Dimensional Synthesis Approach for Planar Open Path Based on Chord Angle Descriptor
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摘要:
针对数值图谱法提取平面开环轨迹特征计算量大、检索效率低的问题，提出一种基于弦角描述符的平面开环轨迹尺寸综合方法. 首先，利用弦角描述符的平移、旋转、缩放不变性，得到与机构机架位置、机架偏转角度和整体缩放均无关的开环轨迹形状特征；其次，基于弦角描述符自包含属性提出不受采样分辨率影响的开环轨迹部分匹配算法；然后，通过多维尺度缩放法将弦角描述符压缩为2维特征，并结合层次聚类算法，建立16 000组平面四杆机构的数值图谱库；在此基础上，根据弦角描述符与图谱库中弦角描述符的相似程度，检索出满足设计要求的机构尺寸型；最后，通过2个平面开环轨迹尺寸综合算例，验证该方法的有效性. 研究结果表明：所提尺寸综合方法能够得到满足设计要求的平面开环轨迹尺寸综合结果，且无需对轨迹进行归一化处理；与B样条曲线描述符、曲率描述符、傅里叶描述符相比，弦角描述符的开环轨迹匹配总时间分别减少了43%、35%、38%.
Abstract:To deal with huge computation and low retrieval efficiency in extracting planar open path features by a numerical atlas method, a dimensional synthesis approach was proposed for planar open paths based on chord angle descriptors. Firstly, by using the invariance of translation, rotation, and scaling of chord angle descriptors, the open path shape features independent of the frame position, orientation, and overall scaling of linkages were obtained. Secondly, based on the self-contained properties of chord angle descriptors, a partial matching algorithm not affected by the sampling resolution was proposed for open paths. Furthermore, the chord angle descriptors were compressed into two-dimensional features by multi-dimensional scaling. The numerical atlases of 16 000 groups of planar four-bar linkages were established by combining the hierarchical clustering algorithm. On this basis, the linkage dimension type suitable for design requirements was retrieved according to the similarity between the chord angle descriptors and those in the atlases. Finally, this method was validated by two-dimensional synthesis examples of planar open paths. The research findings show that overall dimensional results of planar open paths satisfying the design requirements can be achieved through the dimensional synthesis approach, and normalization is not required for paths. Compared with that of the curve descriptor, the curvature descriptor, and the Fourier descriptor of the B sample, the total open path matching time of the chord angle descriptors decreases by 43%, 35%, and 38%, respectively.
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图 12 表3机构的运动轨迹及其匹配轮廓段
Figure 12. Motion paths of linkages in Tab.3 and their matching segments
图 13 肩关节运动轨迹及坐标
Figure 13. Motion path of shoulder joint coordinate graphalong with coordinate
图 14 表4机构的运动轨迹及其匹配轮廓段
Figure 14. Motion paths of linkages in Tab.4 and their matching segments
表 1 轨迹匹配过程所用时间
Table 1. Time consumption of path matching process
描述符 归一化
时间/ms相似度
计算时间/ms总时
间/ms减少时间
百分比/%弦角描述符 13 13 B 样条曲线描述符 12 11 23 43 曲率描述符 12 8 20 35 傅里叶描述符 12 9 21 38 
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表 2 ${\boldsymbol{\varLambda }}$中最大的前10个特征值
Table 2. Top 10 largest eigenvalues in ${\boldsymbol{\varLambda }}$
编号 1 2 3 4 5 6 7 8 9 10 特征值 127984 34480 27102 19090 7166 4120 3694 3068 2586 1847
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表 3 算例1的尺寸综合结果
Table 3. Dimensional synthesis results of example 1
编号 ${r_1}$ ${r_2}$ ${r_3}$ ${r_4}$ $\phi $ ${\rho _{\max }}$ 1 1.3587 3.2236 3.8587 1.2522 − 2.2294 0.9981 2 1.0514 1.1401 2.2410 2.9365 − 0.7038 0.9980 3 0.6868 1.1227 0.7865 0.4678 0.4194 0.9974 4 0.5852 0.8668 0.9214 0.1351 − 0.7159 0.9974 5 0.3917 0.6009 1.5018 1.1475 − 0.2561 0.9973 6 0.7365 1.8271 1.6517 0.3828 0.8777 0.9973
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表 4 算例2的尺寸综合结果
Table 4. Dimensional synthesis results of example 2
编号 ${r_1}$ ${r_2}$ ${r_3}$ ${r_4}$ $\phi $ ${\rho _{\max }}$ 1 0.6485 0.7565 0.7875 2.7891 1.1536 0.9974 2 0.2227 0.7548 0.3097 0.8841 1.2532 0.9969 3 0.6797 0.3256 0.2244 0.3311 − 1.2038 0.9968 4 1.2501 0.6843 0.4309 0.7100 − 1.7501 0.9968 5 0.6241 1.0639 1.1351 1.9733 − 0.6183 0.9967 6 1.3592 1.2685 0.5040 1.8007 − 1.4115 0.9967
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