Trajectory Planning for Overhead Crane With Double Spherical Pendulum and Varying Cable Length Effect
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摘要: 带有双球面摆和变绳长效应的桥式起重机具有多输入多输出以及欠驱动的动力学特性, 目前仍缺乏有效的控制策略. 在台车移动、桥架移动、负载升降同步作业过程中, 吊钩和负载两级球面摆动特性更为复杂, 各状态量之间的非线性耦合关系更强, 桥式起重机的防摆控制更具挑战性. 不仅如此, 现有方法无法保证桥式起重机系统全状态量的暂态控制性能. 为解决上述问题, 提出一种基于多项式的优化轨迹规划方法. 首先, 在未进行近似简化的前提下, 使用拉格朗日方法建立带有双球面摆和变绳长效应的7自由度 (Seven degree-of-freedom, 7-DOF) 桥式起重机的精确动力学模型. 在此基础上, 构造一组包含各状态量的辅助信号, 将施加在台车、桥架、绳长以及吊钩、负载摆动上的约束转化为对辅助信号的约束, 从而将桥式起重机的轨迹规划问题转化为与辅助信号相关的时间优化问题, 并使用二分法求解. 该轨迹规划方法不仅缩短了吊运时间, 而且确保了全状态量满足约束条件. 最后, 仿真结果证明了动力学模型的准确性和轨迹规划方法的有效性.Abstract: Due to the multi-input multi-output and underactuated dynamic characteristics, the overhead crane with double spherical pendulum and varying cable length effect still lacks efficient control approaches. During the simultaneous operation of trolley moving, bridge moving and payload hoisting/lowing, the double spherical pendulum of hook and payload is more complex, and the nonlinear coupling characteristics of various state variables are stronger. It makes anti-swing control of overhead crane much more challenging. Moreover, the existing control methods cannot guarantee the desired transient control performance of the overhead crane system. To address the abovementioned issues, this paper proposes a novel polynomial-based optimal trajectory planning approach. Firstly, an accurate dynamic model for seven degree-of-freedom (7-DOF) overhead crane with double spherical pendulum and varying cable length effects is established by Lagrangian method without any simplifications. Based on this, a group of auxiliary signals containing various states are constructed. Then, the constraints imposed on the trolley/bridge moving, cable length varying, hook/payload swing are equivalently transformed to some new constraints on the auxiliary signals. Therefore, the trajectory planning problem of the overhead crane is transformed into a time optimization problem related to the auxiliary signals and solved by using the bisection method. The proposed trajectory planner not only makes transportation time as short as possible, but also ensures that full-states constraints are satisfied. At last, the simulation results prove the accuracy of the dynamic model and the effectiveness of the trajectory planning method.
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图 7 不同索具绳长度$ l_2$的仿真结果
Fig. 7 Simulation results of different rigging cable lengths $ l_2$
表 1 本文与桥式起重机相关工作的比较
Table 1 Comparison between our work and the relevant work about overhead crane
自由度 参考文献 台车移动 桥架移动 变绳长 吊钩摇摆 负载摇摆 2 [7−8] $\checkmark$ $\times$ $\times$ $\times$ $\checkmark$ 3 [9−11] $\checkmark$ $\times$ $\times$ $\checkmark$ $\checkmark$ 3 [12−14] $\checkmark$ $\times$ $\checkmark$ $\times$ $\checkmark$ 4 [15−17] $\checkmark$ $\times$ $\checkmark$ $\checkmark$ $\checkmark$ 4 [18−19] $\checkmark$ $\checkmark$ $\times$ $\times$ $\checkmark$ 5 [20−21] $\checkmark$ $\checkmark$ $\checkmark$ $\times$ $\checkmark$ 6 [22−24] $\checkmark$ $\checkmark$ $\times$ $\checkmark$ $\checkmark$ 7 本文 $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ 
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表 2 系统参数
Table 2 System parameters
参数 物理意义 单位 $M_1$ 台车质量 kg $M_2$ 台车和桥架质量之和 kg $m_1,m_2$ 吊钩、负载质量 kg $x,y$ 台车、桥架位移 m $l_1,l_2$ 吊绳、索具绳长度 m $\theta_1,\theta_2,\theta_3,\theta_4$ 吊钩、负载三维空间摆角 deg $F_x,F_y,F_z$ 台车、桥架、吊绳驱动力 N $g$ 重力加速度 m/s2
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表 3 量化指标对比结果
Table 3 Comparison results of quantitative indices
控制方法 $t_s$ (s) ${\theta _{1\max }}$ (deg) ${\theta _{1{\rm{res}}}}$ (deg) ${\theta _{3\max }}$ (deg) ${\theta _{3{\rm{res}}}}$ (deg) EI $>15.00$ 2.69 0.79 3.63 1.46 光滑器 $>15.00$ 2.29 0.23 2.46 0.35 EAB $>15.00$ 3.14 0.22 5.39 1.21 CTP 4.82 3.03 0.01 3.24 0.02 本文方法 4.91 1.92 0 1.97 0
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