Pseudospectral Method Based Time Optimal Anti-swing Trajectory Planning for Double Pendulum Crane Systems
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摘要: 在工业生产过程中,桥式吊车系统经常会体现出双摆系统的特性,导致更多欠驱动状态量的出现,增大控制难度.基于此,论文提出了一种针对双摆桥式吊车系统的时间最优轨迹规划方法,可以得到全局时间最优且具有消摆能力的轨迹.具体而言,为方便地构造以时间为代价函数的优化问题,首先对系统运动学模型进行相应的变换;在此基础上,考虑包括两级摆角及台车速度和加速度上限值在内的多种约束,构造出相应的优化问题;然后,利用高斯伪谱法(Gauss-pseudospectral method, GPM)将该带约束的优化问题转化为更易于求解的非线性规划问题,且在转化过程中,可以非常方便地考虑轨迹约束.求解该非线性规划问题,即可得到时间最优的台车轨迹.不同于已有的大多数方法,该方法可获得全局时间最优的结果.最后,通过仿真与实验结果验证了这种时间最优轨迹规划方法具有满意的控制性能.Abstract: In practice, an overhead crane system may behave like a double pendulum, which has more unactuated states and is more difficult to be controlled properly. Motivated by this observation, we present a time optimal trajectory planning scheme for double pendulum crane systems, which can yield a global time-optimal swing-free trajectory. Specifically, to facilitate the optimization problem creation process, we first implement some basic transformations on the system kinematics. Then, various constraints, including upper and lower bounds of the two pendulum angles and upper bounds of the trolley velocity and acceleration, are taken into consideration to set up the optimization problem. After that, the Gauss-pseudospectral method(GPM) is utilized to convert the constrained optimization problem into a nonlinear programming problem, which can be solved more conveniently, while the trajectory constraints are also considered during the transformation. By solving the constructed nonlinear programming problem, a global time-optimal result is obtained, which is different from most existing methods. Finally, numerical simulation and experimental results are given to illustrate the satisfactory performance of the proposed method.
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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 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})
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