# Phase Plane Trajectory Planning for Double Pendulum Crane Anti-Sway Control
## 1 Introduction  
As typical underactuated mechanical pieces of equipment, cranes are widely used in the fields of construction, ships, ports, etc. Due to their structural characteristics and operating conditions [1], cranes are very prone to swaying during operation. Suppressing this swaying phenomenon is the key to improving the efficiency and safety of cranes [2, 3]. Therefore, it is necessary to design an efficient anti-sway control method [4]. On the two-dimensional physical modeling of cranes, the single pendulum model and the double pendulum model are adopted widely. Some scholars choose to ignore the mass of the hook and simplify the crane into a single pendulum model for analysis [5, 6]. However, although this simplification reduces the complexity of the model, it fails to fully capture the dynamic behavior of the system. The crane system will exhibit double pendulum characteristics during operation. This model takes into account more dynamic interactions and system complexity, especially the impact of the hook mass on the overall system. The model that includes the hook mass can more accurately reflect the dynamic characteristics of the crane system. In the field of single pendulum model anti-sway control, [PERSON] et al. [7] proposed a fuzzy sliding mode controller, which integrated the trolley displacement and load swingangle into the same sliding mode surface, introduced fuzzy rules to adjust the control quantity, and ensured that the system state was always on the sliding mode surface. [PERSON] et al. [8] introduced a composite signal to enhance the coupling relationship between the driving variable and the underactuated variable. The stability analysis was based on the original nonlinear model of the system, rather than the linearized model, to ensure the accuracy of the system analysis. [PERSON] et al. [9] proposed a method to achieve more accurate system control by coupling the trolley displacement coordinates with the effective load swing angle coordinates. In addition, the method combined the S-curve based trajectory planning technology to smooth the motion path and reduce the oscillation effect of the system. [PERSON] et al. [10] designed a predictive control method, which used the introduced constraint method to determine that the output constraint of the underactuated system was limited to a preset range. However, the high computational complexity of model predictive control makes it difficult to apply in practice. [PERSON] et al. [11] proposed a nonlinear coupling tracking anti-sway controller. A composite signal was constructed to suppress the swing angle of the system, and a suitable switching function was established to solve the chattering problem existing in the sliding mode control system. A fuzzy controller was designed to improve the positioning accuracy of the system. [PERSON] et al. [12] proposed a PD-SMC hybrid controller. The PD part is used to stabilize the controlled system and the SMC part has strong robustness to external disturbances, parameters and unmodeled uncertainties. In order to improve the control performance of anti-sway, [PERSON] et al. [13] added more swing information to the control input and designed an adaptive law to accurately identify the load mass online, further improving the robustness of the method. The open-loop anti-sway control method is very popular. Among them, the input shaping controller has a simple structure. This technology uses a series of specific pulse signals to convolve with a unique input signal to obtain the final synthetic input signal for control. [PERSON] [14] proposed an adjustable maneuvering waveform command shaper. The initial parameter approximation is taken, and the shaper parameters are optimized using a genetic algorithm to determine the optimal acceleration parameters. The proposed command shaper can effectively eliminate the residual vibration of a single continuous waveform command. [PERSON] et al. [15] proposed a parameter tuning algorithm for a multivariable PID controller based on generalized predictive control (GPC).