# Phase Plane Trajectory Planning for Double Pendulum Crane Anti-Sway Control
## 4 Simulation Verification  
This section verifies the effectiveness of the proposed method through simulation. The numerical simulation process is divided into two groups. In the first group, the proposed method is compared under different constraints, including simulation comparisons under different maximum constraint swing angles and different maximum constraint speeds. In the second group, the proposed method is compared with the sliding mode method [19] and the nonlinear coupling tracking controller (NCTR) [16]. For the simulation of the crane system, accurate setting of various parameters is crucial to ensure the reliability and effectiveness of the simulation. The following are the specific parameters required for the crane system simulation: rope length \(l_{1}\) is 3 m, the distance between the hook and the load \(l_{2}\) is 0.3 m, the hook mass \(m_{1}\) is 2 kg, the load mass \(m_{2}\) is 10 kg, the trolley target distance \(x_{d}\) is 4 m, the load maximum swing angle constraints are 2\({}^{\circ}\), 3\({}^{\circ}\), and 4\({}^{\circ}\), and the trolley maximum speed constraints are 0.6 m/s, 0.8 m/s and 1 m/s. According to the constraints, a series of corresponding solutions are obtained as shown in Table 1, and the simulation results are shown in Figures 4-6, respectively. It can be seen from simulation Figures 4-6 that when the trolley travels the same displacement, when the swing angle constraint and the maximum speed of the trolley increase, the acceleration amplitude also increases, and the shorter the uniform speed time of the trolley, the shorter the total running time of the trolley. The swing angle of the load is controlled within the constrained swing angle range, the phase plane trajectory of the load swing angle is in line with expectations, and the load swing angle eventually reaches a relatively stable state. There is no residual vibration of the load when the trolley stops. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(x_{d}\)(m) & \(v_{\text{max}}\)(m/s) & \(\theta_{\text{max}}\)(deg) & \(a_{1}\)(m/s\({}^{2}\)) & \(t_{1}\)(s) & \(t_{2}\)(s) & \(t_{3}\)(s) & \(t_{4}\)(s) \\ \hline
4 & 0.6 & 2 & 0.1714 & 1.7503 & 1.8081 & 3.5584 & 6.6665 \\
4 & 0.8 & 3 & 0.2664 & 1.5013 & 1.8083 & 3.3096 & 5.0001 \\
4 & 1 & 4 & 0.3719 & 1.3446 & 1.8092 & 3.1538 & 4.0001 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Amplitude and switching time of the trolley constraint displacement and load constraint swing angle, acceleration. Figure 4: Swing angle, trolley velocity, acceleration and phase plane trajectory of the load at maximum swing angle of \(2^{\circ}\): (**a**) trolley velocity, acceleration and load swing angle; (**b**) phase plane trajectory of the load. Figure 5: Swing angle, trolley velocity, acceleration and phase plane trajectory of the load at maximum swing angle of \(3^{\circ}\): (**a**) trolley velocity, acceleration and load swing angle; (**b**) phase plane trajectory of the load. From (10) and (11), we obtain:  
\[\bar{\theta}_{1}=-\frac{\ddot{x}}{l_{1}}-\frac{m_{1}+m_{2}}{m_{1}l_{1}}g\theta_{1 }+\frac{m_{2}}{m_{1}l_{1}}g\theta_{2} \tag{39}\]  
\[\bar{\theta}_{2}=\frac{m_{1}+m_{2}}{m_{1}l_{2}}g\theta_{1}-\frac{m_{1}+m_{2}}{m _{1}l_{2}}g\theta_{2} \tag{40}\]  
To show the reliability of the method, the following system model verification is carried out. The approximate acceleration is substituted into the original model simulation to obtain the images of the hook and load swing angles \(\theta_{1}\) and \(\theta_{2}\). The simulation results of this method are compared with those of VP-TVSMC and NCTR. The displacement image of the trolley is shown in Figure 7. It can be seen from the figure that within the time range of 0-20 s, the displacement growth rate of the black line (Proposed method) is the fastest (about 7 s to reach 4 m), and there is no oscillation in the later period. Compared with VP-TVSMC and NCTR, it can achieve the target displacement more quickly and stably, showing higher control efficiency.