research article a simple perturbation algorithm for...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 712729, 5 pageshttp://dx.doi.org/10.1155/2013/712729

Research ArticleA Simple Perturbation Algorithm for Inverting the Cartesian toGeodetic Transformation

James D. Turner and Tarek Elgohary

Aerospace Engineering Department, Texas A&M University, 701 H.R. Bright Building, 3141 TAMU,College Station, TX 77843-3141, USA

Correspondence should be addressed to Tarek Elgohary; [email protected]

Received 26 March 2013; Accepted 18 June 2013

Academic Editor: Silvia Maria Giuliatti Winter

Copyright © 2013 J. D. Turner and T. Elgohary. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

A singularity-free perturbation solution is presented for inverting the Cartesian to Geodetic transformation. Geocentric latitude isused to model the satellite ground track position vector. A natural geometric perturbation variable is identified as the ratio of themajor and minor Earth ellipse radii minus one. A rapidly converging perturbation solution is developed by expanding the satelliteheight above theEarth and the geocentric latitude as a perturbation power series in the geometric perturbation variable.The solutionavoids the classical problem encountered of having to deal with highly nonlinear solutions for quartic equations. Simulation resultsare presented that compare the solution accuracy and algorithm performance for applications spanning the LEO-to-GEO range ofmissions.

1. Introduction

A frequent calculation for satellites in low Earth orbit (LEO)to geosynchronous Earth orbit (GEO) involves invertingtransformations between 3D satellite Cartesian Earth centredcoordinates and geodetic coordinates. The geodetic coordi-nates consist of 𝜆𝑔, 𝜙𝑔, and ℎ, which denote the geodeticlongitude of the satellite subpoint 𝑔, the geodetic latitudeof the satellite, and the height of the satellite above thereference Earth elliptical surface along the surface normalfrom the geodetic ellipsoid to the satellite position. Referringto Figure 1, the transformation from geodetic coordinates toCartesian (𝑥𝑠, 𝑦𝑠, 𝑧𝑠) coordinates is given by [1]

𝑥𝑠 = (𝑁(𝜙𝑔) + ℎ) cos𝜙𝑔 cos 𝜆𝑔,

𝑦𝑠 = (𝑁(𝜙𝑔) + ℎ) cos𝜙𝑔 sin 𝜆𝑔,

𝑧𝑠 = (𝑁(𝜙𝑔) (1 − 𝑒2) + ℎ) sin𝜙𝑔,

(1)

where𝑁(𝜙𝑔) = 𝑎/√ 1 − 𝑒2sin2𝜙𝑔 denotes the ellipsoid radiusof curvature in the prime vertical plane defined by vectors n(ellipsoid outward normal) and �� (local east), ℎ is assumed

to lie along n, 𝑎 denotes the semimajor axis, 𝑏 denotes thesemiminor axis (Figure 2), and 𝑒 denotes the eccentricityof the Earth’s reference ellipsoid. The solution for 𝜆𝑔 =

tan−1(𝑦𝑠/𝑥𝑠) is obtained by elementary methods.Because of the fundamental problem of nonlinearity,

successive approximation strategies are required for 𝜙𝑔 andℎ. A two-step process is introduced to solve for 𝜙𝑔. First, oneinverts for the geocentric latitude, 𝜙𝑐, Figure 2; second, usingstandard trigonometric identities, 𝜙𝑔 is recovered. A suc-cessive approximation strategy is developed by introducinga naturally available geometric perturbation variable, whichis defined as 𝑝 = 𝑎/𝑏 − 1 ∼ 0.0034. Rapidly convergentapproximations are obtained for 𝜙𝑐 and ℎ in the 𝜏 − 𝑧

plane by developing power series in the expansion variable𝑝. Elementary vector methods are introduced for invertingfor the satellite height, ℎ. The resulting analytic perturbationsolutions are remarkably simple and computationally effi-cient.

Many methods have been proposed for implementingthe inverse of the transformation presented in (1). Thenonlinear Cartesian-to-Geodetic transformation problem ischallenging, as geometrical singularities plague many solu-tion strategies. The solution for the geodetic longitude,

2 Mathematical Problems in Engineering

z

y

x

North

East

𝜆g

𝜙g

𝜏

n

Figure 1: Geodetic and Cartesian coordinates.

Zs

Xs

𝜙c 𝜙g

h

b

a

n

rrg

Figure 2: Geodetic versus geocentric latitude.

however, is elementary and noniterative. The most commonproblem encountered is the need for handling sensitivequartic polynomial solutions [2–5]. The analytic complexityof the problem arises because the geodetic latitude andsatellite height solution algorithms are coupled and highlynonlinear. Three classes of methods have been proposed:(i) closed-form solutions for cubic and quartic polynomials,(ii) perturbationmethods, and (iii) successive approximationalgorithms. The closed-form class of solution algorithmstypically introduces sequences of trigonometric transfor-mations that exploit identities to simplify the governingequation. Important examples of this approach include thafollowing: (i) the very well-known solution in [6], wherethe reduced latitude is iterated in Newton’s method; (ii)a closed-form solution for a high-order algebraic equation[7]; (iii) introducing the geodetic height of the satellite todevelop an elliptic integral-based arc-length solution [8]; (iv)development of an approximate closed-form solution [9]; and(v) introduction of complicated algebraic transformations todevelop a series solution [10]. Not only are the proposedclosed-form solutions highly accurate, but they are alsocomputationally expensive to perform.

Many iterative techniques have been proposed. Earlyexamples of this approach include thework of [1] which influ-enced the GPS-based need for the geodetic transformationmethods developed by [11–13]. Several innovative problemformulations have been proposed, including the work of [3, 5,14, 15]. Unfortunately, geometric singularities plague many ofthese iterative strategies. To avoid troublesome singularities,several authors have investigated vector methods, includingthe work of [4, 16]. In [17], an elegant optimization-basedstrategy is presented. Accelerated convergence techniques areconsidered by [18] who has presented a third-order version ofNewton’s method (known as Halley’s method). Recently, [19]has presented a very fast singularity-free second-order per-turbation solution that introduces an artificial perturbationvariable to transform the classical quartic solution probleminto a singularity-free noniterative quadratic equation prob-lem. In a more recent addition to iterative methods [20], theprojection of a point on the reference ellipsoid is used to solvea system of nonlinear equations using second- and third-order Newton’s method. The results presented by the authorsshow millimeter accuracy in height and 10−8 degree accuracyin latitude with the third-order approach.

Themain contribution of this paper is the presentation ofa noniterative series-based solution algorithm that effectivelyprovides a closed-form solution for the Cartesian-to-Geodetictransformation throughout the LEO-to-GEO range of appli-cations.

2. Mathematical Formulation

The problem is formulated by introducing a local coordinatesystem that tracks the local 𝑥-𝑦 axis motion of the satellite.In the local coordinate system, a simplified perturbationsolution is developed in the 𝜏 − 𝑧 plane by defining a vectorconstraint of the form

r − r𝑔 − ℎn = 0, (2)

where r = (𝑟𝑥𝑦, 𝑧) denotes the satellite position vector, with

𝑟𝑥𝑦 = √𝑥2 + 𝑦2, r𝑔 = (𝑎 cos(𝜙𝑐), 𝑏 sin(𝜙𝑐)) denotes the satel-lite ground track point, 𝜙𝑐 denotes the geocentric latitude, ℎdenotes the height of the satellite above the Earth’s surface,and n=(cos(𝜙𝑐)/𝑎, sin(𝜙𝑐)/𝑏)/√((cos(𝜙𝑐)/𝑎)

2+ (sin(𝜙𝑐)/𝑏)

2)

denotes the unit vector that is normal to the Earth’s surfaceand points at the satellite. Expanding (2) provides twonecessary conditions:

𝑟𝑥𝑦 − 𝑎 cos (𝜙𝑐) −ℎ cos (𝜙𝑐)

𝑎√((cos (𝜙𝑐) /𝑎)2+ (sin (𝜙𝑐) /𝑏)

2)

= 0,

𝑧 − 𝑏 sin (𝜙𝑐) −ℎ sin (𝜙𝑐)

𝑏√((cos (𝜙𝑐) /𝑎)2+ (sin (𝜙𝑐) /𝑏)

2)

= 0.

(3)

Mathematical Problems in Engineering 3

Clearly, the equations are highly nonlinear. To begin thesimplification process, one replaces 𝑎 in (3) with

𝑎 = 𝑏 (1 + 𝑝) , 𝑝 ≈ 0.00314, (4)

which exploits the natural parameter for the problem andtransforms (3) into

𝑟𝑥𝑦 − 𝑏 (1 + 𝑝) cos (𝜙𝑐)

−ℎ cos (𝜙𝑐)

√((cos (𝜙𝑐))2+ ((1 + 𝑝) sin (𝜙𝑐))

2)

= 0,

𝑧 − 𝑏 sin (𝜙𝑐) −ℎ sin (𝜙𝑐)

√((cos (𝜙𝑐) /(1 + 𝑝))2+ (sin (𝜙𝑐))

2)

= 0.

(5)

An approximate solution is recovered by assuming thatthe geocentric latitude and satellite height are expanded inthe power series representations as follows:

𝜙𝑐 = 𝜙0 + 𝑝𝜙1 + 𝑝2𝜙2 + ⋅ ⋅ ⋅ ,

ℎ = ℎ0 + 𝑝ℎ1 + 𝑝2ℎ2 + ⋅ ⋅ ⋅ .

(6)

Introducing (6) into (5) and collecting terms in powers of𝑝 yield the cascade of necessary conditions as follows:

𝑂(𝑝0) : ℎ0 = √𝑟

2𝑥𝑦+ 𝑧2 − 𝑏,

𝜙0 = 2 tan−1(

√𝑟2𝑥𝑦+ 𝑧2

𝑧−𝑟𝑥𝑦

𝑧) ,

𝑂 (𝑝1) : ℎ1 = −𝑏 cos (𝜙0)

2,

𝜙1 =𝑏 − ℎ0

2 (𝑏 + ℎ0)sin (2𝜙0) ,

𝑂 (𝑝2) : ℎ2 =

𝑏 (3𝑏 − ℎ0)

8 (𝑏 + ℎ0)sin (2𝜙0)

2,

𝜙2 =ℎ2

0− 4𝑏ℎ0 + 3𝑏

2

8(𝑏 + ℎ0)2

sin (4𝜙0) +sin (2𝜙0)

4,

𝑂 (𝑝3) : ℎ3 =

𝑏 sin2 (2𝜙0)8(𝑏 + ℎ0)

2[(ℎ0−3𝑏)

2cos2 (𝜙0)+4𝑏 (ℎ0 − 𝑏)] ,

Table 1: List of polynomial coefficients.

Coefficient Expression𝐶1

−4ℎ3

0+ 37𝑏

3− 66𝑏

2ℎ0+ 33𝑏ℎ

2

0

𝐶2

ℎ3

0− 31𝑏

3+ 75𝑏

2ℎ0− 33𝑏ℎ

2

0

𝐶3

3𝑏 (𝑏2+ 3ℎ2

0− 6𝑏ℎ

0)

𝐶4

139𝑏3− 5ℎ3

0+ 49𝑏ℎ

2

0− 143𝑏

2ℎ0

𝐶5

ℎ3

0− 127𝑏

3+ 163𝑏

2ℎ0− 45𝑏ℎ

2

0

𝐶6

4𝑏 (−10𝑏ℎ0+ ℎ2

0+ 5𝑏2)

𝐶7

4ℎ4

0+ 118𝑏

4+ 198𝑏

2ℎ2

0− 266𝑏

3ℎ0− 54𝑏ℎ

3

0

𝐶8

−155𝑏4− 2ℎ4

0+ 67𝑏ℎ

3

0+ 421𝑏

3ℎ0− 315𝑏

2ℎ2

0

𝐶9

49𝑏4− 15𝑏ℎ

3

0− 185𝑏

3ℎ0+ 135𝑏

3ℎ2

0

𝐶10

2𝑏2(10𝑏ℎ

0− 5ℎ2

0− 𝑏2)

𝜙3 =sin (2𝜙0)6(𝑏 + ℎ0)

3[𝐶1cos

4(𝜙0) + 𝐶2cos

2(𝜙0) + 𝐶3] ,

𝑂 (𝑝4) : ℎ4 =

𝑏 sin2 (2𝜙0)32(𝑏 + ℎ0)

3[𝐶4cos

4(𝜙0)+𝐶5cos

2(𝜙0)+𝐶6] ,

𝜙4 =sin (2𝜙0)4(𝑏 + ℎ0)

4[𝐶7cos

6(𝜙0) + 𝐶8cos

4(𝜙0)

+𝐶9cos2(𝜙0) + 𝐶10] .

(7)

Simple algebraic manipulations yield the ten coefficientsappearing in (7) as the polynomials presented in Table 1.

These analytic results are very compact for a fourth-orderperturbation expansion. The conversion from the geocentricto the geodetic latitude is given by

𝜙𝑔 = tan−1 (cos𝜙𝑐

𝑎/𝑏 sin𝜙𝑐) . (8)

3. Numerical Results

The perturbation expansion method is used to carry outthe coordinate transformation for several cases of LEO-to-GEO orbits. Using the WGS84, the forward transformationis carried out first, then the perturbation solution is applied,and the results are compared with the original values, whichrepresent exact values for the inverse solution. For the sakeof demonstration a longitude angle of 30∘ is utilized. Thegeodetic latitude, 𝜙𝑔, is swept for angles from −90 to 90degrees and the height is swept from 200KM (LEO) to35,000KM (GEO). First, the expansion is carried to secondorder, and the errors in latitude and height are plotted asfunctions of the true latitudes and heights as shown in Figures3 and 4, respectively. The expansion is then carried out tothird order, and the errors in latitude and height are plottedin Figures 5 and 6, respectively. Finally, the fourth-orderexpansion is utilized, and results are shown in Figures 7 and8, respectively.

The improvement of accuracy is quite obvious as theorder of expansion is increased. A two-order-of-magnitude

4 Mathematical Problems in Engineering

050

100

01

23

4

0

0.5

1

1.5

−0.5

−1

−1.5

−100−50

×10−6

×104

h (km)𝜙g (deg)

Δ𝜙g(deg)

Figure 3: Errors in latitude, second-order expansion.

050

100

01

23

4

0

0.02

0.04

0.06

−0.02

−0.04

−100−50h (km)

Δh(m

)

𝜙g (deg)

×104

Figure 4: Errors in height, second-order expansion.

050

100

01

23

4

0

5

−5

−100−50

×10−9

×104

h (km)

Δ𝜙g(deg)

𝜙g (deg)

Figure 5: Errors in latitude, third-order expansion.

improvement is achieved by adding the third-order termsto each of the coordinates. Another two-order-of-magnitudeimprovement is achieved with the fourth-order terms. Inheight, millimeter accuracy is achieved at the fourth-orderexpansion level. This shows the fast convergence natureand the accuracy of the perturbation solution. These results

050

100

01

23

4

0

1

2

3

−1

−2

−3

−100−50

×10−4

×104

h (km)

Δh(m

)

𝜙g (deg)

Figure 6: Errors in height, third-order expansion.

050

100

01

23

4

0

1

2

3

−1

−2

−100−50

×10−11

×104

h (km)

Δ𝜙g(deg)

𝜙g (deg)

3

Figure 7: Errors in latitude, fourth-order expansion.

demonstrate that higher-order approximations do not pro-vide additional useful information for the inversion process.

4. Conclusion

Earth-Centered Earth-Fixed (ECEF) to geodetic coordinatetransformation has been examined with several numericaland analytical approaches throughout the literature. A non-iterative expansion-based approach inspired by the Earth’sperturbed geometry is introduced in this work, where theexpansion parameter is nothing but the ratio of the Earthsemimajor axis and semi-minor axis subtracted from 1. Theexpansion is carried out to second, third, and fourth orders.A numerical example is introduced to compare the accuraciesat each order of expansion. Accuracies showed significantimprovements as the order of expansion is increased, andthe at fourth order, millimeter accuracy is achieved in heightand 10−11 degree error in latitude. Those errors at such loworders of the expansion are proof of the effectiveness of themethod and its potential in solving such a highly nonlineartransformation noniteratively. The method can be furtherstreamlined for timing studies, but in general it is a cleanstraightforward approach to the coordinate transformation

Mathematical Problems in Engineering 5

050

100

01

23

4

0246

−8−6−4−2

−100−50

×10−7

×104

h (km)

Δh(m

)

𝜙g (deg)

Figure 8: Errors in height, fourth-order expansion.

problem that utilizes a physical perturbation parameter andthat proved to be very accurate and efficient.

References

[1] W. A. Heiskanen and H. Moritz, Physical Geodesy, W. H.Freeman, San Francisco, Calif, USA, 1967.

[2] T. Fukushima, “Fast transform from geocentric to geodeticcoordinates,” Journal of Geodesy, vol. 73, no. 11, pp. 603–610,1999.

[3] K. M. Borkowski, “Accurate algorithms to transform geocentricto geodetic coordinates,” Bulletin Geodesique, vol. 63, no. 1, pp.50–56, 1989.

[4] J. Pollard, “Iterative vector methods for computing geodeticlatitude and height from rectangular coordinates,” Journal ofGeodesy, vol. 76, no. 1, pp. 36–40, 2002.

[5] K.-C. Lin and J. Wang, “Transformation from geocentricto geodetic coordinates using Newton’s iteration,” BulletinGeodesique, vol. 69, no. 4, pp. 300–303, 1995.

[6] B. R. Bowring, “Transformation from spatial to geographicalcoordinates,” Survey Review, vol. 23, no. 181, pp. 323–327, 1976.

[7] V. C. Petr and P. Vanıcek, Geodesy: The Concepts, Elsevier, NewYork, NY, USA, 1982.

[8] M. Pick and Z. Simon, “Closed formulae for transformationof the cartesian coordinate system into a system of geodeticcoordinates,” Studia Geophysica et Geodaetica, vol. 29, no. 2, pp.112–119, 1985.

[9] A. Fotiou, “A pair of closed expressions to transform geocentricto geodetic coordinates,” Zeitschrift fur Vermessungswesen, vol.123, no. 4, pp. 133–135, 1998.

[10] H. Vermeille, “Computing geodetic coordinates from geocen-tric coordinates,” Journal of Geodesy, vol. 78, no. 1-2, pp. 94–95,2004.

[11] A.Kleusberg andP. J. Teunissen,GPS forGeodesy, Springer,NewYork, NY, USA, 1998.

[12] B. Hofmann-Wellenhof, H. Lichtenegger, and J. Collins, GlobalPositioning System: Theory and Practice, Springer, New York,NY, USA, 1997.

[13] G. Strang and K. Borre, Linear Algebra, Geodesy, and GPS,Wellesley Cambridge Press, 1997.

[14] W. Torge, “Regional gravimetric geoid calculations in the NorthSea test area,”Marine Geodesy, vol. 3, no. 1–4, pp. 257–271, 1980.

[15] L. O. Lupash, “A new algorithm for the computation of thegeodetic coordinates as a function of earth-centered earth-fixedcoordinates,” Journal of Guidance, Control, and Dynamics, vol.8, no. 6, pp. 787–789, 1985.

[16] J. Feltens, “Vector method to compute the Cartesian (X, Y, Z) togeodetic (Φ, 𝜆, h) transformation on a triaxial ellipsoid,” Journalof Geodesy, vol. 83, no. 2, pp. 129–137, 2009.

[17] C.-D. Zhang, H. T. Hsu, X. P.Wu et al., “An alternative algebraicalgorithm to transform Cartesian to geodetic coordinates,”Journal of Geodesy, vol. 79, no. 8, pp. 413–420, 2005.

[18] T. Fukushima, “Transformation from Cartesian to geodeticcoordinates accelerated byHalley’smethod,” Journal of Geodesy,vol. 79, no. 12, pp. 689–693, 2006.

[19] J. D. Turner, “A non-iterative and non-singular perturbationsolution for transforming Cartesian to geodetic coordinates,”Journal of Geodesy, vol. 83, no. 2, pp. 139–145, 2009.

[20] M. Ligas and P. Banasik, “Conversion between Cartesian andgeodetic coordinates on a rotational ellipsoid by solving asystem of nonlinear equations,” Geodesy and Cartography, vol.60, no. 2, pp. 145–159, 2011.

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