图  1  双摆桥式吊车模型示意图

Fig.  1  Schematic diagram of double pendulum crane

图  2  高斯伪谱法流程图

Fig.  2  Flowchart of the Gauss-pseudospectral method

图  3  本文方法仿真结果(台车位置以及速度) (实线:仿真结果; 虚线: 目标位置 $x_f=0. 6~{\rm{m}}$ ; 点画线:台车速度约束 $v_{\max}=0. 3~{\rm{m/s}})$

Fig.  3  Simulation results (trolley position and velocity)(Solid line: simulation results; Dashed line: target position $x_f=0. 6~{\rm{m}}$ ; Dotted-dashed line: trolley velocity constraint $v_{\max}=0. 3~{\rm{m/s}}$ )

图  4  本文方法仿真结果(两级摆动对应的摆角及角速度) (实线: 仿真结果;虚线: 摆角约束 $\theta_{1\max}=\theta_{2\max}=2~{\rm{deg}}$ ;点画线: 角速度约束 $\omega_{1\max}=\omega_{2\max}=5~{\rm{deg/s}})$

Fig.  4  Simulation results (first and second order swing angles and angular velocities) (Solid line: simulation results;Dashed line: swing angle constraint $\theta_{1\max}=\theta_{2\max}=2~{\rm{deg}}$ ; Dotted-dashed line:angular velocity constraint $\omega_{1\max}=\omega_{2\max}=5~{\rm{deg/s})}$

图  5  轨迹规划与控制流程图

Fig.  5  Flowchart of trajectory planning and tracking control

图  6  本文方法实验结果(台车位置、 一级摆角、二级摆角) (实线: 实验结果; 虚线: 待跟踪最优轨迹; 点画线:摆角约束 $\theta_{1\max}=\theta_{2\max}=2~{\rm{deg})}$

Fig.  6  Experimental results of proposed method (trolley position,first and second order swing angles) (Solid line:experimental results; Dashed line: planned trajectory; Dotted-dashed line: swing angle constraint $\theta_{1\max}=\theta_{2\max}=2~{\rm{deg}}$ )

图  7  LQR方法实验结果(台车位置、 一级摆角、二级摆角) (实线:实验结果)

Fig.  7  Experimental results of the LQR method (trolley position,first and second order swing angles) (Solid line:experimental results)

图  8  文献[21]方法实验结果(台车位置、一级摆角、 二级摆角) (实线: 实验结果; 虚线: 待跟踪轨迹; 点画线:摆角约束 $\theta_{1\max}=\theta_{2\max}=2~{\rm{deg}}$ )

Fig.  8  Experimental results of the method in [21] (trolley position,first and second order swing angles) (Solid line:experimental results; Dashed line: planned trajectory; Dotted-dashed line: swing angle constraint$\theta_{1\max}=\theta_{2\max}=2~{\rm{deg})