# Multi-Objective Motion Control Optimization for the Bridge Crane System
## 2 Dynamic Modeling of the Crane System  
In order to design an algorithm with the target of precise position control and anti-disturbance, the physical model of the two-dimensional (2D) underactuated system is established, as shown in Figure 1. In order to simplify the modeling, some assumptions are made.  
**Assumption 1**.: _The crane is a rigid driving system, and the payload is assumed to be a rigid body. Additionally, the air friction is neglected._  
**Assumption 2**.: _The cable length is unchanged._  
**Assumption 3**.: _The payload swing angle \(\theta\) satisfies \(-\pi/2<\theta(t)<\pi/2\)._  
With these assumptions, dynamic equations can be formulated [32]:  
\[(m_{T}+m_{P})\ddot{x}_{T}+m_{P}L\ddot{\theta}\cos\theta-m_{P}L\dot{\theta}^{2 }\sin\theta=F \tag{1}\]  
\[\ddot{x}_{T}\cos\theta+L\ddot{\theta}+g\sin\theta=0 \tag{2}\]  
where \(x_{P}\) is the horizontal position of the payload, \(x_{T}\) is the crane horizontal position, \(L\) is the cable length, \(m_{T}\) and \(m_{P}\) are the mass of the crane and payload, respectively, \(g\) presents the gravity factor, and \(F\) is the driving force. Through the geometric operation and analysis, the payload position \(x_{P}\) can be obtained:  
\[x_{P}=x_{T}+l\sin\theta \tag{3}\]  
It can be found that due to the underactuated characteristics, the payload swing cannot be designed directly, and instead it can be planned by the coupling relationship between the crane and the payload. From Equation (2), we can provide the theoretical analysis for the subsequent control methods.  
During the process of the actual transportation, the payload swing should be imposed within the small angle, i.e.,:  
\[\left\{\begin{array}{l}\sin\theta\approx\theta\\ \cos\theta\approx 1\\ \dot{L}=\ddot{L}\approx 0\end{array}\right. \tag{4}\]  
Figure 1: Crane physical model.  
According to Equations (2) and (4), the dynamic equations can be simplified as:  
\[\ddot{x}_{T}+L\ddot{\theta}+g\theta=0 \tag{5}\]  
In order to fully consider the Stribeck effect, the friction, and the crane mass, \(\ddot{x}_{T}\) is selected as the control command. Now, the matrix equation of model can be established according to Equation (5):  
\[\dot{X}=\left[\begin{array}{cccc}0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&-g/L&0\end{array}\right]X+\left[\begin{array}{c}0\\ 1\\ 0\\ -1/L\end{array}\right]u \tag{6}\] \[\qquad\qquad Y=\left[\begin{array}{cccc}1&1&1&1\end{array} \right]X\]  
where \(X=\left[x_{T},\dot{x}_{T},\theta,\dot{\theta}\right]^{T}\) is the state matrix of the bridge crane, \(T\) states the matrix transposition, and \(u(t)=\ddot{x}_{T}\) denotes the crane acceleration.  
Aiming to satisfy the real-world requirement, three constraints are considered.  
**Constraint 1**.: _The crane is required to reach the target position \(x_{d}\in R\) within a limited time \(t_{total}\):_  
\[x_{T}(t)=x_{d},\hskip 14.226378 pt\forall t\leq t_{total} \tag{7}\]  
**Constraint 2**.: _The velocity and acceleration of the crane should be restricted due to the actual limitation of actuators:_  
\[\left|\dot{x}_{T}(t)\right|\leq v_{\text{lim}},\hskip 14.226378 pt\left|\ddot{x }_{T}(t)\right|\ \leq a_{\text{lim}} \tag{8}\]  
**Constraint 3**.: _The maximum swing angle of the payload should be bounded:_  
\[\left|\theta(t)\right|\leq\theta_{\text{lim}} \tag{9}\]  
where \(v_{\text{lim}},a_{\text{lim}}\in R\) are defined as the maximum velocity and maximum acceleration, respectively, and \(\theta_{\text{lim}}\in R^{+}\) is the limited maximum swing angle.  
Next, the corresponding controller design is detailed in the following section.