A simplex method for the orbit determination of maneuvering satellites

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SCIENCE CHINA Physics, Mechanics & Astronomy
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SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 61, Issue 2: 024511(2018) https://doi.org/10.1007/s11433-017-9102-1

A simplex method for the orbit determination of maneuvering satellites

JianRong Chen1,2, JunFeng Li1,*, XiJing Wang2, Jun Zhu2, DanNa Wang3
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  • ReceivedMay 31, 2017
  • AcceptedSep 13, 2017
  • PublishedNov 9, 2017
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Abstract

A simplex method of orbit determination (SMOD) is presented to solve the problem of orbit determination for maneuvering satellites subject to small and continuous thrust. The objective function is established as the sum of the nth powers of the observation errors based on global positioning satellite (GPS) data. The convergence behavior of the proposed method is analyzed using a range of initial orbital parameter errors and n values to ensure the rapid and accurate convergence of the SMOD. For an uncontrolled satellite, the orbit obtained by the SMOD provides a position error compared with GPS data that is commensurate with that obtained by the least squares technique. For low Earth orbit satellite control, the precision of the acceleration produced by a small pulse thrust is less than 0.1% compared with the calibrated value. The orbit obtained by the SMOD is also compared with weak GPS data for a geostationary Earth orbit satellite over several days. The results show that the position accuracy is within 12.0?m. The working efficiency of the electric propulsion is about 67% compared with the designed value. The analyses provide the guidance for subsequent satellite control. The method is suitable for orbit determination of maneuvering satellites subject to small and continuous thrust.


Funded by

National Natural Science Foundation of China(11503096)

State Key Laboratory of Geo-information Engineering(SKLGIE2014-M-2-3)


Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant No. 11503096), and the State Key Laboratory of Geo-information Engineering (Grant No. SKLGIE2014-M-2-3).


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  • Click to view Download high-quality image

    Figure 1

    Position errors obtained for the SMOD and the LST for a satellite in LEO.

  • Click to view Download high-quality image

    Figure 2

    Position errors obtained for the SMOD and the LST for a satellite in GEO.

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    Figure 3

    Position errors obtained for the SMOD for a satellite in LEO under pulse control for 60.0?s.

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    Figure 4

    Variation of the semi-major axis, eccentricity, and inclination of the mean orbit obtained by the SMOD and GPS for the Shijian-17 satellite in GEO during the 2?h process of electric propulsion over four days. December 3, 2016 (a), December 4, 2016 (b), December 5, 2016 (c), December 6, 2016 (d).

  • Table 1   Selection of the perturbations adopted in orbit determination

    Perturbation types

    Perturbation source

    Conservative perturbation

    Earth gravity

    N-body, including Solar, lunar, Venus and Jupiter gravitations

    Solid Earth tides

    Ocean tides

    Earth rotation parameter

    Relativistic perturbation

    Non-conservative perturbation

    Atmospheric drag (for LEO satellite)

    Earth albedo radiation

    Solar radiation pressure (for GEO satellite)
  • Table 2   Errors of initial orbital elements, number of iterations (NUM), and final objective function value (OFV)

    Initial orbital errors

    Output parameters

    a

    (km)

    e

    i (°)

    Ω, ω, M (°)

    NUM

    OFV

    Error 1

    10.0

    0.0001

    0.001

    0.01

    526

    912880

    Error 2

    5.0

    0.00015

    0.0015

    0.015

    1336

    0.086

    Error 3

    1.0

    0.0002

    0.002

    0.02

    593

    0.0043

    Error 4

    ?0.5

    0.0002

    0.002

    0.02

    1105

    0.16

    Error 5

    0.1

    0.0005

    0.005

    0.05

    735

    59.53

  • Table 3   Number of iterations (NUM), running time and final objective function value (OFV) with respect to different exponents n

    n

    Running time (min)a)

    NUM

    OFV

    1

    3.32

    197

    3.44

    2

    10.51

    711

    0.36

    3

    2.92

    159

    0.51

    4

    8.63

    584

    0.07

    5

    14.58

    991

    0.08

    6

    21.58

    1486

    0.02

    7

    19.12

    1317

    0.002

    8

    20.55

    1418

    0.03

    a)Here givens the operating environment for the program. The computer is configured with an Intel(R) Core (TM) processor, 3.4?GHz CPU and 3.4 GB of RAM. The operating system is Windows XP and the program language is Fortran.

  • Table 4   The orbital accuracy and acceleration increment estimated by two methods

    Method

    Date

    Position error (m)

    Estimated electronic propulsive acceleration/empirical acceleration (10?5?m/s2)

    R

    T

    N

    Total

    R

    T

    N

    SMOD

    3rd

    9.45

    0.99

    6.12

    11.31

    ?1.40

    0.10

    0.20

    4th

    8.61

    4.39

    4.00

    10.46

    ?1.40

    0.30

    0.60

    5th

    4.04

    1.02

    6.95

    8.10

    –

    –

    –

    6th

    3.00

    1.41

    2.74

    4.30

    ?1.30

    0.30

    0.30

    LST

    3rd

    19.69

    1.89

    4.53

    20.29

    ?7.84

    0.74

    0.32

    4th

    16.57

    6.24

    4.67

    18.31

    ?2.35

    0.84

    1.94

    5th

    7.05

    1.41

    3.64

    8.06

    –

    –

    –

    6th

    6.84

    1.68

    7.68

    10.43

    ?1.24

    0.59

    0.15

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