Interpolation and fitting algorithms for GPS satellite orbit

Hui Hu*, Lian Fang School of Information Engineering, East China JiaoTong University, Nanchang, Jiangxi, China

330013

ABSTRACT

The paper used the fitting methods of Lagrange interpolation, Chebyshev polynomial fitting and Hermite interpolation to fitting the orbit based on broadcast ephemeris, then compared these results with the orbit of extrapolation, which indicated the Chebyshev polynomial fitting got the best precision, the Lagrange interpolation was inferior and the Hermite Interpolation was worst in the condition of same order. And the most precisions of the three above-mentioned algorithms were amount to the meter levels of 10-6, 10-5, and 10-3 respectively. However in consideration of operating time, the Lagrange Interpolation could get the fastest, the Chebyshev polynomial fitting was inferior and the Hermite interpolation was the slowest. And the average operating time of the three above-mentioned algorithms were 6 seconds, 146 seconds, and 260 seconds respectively.

Keywords: Lagrange interpolation, Chebyshev polynomial fitting, Hermite interpolation, broadcast ephemeris

1. INTRODUCTION It is one of the key elements for developing GPS receiver to solve the satellite position in real time [1]. All of the information about the satellite orbital parameters in the GPS navigation data is about the satellite position on one point. So it is necessary to make the GPS receiver fit the satellite orbit in a period of time according to the current parameters of the satellite orbit. The GPS ephemeris includes broadcast ephemeris and precise ephemeris. While using broadcast ephemeris to calculate the satellite coordinates, it needs to call ephemeris data repeatedly. Although the precision of position is low, it could be used for navigating and positioning at real time. The precise ephemeris can figure out the satellite coordinates with high precision. However, the time interval of the precise ephemeris is fifteen minutes, which isn’t conducive to navigating and positioning at real time, it is mainly used for subsequent processing. So the GPS receiver needs to fit the satellite orbit. And it is important significance to choose appropriate algorithm and order for improving the speed and precision on the positioning solution of the GPS receiver.

2. ALGORITHM PRINCIPLE The GPS satellite position is changing continuously, while the GPS broadcast ephemeris updates every two hours. So it only needs to call the polynomial coefficients to calculate the satellite coordinates. And it is more greatly to improve the efficiency of the data processing than the method lay in the literature [2]. The paper uses three methods such as Lagrange interpolation [3], Chebyshev polynomial fitting [4] and Hermite interpolation to interpolate and fit the satellite orbit.

2.1 Lagrange interpolation

The function of Lagrange interpolation is defined as:

)()()()()( 1100n xlyxlyxlyxlyxP nnkk +++== ∑ L (1)

Where )(xlk is the basis function of Lagrange defined as:

*hu_hui@ecjtu.jx.cn; phone 86 0791-7046273; fax 86 0791-7046273

PIAGENG 2009: Remote Sensing and Geoscience for Agricultural Engineering, edited by Honghua Tan, Qi Luo, Proc. of SPIE Vol. 7491, 749109 · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.836851

Proc. of SPIE Vol. 7491 749109-1

),,2,1,0()())(()(

)())(()()(

110

110 nkxxxxxxxx

xxxxxxxxxl

nkkkkkk

nkk LLL

LL=

−−−−−−−−

=+−

+− (2)

⎩⎨⎧

≠=

==kiki

xl kikk 01

)( ,δ (3)

Therefore, the satellite coordinates in the reference frame of WGS-84 could be calculated by the Lagrange Interpolation function as follows:

)()()(),()()(),()()(000

k

n

kkkk

n

kkkk

n

kkk zfzltzyfyltyxfxltx ∑∑∑

===

=== (4)

2.2 Chebyshev Polynomial Fitting

While calculating the coefficients of n-order Chebyshev polynomial in the time interval of ],[ 00 tttt Δ+=Δ [5], and transforming the time interval as ]1,1[−∈τ , the function of Chebyshev polynomial fitting is:

∑−

=n

iiiTCtf

0)()( τ (5)

Where n is the order of Chebyshev polynomial and iC is the coefficient of Chebyshev polynomial. And the Chebyshev polynomial iT is determined as:

2,1),()(2)(,,)(,1)( 2110 ≥≤−=== −− nTTTTT nnN ττττττττ L (6)

Assume that the error formula is FBCV −= , and take the X coordinate for example, what’s more:

[ ][ ][ ]Txmxxx

Txmxxx

Txmxxx

mnmmm

n

n

n

FFFFF

CC

VVVVV

TTTT

TTTTTTTTTTTT

B

C C C

)( )( )( )( )( )( )( )()( )( )( )()( )( )( )(

321

321

321

210

3323130

2222120

1121110

L

L

L

L

MMMMM

L

L

L

=

=

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

ττττ

ττττττττττττ

(7)

Where C are the Chebyshev coefficients, and FBBBC TT 1)( −= . So according to the formula (6) the coordinate function fitted in coordinate system of WGS-84 is:

∑∑∑===

===n

iizi

n

iiyi

n

iixi TCtzTCtyTCtx

000)()( ,)()( ,)()( τττ (8)

2.3 Hermite Interpolation

The recursive formula of Hermite interpolation is:

),2,1( ),(2)(2)(,,)(,1)( 21n10 LL =−=== −− nnHHHHH nn τττττττ (9)

Where t=τ and ],[ 00 tttt Δ+∈ . The rest are similar to Chebyshev polynomial fitting, such as the formula (7) and (8) [6].

Proc. of SPIE Vol. 7491 749109-2

o4otp

Ip otuWoi

uq iLL!tJ

cowbiuuEbPUJLP

3. OPERATING TIME ANALYZING Hereinafter, the operating time for calculating the X-axis is taken for example to analyze, because the operating time of calculating y(t) or z(t) are the same as x(t). Suppose that time for one time addition and one time multiplication are t1 and t2 respectively. While calculating Lagrange interpolation of n-order, the computation of the basis function is 2n times of addition and 2n-1 times of multiplication. Then the operating time of this interpolation is: nttnntnA +−++= ])1(22)[1( .

Suppose the control point is m, and according to the formula (5) the required operating time of completing one recursion is

121C 2 ttA += . In addition, while solving the coefficients of Chebyshev polynomial, the operating time of solving the inverse matrix *1 )(1)( BBBBBB TTT =− is:

22

2

321

2

3222 ])2()1()1[(])1()1()1()1[( tmnnniintnnnmiinA

n

i

n

iC +−−−++−−−+−+= ∑∑

==

(10)

Then the operating time of TT BBB 1)( − , xTT FBBB 1)( − and )()( 1 τix

TT TFBBB − are nmtnntA ])1([ 123C −+= ,

nmtnntAC ])1([ 123 −+= , nmttmAC ])1[( 214 +−= and 125 7201tntAC += respectively. The normalized time

is 1)( 02 −−= Δ tttτ , and the operating time of ],[ 00 tttt Δ+∈ is 216 22 ntntAC += . So the total operating time of

calculating ∑=

−=n

iix

TT TFBBBtx0

1 )()()( τ is 1554321)2( ntAAAAAAnA CCCCCCC ++++++−= .

According to the formula (10) the required time for completing the calculation of Hermite polynomial once is 121H 4 ttA += . And that the AH2, AH3, AH4 and AH5 of Hermite interpolation are the same as Chebyshev polynomial

fitting. So the total operating time of calculating ∑=

−=n

iix

TT TFBBBtx0

1 )()()( τ is 4321)2( HHHHH AAAAnA +++−=

155 ntAA HH +++ .

4. EXPERIMENT ANALYZING The paper gathered the navigation data on the roof of the NO.4 Building at the school of Information Engineering in the East China JiaoTong University at 18: 00 o'clock on January 7, 2008. And it uses the navigation data of the NO.10 satellite to extrapolate each an hour forward and afterward [7]. The paper chooses one second as the time interval, divides the coordinates into N-1 paragraph and acquires the reference point of this satellite. Then it uses the navigation data to fit and interpolate the orbit. According to the Fig 1, the paper compares the results of extrapolation, fitting and interpolation at the corresponding sampling points to study the precision of each algorithm.

Fig. 1. The flow chart of verifying the accuracy of fitting algorithm

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What shown in Fig 2 are the partial results about the error residuals of X direction in Lagrange interpolation, Chebyshev fitting and Hermite interpolation. According to Fig 2 (a) and (b), the largest interpolation error of 8-order is 3.5 meters, the largest interpolation error of 9-order is 0.08 meter, and the largest interpolation error of 10-order to 16-order is amount to the meter level of 10-4. Thereafter, the largest interpolation error drop continuously along the increasing order. Such as the biggest interpolation error of 19-order is 5 meters, and the biggest interpolation error of 21-order reach 16 meters. According to Fig 2 (c) and (d), the error of Chebyshev polynomial fitting decreases along the increasing order when the fitting order is less than 9-order. The largest fitting error of 5-order, 7-order and 8-order are 500 meters, 2.5 meters and 0.5 meters respectively, and the largest fitting error of 9-order to 20-order is amount to the meter levels of 10-6 to 10-3. Thereafter, the largest fitting error increases more and more along the increasing order. According to Fig 2 (e) and (f), the error of Hermite interpolation decreases along with the increasing order when the fitting order is less than 12-order. The largest interpolation error of 7-order is 2 meters, the largest interpolation error of 8-order is 0.25 meters and the largest interpolation error of 9-order to 11-order is amount to the meter level of 10-4 to 10-3. Thereafter, the largest interpolation error doesn’t decrease along the increasing order. Such as the largest interpolation error of 13-order to 14-order is amount to the meter level of 10-1.

(a) the residuals of X direction for Lagrange (b) the residuals of X direction for Lagrange

interpolation of 13-order interpolation of 21-order

(c) the residuals of X direction for Chebyshev (d) the residuals of X direction for Chebyshev

polynomial fitting of 11-order polynomial fitting of 19-order

(e) the residuals of X direction for Hermite (f) the residuals of X direction for Hermite

interpolation of 10-order interpolation of 13-order

Fig. 2. The error residuals of X direction for Lagrange interpolation, Chebyshev fitting and Hermite interpolation

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The paper interpolates and fits the satellite orbits of NO.6, NO.10 and NO.29 satellites respectively, and shows the operating times in Table 1. According to Table 1, within the same order, the operating time of the algorithm of Lagrange interpolation is the fastest, the algorithm of Chebyshev polynomial fitting is inferior, and the algorithm of Hermite interpolation is the slowest.

Table. 1. The comparison of operating time.

Algorithm Lagrange Interpolation Chebyshev polynomial Fitting Hermite interpolation

Time (s) 8-order 11-order 13-order 8-order 11-order 13-order 8-order 11-order 13-order

PRN=6 4.6818 5.4780 6.0050 134.9320 160.5470 161.9570 165.9420 257.4820 389.2890

PRN=10 5.1007 5.8152 6.4360 125.5180 156.3310 160.5280 133.3650 255.6680 391.4230

PRN=29 5.2169 5.5980 6.5390 120.3730 153.5010 154.0120 129.9970 264.4700 387.8170

5. CONCLUSIONS It is the key for location and navigation of the GPS receiver to use the orbit parameters to interpolate and fit the satellite orbit. Based on the self-developed GPS receiver, the paper uses three algorithms such as Lagrange interpolation, Chebyshev polynomial fitting and Hermite interpolation to study the orbital interpolation and fitting. The experiment shows that the Chebyshev polynomial fitting can achieve great fitting result on the case of the excellent performance of processor and the high requirement of precision. And the Lagrange interpolation can be the achievement on case of low requirement of precision and fast speed of operation. In addition, the fitting precision of Hermite interpolation is inferior to the algorithm of Chebyshev polynomial fitting, and its operating quantum is too high. In other words, both the fitting precision and the operating quantum of this algorithm are inferior to the two former.

ACKNOWLEDGMENTS The work was supported by the Scientific Research Foundation with Jiangxi Provincial Department of Education in China (Grant No. GJJ08243).

REFERENCES

[1] Elliott D.Kaplan and Christopher J.Hegarty, [Understanding Principles and Applications of the Global Positioning System], Editions 2, Artech House, Norwood, 275-279 ( 2006).

[2] ICD-GPS-200D, "Interface Specification of the Navstar Global Positioning System, " Navstar GPS Joint Program Office, USA, 151-156 (2004).

[3] Mario Juric and Scott Tremarine, "Dynamical Origin of Extrasolar Planet Eccentric contribution, " Journal of Astrophysical, Papers 686, 603-620 (2008).

[4] Liu-cheng Chen and Xiao-lin Jia, "Fitting the Broadcast Ephemeris of Navigation Satellites by Chebshev Approximation, " Progress in Astronomy, China, Papers 24(2), 167-173 (2006).

[5] Rodolpho V. de Moraes, Aurea A. da Silva and Helio K. Kuga, "Simple Orbit Determination Using GPS Based on a Least-Squares Algorithm Employing Seqential Givens Rotations, " Mathematical Problems in Engineering, Papers 10, 49781-49788 (2007).

[6] Yue-liang Xu, [Numerical Analysis], Southwest Jiaotong University, Chengdu, China, 40-48 (2005). [7] Hai-feng Yang and Chao-zhen Hou, "Orbit Extrapolation Method of Remote-Sensing Satellite, " Gnss World of

China, Papers 6, 1-4 (2007).

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