Abstract
A finite algorithm for determining a vehicle’s position by differences in measured pseudoranges to known reference points is considered. Equations are derived for the case of excessive number of reference points and for coplanar reference points. A convenient complex-valued form of the problem solution is obtained for the coplanar case.
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References
Barabanov, O.O. and Barabanova, L.P., Algorithms for solving the navigation range-difference problem: from Apollonius to Cauchy, Istoriya nauki i tekhniki, 2008, no. 11, pp. 3–21.
Shebshaevich, V.S., Preliminary assessment of the possibilities of using artificial satellites for navigation, LKVVIA im. A.F. Mozhaiskogo, 1958, no. 33, pp. 5–38.
Shebshaevich, V.S., Dmitriev, P.P., Ivantsevich, N.V., et al. Setevye sputnikovye radionavigatsionnye sistemy (Network Satellite Radio Navigation Systems), Moscow: Radio i svyaz’, 1993.
Krause, L.O., A direct solution to GPS-type navigation equations, IEEE Trans. on Aerospace and Electronic Systems, vol. AES-23, no. 2, 1985, pp. 225–232.
Bancroft, S., An algebraic solution of the GPS equations, IEEE Trans. on Aerospace and Electronic Systems, vol. AES-21, no. 6, 1985, pp. 56–59.
Abel, J.S. and Chaffee, J.W., Existence and uniqueness of GPS solutions, IEEE Trans. on Aerospace and Electronic Systems, vol. 27, no. 6, 1991, pp. 952–956.
Chaffee, J.W. and Abel, J.S., On the exact solutions of pseudorange equations, IEEE Trans. on Aerospace and Electronic Systems, vol. 30, no. 4, 1994, pp. 1021–1030.
Hoshen, J., The GPS equations and the problem of Apollonius, IEEE Trans. on Aerospace and Electronic Systems, vol. 32, no. 3, 1996, pp. 1116–1124.
Grafarend, E.W. and Shan, J., A closed-form solution of the nonlinear pseudo-ranging equations (GPS), Artificial Satellites, Planetary Geodesy, no. 28, vol. 31, no. 3, 1996. pp. 133–147.
Grafarend, E.W. and Shan, J., GPS Solutions: closed forms, critical and special configurations of P4P, GPS Solutions, vol. 5, no. 3, 2002, pp. 29–41.
Leva, J.L., An alternative to closed-form solution of the GPS pseudo-range equations, IEEE Trans. on Aerospace and Electronic Systems, vol. 32, no. 4, 1996, pp. 1430–1439.
Kleusberg, A., Analytical GPS navigation solution, Festshrift for E. W. Grafarend on the occasion of his 60th birthday, Report no. 1999.6.1, 1999, pp. 247–251.
Awange, J.L. and Grafarend, E.W., Algebraic solution of GPS pseudo-ranging equations, GPS Solutions, vol. 5, no. 4, 2002, pp. 20–32.
Barabanov, O.O. and Barabanova, L.P., Matematicheskie zadachi dal’nomernoi navigatsii (Mathematical Problems of Range-measuring Navigation), Moscow: Fizmatlit, 2007.
Babich, O.A., Obrabotka informatsii v navigatsionnykh kompleksakh (Information Processing in Navigation Integrated Systems), Moscow: Mashinostroenie, 1991.
Magaril-Il’yaev, G.G. and Tikhomirov, V.M., Vypuklyi analiz i ego prilozheniya (Convex analysis and its Applications), Moscow: Editorial URSS, 2000.
Voevodin, V.V. and Kuznetsov, Yu.A., Matritsy i vychisleniya (Matrices and Calulations), Moscow: Nauka, 1984.
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Published in Giroskopiya i Navigatsiya, 2015, No. 2, pp. 106–117.
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Derevyankin, A.V., Matasov, A.I. Finite algorithm for determining a vehicle’s position by differences in measured pseudoranges. Gyroscopy Navig. 7, 100–106 (2016). https://doi.org/10.1134/S2075108716010041
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DOI: https://doi.org/10.1134/S2075108716010041