# Phase Plane Trajectory Planning for Double Pendulum Crane Anti-Sway Control
## 6 Conclusions  
This paper proposes an input shaping double pendulum anti-sway control method based on phase plane trajectory planning. This method derives the acceleration switching time and amplitude by analyzing the phase plane swing angle and the physical constraints of the system, ensuring that the displacement, speed, hook and load swing angles of the trolley are kept within the constraints. In addition, considering that the oscillation frequency of the actual system is affected by multiple parameters such as hook mass, load mass and rope length, the model introducing these parameters can more accurately describe the dynamic behavior of the crane system and improve the accuracy and reliability of system modeling. The proposed control method is compared with the sliding mode controller. The numerical simulation results show that the proposed method not only achieves a trolley displacement trajectory without overshoot but also significantly reduces the load swing angle and maintains a stable swing angle for a longer time during the movement. In addition, numerical simulation and physical experimental verification also indicate that the experimental curve is highly consistent with the simulation curve, further showing the reliability and effectiveness of the control method. Although this study mainly focuses on the two-dimensional physical model of the crane, the results of the numerical simulation and the physical experiments both show the good performance of this method. Future research will be further extended to three-dimensional physical models to further improve the applicability and control accuracy of this method. [PERSON] [21] introduces a system combining a laser radar rangefinder and an ultrasonic rangefinder used in autonomous vehicles, which can help overcome the obstacles of the traumatic laser radar system and have better target detection capabilities. In the future, it can be considered to be applied to the problem of measuring the distance of trolley. First, an ultrasonic sensor matching the working conditions is selected and installed side by side with the laser sensor to cover the target area; secondly, the controller interface is expanded and the ultrasonic drive circuit is integrated, and then the ranging code is written and the laser and ultrasonic data are integrated. Finally, the accuracy is verified through static calibration and dynamic testing so as to achieve dual sensor complementarity and improve the measurement reliability and anti-interference ability in complex environments. Some shortcomings of this method and suggestions for improvement are that at the current stage, this research method is mainly applicable to controlled environments with weak or no interference. Subsequent research plans will focus on the development of dynamic disturbance suppression strategies, with an emphasis on breakthroughs in modeling and compensation techniques for typical external disturbance sources such as wind-induced  
Figure 12: (**a**) Comparison of hook swing angle results for different load masses; (**b**) comparison of load swing angle results for different load masses. interference in order to achieve enhanced robustness in complex open environments. The trajectory planning scheme used in this paper relies on offline calculations to obtain the optimal parameters. Therefore, when the crane faces different work requirements (such as changing the cargo handling distance or transportation time), the parameters need to be recalculated, which is time-consuming and reduces production efficiency. In order to improve the flexibility and response speed of the system, future research can focus on the online trajectory planning problem of the crane. The crane model in this study only considers the two-dimensional anti-sway problem and has not yet covered the scenario of three-dimensional anti-sway. In order to improve the integrity of the system and the breadth of practical applications, future related research should consider incorporating three-dimensional anti-sway problems into the analysis.