# Phase Plane Trajectory Planning for Double Pendulum Crane Anti-Sway Control
## 1 Introduction
The designed multivariable PID controller has the same structural mathematical expression as the GPC law, making the transfer and calculation of the three parameters P, I, and D simpler. The above studies assume that the mass of the hook can be ignored, but in actual applications, the mass of the hook cannot be simply ignored. In this case, the system will produce a double pendulum effect; therefore, the control performance of the control algorithm based on the single-stage swing assumption will be greatly reduced [16]. In response to the anti-sway control problem of the double pendulum model of the crane, several innovative solutions have been proposed. [PERSON] et al. [17] proposed a frequency-modulation (FM) input-shaping strategy for a double pendulum crane. FM technology uses model-based feedback and partial feedback linearization. FM system parameters are time-independent and do not need to change when the system frequency changes. [PERSON] [18] proposed a composite trajectory planning method for a double pendulum bridge crane without advance or offline planning. This method keeps the load swing angle within the constraint range by adjusting the gain of the damping term. However, the above two methods [17; 18] both have residual oscillations of different magnitudes. [PERSON] et al. [19] proposed a variable parameter time-varying sliding mode control (VP-TVSMC) method, designed time-varying parameters, realized dynamic adjustment of sliding surface, and enhanced the adjustment ability of sliding surface. The existing anti-sway control strategies [17; 18; 19] generally adopt the \"trajectory-velocity\" cascade control framework: first plan the displacement trajectory of the trolley, and then indirectly suppress the load swing by adjusting the trolley acceleration/velocity. Although this method can achieve basic anti-sway effects, its essence is to passively weaken the swing angle through kinematic constraints, and there are core limitations: the swing angle control depends on the indirect transmission of the trolley motion, and the dynamic response has a lag. In response to the above problems, this paper proposes an innovative control of \"active swing angle planning-inverse solution of trolley motion\", taking the swing angle as the direct planning object and establishing an explicit mapping relationship with the trolley displacement and acceleration to achieve a fundamental reconstruction of anti-sway control. Specific innovations include constructing the coordinated trajectory of the load swing angle in the phase plane, suppressing high-order oscillation modes through input shaping technology, and ensuring that the double pendulum angle is controlled throughout the process. In view of the under-actuated problem that the double pendulum system cannot independently plan the double pendulum angle, the parameter approximate equivalence hypothesis is proposed, and the trajectory is solved after the double pendulum dynamics is approximately simplified, which significantly reduces the planning complexity. The acceleration trajectory obtained by phase plane trajectory planning is substituted into the original double pendulum nonlinear model, abandoning the traditional linearization/order reduction simplification operation and retaining the complete dynamic characteristics. The remainder of this paper is organized as follows. Section 2 discusses the modeling of the crane double pendulum system. Section 3 performs phase plane trajectory planning for the trolley and the load. Section 4 performs numerical simulations to verify the proposed method. Section 5 performs physical experiments to verify the proposed method. Section 6 provides some conclusions.