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缓和曲线正交拟合的Levenberg-Marquardt算法

宋占峰 王健 李军

宋占峰, 王健, 李军. 缓和曲线正交拟合的Levenberg-Marquardt算法[J]. 西南交通大学学报, 2020, 55(1): 144-149. doi: 10.3969/j.issn.0258-2724.20190130
引用本文: 宋占峰, 王健, 李军. 缓和曲线正交拟合的Levenberg-Marquardt算法[J]. 西南交通大学学报, 2020, 55(1): 144-149. doi: 10.3969/j.issn.0258-2724.20190130
SONG Zhanfeng, WANG Jian, LI Jun. Levenberg-Marquardt Algorithm for Orthogonal Fitting of Transition Curves[J]. Journal of Southwest Jiaotong University, 2020, 55(1): 144-149. doi: 10.3969/j.issn.0258-2724.20190130
Citation: SONG Zhanfeng, WANG Jian, LI Jun. Levenberg-Marquardt Algorithm for Orthogonal Fitting of Transition Curves[J]. Journal of Southwest Jiaotong University, 2020, 55(1): 144-149. doi: 10.3969/j.issn.0258-2724.20190130

缓和曲线正交拟合的Levenberg-Marquardt算法

doi: 10.3969/j.issn.0258-2724.20190130
基金项目: 国家自然科学基金资助项目(51678574)
详细信息
    作者简介:

    宋占峰(1973—),男,副教授,博士,研究方向为道路与铁道线路优化设计方法,E-mail:songzhanfeng@csu.edu.cn

    通讯作者:

    李军(1973—),男,副教授,博士,研究方向为道路与铁道线路优化设计方法,E-mail:lijun_csu@csu.edu.cn

  • 中图分类号: U212.3

Levenberg-Marquardt Algorithm for Orthogonal Fitting of Transition Curves

  • 摘要: 为了由测量点识别既有线路中的缓和曲线参数,研究了基于参数方程的缓和曲线正交拟合迭代优化方法. 首先,通过特征值分析,阐明了由于病态性的存在,在迭代过程中,常规的Gauss-Newton (GN)算法会发散. 其次,提出了双目标优化模型,将GN算法与最速下降法结合,确定了正交拟合缓和曲线的Levenberg-Marquardt (LM)算法. 同时提出了在寻优过程中,评估当前迭代位置距离最优位置的远近来动态设置LM参数. 最后以一段缓和曲线的实测点为例,随机取样了5 000例初值,采用蒙特卡罗方法对比了GN算法和LM算法拟合缓合曲线的性能. 试验结果表明:GN算法拟合缓合曲线不收敛;对于不同的初始值,LM算法都收敛到相同的最优值,体现了LM算法具有良好的稳健性;LM算法的迭代次数最少为5次,最大为50次,平均为16.8次,迭代次数和初值与最优值位置的远近相关.

     

  • 图 1  缓和曲线正交拟合

    Figure 1.  Details for transition curve orthogonal fitting

    图 2  LM算法流程

    Figure 2.  Flowchart of the LM algorithm

    图 3  等高线图中的迭代搜索路径

    Figure 3.  Search paths in contour maps

    表  1  测量点坐标

    Table  1.   Coordinates of measured points m

    测量点编号xy
    1865.463134.175
    2846.063149.943
    3826.580165.608
    4807.136181.321
    5791.543193.847
    6771.752209.123
    7759.529218.475
    下载: 导出CSV

    表  2  Gauss-Newton算法迭代过程中的参数和相应指标

    Table  2.   Parameters and indexes of Gauss-Newton fitting processes

    $k$xo /myo /mα/(º)A/m${\rm{lg}}\;F $${\rm{lg}}\left( {\left\| {{{J}}_k^{\rm{T}}{{r}}\left( {{{{\varTheta}} _k}} \right)} \right\|} \right)$${\rm{lg}}\left( {\left\| {{{{d}}_{k,\;{\rm{L}}}}} \right\|} \right)$
    0 900.000 100.000 140.249 600.000 2.072 3.537 −∞
    1 735.480 243.913 142.385 108.016 3.980 4.404 2.898
    2 879.610 110.057 176.670 26.008 8.952 8.327 2.763
    3 802.088 131.635 176.339 −3.367 7.878 8.138 2.520
    4 802.088 155.634 179.746 −4.745 9.416 9.523 1.709
    5 775.148 160.986 181.215 −3.605 12.229 12.451 1.803
    下载: 导出CSV

    表  3  缓和曲线拟合LM算法性能及参数识别统计结果

    Table  3.   Statistical results of the LM algorithm performance and parameter identification

    E(xo)/mE(yo)/mE(A)/mE(α)/(º)E(k)/次σ(xo)/mmσ(yo)/mmσ(A)/mmσ(α)/(″)σ(k)/次
    838.843155.731369.277140.94416.80.0010.0010.0070.0109.2
    下载: 导出CSV

    表  4  LM算法迭代过程中的参数和相应指标

    Table  4.   Parameters and indexes of LM fitting processes

    $k$xo /myo /mα/(º)A/m${\rm{lg}}\;F$${\rm{lg}}\left( {\left\| {{{J}}_k^{\rm{T}}{{r}}\left( {{{{\varTheta}} _k}} \right)} \right\|} \right)$$\lg\left( {\left\| { { {{d} }_{k,\;{\rm{L} } } }} \right\|} \right)$
    0 900.000 100.000 140.249 600.00 2.072 3.537 −∞
    1 902.656 103.143 140.258 600.370 −1.216 −1.895 0.618
    2 893.401 111.244 140.586 625.124 −1.391 0.825 1.630
    3 889.194 114.670 140.635 625.146 −1.519 −0.964 1.186
    $\boxed4$ 869.025 131.312 140.810 546.664 −1.291 1.530 2.033
    4 886.501 116.895 140.664 617.487 −1.528 −0.081 1.097
    $\boxed5$ 866.523 133.349 140.827 534.455 −1.326 1.488 2.044
    5 884.137 118.844 140.684 608.059 −1.536 −0.232 1.107
    $\boxed6$ 863.879 135.507 140.846 521.855 −1.329 1.474 2.056
    6 881.760 120.796 140.703 597.916 −1.546 −0.260 1.126
    $\boxed7$ 861.167 137.718 140.865 508.737 −1.326 1.462 2.067
    7 879.306 122.812 140.722 587.188 −1.557 −0.255 1.146
    $\boxed8$ 858.414 139.961 140.885 495.097 −1.326 1.446 2.078
    8 876.759 124.902 140.742 575.845 −1.567 −0.244 1.167
    $\boxed9$ 855.641 142.217 140.903 480.977 −1.329 1.425 2.088
    9 874.114 127.070 140.762 563.828 −1.582 −0.234 1.188
    $\boxed{10}$ 852.880 144.460 140.921 466.465 −1.338 1.396 2.095
    10 871.363 129.321 140.782 551.071 −1.597 −0.226 1.211
    $\boxed{11}$ 850.172 146.655 140.938 451.716 −1.355 1.356 2.100
    11 868.503 131.659 140.803 537.503 −1.615 −0.221 1.234
    12 865.534 134.084 140.824 523.060 −1.634 −0.223 1.257
    13 862.459 136.591 140.844 507.692 −1.657 −0.233 1.279
    14 859.294 139.168 140.865 491.390 −1.684 −0.257 1.301
    15 856.090 141.790 140.884 474.232 −1.714 −0.303 1.318
    16 852.847 144.407 140.902 456.449 −1.747 −0.389 1.329
    17 839.303 155.394 140.971 379.957 −1.767 0.556 1.959
    18 839.639 155.090 140.944 374.792 −1.881 −0.781 0.726
    19 838.843 155.731 140.944 369.277 −1.882 −7.307 −2.208
    下载: 导出CSV
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出版历程
  • 收稿日期:  2019-03-05
  • 修回日期:  2019-06-11
  • 网络出版日期:  2019-09-18
  • 刊出日期:  2020-02-01

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