[10]
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Kirubarajan, T., Bar-Shalom, Y., Blair, W. D., and Watson, G. A. (1998) IMMPDA solution to benchmark for radar resource allocation and tracking in the presence of ECM. IEEE Transactions on Aerospace and Electronic Systems, 34, 3 (Oct. 1998), 1023–1036. Li, X. R. (1996) Hybrid estimation techniques. In C. T. Leondes (Ed.), Control and Dynamic Systems, Vol. 76, San Diego, CA: Academic Press, 1996. Mazor, E., Averbuch, A., Bar-Shalom, Y., and Dayan, J. (1998) Interacting multiple model methods in target tracking: A survey. IEEE Transactions on Aerospace and Electronic Systems, 34 (Jan. 1998), 103–123. Wang, H., Kirubarajan, T., and Bar-Shalom, Y. (1999) Large scale air traffic surveillance using IMM estimators with assignment. IEEE Transactions on Aerospace and Electronic Systems, 35, 1 (Jan. 1999), 255–266. Yeddanapudi, M., Bar-Shalom, Y., and Pattipati, K. R. (1997) IMM estimation for multitarget-multisensor air traffic surveillance. IEEE Proceedings, 85, 1 (Jan. 1997), 80–94.
Algorithm of Earth-Centered Earth-Fixed Coordinates to Geodetic Coordinates
The transformation between Earth-centered Earth-fixed (ECEF) coordinates and geodetic coordinates is required in many applications. This correspondence presents an iterative approach using the Newton-Raphson (NR) method that has good efficiency and accuracy and is free from singularity and nonconvergence except in a small region near the center of the
Fig. 1. Reference ellipsoid and geodetic parameters.
divided into two categories: exact approaches and approximations. After comparisons among several exact transformations, Zhu [4] pointed out that at least one cubic root had to be calculated for exact approaches. It is highly desirable to find efficient algorithms that can be used in real-time applications, so the transformation is usually carried out by approximation in practice. Presently, there have been many different methods to approximate the exact transformation, e.g., Lupash [1], Nautiyal [2], and Olson [3]. Unaware of the existence of Nautiyal [2], we developed a similar but more efficient iterative algorithm, of good efficiency and accuracy and free from singularity and nonconvergence for all points excluding a kernel region of the Earth of radius about 50 km. In the next section, mathematical formulation is provided concerning the transformations from ECEF coordinates to geodetic coordinates. In Section III, the algorithm is presented in a concise manner and in Section IV, the algorithm is investigated on computer and some graphical results are presented. Comparisons with other well-known efficient approximations are made in Section V. Conclusions are drawn in Section VI.
Earth. Comparisons with other well-known approximate methods are made in complexity and performance.
I.
INTRODUCTION
The transformation between Earth-centered Earth-fixed (ECEF) coordinates and geodetic coordinates is required in many applications, for example in GPS navigation and geodesy. Many papers have dwelled upon this problem in early decades and proposed a number of methods that can be Manuscript received June 2, 2002; revised April 7 and July 11, 2003; released for publication September 5, 2003. IEEE Log No. T-AES/39/4/822072. Refereeing of this contribution was handled by J. L. Leva. This work was supported in part by the National Natural Science Foundation of China under Grants 60374006 and 60171003. c 2003 IEEE 0018-9251/03/$17.00 CORRESPONDENCE
II.
MATHEMATICAL FORMULATION
The relationship between the reference ellipsoid and geodetic parameters of points in the ECEF frame are depicted in Fig. 1. For the reference ellipsoid, R length of the semimajor axis (R = 6378:137 km based on Model WGS-84), r length of the semiminor axis (r = 6356:7523142 km based on Model WGS-84), f flattening of the ellipsoid (f = (R r)=R). For the point P, L geodetic latitude, ¸ geodetic longitude, h altitude normal to reference ellipsoid.
In Fig. 1, P1 is the intersecting point of the reference ellipsoid and the normal through P, and P2 is the intersecting point of the reference ellipsoid and the line passing through P and O, the center of the Earth. Let the ECEF coordinates of P be (Xe Ye Ze ). 1457
satisfy the condition f(t) = 0 where f(t) = At4 + (B + C)t3 + (B C)t A
(7)
and A = rz0 ,
(1)
where atan 2(y, x) is the four-quadrant inverse tangent, which returns the arctangent of y=x in the range ¼ to ¼ rad. We now consider the intersecting part of the reference ellipsoid and the plane passing through the point P, the point O, and the Ze axis. In Fig. 2, we see that the intersection is an ellipse that can be represented as z2 x2 + 2 = 1: (2) 2 R r The coordinates of P in the OXZ frame are (x0 z0 ), which are given by T z0 = Ze : x0 = Xe2 + Ye2 ,
(3)
Next we parameterize, in terms of parameter µ, the ellipse equation as x = R cos µ ¼ µ < ¼: , (4) z = r sin µ Accordingly, coordinates of P1 can be represented as (R cos µ r sin µ) and the vector normal to the ellipse is (cos µ=R sin µ=r). In fact from the formation of intersecting ellipse, we are certain that parameters µ of points P1 and P2 are confined to the range [¼=2 ¼=2], that is, by construction points P1 and P2 lie on the right semiellipse with x 0. Therefore the geodetic latitude is related to the parameter µ associated with P1 by tan L =
sin µ=r cos µ=R
or cot L = (1 f) cot µ:
(8)
(6)
k = 0, 1, 2, : : :
(9)
in which t0 is the initial guess value for t and f (t) = 4At3 + 3(B + C)t2 + (B C). As we all know, the initial value is of vital importance to the convergence of NR method, so great care must be taken choosing t0 . Having noticed that the semimajor and semiminor axes are almost equal in length, we naturally take the geodetic latitude of point P2 as that of point P and further calculate t0 on this hypothesis. In view of Fig. 2, it appears that there are just two real roots for t (one related with P1 , the other related with P1 ) for the quartic (7), one of which has different sign from the other. This fact is greatly favorable to us, because the NR method would quadratically converge nearby a distinct root if the initial estimate was reasonable. Luckily, it is quite clear in Fig. 2 that our initial value t0 is of the same sign with the real root we want, the one related with P1 . In Section IV, we show numerical results that the selection of t0 in this way is quite ideal for the iterative process. And furthermore, the root related with P1 , normally being distinct, ensures f (t) = 0 nearby the root, which is advantageous when using NR method. However, the above statements do not hold true when the height takes large negative values and point P is near the center of the Earth. When a point is deep within the Earth (see P , P2 and h in Fig. 2) point P2 departs from point P1 violating the rationality of approximating the geodetic latitude of point P by that of point P2 , which can lead to nonconvergence of the NR method. From (4), t0 would be easily acquired. Let coordinates of point P1 and point P2 in the OXZ frame be (x1 z1 ) and (x2 z2 ), respectively. Using tan µ = 2t=(1 t2 ) we obtain
With a second parameter t = tan(µ=2), (6) can be rewritten in terms of the variable t. After rearranging the resultant equation, the parameter t is observed to 1458
tk+1 = tk f(tk )=f (tk ),
(5)
The line normal to the ellipse at point P1 , which passes through P, is specified as r cos µ x0 R cos µ : = z0 r sin µ R sin µ
C = 2(R 2 r2 ):
Equation (7) is a quartic equation whose analytic roots could be acquired after tedious development, in which at least one cubic root operation is needed (see Zhu [4]). However, the roots of (7) can also be readily approximated by the powerful Newton-Raphson (NR) method, that is
Fig. 2. Ellipse and geodetic latitude.
The longitude of the point P can be directly determined by ¸ = atan 2(Ye , Xe )
B = 2Rx0 ,
(10)
z r z z tan µ0 = 1 2 = 0 R x1 x2 x0
or z0 t2 + 2(1 f)x0 t z0 = 0
(11)
where µ0 is the parameter µ of the intersecting ellipse that is related with point P1 . Equation (11) has two
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roots, one of which can be abandoned under the condition ¼=4 µ=2 ¼=4, i.e., MtM 1. The desired root is V x0 x 2 t0 = (1 f) + sgn(z0 ) 1 + (1 f) 0 (12) z0 z0 where sgn(x) is the sign function, which gives +1 when x is greater than zero, 1 when less than zero. Having determined t, we calculate the geodetic latitude from (5) and (10), obtaining 1 t2 : L = acot (1 f) (13) 2t In Fig. 2, the relationship between coordinates of point P and point P1 is
Fig. 3. Iteration numbers as function of height and geodetic latitude.
z0 = z1 + h sin L: Using (4) and sin µ = 2t=(1 + t2 ) we obtain the geodetic height through the equation V 2 1 t2 2t f) : h = sgn(L) z0 r 1 + (1 1 + t2 2t
8) If k < kmax , go to Step 5; else the algorithm does NOT converge, go to Step 12. 9) Determine geodetic latitude L (see (13)). 10) Determine geodetic longitude ¸ (see (1)). 11) Determine geodetic height h (see (14)). 12) End.
(14) Before the end of this section, we present, in the context of this correspondence, the inverse transformation formulas that include (5) and
IV.
M = x1 + h cos L = R cos µ + h cos L Xe = M cos ¸ Ye = M sin ¸
(15)
Ze = z1 + h sin L = r sin µ + h sin L which transform geodetic coordinates to ECEF coordinates, for the purpose of simulation and comparisons in Sections IV and V. III.
ALGORITHM
Basing on the preceding discussion, we now implement our algorithm in a concise manner as follows. Given coordinates Xe , Ye , Ze , ellipsoid parameters R, r, f, iterative accuracy ±, and maximum limit of iteration kmax : 1) Start. 2) Compute x0 , z0 (see (3)) and coefficients: A, B, C (see (8)). 3) If z0 = 0, let L = 0, h = x0 R and go to Step 10; else go to Step 4. 4) Compute initial value t0 for iteration (see (12)) and initialize k = 0. 5) Set k = k + 1. 6) Compute tk as the function of tk1 (see (9)). 7) If Mtk tk1 M ±, go to Step 9; else go to Step 8. CORRESPONDENCE
NUMERICAL RESULTS
According to the algorithm outlined in last section, we now do some simulations on a computer with PII 450 MHz processor and 256 M memory. The code is programmed using Matlab 6.1. In this section, 18,000 test points are considered using (15): for height, h takes 20 points from 6, 300 km to 30,000 km; for geodetic latitude, L takes 30 points in the range [90 90 ]; and for geodetic longitude, ¸ takes 30 points in the range [180 180 ]. Iterative accuracy ± is set to 109 . Similar with Zhu [4], the error in our algorithm is estimated as the Euclid distance between the given coordinates and calculated coordinates, which are results of (15) using estimated geodetic parameters obtained by our algorithm. The reported errors in the following text are the maximum with respect to the geodetic longitude ¸, if not specifically mentioned. The results are graphically shown is Figs. 3 and 4, in which Fig. 3 gives the iteration numbers and Fig. 4 gives the Euclid distance errors. Except for the range of height from 6, 300 km to 5, 000 km in which iteration numbers are mostly between 4 and 6, the iteration numbers are less than 3. The Euclid distance errors are all below 2
108 m. Since most applications of this algorithm are near the surface of the Earth, let us examine the results with smaller range of height. For height, h takes 20 points from 50 km to 1,000 km; and geodetic latitude and geodetic longitude are unchanged. The results are shown in Figs. 5 and 6. In this special section of space, smaller errors are obtained. 1459
Fig. 4. Euclid distance errors as function of height and geodetic latitude.
Fig. 7. Nonconvergences in ECEF frame when test points come close to center of Earth. TABLE I Comparison of Arithmetic Operations
Fig. 5. Iteration numbers as function of height and geodetic latitude.
Algorithms
Cubic Roots
Square Roots
Mul/Div
Add/Sub
Trigonometric
Lupash A Lupash B Nautiyal Olson This paper
0 0 0 0 0
3 3 4 4 3
24 + 11k 1 + 9k 18 + 11k 33 16 + 9k
16 + 10k 7k 9 + 10k 23 7 + 10k
2 2 2 2 2
Nonconvergent points in ECEF frame are graphically shown in Fig. 7. The allowed maximum iteration number kmax is set to 20. Whether the maximum iteration number is reached (denoted by ‘+’) or the algorithm converges to a wrong solution (denoted by ‘o’), it is assumed that nonconvergences occur for the corresponding test points. It is demonstrated in Fig. 7 that all nonconvergent test points are wrapped up in a sphere of radius about 50 km centered at the center of the Earth. Although nonconvergent problems do exist, it is only of significance for theoretical discussions. After all, no vehicle could approach so close to the center of the Earth. V.
Fig. 6. Euclid distance errors as function of height and geodetic latitude.
In order to evaluate the new algorithm in the whole space, simulation with test points close to the center of the Earth is carried out. For height, h takes 20 points from 6, 378 km to 6, 300 km; and geodetic latitude and geodetic longitude are as before. 1460
COMPARISONS
This section contains the comparisons of several well-known approximate algorithms, which include Lupash A(see algorithm A in Lupash [1]), Lupash B (see algorithm B in Lupash [1]), Nautiyal [2], and Olson [3]. For each algorithm, numerical simulations were performed as in the previous section. First, let’s consider the arithmetic operations needed to carry out each algorithm. The operations between constants are not counted. Table I is the result of comparison in operating complexity, in which k denotes the iteration number. Second, we compare the accuracy of the different algorithms. Coincidentally, all of the iterative algorithms mentioned here use the NR method in the
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TABLE II Comparison of Speed and Accuracy of Algorithms Near the Earth Maximal Point Algorithms
Max. Error (nm)
Average Error (nm)
Height (km)
Latitude (degree)
Points Nonconvergent
Average Iterations k
Average Time (ms)
Lupash A Lupash B Nautiyal Olson This paper
1:6942
1016 2.7815 659.01 1:3794
1016 3.2596
1:5852
1016 2.7152 623.06 1:2713
1016 3.1262
527.5 475 1000 1000 737.5
89:9 41:95 89:9 89:9 17:98
490 658 0
3.4867 7.94 2.7983
1.4 1.7 1.3 1.1 1.1
process of iteration. Since there exit singularities in some algorithms, we make this comparison near the surface of the Earth. For height, h takes 20 points from 50 km to 1,000 km; for geodetic latitude, L takes 30 points in the range [89:9 89:9 ]; and for geodetic longitude, ¸ takes 30 points in the range [180 180 ]. The comparison of the accuracy of algorithms is shown in Table II. The symbol indicates Olson’s algorithm is not iterative; the item labeled average iterations is the average iterative numbers of test points excluding nonconvergent points. From Table I and Table II, several observations can be made. First, unlike the exact transformations, the approximate algorithms do NOT have to solve cubic polynomials, which is also stated by Zhu [4]. Considering Table I and average iterations k in Table II, the algorithm presented in this correspondence is the most efficient one of iterative approximations, which can be proved, to some extent, by average time in Table II. Since Olson’s algorithm is not iterative, its average time is also the least. Second, in view of Euclid distance error, singularities do exist for Lupash A and Olson, which are not pointed out by Lupash [1] and Olson [3]; as geodetic latitude L approaches 90 , the accuracy of Nautiyal’s algorithm also deteriorates. Third, both Nautiyal’s and our algorithm are free from nonconvergent points. Unfortunately Lupash A and Lupash B are both obsessed by non-convergence problem. Finally as far as average iterations is concerned, Nautiyal’s algorithm is slightly better than ours. CONCLUSION
The approximate transformation from ECEF coordinates to geodetic coordinates has been considered. After comprehensive comparisons with other well-known algorithms, it is shown that the algorithm developed in this correspondence has good efficiency and accuracy. It is free from singularity and nonconvergence for all points in space excluding a region near the center of the Earth of radius about 50 km. The algorithm outperforms other approximate algorithms and is a good candidate for real-time applications. CORRESPONDENCE
2.8033
ACKNOWLEDGMENT The authors would like to thank the editor Joseph L. Leva and anonymous reviewers for their useful suggestions and detailed comments. We also are grateful to Liangqing Lu, a doctor candidate at the same laboratory, for his discussions. YUANXIN WU PING WANG XIAOPING HU Laboratory of Inertial Technology Dept. of Automatic Control National University of Defense Technology Changsha, Hunan PR China 410073 E-mail: (yuanx
[email protected] or yuanx
[email protected]) REFERENCES [1]
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[4]
VI.
0
Lupah, L. O. (1985) A new algorithm for the computation of the Geodetic coordinates as a function of Earth-centered Earth-fixed coordinates. Journal of Guidance, Control, and Dynamics, 8 (Nov.–Dec. 1985), 787–789. Nautiyal, A. (1988) Algorithm to generate geodetic coordinates from Earth-centered Earth-fixed coordinates. Journal of Guidance, Control, and Dynamics, 11, 3 (1988), 281–283. Olson, D. K. (1988) Calculation of geodetic coordinates from Earth-centered Earth-fixed coordinates. Journal of Guidance, Control, and Dynamics, 11, 2 (1988), 188–190. Zhu, J. (1994) Conversion of Earth-centered Earth-fixed coordinates to geodetic coordinates. IEEE Transactions on Aerospace and Electronic Systems, 30, 3 (1994), 957–961.
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