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摘要:
大型工件加工时执行机构伸展空间不足是目前航天制造业一个亟待解决的问题。为增大执行机构末端的工作空间，并为其提供稳定可靠的伸展基础，提出一类具备高刚度、大伸展特性的并联可伸展机构。重点针对并联可伸展机构展开设计与分析，基于图论进行可伸展机构构型综合，利用构型演化得到PRRR、PRRR-3R和PRRR-6R（P表示移动副，R表示转动副）这3种可伸展支链单元构型，在此基础上，对3种可伸展机构支链单元的伸展性能、刚度进行对比分析，最终优选出满足要求的PRRR支链单元配置成并联可伸展机构。该并联可伸展机构可应用于需要提供大伸展及大刚度的场合，实现对其他执行机构运动空间的拓展。
Abstract:In the process of large-scale workpiece machining, the insufficient extension space of the executive mechanism is an urgent problem to be solved in the aerospace manufacturing industry. To increase the working space at the end of the executive mechanism, and to provide the mechanism with a stable and reliable extension foundation, a parallel extendable mechanism with high stiffness and large extension is proposed. This paper focuses on the design and analysis of parallel extensible mechanisms. Based on graph theory, the configuration of the extendable mechanism was synthesized, and the three extendable branch unit configurations, PRRR, PRRR-3R and PRRR-6R (P represents prismatic pair, R represents rotation pair) were obtained by using the configuration evolution. On this basis, the extensibility and stiffness of the branch chain unit of the three extensible mechanisms were compared and analyzed. And finally, the PRRR branch chain unit that satisfies the requirements was configured as a parallel extensible mechanism. The parallel extendable mechanism can be applied to occasions that need to provide large extension and large stiffness and realize the expansion of the movement space of other actuators.
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图 4 四杆机构拓扑图Ⅳ的拓扑子图
Figure 4. Topological subgraph of topological figure Ⅳ of four-bar mechanism
图 11 PRRR支链伸展性能变化曲线
Figure 11. Stretching performance change curve of PRRR branch chain
图 12 建立PRRR-3R支链坐标系
Figure 12. Establishment of PRRR-3R branch chain coordinate system
图 13 PRRR-3R支链伸展性能变化曲线
Figure 13. Stretching performance change curve of PRRR-3R branch chain
图 14 建立PRRR-6R支链坐标系
Figure 14. Establishment of PRRR-6R branch chain coordinate system
图 15 PRRR-6R支链伸展性能变化曲线
Figure 15. Stretching performance change curve of PRRR-6R branch chain
表 1 构件数N、运动副数M和运动链闭环数L的对应关系
Table 1. Corresponding relationship between components N,motion pairs M and closed loops L
构件数N 运动副数量M 运动链的闭环数L 2 1 0 4 4 1 6 7 2 8 10 3 $ \vdots $ $ \vdots $ $ \vdots $ 
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表 2 四杆机构拓扑图Ⅳ与邻接矩阵的对应关系
Table 2. Correspondence between topological diagram Ⅳ of four-bar mechanism and adjacency matrix
顶点 1 2 3 4 1 0 1 0 1 2 1 0 1 0 3 0 1 0 1 4 1 0 1 0
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表 3 四杆机构拓扑图部分构型演化
Table 3. Partial configuration evolution of topological diagram for four-bar mechanism
运动副情况 拓扑图 演化构型 1个P副 
1个P副 
全R副 
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表 4 六杆机构拓扑VI-1的构型总数
Table 4. Total number of configurations of topological diagram for six-bar mechanism Ⅵ-1
机架 拓扑图 1个P副 2个P副 全R副 构型总数 构件1 
$ {}_1^1{M_6} = 7 $ $ {}_1^2{M_6} = 21 $ $ {}_1^0{M_6} = 1 $ 29 构件2 
$ {}_2^1{M_6} = 4 $ $ {}_2^2{M_6} = 12 $ $ {}_2^0{M_6} = 1 $ 17
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表 5 六杆机构拓扑VI-2的构型总数
Table 5. Total number of configurations of topological diagram for six-bar mechanism Ⅵ-2
机架 拓扑图 1个P副 2个P副 全R副 构型总数 构件1 
$ {}_1^1{M_6} = 5 $ $ {}_1^2{M_6} = 13 $ $ {}_1^0{M_6} = 1 $ 19 构件2 
$ {}_2^1{M_6} = 4 $ $ {}_2^2{M_6} = 12 $ $ {}_2^0{M_6} = 1 $ 17 构件5 
$ {}_5^1{M_6} = 5 $ $ {}_5^2{M_6} = 13 $ $ {}_5^0{M_6} = 1 $ 19
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表 6 六杆机构拓扑图部分构型演化
Table 6. Partial configuration evolution of topological diagram for six-bar mechanism
序号 运动副情况 拓扑图 演化构型 1 1个P副 
2 1个P副 
3 1个P副 
4 1个P副 
5 全R副 
6 全R副 
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表 7 八杆机构拓扑图部分构型演化
Table 7. Partial configuration evolution of topological diagram of eight-bar mechanism
运动副情况 构型拓扑图 演化构型 1个P副 
全R副 
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表 8 可伸展支链构型拓扑图及其演化构型
Table 8. Topological diagram of extendable branched chain configuration and its evolutionary configuration
支链单元 机构 构型拓扑图 演化构型 PRRR
支链四杆机构 
PRRR-3R
支链六杆机构 
PRRR-6R
支链八杆机构 
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表 9 不同可伸展机构的伸展性能比较
Table 9. Comparison of stretching performance of different extendable mechanisms
序
号基本支
链单元移动距离
$\Delta y$/mm起始高度/
mm终止高度/
mm高度差
$\Delta h$/mm1 PRRR 支链 1000 1636.3068 421.3075 1214.9993 2 PRRR-3R
支链1000 2545.3662 655.3672 1889.9990 3 PRRR-6R
支链1000 2454.4602 631.9612 1822.4990
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表 10 3种支链构型的线伸展比
Table 10. Linear extension ratio of three branched chain configurations
基本支链单元 支链尺寸参数 高度差$\Delta h$/mm 线伸展比 PRRR
支链
1214.9993 1.215 PRRR-3R
支链
1889.9990 1.890 PRRR-6R
支链
1822.4990 1.822
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表 11 45号钢的相关性能参数
Table 11. Relevant performance parameters of 45# steel
材料名称 弹性模量/
(N·m−2)泊松比 质量密度/
(kg·m−3)屈服强度/
(N·m−2)45号钢 2.09×1011 0.269 7.89×103 3.55×108
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表 12 3种可伸展支链构型的性能比较
Table 12. Stiffness comparison of three extendable branched chain configurations
基本支链
单元可伸展支链
结构可伸展支链
应力云图最大应
力/ MPa占[σ]
百分比/%伸展
性能Θ并联可伸展机构 PRRR
支链
22.5 15.8 1.215 
PRRR-3R
支链
98.4 69.3 1.890 
PRRR-6R
支链
94.8 66.8 1.822 
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