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SCIENCE CHINA Information Sciences, Volume 61, Issue 11: 112203(2018) https://doi.org/10.1007/s11432-017-9338-8

Spacecraft angular velocity estimation method using optical flow of stars

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  • ReceivedNov 1, 2017
  • AcceptedJan 2, 2018
  • PublishedSep 25, 2018

Abstract

Angular velocity is a crucial parameter for spacecraft navigation, which, at present, is mainly obtained by using gyroscopes. Several studies have been performed on angular velocity estimation using star sensors when data from gyroscopes are not available. Most of the previous angular velocity estimation methods using star sensors are based on information on the attitude or star vector measurements. In this paper, an angular velocity estimation method using the optical flow (OF) directly from the star images is proposed, which, unlike the previous methods, requires star coordinates in two consecutive images only. Because the procedure of star identification is eliminated, the corresponding high computation requirement is reduced. Simulations demonstrate that the proposed method has a robust performance in terms of computational cost and the number of stars in the field of view (FOV) compared with the previous methods. Finally, certain affecting factors are analyzed, including lens distortion, star senor instantaneous FOV (IFOV), star number, and accuracy of the star sensor.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61503013, 61722301) and National Basic Research Program of China (973 Program) (Grant No. 2014CB744206). The authors express their gratitude to all members of Science Technology on Inertial Laboratory, Fundamental Science on Novel Inertial Instrument Navigation System Technology Laboratory, and Key Laboratory of Ministry of Industry and Information Technology on Quantum Sensing Technology for their valuable comments.


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  • Figure 1

    Estimation approach of the angular velocity.

  • Figure 2

    OF acquisition.

  • Figure 3

    (Color online) Reference frames.

  • Figure 4

    Determination of the angular velocity using the OF of stars.

  • Figure 5

    (Color online) Simulated spacecraft angular velocity in (a) case 1 and (b) case 2.

  • Figure 6

    (Color online) Estimated angular velocity errors in case 1 of (a) the VDLSKF, (b) the VDKF, (c) the OFLSKF, and (d) the OFKF method.

  • Figure 8

    (Color online) Estimated angular velocity errors in case 1 of (a) the VDLSKF, (b) the VDKF, (c) the OFLSKF, and (d) the OFKF method, when there are less than two stars in the FOV.

  • Figure 9

    (Color online) Angular velocity estimation errors around the (a) $x$-axis, (b) $y$-axis, and (c) $z$-axis, under different star sensor IFOVs.

  • Figure 10

    (Color online) Angular velocity estimation errors around the (a) $x$-axis, (b) $y$-axis, and (c) $z$-axis, under different numbers of matching stars in the FOV.

  • Figure 11

    (Color online) Angular velocity estimation errors around the (a) $x$-axis, (b) $y$-axis, and (c) $z$-axis, under different star sensor accuracies.

  • Table 1   Intrinsic parameters of the star sensor
    Value $f$ (mm) $u_0$ $v_0$ $k_1$ $k_2$ $k_3$ $p_1$ $p_2$ $p_3$
    Simulation value 73.0703 510 518 2E$-$4 $-$4E$-$7 1E$-$8 2E$-$4 2E$-$4 4E$-$6
    Calibration value 73.0703 510.05 517.97 2.00E$-$4 $-$4.00E$-$7 1.00E$-$8 2.00E$-$4 2.00E$-$4 1.11E$-$6
  • Table 2   Estimated angular velocity errors and computation time in case 1
    Method Mean angular velocity error ($^\circ$/s) Root mean square error ($^\circ$/s) Computation
    $x$-axis $y$-axis $z$-axis $x$-axis $y$-axis $z$-axis time (s)
    VDLSKF 7.35E$-$4 7.08E$-$4 2.42E$-$3 9.17E$-$4 8.87E$-$4 3.06E$-$3 2.6412
    VDKF 7.35E$-$4 7.08E$-$4 2.42E$-$3 9.17E$-$4 8.87E$-$4 3.06E$-$3 2.0950
    OFLSKF 6.40E$-$4 6.37E$-$4 2.36E$-$3 8.00E$-$4 7.94E$-$4 2.97E$-$3 2.1840
    OFKF 6.40E$-$4 6.37E$-$4 2.36E$-$3 8.00E$-$4 7.94E$-$4 2.97E$-$3 1.7990
  • Table 3   Estimated angular velocity errors and computation time in case 2
    Method Mean angular velocity error ($^\circ$/s) Root mean square error ($^\circ$/s) Computation
    $x$-axis $y$-axis $z$-axis $x$-axis $y$-axis $z$-axis time (s)
    VDLSKF 7.15E$-$4 6.91E$-$4 2.24E$-$3 8.95E$-$4 8.62E$-$4 2.81E$-$3 2.6633
    VDKF 7.15E$-$4 6.91E$-$4 2.24E$-$3 8.95E$-$4 8.62E$-$4 2.81E$-$3 2.0658
    OFLSKF 6.19E$-$4 6.20E$-$4 2.21E$-$3 7.79E$-$4 7.71E$-$4 2.76E$-$3 2.1956
    OFKF 6.19E$-$4 6.20E$-$4 2.21E$-$3 7.79E$-$4 7.71E$-$4 2.76E$-$3 1.7718
  • Table 4   Proportions of a single star in a $12^\circ~\times~12^\circ$ FOV under different Mvs
    Mv 4.5 5 5.5 6
    Proportion (%) 26.3 7.1 0.8 0
  • Table 5   Proportions of a single star in different FOVs when the Mv of the star sensor is 5
    FOV 8$^\circ~\times~8^\circ$ 10$^\circ~\times~10^\circ$ 12$^\circ~\times~12^\circ$ 14$^\circ~\times~14^\circ$ 16$^\circ~\times~16^\circ$
    Proportion (%) 29 16.6 7.1 3.3 1.2
  • Table 6   Estimated angular velocity errors in case 1 when there are less than two stars in the FOV
    Method Mean angular velocity error ($^\circ$/s) Root mean square error ($^\circ$/s)
    $x$-axis $y$-axis $z$-axis $x$-axis $y$-axis $z$-axis
    VDLSKF 7.65E$-$3 3.83E$-$3 1.09E$-$2 1.85E$-$2 7.92E$-$3 1.79E$-$2
    VDKF 1.30E$-$3 1.09E$-$3 8.11E$-$3 1.72E$-$3 1.37E$-$3 1.24E$-$2
    OFLSKF 7.33E$-$3 3.01E$-$3 9.15E$-$3 1.84E$-$2 6.42E$-$3 1.51E$-$2
    OFKF 1.11E$-$3 9.32E$-$4 6.70E$-$3 1.45E$-$3 1.17E$-$3 9.98E$-$3
  • Table 7   Estimated angular velocity errors under different calibration errors
    Calibration Mean angular velocity error ($^\circ$/s) Root mean square error ($^\circ$/s)
    error (pixel) $x$-axis $y$-axis $z$-axis $x$-axis $y$-axis $z$-axis
    0.05 6.1902E$-$4 6.1980E$-$4 2.2055E$-$3 7.7883E$-$4 7.7115E$-$4 2.7601E$-$3
    0.10 6.1957E$-$4 6.2023E$-$4 2.2060E$-$3 7.7952E$-$4 7.7169E$-$4 2.7608E$-$3
    0.15 6.1988E$-$4 6.2048E$-$4 2.2063E$-$3 7.7991E$-$4 7.7200E$-$4 2.7611E$-$3
    0.20 6.2021E$-$4 6.2074E$-$4 2.2066E$-$3 7.8033E$-$4 7.7232E$-$4 2.7615E$-$3
  • Table 8   Number of stars in the FOV under different star Mvs
    Mv 5 5.5 5.75 6.0 6.25 6.5 6.75
    Number of matching stars in FOV 2 3 6 7 11 23 36

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