Quaternion Form of Longitude Latitude and Heading Kinematics Equations

Gan Xin

doi:doi:10.11648/j.ajtte.20230804.13

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Quaternion Form of Longitude Latitude and Heading Kinematics Equations

Gan Xin

Published in American Journal of Traffic and Transportation Engineering (Volume 8, Issue 4)

Received: 20 July 2023 Accepted: 8 August 2023 Published: 22 August 2023

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Abstract

The kinematics equations of longitude and latitude have singularities in polar of the earth. To solve the problem, a solution called the quaternion form of longitude, latitude and heading kinematics equations was created and introduced in the paper. The key point of the solution is to define an instantaneous great circle for a moving particle. To a moving particle, it is impossible to define three definite Euler angles, thus the definite quaternion to it does not exist. But to the instantaneous great circle, three definite Euler angles can be defined. Meanwhile, the instantaneous great circle is rotating by driving of the moving particle, thus quaternion can be used to model the instantaneous great circle. The model is the kinematics equations of longitude, latitude and heading in quaternion form. This form of equations can be used all over of the earth. It works well on the polar of the earth automatically. Verifying by mathematics simulation has been designed and practiced. The simulation includes some flights around the earth with flying by the polar and turning in polar region. The results of simulation suggest that the flight plan can be executed precisely by the algorithm. The solution can be applied in fields of flight simulation and inertial navigation.

Published in	American Journal of Traffic and Transportation Engineering (Volume 8, Issue 4)
DOI	10.11648/j.ajtte.20230804.13
Page(s)	99-104
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Navigation, Flight Simulation, Polar, Quaternion, Longitude, Latitude, Heading

References

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[3]	Titterton D H. (1997). J L Weston, Strapdown Inertial Navigation Technology [M], on behalf of the Institution of Electrical Engineers, London.
[4]	Esmat Berkir. (2007). Introduction to modern navigation system [M]. New Jersey, USA, World Scientific.
[5]	Aboelmagd Noureldin, Tashfeen B. Karamat, Jacques Georgy. (2013). Fundamentals of Inertia Navigation, Satellite-based Positioning and their Integration. Srpinger. DOI: 10.1007/978-3-642-30466-8.
[6]	Savage P G. (1998). Strapdown inertial navigation integration algorithm design part 2: velocity and position Algorithm [J]. Journal of guidance, Control and dynamics, vol. 21 (2), pp. 208-229.
[7]	Dyer G C. (1971). "Polar navigation - A new transverse Mercator technique [J]". Journal of Navigation-, vol. 24 (4), pp. 484-495.
[8]	Davydenko L. (2015). "Arctic navigation [J]", Polar Navigation. vol. 2 (1), pp. 173-181.
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[10]	Gade K. (2010). "A non-singular horizontal position representation [J]". The Journal of Navigation, vol. 63, pp. 395-417.
[11]	Yongyuan Qin. (2020) Inertia Navigation System. Beijing. Science Press. (in Chinese).
[12]	Zhou Q, Qin Y Y, Fu Q W. (2013). Grid Mechanization in Inertial Navigation Systems for Transpolar Aircraft. Journal of Northwestern Polytechnical University, vol. 31 (2), pp. 210-217. (in Chinese).
[13]	Qi Zhou, Yazhou Yue, Xiaodong Zhang & Yu Tian. (2014). Indirect Grid Inertial Navigation Mechanization for Transpolar Aircraft. Journal of Chinese Inertial Technology. doi: 10.13695/j.cnki.12-1222/o3.2014.01.005.
[14]	Yiqing Yao, Xiaosu Xu & Jinwu Tong. (2015). Indirect Transverse Inertial Navigation Algorithm in Polar Region. Journal of Chinese Inertial Technology. doi: 10.13695/j.cnki.12-1222/o3.2015.01.007.
[15]	Meng Liu, Guangchun Li, Yanbin Gao, Ying Li & Di Wang. (2017). Algorithm of Global Navigation for Inertial Navigation System Based on Pseudo-Earth Frame. Journal of Chinese Inertial Technology. doi: 10.13695/j.cnki.12-1222/o3.2017.04.004.
[16]	Chan Liu, Wenqi, Wu, Guohu Feng & Maosong Wang. (2020). Polar Navigation Algorithm for INS Based on Virtual Sphere n-vector. Journal of Chinese Inertial Technology. doi: 10.13695/j.cnki.12-1222/o3.2020.04.001.

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Gan Xin. (2023). Quaternion Form of Longitude Latitude and Heading Kinematics Equations. American Journal of Traffic and Transportation Engineering, 8(4), 99-104. https://doi.org/10.11648/j.ajtte.20230804.13

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ACS Style

Gan Xin. Quaternion Form of Longitude Latitude and Heading Kinematics Equations. Am. J. Traffic Transp. Eng. 2023, 8(4), 99-104. doi: 10.11648/j.ajtte.20230804.13

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AMA Style

Gan Xin. Quaternion Form of Longitude Latitude and Heading Kinematics Equations. Am J Traffic Transp Eng. 2023;8(4):99-104. doi: 10.11648/j.ajtte.20230804.13

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@article{10.11648/j.ajtte.20230804.13,
  author = {Gan Xin},
  title = {Quaternion Form of Longitude Latitude and Heading Kinematics Equations},
  journal = {American Journal of Traffic and Transportation Engineering},
  volume = {8},
  number = {4},
  pages = {99-104},
  doi = {10.11648/j.ajtte.20230804.13},
  url = {https://doi.org/10.11648/j.ajtte.20230804.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtte.20230804.13},
  abstract = {The kinematics equations of longitude and latitude have singularities in polar of the earth. To solve the problem, a solution called the quaternion form of longitude, latitude and heading kinematics equations was created and introduced in the paper. The key point of the solution is to define an instantaneous great circle for a moving particle. To a moving particle, it is impossible to define three definite Euler angles, thus the definite quaternion to it does not exist. But to the instantaneous great circle, three definite Euler angles can be defined. Meanwhile, the instantaneous great circle is rotating by driving of the moving particle, thus quaternion can be used to model the instantaneous great circle. The model is the kinematics equations of longitude, latitude and heading in quaternion form. This form of equations can be used all over of the earth. It works well on the polar of the earth automatically. Verifying by mathematics simulation has been designed and practiced. The simulation includes some flights around the earth with flying by the polar and turning in polar region. The results of simulation suggest that the flight plan can be executed precisely by the algorithm. The solution can be applied in fields of flight simulation and inertial navigation.},
 year = {2023}
}

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TY - JOUR
T1 - Quaternion Form of Longitude Latitude and Heading Kinematics Equations
AU - Gan Xin
Y1 - 2023/08/22
PY - 2023
N1 - https://doi.org/10.11648/j.ajtte.20230804.13
DO - 10.11648/j.ajtte.20230804.13
T2 - American Journal of Traffic and Transportation Engineering
JF - American Journal of Traffic and Transportation Engineering
JO - American Journal of Traffic and Transportation Engineering
SP - 99
EP - 104
PB - Science Publishing Group
SN - 2578-8604
UR - https://doi.org/10.11648/j.ajtte.20230804.13
AB - The kinematics equations of longitude and latitude have singularities in polar of the earth. To solve the problem, a solution called the quaternion form of longitude, latitude and heading kinematics equations was created and introduced in the paper. The key point of the solution is to define an instantaneous great circle for a moving particle. To a moving particle, it is impossible to define three definite Euler angles, thus the definite quaternion to it does not exist. But to the instantaneous great circle, three definite Euler angles can be defined. Meanwhile, the instantaneous great circle is rotating by driving of the moving particle, thus quaternion can be used to model the instantaneous great circle. The model is the kinematics equations of longitude, latitude and heading in quaternion form. This form of equations can be used all over of the earth. It works well on the polar of the earth automatically. Verifying by mathematics simulation has been designed and practiced. The simulation includes some flights around the earth with flying by the polar and turning in polar region. The results of simulation suggest that the flight plan can be executed precisely by the algorithm. The solution can be applied in fields of flight simulation and inertial navigation.
VL - 8
IS - 4
ER -

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Gan Xin

Xi’an Aircraft Industry (Group) Hangdian Technology Co. Limited, Xi’an, China

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APA Style

Gan Xin. (2023). Quaternion Form of Longitude Latitude and Heading Kinematics Equations. American Journal of Traffic and Transportation Engineering, 8(4), 99-104. https://doi.org/10.11648/j.ajtte.20230804.13

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ACS Style

Gan Xin. Quaternion Form of Longitude Latitude and Heading Kinematics Equations. Am. J. Traffic Transp. Eng. 2023, 8(4), 99-104. doi: 10.11648/j.ajtte.20230804.13

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AMA Style

Gan Xin. Quaternion Form of Longitude Latitude and Heading Kinematics Equations. Am J Traffic Transp Eng. 2023;8(4):99-104. doi: 10.11648/j.ajtte.20230804.13

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@article{10.11648/j.ajtte.20230804.13,
  author = {Gan Xin},
  title = {Quaternion Form of Longitude Latitude and Heading Kinematics Equations},
  journal = {American Journal of Traffic and Transportation Engineering},
  volume = {8},
  number = {4},
  pages = {99-104},
  doi = {10.11648/j.ajtte.20230804.13},
  url = {https://doi.org/10.11648/j.ajtte.20230804.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtte.20230804.13},
  abstract = {The kinematics equations of longitude and latitude have singularities in polar of the earth. To solve the problem, a solution called the quaternion form of longitude, latitude and heading kinematics equations was created and introduced in the paper. The key point of the solution is to define an instantaneous great circle for a moving particle. To a moving particle, it is impossible to define three definite Euler angles, thus the definite quaternion to it does not exist. But to the instantaneous great circle, three definite Euler angles can be defined. Meanwhile, the instantaneous great circle is rotating by driving of the moving particle, thus quaternion can be used to model the instantaneous great circle. The model is the kinematics equations of longitude, latitude and heading in quaternion form. This form of equations can be used all over of the earth. It works well on the polar of the earth automatically. Verifying by mathematics simulation has been designed and practiced. The simulation includes some flights around the earth with flying by the polar and turning in polar region. The results of simulation suggest that the flight plan can be executed precisely by the algorithm. The solution can be applied in fields of flight simulation and inertial navigation.},
 year = {2023}
}

Copy | Download