﻿ 新型2-2PRUR并联机构运动学分析及工作空间优化<sup>*</sup>
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Kinematic analysis and workspace optimization of a novel 2-2PRUR parallel mechanism
LI Dian, GUO Sheng, HUANG Guanyu, ZHAO Fuqun
Robotics Institute, School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China
Received: 2016-10-19; Accepted: 2016-12-16; Published online: 2017-01-10 15:04
Foundation item: National Natural Science Foundation of China (51475035)
Corresponding author. GUO Sheng, E-mail: shguo@bjtu.edu.cn
Abstract: A novel 2-2PRUR parallel mechanism is proposed to fit the actual demand of industrial production line. Kinematics analysis and workspace optimization are performed based on scatter plot. A method is presented by splitting the moving platform into two parts and adding planetary gear train, which could solve the over-constraints of four degrees of freedom parallel mechanism and increase the turning angle. The constraint equations are obtained using the coordinate method. The inverse-forward solutions of the mechanism are worked out. And the workspace is obtained by using the scatter plot of inverse kinematic solutions. Meanwhile, reasonable structural parameters are obtained by optimizing workspace with the principle of maximizing scattered points based on genetic algorithm. This work would lay the foundation for the future research and application of this type of parallel mechanism.
Key words: parallel mechanism     non-overconstrained     degree of freedom     kinematic analysis     workspace optimization

Guo等提出了一种设计非过约束四自由度(3T1R)并联机构的构型综合方法[15]，解决了驱动和约束的冲突，实现了三平动一转动机构的设计综合方法，但机构所输出的转动运动十分有限，不能充分满足工业生产的需要。

1 机构的提出及自由度分析 1.1 新型并联机构动平台的构建

 图 1 动平台拆分过程 Fig. 1 Moving platform split process

 图 2 行星轮系结构示意图 Fig. 2 Schematic diagram of planetary gear train structure
 图 3 动平台结构示意图 Fig. 3 Schematic diagram of moving platform structure
1.2 具新型动平台结构并联机构的提出

 图 4 2-2PRUR机构及其简图 Fig. 4 2-2PRUR mechanism and its diagrams
1.3 新型并联机构自由度分析

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1.3.1 4PRUR并联机构自由度分析

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 图 5 4PRUR与2-2PRUR机构构型简图 Fig. 5 4PRUR and 2-2PRUR mechanism configurations' diagrams

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1.3.2 2-2PRUR并联机构自由度分析

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Pab(a=1, 2;b=1, 2, 3, 4) 的形式说明当动平台被拆分且未连接时，在支链的作用下，副动平台P1可实现沿xyz轴的移动和沿y轴的转动，副动平台P2可实现沿xyz轴的移动和沿x轴的转动。

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2 机构运动学分析 2.1 2-2PRUR并联机构分析模型建立

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D为中心齿轮的齿距，则中心齿轮转过的齿数为：Z′=Y/D，其中：

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2.2 机构杆长约束方程

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2.3 运动学位置反解

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2.4 运动学位置正解

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2.5 位置正反解计算实例

 mm x y1 y2 z l1 l2 l3 l4 160 100 200 -150 106.0144 66.5415 21.0464 95.1702 140 100 200 -150 105.2031 21.0464 66.5415 108.8730 150 90 190 -150 105.9616 66.5415 21.0464 103.9075 150 110 210 -150 105.9616 21.0464 66.5415 99.04466 160 110 210 -120 136.0012 79.3318 79.3318 120.7044 140 100 200 -120 135.2031 51.0464 96.5415 138.8730

 mm l1 l2 l3 l4 x y1 y2 z 106 66 21 95 160.0099 100.0035 200.3694 -150.0141 105 21 66 108 146.8173 107.2413 206.5883 -148.6636 105 66 21 103 149.9972 90.1848 189.6942 -150.9609 105 21 66 99 149.3046 109.5547 209.1319 -150.9323 136 79 79 120 160.5072 110.3637 210.6410 -119.9843 135 51 96 138 146.8173 107.2413 206.5883 -178.6636

3 机构工作空间分析

 参数 初始值 kij/mm 150 a/mm 300 l/mm 100 r/mm 50 li/mm 0≤li≤100 P(x，y，z)运动范围/mm 50≤x≤250 50≤y≤250 -400≤z≤-100

 图 6 姿态角为0°时的工作空间 Fig. 6 Workspace with posture angle values 0°

 图 7 姿态角为±45°时的工作空间 Fig. 7 Workspace with posture angle values ±45°

4 基于遗传算法的工作空间优化

 图 8 各代适应度函数m值 Fig. 8 m values of each generation of fitness function

 参数 初始值 优化值 k11/mm 150 58.21356151 k12/mm 150 128.4783055 k21/mm 150 57.12848677 k22/mm 150 116.0951255 a/mm 300 226.6077689 l/mm 100 148.1780424 r/mm 50 149.8667562 m 37852×3 161928×3

 图 9 优化后的工作空间视图 Fig. 9 Optimized workspace view

 图 10 优化前后的工作空间截面边界图 Fig. 10 Cross section boundary of workspace before and after optimization

5 结论

1) 提出了新型四自由度并联机构与消除过约束、增大转动自由度角度的方法，确定四自由度的运动性质为三平动一绕z轴的转动。

2) 建立了新型四自由度并联机构约束方程，计算位置正反解，并以位置反解为基础绘制机构工作空间。

3) 提出以散点数目代替体积表达式优化工作空间体积的新方法，基于遗传算法优化机构工作空间，给出优化机构尺寸。对此类机构进一步研究和发展应用有重要意义。

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文章信息

LI Dian, GUO Sheng, HUANG Guanyu, ZHAO Fuqun

Kinematic analysis and workspace optimization of a novel 2-2PRUR parallel mechanism

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(10): 2011-2020
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0816