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PSO-based time optimal trajectory planning for six degrees of freedom robot manipulators with speed constraints
LI Xiaowei , HU Likun, WANG Hu
College of Electrical Engineering, Guangxi University, Nanning 530004, China
Abstract:The trajectory planning is designed for robot manipulators in the joint space according to interpolation points on the premise of the solutions to the forward and inverse kinematics problems of 6-DOF (depth of field) manipulators. This paper puts forward the particle swarm optimization (PSO)-based time optimal trajectory planning of the 3-5-3 polynomial interpolation method in order to make mechanical arms run in the shortest time at a constrained speed. The PSO is proposed to optimize run time due to its simple structure and easily adjustable parameters. The polynomial interpolation time rather than coefficient is selected as searching variable in PSO optimization. If the interpolation time of three polynomials doesn't meet the constraints, the particle will be excluded by comparison. The shortest time of the 6-DOF manipulators running at different speeds is obtained by offline PSO. Real-time experiments are conducted on the robot control platform. It shows that this method can accurately realize time optimal trajectory planning at any speed through its position, velocity and acceleration curves.
Key words: robot     6 DOF manipulators     particle swarm optimization     trajectory planning     polynomial interpolation     speed constraints     time optimal

1 多项式插值函数的构造

i段关节3-5-3样条多项式的通式为

2 PSO求解速度约束下的最优时间

1)选定种群的规模m(一般为20)，在插值时间的3维搜索空间中随机产生m个粒子构成初始种群，并初始化粒子的位置和速度。

2)由m组时间变量ti1ti2ti3代入式(2)~(4)中可得出3-5-3多项式的未知系数aij

3)将3-5-3多项式的系数aij代入式(1)并对时间求导，得到关节角度的速度函数，判断实时速度是在否满足式(8)。

4)计算每个微粒的适应度值。对步骤3)的计算结果进行筛选，如果3段中的任何一段速度不符合式(8)，则将该粒子的适应度值设置为极大的常数，在寻找最优粒子时，通过适应度值的比较，将会排除这个适应度值较大的粒子，不被筛选为最优粒子。而这个粒子本身也会慢慢向最佳值进行靠拢，直到满足速度的约束。如果3段的最大速度都符合式(8)，则采用式(7)作为适应度函数，粒子群算法迭代以获得最小插值时间为目标。

5)对每个微粒，将其适应度值与其经历过的最好位置pi的适应度值作比较，如果较好，则将其替换为当前的最好位置pi

6)比较每个微粒当前最好位置的适应度值，得到当前整体最优粒子，再与群体所经历的全局最好位置pg作比较，如果较好，则替换pg

7)根据式(5)~(6)变化微粒的速度和位置，重新整合成新的由m个粒子构成的种群。

8)如满足终止条件(通常为足够好的适应值或达到一个预设最大迭代次数(Nmax)则算法结束，否则返回步骤2)。

3 机器人建模与PSO仿真

 图 1 机械臂的D-H坐标系 Fig. 1 The D-H coordinates of robot manipulators

 关节i di/mm ai/mm ai/(°) θi/(°) 关节变量范围/(°) 1 0 150 90 0 [-150,150] 2 0 570 0 90 [-80,65] 3 0 150 90 0 [-80,80 4 650 0 -90 0 [-175,175] 5 0 0 90 0 [-110, 110] 6 0 105 0 0 [-200, 200]

 起始点 路径点1 路径点2 终点 (800,0,615) (950,100,560) (750,300,560) (550,200,700)

 关节i θj0 θj1 θj2 θj3 关节1 0 －6.009 -21.803 -19.983 关节2 0 15.053 -0.851 -18.161 关节3 0 11.948 -10.833 -7.900 关节4 0 0 0 0 关节5 90 86.871 79.975 100.240 关节6 0 -6.009 -21.803 -19.983

 图 2 关节1的最优粒子pg位置进化 Fig. 2 The optimal particle pgof joints 1 evolution

 速度范围(°/s) t11/s t12/s t13/s (-115,115) 0.222 9 0.294 9 0.138 4 (-57,57) 0.444 9 0.592 4 0.285 6 (-20,20) 1.274 7 1.636 1 0.848 5 (-10,10) 2.567 1 3.245 8 1.720 9

 关节i ti1/s ti2/s ti3/s 关节1 1.274 7 1.636 1 0.848 5 关节2 3.116 2 2.188 8 3.280 0 关节3 3.107 0 3.997 7 1.605 0 关节4 0 0 0 关节5 0.813 1 1.170 2 3.062 0 关节6 1.274 7 1.636 1 0.848 5
4 实验结果与分析

 图 3 粒子群优化的机械臂关节位置曲线 Fig. 3 Mechanical arms’ joint position curves of PSO

 图 4 粒子群优化的机械臂关节速度曲线 Fig. 4 Mechanical arms’ joint speed curves of PSO

 图 5 粒子群优化的机械臂关节加速度曲线 Fig. 5 Mechanical arms’ joint acceleration curves of PSO

 图 6 粒子群优化的机械臂关节脉动曲线 Fig. 6 Mechanical arms’ joint pulsation curves of PSO

 图 7 基于粒子群优化的机械臂轨迹曲线 Fig. 7 Mechanical arms’ joint trajectory curve of PSO
5 结束语

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DOI: 10.3969/j.issn.1673-4785.201404035

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#### 文章信息

LI Xiaowei, HU Likun, WANG Hu

PSO-based time optimal trajectory planning for six degrees of freedom robot manipulators with speed constraints

CAAI Transactions on Intelligent Systems, 2015, 10(03): 393-398.
DOI: 10.3969/j.issn.1673-4785.201404035