A non-iterative and non-singular perturbation solution for transforming ...

J Geod (2009) 83:139–145 DOI 10.1007/s00190-008-0247-4

ORIGINAL ARTICLE

A non-iterative and non-singular perturbation solution for transforming Cartesian to geodetic coordinates James D. Turner

Received: 5 September 2007 / Accepted: 10 July 2008 / Published online: 6 August 2008 © Springer-Verlag 2008

Abstract The Cartesian-to-Geodetic coordinate transformation is re-cast as a minimization algorithm for the height of the Satellite above the reference Earth surface. Optimal necessary conditions are obtained that fix the satellite ground track vector components in the Earth surface. The introduction of an artificial perturbation variable yields a rapidly converging second order power series solution. The initial starting guess for the solution provides 3–4 digits of precision. Convergence of the perturbation series expansion is accelerated by replacing the series solution with a Padé approximation. For satellites with heights λ = λ0 − ( pλ2 −λ 1)

Padé

Step 5: Compute non-dimensional satellite sub-point ground track coordinates ⎞ ⎛ ⎞ ⎛ ρx u s,x / (1 + 2λ) Step 5.1: ⎝ ρ y ⎠ = ⎝ u s,y / (1 + 2λ)  ⎠ u s,z / 1 + 2k 2 λ ρz Step 6: Compute physical domain ground track coordinates r g = a(ρx ρ y ρz )T Step 7: Compute geodetic longitude from Eq. (1) as λg = atan2(y, x) for |x| > 0.5 or λg π = − atan2(x, y) for |x| < 0.5 2 Step 8: Compute the height of the satellite above the reference earth ellipse H =h=


(r s − r g )T (r s − r g )

Step 9: Compute the geodetic latitude
  2 + ρ2 φg = atan2 k 2 ρg,z , ρg,x for g,y |r g,z | > 0.5 or 
 π 2 + ρ2 , k2ρ for φg = −atan2 ρg,x g,z g,y 2 |r g,z | < 0.5.

References Appendix Appendix A: Summary of computations The following steps in order of solution are required for completing the Cartesian-to-Geodetic transformation. Step 1: Obtain satellite position vector in earth-centered earth-fixed coordinates Step 2: Compute non-dimensional form of satellite position vector coordinates

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Allan DW, Weiss MA (1980) Accurate time and frequency transfer during common-view of a GPS satellite. In: Proc. 34th ann. freq. control symposium, USAERADCOM, FT. Monmouth, NJ Awange JL, Grafarend EW (2005) Solving algebraic computational problems in geodesy and geoinformatics. In: Earth and environmental sciences series, chap 2. Springer, Berlin, pp 147–164 Baker GA, Graves-Morris P (1981a) Pade approximants. Part I: Basic theory. Addison-Wesley, Reading Baker GA, Graves-Morris P (1981b) Pade approximants. Part II: Extensions and applications. Addison-Wesley, Reading Baker GA, Graves-Morris P (1996) Padé approximants encyclopedia of mathematics and its applications 2nd edn, vol 59. Cambridge University Press, Cambridge

A non-iterative and non-singular perturbation solution Borkowski KM (1989) Accurate algorithms to transform geocentric to geodetic coordinates. Bull Geod 63:50–56. doi:10.1007/ BF02520228 Bowring BR (1976) Transformation from spatial to geographical coordinates. Surv Rev 181:323–327 Bruno OP, Reitich F (1994) Approximation of analytic functions: a method of enhanced convergence. Math Comput 63(207):195– 213. doi:10.2307/2153569 Featherstone WE, Claessens SJ (2008) Closed-form transformation between geodetic and ellipsoidal coordinates. Stud Geophys Geod (to appear) Feltens J (2007) Vector methods to compute azimuth, elevation, ellipsoidal normal, and the Cartesian (X, Y, Z) to geodetic (φ, λ, h) transformation. J Geod. doi:10.1007/s00190-007-0198-1 Fotiou A (1998) A pair of closed expressions to transform geocentric to geodetic coordinates. Zeitschrift f¨ur Vermessungswesen 123(4):133–135 Fukushima T (1999) Fast transform from geocentric to geodetic coordinates. J. Geod. 73:603–610. doi:10.1007/s001900050271 Fukushima T (2006) Tansformation from Cartesian to geodetic coordinates accelerated by Halley’s method. J Geod 79(12):689–693. doi:10.1007/s00190-006-0023-2 Heiskanen WA, Moritz H (1967) Physical geodesy. W. H. Freeman and Co., San Francisco 181 Hofmann-Wellenhof B, Lichtenegger H, Collins J (1997) GPS theory and practice. Springer, Berlin Jentschura UD, Becher J, Weniger EJ, Scoff G (2000) Resummantion of QED perturbation series by sequence transformations and the prediction of perturbative coefficients. Phys Rev Lett 85(12): 2446–2449. doi:10.1103/PhysRevLett.85.2446 Jones GC (2002) New solutions for the geodetic coordinate transformation. J Geod 76(8):437–446. doi:10.1007/s00190-002-0267-4 Junkins JL, Turner JD (1977) Development of a Space Oblique Mercator Map Projection. In: Symposium presented to the US Coast and Geodetic Survey, McLean, VA Junkins JL, Turner JD (1978) A distortion-free map projection for analysis of satellite imagery. J Astronaut Sci XXVI(3): 211–243 Kleusber A, Teunissen P (eds) (1996) GPS for geodesy. Springer, Berlin Lin KC, Wang J (1995) Transformations from geocentric coordinates using Newton’s iteration. Bull Geod 69:14–17. doi:10.1007/ BF00806742

145 Lin X, Kirubarajan T, Bar-Shalom Y (2001) Enhanced accuracy GPS navigation using the interactiing multiple model estimator. In: Aerospace conf., IEEE proceeding, pp 4–1911, 4–1923 Lupash LO (1985) A new algorithm for the computation of the geodetic coordinates as a function of earth-centered earth-fixed coordinates. J Guidance Dyn Control Nov–Dec:787–789 MACSYMA (1996) Symbolic/numeric/graphical mathematics software: mathematics and system reference manual, 16th edn. Macsyma, Inc. Arlington Meriovitch L (1970) Methods of analytical dynamics. McGraw-Hill, New York Nayfeh AH (1985) Problems in perturbation. Wiley, London Pick M (1985) Closed fomulae for transformation of the Cartesian coordinate system into a system of geodetic coordinates. Stud Geophys Geod 29(2). doi:10.1007/BF01585714 Pollard J (2002) Iterative vector methods for computing geodetic latitude and height from rectangular coordinates. J Geod 76:36–40. doi:10.1007/s001900100220 Samuel MA, Ellis J, Karliner M (1995) Comparison of the pade approximation method to perturbative QCD calculations. Phys Rev Lett 74(22):4380–4383. doi:10.1103/PhysRevLett.74.4380 Sjoberg LE (1999) An efficient iterative solution to transform rectangular geocentric coordinates to geodetic coordinates. ZfV 9:295–29 Strang G, Borre K (1997) Linear algebra, geodesy, and GPS. WellesleyCambridge Press, Wellesley Torge W (1980) Geodesy, 3rd edn. de Gruyter, ISBN 3110170728 Vaníˇcek P, Krakiwski EJ (1982) Geodesy: the concepts. North Holland, Amsterdam, p 324 Vermeille H (2002) Direct transformation from geocentric coordinates to geodetic coordinates. J Geod 76:451–454. doi:10.1007/ s00190-002-0273-6 Vermeille H (2004) Computing geodetic coordinates from geocentric coordinates. J Geod 78(1–2):94–95. doi:10.1007/ s00190-004-0375-4 You RJ (2000) Transformation of Cartesian to geodetic coordinates without iterations. J Surv Eng Feb:1, 7 Zhang CD, Hsu HT, Wu XP, Li SS, Wang QB, Chai HZ, Du L (2005) An alternative algebraic algorithm to transform Cartesian to geodetic coordinates. J Geod 79:413–420. doi:10.1007/ s00190-005-0487-5

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