Density-Reducing Monte Carlo Method for 7 Degrees of Freedom Humanoid Robot Arm Workspace Solution
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摘要:
针对蒙特卡洛法和改进蒙特卡洛法在求解机械臂工作空间时存在精度不够准确和加密点云浪费的问题，提出一种降密蒙特卡洛法. 首先，基于蒙特卡洛法中随机点分布不均的特性，对机械臂初始工作空间进行均匀加密，使空间的内部与边界区域分明；然后，采用扩展关节角度和循环加密随机点的方式，只对边界区域进行加密，达到降低工作空间随机点云密度的目的；同时，还研究了该方法中初始点云数量、各轴向分割体素数量、精度阈值、扩展关节角度和循环次数等参数对工作空间精度的影响；最后，通过仿真分析对降密蒙特卡洛法的有效性进行验证. 结果表明：相比于蒙特卡洛法，降密蒙特卡洛法在工作空间平均误差率为0.02242%时，总随机点云数量降幅为93.89%；相比于改进蒙特卡洛法，在循环次数为2次和4次时，降密蒙特卡洛法工作空间的平均误差率分别降低0.13853%和0.11329%，总随机点云数量降幅分别为44.83%和64.52%.
Abstract:A density-reducing Monte Carlo method was proposed to address the problems of inaccurate precision and waste of encrypted point cloud in the Monte Carlo method and the improved Monte Carlo method for solving robot arm workspace. Firstly, based on the characteristic of uneven distribution of random points in the Monte Carlo method, the initial workspace of the robot arm was uniformly densified to make the inner and boundary regions of the space clear. Then, only the boundary region was encrypted by adopting the extended joint angle and the cyclic encryption of random points, so as to reduce the density of the random point cloud in the workspace. Meanwhile, the influence of initial point cloud quantity, axial segmentation voxel quantity, precision threshold, extended joint angle, and cycle number on the precision of the workspace was studied. Finally, the effectiveness of the density-reducing Monte Carlo method was verified by simulation analysis. The results show that compared with the Monte Carlo method, the total number of random point clouds of the density-reducing Monte Carlo method decreases by 93.89% when the average error rate of the workspace is 0.022 42%. In addition, compared with the improved Monte Carlo method, the density-reducing Monte Carlo method reduces the average error rate of the workspace by 0.138 53% and 0.113 29% when the number of cycles is 2 and 4, and the total number of random point clouds decreases by 44.83% and 64.52%.
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图 5 扩展关节角度对工作空间精度的影响
Figure 5. Influence of extended joint angle on workspace precision
表 1 仿人机械臂关节限位
Table 1. Joint limit of humanoid robot arm
关节角度 最大值/（°） 最小值/（°） θ1 180 −90 θ2 10 −180 θ3 120 −90 θ4 130 −60 θ5 180 −180 θ6 80 −80 θ7 90 −90 
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表 2 仿人机械臂D-H参数
Table 2. D-H parameters for humanoid robot arm
i θi /（º） di /m ai /m αi / （º） 1 90 dbs 0 −90 2 0 0 0 90 3 0 dse 0 −90 4 0 0 0 90 5 0 dew 0 −90 6 −90 0 0 90 7 0 0 awt 0
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表 3 机械臂实际工作空间范围
Table 3. Actual workspace range of robot arm
m 轴向 最大值 最小值 x 0.63975 −0.63979 y 0.63982 −0.63978 z 0.76096 −0.31102
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表 4 初始点云数量对工作空间精度的影响
Table 4. Influence of initial point cloud quantity on workspace precision
Pinit/个 xmin/m xmax/m ymin/m ymax/m zmin/m zmax/m εa/% 5000 −0.63738 0.63639 −0.63458 0.63739 −0.29514 0.76061 1.20751 10000 −0.63909 0.63447 −0.63770 0.63850 −0.30532 0.76083 0.55259 50000 −0.63939 0.63772 −0.63644 0.63939 −0.30078 0.76069 0.71627 100000 −0.63960 0.63689 −0.63531 0.63619 −0.29769 0.76089 1.00622
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表 5 体素数量对工作空间精度的影响
Table 5. Influence of voxel quantity on workspace precision
各轴向体素数量/个 xmin/m xmax/m ymin/m ymax/m zmin/m zmax/m εa/% 6 −0.63730 0.63468 −0.63729 0.63618 −0.29272 0.76062 1.34452 10 −0.63939 0.63772 −0.63644 0.63939 −0.30078 0.76069 0.71627 14 −0.63781 0.63859 −0.63663 0.63872 −0.30624 0.76076 0.45310 18 −0.63839 0.63899 −0.63931 0.63895 −0.30600 0.76087 0.36249
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表 6 精度阈值对工作空间精度的影响
Table 6. Influence of precision threshold on workspace precision
Nε xmin/m xmax/m ymin/m ymax/m zmin/m zmax/m εa/% 300 −0.63789 0.63678 −0.63716 0.63912 −0.30574 0.76089 0.49774 600 −0.63839 0.63899 −0.63931 0.63895 −0.30600 0.76087 0.36249 900 −0.63864 0.63646 −0.63758 0.63900 −0.30816 0.76086 0.35013
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表 7 最大循环次数对工作空间精度的影响
Table 7. Influence of maximum cycle number on workspace precision
Cm/次 xmin/m xmax/m ymin/m ymax/m zmin/m zmax/m εa/% 5 −0.63839 0.63899 −0.63931 0.63895 −0.30600 0.76087 0.36249 8 −0.63940 0.63868 −0.63974 0.63907 −0.30554 0.76078 0.35615 16 −0.63945 0.63785 −0.63838 0.63973 −0.30664 0.76093 0.33245
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表 8 降密蒙特卡洛法中工作空间范围及其误差率
Table 8. Workspace range and error rate by density-reducing Monte Carlo method
项目 xmin/m xmax/m ymin/m ymax/m zmin/m zmax/m 极值 −0.63961 0.63937 −0.63900 0.63956 −0.31113 0.76091 误差率 0.00018 0.00038 −0.00078 0.00026 −0.00011 0.00005
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表 9 降密蒙特卡洛法与改进蒙特卡洛法对比
Table 9. Comparison between density-reducing Monte Carlo method and improved Monte Carlo method
方法 循环
次数/次总点云数/
（×105 个）εa/% 耗时/h 改进蒙特卡洛法 2 5.8 0.68532 4.591 4 9.3 0.55971 13.841 降密蒙特卡洛法 2 3.2 0.54679 0.997 4 3.3 0.44642 1.051
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