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Great Ellipse Route Planning Based on Space Vector
LIU Wenchao, BIAN Hongwei, WANG Rongying, WEN Chaojiang
Department of Navigation Engineering, Naval University of Engineering, Wuhan 430033, China
First author: LIU Wenchao(1988-)，male，PhD candidate，majors in ship navigation technology. E-mail： wenchao19880102@126.com
Abstract: Aiming at the problem of navigation error caused by unified earth model in great circle route planning using sphere model and modern navigation equipment using ellipsoid mode, a method of great ellipse route planning based on space vector is studied. By using space vector algebra method, the vertex of great ellipse is solved directly, and description of great ellipse based on major-axis vector and minor-axis vector is presented. Then calculation formulas of great ellipse azimuth and distance are deduced using two basic vectors. Finally, algorithms of great ellipse route planning are studied, especially equal distance route planning algorithm based on Newton-Raphson(N-R) method. Comparative examples show that the difference of route planning between great circle and great ellipse is significant, using algorithms of great ellipse route planning can eliminate the navigation error caused by the great circle route planning, and effectively improve the accuracy of navigation calculation.
Key words: great circle     great ellipse     route planning     space vector     Newton-Raphson method

﻿1 引 言

2 大椭圆描述方法 2.1 位置相关矢量[12]

 图 1 地球椭球体 Fig. 1 Earth ellipsoid

2.2 大椭圆顶点、长轴矢量和短轴矢量

 图 2 大椭圆顶点、长轴矢量和短轴矢量 Fig. 2 Vertex,major-axis vector and minor-axis vector of great ellipse

φV代入式(3)，易知大椭圆短半径b

3 大椭圆航线航程和方位角 3.1 大椭圆航线航程计算

 图 3 大椭圆剖面图 Fig. 3 Section of Great ellipse

PQ分别为大椭圆航线的始点和终点，应用式(18)求出相应的θ角，分别为θPθQ，考虑参考椭球上两点之间的大椭圆航线为劣弧，因此PQ两点之间的航程为

3.2 大椭圆航线方位角计算

 图 4 大椭圆航线方位角 Fig. 4 Azimuth of Great ellipse

4 大椭圆航线设计算法 4.1 等经差大椭圆航线设计算法

4.2 等距离大椭圆航线设计方法

5 算例分析

 大椭圆航线 大圆航线 总航程/n mile 5 474.484 5 458.840 初始方位角 139°39′44″ 139°47′23″

 大椭圆航线 大圆航线 分点经度/(°) 分点纬度/(°) 分段大椭圆航程/n mile 分段恒向线航程/n mile 分段恒向线方位/(°) 分点经度/(°) 分点纬度/(°) 分段大圆航程/n mile 分段恒向线航程/n mile 分段恒向线方位/(°) 20.000 -35.000 825.335 826.015 135.875 20.000 -35.000 824.402 826.654 135.875 32.500 -44.890 629.327 630.021 127.141 32.500 -44.890 627.727 629.616 127.141 45.000 -51.226 507.419 508.066 117.436 45.000 -51.226 505.684 507.292 117.436 57.500 -55.121 437.811 438.412 107.197 57.500 -55.121 436.096 437.525 107.197 70.000 -57.277 404.946 405.521 96.686 70.000 -57.277 403.263 404.604 96.686 82.500 -58.062 401.322 401.894 86.080 82.500 -58.062 399.645 400.975 86.080 95.000 -57.606 426.120 426.712 75.529 95.000 -57.606 424.413 425.812 75.529 107.500 -55.832 484.985 485.618 65.199 107.500 -55.832 483.248 484.801 65.199 120.000 -52.443 591.257 591.941 55.321 120.000 -52.443 589.592 591.398 55.321 132.500 -46.835 765.961 766.656 46.277 132.500 -46.835 764.768 766.921 46.277 145.000 -38.000 - - - 145.000 -38.000 - - - 总计/n mile 5 474.484 5 480.856 5 458.840 5 475.599

 大椭圆航线 大圆航线 分点经度/(°) 分点纬度/(°) 分段大椭圆航程/n mile 分段恒向线航程/n mile 分段恒向线方位/(°) 分点经度/(°) 分点纬度/(°) 分段大圆航程/n mile 分段恒向线航程/n mile 分段恒向线方位/(°) 20.000 -35.000 547.448 547.615 137.320 20.000 -35.000 545.884 547.084 137.325 27.876 -41.717 547.448 547.767 131.523 27.858 -41.704 545.884 547.237 131.540 37.480 -47.768 547.448 548.031 123.595 37.448 -47.752 545.884 547.504 123.621 49.359 -52.817 547.448 548.418 113.123 49.321 -52.805 545.884 547.896 113.152 63.826 -56.400 547.448 548.800 100.208 63.794 -56.395 545.884 548.282 100.227 80.382 -58.018 547.448 548.890 85.958 80.368 -58.017 545.884 548.374 85.960 97.387 -57.374 547.448 548.602 72.258 97.395 -57.373 545.884 548.081 72.245 112.898 -54.594 547.448 548.188 60.655 112.921 -54.588 545.884 547.663 60.637 125.903 -50.122 547.448 547.866 51.650 125.926 -50.112 545.884 547.338 51.636 136.437 -44.460 547.448 547.670 44.984 136.451 -44.450 545.884 547.140 44.979 145.000 -38.000 - - - 145.000 -38.000 - - - 总计/n mile 5 474.484 5 481.847 5 458.840 5 476.598

6 结 论

http://dx.doi.org/10.11947/j.AGCS.2015.20130799

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#### 文章信息

LIU Wenchao, BIAN Hongwei, WANG Rongying, WEN Chaojiang

Great Ellipse Route Planning Based on Space Vector

Acta Geodaeticaet Cartographica Sinica, 2015, 44(7): 741-746.
http://dx.doi.org/10.11947/j.AGCS.2015.20130799