta\\ z_{4}=\dot{\theta}\end{array}\right. \tag{2}\]  
where \(L(x,\theta)\) is the Lagrange equation of the 2-D overhead crane system.  
Simultaneously from (1), (2) is obtained:  
\[\dot{z}=f(z)+Bu \tag{3}\]  
where:  
\[f_{1}(z)=[\frac{z_{2}}{m_{11}}+(\frac{d\alpha(\theta)}{d\theta}-\frac{m_{11}} {m_{12}})z_{4}]\Big{|}_{\xi=\Gamma^{-1}(z)}\]  
\[f_{2}(z)=[\frac{1}{2}[\dot{x},\dot{\theta}]^{\top}\frac{\partial M(x,\theta)} {\partial x}-mgl\sin\theta]\Big{|}_{\xi=\Gamma^{-1}(z)}\]  
\[f(z)=[f_{1}(z),f_{2}(z),z_{4},0]^{\top}\]  
\[B=[0,0,0,1]^{\top}\]  
\[\zeta=[x,\theta,\dot{x},\dot{\theta}]^{\top}\]  
The value of the costate coefficient \(\alpha(\theta)\) is set 0 in this paper to ensure the conciseness of the transformed system structure. It is proved later that the controllability and observability of the system is not affected by the choice of the value of \(\alpha(\theta)\).  
Define:  
\[\sigma=f(z)-Az \tag{4}\]  
Figure 2: Two-dimensional plane model of an overhead crane.  
The system (1) is equivalent to the following mathematical model:  
\[\left\{\begin{array}{l}\dot{z}=Az+Bu+\sigma\\ y=Cz\end{array}\right. \tag{5}\]  
where:  
\[A =\frac{\partial f(z)}{\partial z}|_{z=0}=\left[\begin{array}{ cccc}a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ 0&0&0&1\\ 0&0&0&0\end{array}\right],\;a_{ij}=\frac{\partial f_{i}(z)}{\partial z_{j}}|_{ z=0},\;i=1,2,\;j=1,2,3,4\] \[C =\left[\begin{array}{ccc}1&0&0&0\\ 0&0&1&0\end{array}\right]\]  
From mathematical description (8), a conclusion can be made that the nonlinear crane system (1) can be equivalent to a linear combination of a linear subsystem and a nonlinear term. The nonlinear term \(\sigma\) can be treated as a virtual disturbance. For the linear subsystem, it can be controlled and observed if and only if the following inequality is satisfied:  
\[\left[\frac{m_{12}}{m_{11}}\frac{\partial(mgl\sin\theta)}{\partial x}-\frac{ \partial(mgl\sin\theta)}{\partial\theta}\right]|_{x,\theta=0}\
eq 0,\;-\frac{1}{m_{11} \;(x=0)}<0 \tag{6}\]