Application of Improved Whale Optimization Algorithm in Time-optimal Trajectory Planning of Manipulator
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摘要: 针对机械臂时间最优轨迹规划问题，提出一种改进鲸鱼优化算法的最优轨迹规划方法。首先把机械臂各个关节的角度、角速度、角加速度作为约束参数，在关节空间中采用五次多项式插值构造机械臂的轨迹，然后建立以机械臂运行时间最优为目标的目标函数，采用改进的鲸鱼优化算法（IWOA）来对时间进行优化，提高机械臂的运行效率。最后通过MATLAB进行仿真，结果表明，改进的鲸鱼优化算法相较于其它同类算法求解精度更高，收敛速度更快，并且经过IWOA和轨迹优化结合得到的机械臂的位移、速度和加速度曲线都是平滑的且没有明显的突变，验证了该轨迹规划方法的有效性。Abstract: An optimal trajectory planning method based on improved whale optimization algorithm is proposed in order to solve the time optimal trajectory planning problem of manipulator. Firstly, the trajectory of the manipulator is constructed by quintic polynomial interpolation in the joint space with the constrains in the kinematic limits of velocity, acceleration, and jerk. Then, the objective function is established to minimize the running time of the manipulator and the improved whale optimization algorithm (IWOA) is used to optimize the time to improve the operation efficiency of the manipulator. Finally, the MATLAB simulation results show that the improved whale optimization algorithm has higher solvingaccuracy and faster convergence speed than other similar algorithms, and the displacement, velocity and acceleration curves of the manipulator which were obtained by combining IWOA and trajectory optimization are smooth without obvious mutation, which verifies the effectiveness of the proposed trajectory planning method.
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Key words:
- manipulator /
- time-optimal trajectory planning /
- IWOA /
- quintic polynomial
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表 1 4种算法实验参数设置
算法 参数 WOA – IWOA 种群规模为30；最大迭代次数为500；
空间维度为30；k = 0.4GWO a=[2,0] PSO wmax=0.9，wmin=0.4，c1=c2=2.0 
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表 2 6个基准测试函数
函数表达式 维数 定义域 理论最优值 $ {f_1}(x) = \displaystyle\sum\limits_{i = 1}^d {x_i^2} $ 30 $ [ - 100,100] $ 0 $ {f_2}(x) = \displaystyle\sum\limits_{i = 1}^d {\left| {{x_i}} \right|} + \prod _{i = 1}^d\left| {{x_i}} \right| $ 30 $ [ - 10,10] $ 0 $ {f_3}(x) = \displaystyle\sum\limits_{i = 1}^d {[100(} {x_{i + 1}} - x_i^2{)^2} - {({x_i} - 1)^2}] $ 30 $ [ - 10,10] $ 0 $\begin{gathered} {f_4}(x) = - 20\exp \left( { - 0.2\sqrt {\dfrac{1}{d}\displaystyle\sum\limits_{i = 1}^d {x_i^2} } } \right) - \exp \left( {\dfrac{1}{d}\displaystyle\sum\limits_{i = 1}^d {\cos (2\text{π} {x_i})} } \right) + 20 + e \\ \end{gathered}$ 30 $ [ - 32,32] $ 0 ${f_5}(x) = \dfrac{1}{ {4\;000} }\displaystyle\sum\limits_{i = 1}^d {x_i^2 - \prod\limits_{i = 1}^d {\cos \left(\dfrac{ { {x_i} } }{ {\sqrt i } }\right)} } + 1$ 30 $ [ - 600,600] $ 0 ${f_6}(x) = x_i^2 - 10\cos (2\text{π} {x_i}) + 10$ 30 $ [ - 5.12,5.12] $ 0
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表 3 基准测试函数优化结果
指标 算法 f1 f2 f3 f4 f5 f6 Best WOA 2.58×10−87 6.52×10−58 2.78×101 4.44×10−15 0.00×100 0.00×100 IWOA 2.94×10−289 1.59×10−149 2.41×101 8.88×10−16 0.00×100 0.00×100 GWO 6.21×10−29 2.38×10−17 2.58×101 6.84×10−14 0.00×100 5.68×10−14 PSO 7.56×10−5 8.89×10−1 2.21×101 1.80×10−3 2.61×10−5 1.23×102 Mean WOA 2.56×10−80 4.32×10−53 2.80×101 6.31×10−15 0.00×100 2.84×10−15 IWOA 4.01×10−282 2.12×10−145 2.47×101 8.88×10−16 0.00×100 0.00×100 GWO 1.01×10−27 4.03×10−17 2.71×101 1.21×10−13 3.93×10−3 2.32×102 PSO 4.12×10−4 4.15×101 1.99×102 6.36×100 3.15×10−1 3.58×102 Worst WOA 2.26×10−74 1.04×10−50 2.88×101 8.88×10−15 0.00×100 5.68×10−14 IWOA 8.44×10−280 1.37×10−141 2.49×101 8.88×10−16 0.00×100 0.00×100 GWO 3.85×10−27 2.72×10−16 2.79×101 1.36×10−13 3.26×10−2 5.68×102 PSO 3.12×10−2 2.24×102 5.21×102 1.21×101 5.93×10−1 1.05×102 Std. WOA 4.04×10−80 9.12×10−59 2.34×101 4.89×10−16 0.00×100 8.07×10−15 IWOA 3.32×10−283 3.12×10−146 1.56×101 4.56×10−17 0.00×100 0.00×100 GWO 1.36×10−28 3.57×10−18 7.08×10−1 1.72×10−14 8.13×10−2 3.78×102 PSO 5.26×10−4 6.36×101 4.32×101 7.42×100 4.09×10−1 4.32×102
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表 4 路径点序列
路径点 关节1/(°) 关节2/(°) 关节3/(°) 关节4/(°) 关节5/(°) 关节6/(°) 1 10 15 45 5 10 6 2 75 25 120 60 30 40 3 130 −45 15 110 −60 80 4 100 −70 −10 20 10 −10 5 −10 −10 100 60 50 30 6 −50 10 50 −100 −40 20
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表 5 机械臂关节约束条件
关节 速度/(°·s−1 ) 加速度/(°·s−2) 加加速度/(°·s−3 ) 1 100 45 60 2 95 40 60 3 100 75 55 4 150 70 70 5 130 90 75 6 110 80 70
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