Abstract

With the expansion of applications, star trackers break through the domain of traditional applications and operate under high dynamic environments. These new applications require high maneuverability and render the analyses and conclusions of previous traditional star trackers unsuitable. In order to resolve the limitation of the previous studies, we focus on the global field-of-view (GFOV) imaging performance of a high dynamic star tracker (HDST) in this paper. A GFOV imaging trajectory model is derived to correctly describe the different motions of stars imaged at different positions of focal plane. A comprehensive positional accuracy expression is obtained by analyzing the centroiding errors of stars in GFOV. On the basis of the proposed trajectory model and positional accuracy expression, a solution of GFOV optimal parameters is presented for the best performance in centroid estimation. Finally, comparative evaluations, numerical simulations, and a night sky experiment support the conclusions.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (5)

J. Li, G. Wang, and X. Wei, “Generation of guide star catalogue for star trackers,” IEEE Sensors J. 18(11), 4592–4601 (2018).
[Crossref]

Z. Wang, J. Jiang, and G. Zhang, “Distributed parallel super-block-based star detection and centroid calculation,” IEEE Sensors J. 18(19), 8096–8107 (2018).
[Crossref]

M. S. Wei, F. Xing, and Z. You, “A real-time detection and positioning method for small and weak targets using a 1d morphology-based approach in 2d images,” Light. Sci. Appl. 7(5), 18006 (2018).
[Crossref]

G. Wang, J. Li, and X. Wei, “Star identification based on hash map,” IEEE Sensors J. 18(4), 1591–1599 (2018).
[Crossref]

S. Wang, S. Zhang, M. Ning, and B. Zhou, “Motion blurred star image restoration based on mems gyroscope aid and blur kernel correction,” Sensors 18(8), 2662 (2018).

2017 (2)

G. Wang, F. Xing, M. Wei, and Z. You, “Rapid optimization method of the strong stray light elimination for extremely weak light signal detection,” Opt. Express 25(21), 26175–26185 (2017).
[Crossref] [PubMed]

C. Zhang, J. Zhao, T. Yu, H. Yuan, and F. Li, “Fast restoration of star image under dynamic conditions via lp regularized intensity prior,” Aerospaceence Technol. 61(1), 29–34 (2017).
[Crossref]

2016 (3)

2015 (2)

2014 (3)

T. Sun, F. Xing, Z. You, X. Wang, and B. Li, “Smearing model and restoration of star image under conditions of variable angular velocity and long exposure time,” Opt. Express 22(5), 6009–6024 (2014).
[Crossref] [PubMed]

X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure time optimization for highly dynamic star trackers,” Sensors 14(3), 4914–4931 (2014).
[Crossref] [PubMed]

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99(1), 231–241 (2014).
[Crossref]

2013 (1)

2012 (2)

WANG Haiyong, ZHOU Wenrui, CHENG Xuan, and Haoyu, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronaut. 25(1), 115–123 (2012).
[Crossref]

Z. Weina, Q. Wei, and G. Lei, “Blurred star image processing for star sensors under dynamic conditions,” Sensors 12(5), 6712–6726 (2012).
[Crossref]

2011 (2)

J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors 11(8), 7341–7363 (2011).
[Crossref] [PubMed]

X. Wu and X. Wang, “Multiple blur of star image and the restoration under dynamic conditions,” Acta Astronaut. 68(11), 1903–1913 (2011).
[Crossref]

2010 (2)

J. Shen, G. Zhang, and X. Wei, “Simulation analysis of dynamic working performance for star trackers,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 27(12), 2638–2647 (2010).
[Crossref] [PubMed]

A. Katake and C. Bruccoleri, “Starcam sg100: a high-update rate, high-sensitivity stellar gyroscope for spacecraft,” Proc. SPIE 7536, 753608 (2010).
[Crossref]

2009 (2)

B. J. Shen, J. C. Tan, J. K. Yang, and J. L. Liao, “Exposure time optimization of the star sensor,” Opto-Electronic Eng. 36(12), 22–26 (2009).

X. Li and H. Zhao, “Analysis of star image centroid accuracy of an aps star sensor in rotation,” Aerosp. Control. Appl. 35(4), 11–16 (2009).

2004 (1)

C. C. Liebe, K. Gromov, and D. M. Meller, “Toward a stellar gyroscope for spacecraft attitude determination,” J. Guid. Control. Dyn. 27(1), 91–99 (2004).
[Crossref]

2002 (2)

M. A. Samaan, T. C. Pollock, and J. L. Junkins, “Predictive centroiding for star trackers with the effect of image smear,” J. Astronaut. Sci. 50(1), 113–123 (2002).

C. C. Liebe, “Accuracy performance of star trackers-a tutorial,” IEEE Transactions on Aerosp. Electron. Syst. 38(2), 587–599 (2002).
[Crossref]

1996 (1)

R. C. Stone, “An accurate method for computing atmospheric refraction,” Publ. Astron. Soc. Pac. 108(729), 1051–1058 (1996).
[Crossref]

1981 (1)

M. D. Shuster and S. D. Oh, “Three-axis attitude determination from vector observations,” J Guid. Control. Dynam 4(1), 70–77 (1981).
[Crossref]

Aretskinhariton, E. D.

E. D. Aretskinhariton and A. J. Swank, “Star tracker performance estimate with imu,” in AIAA Guidance, Navigation, and Control Conference, (American Institute of Aeronautics and Astronautics, 2015), p. 44135.

Bruccoleri, C.

A. Katake and C. Bruccoleri, “Starcam sg100: a high-update rate, high-sensitivity stellar gyroscope for spacecraft,” Proc. SPIE 7536, 753608 (2010).
[Crossref]

Chu, D.

T. Sun, F. Xing, X. Wang, Z. You, and D. Chu, “An accuracy measurement method for star trackers based on direct astronomic observation,” Sci. Reports 6, 22593 (2016).
[Crossref]

Da, L.

Fu, S.

Gromov, K.

C. C. Liebe, K. Gromov, and D. M. Meller, “Toward a stellar gyroscope for spacecraft attitude determination,” J. Guid. Control. Dyn. 27(1), 91–99 (2004).
[Crossref]

Haiyong, WANG

WANG Haiyong, ZHOU Wenrui, CHENG Xuan, and Haoyu, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronaut. 25(1), 115–123 (2012).
[Crossref]

Hao, Y.

Haoyu,

WANG Haiyong, ZHOU Wenrui, CHENG Xuan, and Haoyu, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronaut. 25(1), 115–123 (2012).
[Crossref]

Hu, F.

Huang, Z.

Jia, H.

Jiang, G.

Jiang, J.

Junkins, J. L.

M. A. Samaan, T. C. Pollock, and J. L. Junkins, “Predictive centroiding for star trackers with the effect of image smear,” J. Astronaut. Sci. 50(1), 113–123 (2002).

Katake, A.

A. Katake and C. Bruccoleri, “Starcam sg100: a high-update rate, high-sensitivity stellar gyroscope for spacecraft,” Proc. SPIE 7536, 753608 (2010).
[Crossref]

Lei, G.

Z. Weina, Q. Wei, and G. Lei, “Blurred star image processing for star sensors under dynamic conditions,” Sensors 12(5), 6712–6726 (2012).
[Crossref]

Li, B.

Li, F.

C. Zhang, J. Zhao, T. Yu, H. Yuan, and F. Li, “Fast restoration of star image under dynamic conditions via lp regularized intensity prior,” Aerospaceence Technol. 61(1), 29–34 (2017).
[Crossref]

Li, J.

G. Wang, J. Li, and X. Wei, “Star identification based on hash map,” IEEE Sensors J. 18(4), 1591–1599 (2018).
[Crossref]

J. Li, G. Wang, and X. Wei, “Generation of guide star catalogue for star trackers,” IEEE Sensors J. 18(11), 4592–4601 (2018).
[Crossref]

X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure time optimization for highly dynamic star trackers,” Sensors 14(3), 4914–4931 (2014).
[Crossref] [PubMed]

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99(1), 231–241 (2014).
[Crossref]

Li, W.

Li, X.

X. Li and H. Zhao, “Analysis of star image centroid accuracy of an aps star sensor in rotation,” Aerosp. Control. Appl. 35(4), 11–16 (2009).

Liang, B.

J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors 11(8), 7341–7363 (2011).
[Crossref] [PubMed]

Liao, J. L.

B. J. Shen, J. C. Tan, J. K. Yang, and J. L. Liao, “Exposure time optimization of the star sensor,” Opto-Electronic Eng. 36(12), 22–26 (2009).

Liebe, C. C.

C. C. Liebe, K. Gromov, and D. M. Meller, “Toward a stellar gyroscope for spacecraft attitude determination,” J. Guid. Control. Dyn. 27(1), 91–99 (2004).
[Crossref]

C. C. Liebe, “Accuracy performance of star trackers-a tutorial,” IEEE Transactions on Aerosp. Electron. Syst. 38(2), 587–599 (2002).
[Crossref]

Ma, L.

Meller, D. M.

C. C. Liebe, K. Gromov, and D. M. Meller, “Toward a stellar gyroscope for spacecraft attitude determination,” J. Guid. Control. Dyn. 27(1), 91–99 (2004).
[Crossref]

Ning, M.

S. Wang, S. Zhang, M. Ning, and B. Zhou, “Motion blurred star image restoration based on mems gyroscope aid and blur kernel correction,” Sensors 18(8), 2662 (2018).

Oh, S. D.

M. D. Shuster and S. D. Oh, “Three-axis attitude determination from vector observations,” J Guid. Control. Dynam 4(1), 70–77 (1981).
[Crossref]

Pollock, T. C.

M. A. Samaan, T. C. Pollock, and J. L. Junkins, “Predictive centroiding for star trackers with the effect of image smear,” J. Astronaut. Sci. 50(1), 113–123 (2002).

Samaan, M. A.

M. A. Samaan, T. C. Pollock, and J. L. Junkins, “Predictive centroiding for star trackers with the effect of image smear,” J. Astronaut. Sci. 50(1), 113–123 (2002).

Shen, B. J.

B. J. Shen, J. C. Tan, J. K. Yang, and J. L. Liao, “Exposure time optimization of the star sensor,” Opto-Electronic Eng. 36(12), 22–26 (2009).

Shen, J.

J. Shen, G. Zhang, and X. Wei, “Simulation analysis of dynamic working performance for star trackers,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 27(12), 2638–2647 (2010).
[Crossref] [PubMed]

Shuster, M. D.

M. D. Shuster and S. D. Oh, “Three-axis attitude determination from vector observations,” J Guid. Control. Dynam 4(1), 70–77 (1981).
[Crossref]

Song, J.

J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors 11(8), 7341–7363 (2011).
[Crossref] [PubMed]

Stone, R. C.

R. C. Stone, “An accurate method for computing atmospheric refraction,” Publ. Astron. Soc. Pac. 108(729), 1051–1058 (1996).
[Crossref]

Sun, T.

Swank, A. J.

E. D. Aretskinhariton and A. J. Swank, “Star tracker performance estimate with imu,” in AIAA Guidance, Navigation, and Control Conference, (American Institute of Aeronautics and Astronautics, 2015), p. 44135.

Tan, J. C.

B. J. Shen, J. C. Tan, J. K. Yang, and J. L. Liao, “Exposure time optimization of the star sensor,” Opto-Electronic Eng. 36(12), 22–26 (2009).

Tan, W.

X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure time optimization for highly dynamic star trackers,” Sensors 14(3), 4914–4931 (2014).
[Crossref] [PubMed]

Wang, G.

J. Li, G. Wang, and X. Wei, “Generation of guide star catalogue for star trackers,” IEEE Sensors J. 18(11), 4592–4601 (2018).
[Crossref]

G. Wang, J. Li, and X. Wei, “Star identification based on hash map,” IEEE Sensors J. 18(4), 1591–1599 (2018).
[Crossref]

G. Wang, F. Xing, M. Wei, and Z. You, “Rapid optimization method of the strong stray light elimination for extremely weak light signal detection,” Opt. Express 25(21), 26175–26185 (2017).
[Crossref] [PubMed]

Wang, S.

S. Wang, S. Zhang, M. Ning, and B. Zhou, “Motion blurred star image restoration based on mems gyroscope aid and blur kernel correction,” Sensors 18(8), 2662 (2018).

Wang, X.

Wang, Z.

Z. Wang, J. Jiang, and G. Zhang, “Distributed parallel super-block-based star detection and centroid calculation,” IEEE Sensors J. 18(19), 8096–8107 (2018).
[Crossref]

Wei, M.

Wei, M. S.

M. S. Wei, F. Xing, and Z. You, “A real-time detection and positioning method for small and weak targets using a 1d morphology-based approach in 2d images,” Light. Sci. Appl. 7(5), 18006 (2018).
[Crossref]

Wei, Q.

Z. Weina, Q. Wei, and G. Lei, “Blurred star image processing for star sensors under dynamic conditions,” Sensors 12(5), 6712–6726 (2012).
[Crossref]

Wei, X.

G. Wang, J. Li, and X. Wei, “Star identification based on hash map,” IEEE Sensors J. 18(4), 1591–1599 (2018).
[Crossref]

J. Li, G. Wang, and X. Wei, “Generation of guide star catalogue for star trackers,” IEEE Sensors J. 18(11), 4592–4601 (2018).
[Crossref]

X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure time optimization for highly dynamic star trackers,” Sensors 14(3), 4914–4931 (2014).
[Crossref] [PubMed]

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99(1), 231–241 (2014).
[Crossref]

J. Shen, G. Zhang, and X. Wei, “Simulation analysis of dynamic working performance for star trackers,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 27(12), 2638–2647 (2010).
[Crossref] [PubMed]

Weina, Z.

Z. Weina, Q. Wei, and G. Lei, “Blurred star image processing for star sensors under dynamic conditions,” Sensors 12(5), 6712–6726 (2012).
[Crossref]

Wenrui, ZHOU

WANG Haiyong, ZHOU Wenrui, CHENG Xuan, and Haoyu, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronaut. 25(1), 115–123 (2012).
[Crossref]

Wu, W.

Wu, X.

X. Wu and X. Wang, “Multiple blur of star image and the restoration under dynamic conditions,” Acta Astronaut. 68(11), 1903–1913 (2011).
[Crossref]

Xing, F.

Xu, J.

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99(1), 231–241 (2014).
[Crossref]

Xuan, CHENG

WANG Haiyong, ZHOU Wenrui, CHENG Xuan, and Haoyu, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronaut. 25(1), 115–123 (2012).
[Crossref]

Yan, J.

G. Zhang, J. Jiang, and J. Yan, “Dynamic imaging model and parameter optimization for a star tracker,” Opt. Express 24(6), 5961–5983 (2016).
[Crossref] [PubMed]

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99(1), 231–241 (2014).
[Crossref]

Yang, J.

J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors 11(8), 7341–7363 (2011).
[Crossref] [PubMed]

Yang, J. K.

B. J. Shen, J. C. Tan, J. K. Yang, and J. L. Liao, “Exposure time optimization of the star sensor,” Opto-Electronic Eng. 36(12), 22–26 (2009).

You, Z.

Yu, T.

C. Zhang, J. Zhao, T. Yu, H. Yuan, and F. Li, “Fast restoration of star image under dynamic conditions via lp regularized intensity prior,” Aerospaceence Technol. 61(1), 29–34 (2017).
[Crossref]

Yu, W.

Yuan, H.

C. Zhang, J. Zhao, T. Yu, H. Yuan, and F. Li, “Fast restoration of star image under dynamic conditions via lp regularized intensity prior,” Aerospaceence Technol. 61(1), 29–34 (2017).
[Crossref]

Zhan, D.

Zhang, C.

C. Zhang, J. Zhao, T. Yu, H. Yuan, and F. Li, “Fast restoration of star image under dynamic conditions via lp regularized intensity prior,” Aerospaceence Technol. 61(1), 29–34 (2017).
[Crossref]

Zhang, G.

Z. Wang, J. Jiang, and G. Zhang, “Distributed parallel super-block-based star detection and centroid calculation,” IEEE Sensors J. 18(19), 8096–8107 (2018).
[Crossref]

W. Yu, J. Jiang, and G. Zhang, “Multi exposure imaging and parameter optimization for intensified star trackers,” Appl. Opt. 55(36), 10187–10197 (2016).
[Crossref]

G. Zhang, J. Jiang, and J. Yan, “Dynamic imaging model and parameter optimization for a star tracker,” Opt. Express 24(6), 5961–5983 (2016).
[Crossref] [PubMed]

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99(1), 231–241 (2014).
[Crossref]

X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure time optimization for highly dynamic star trackers,” Sensors 14(3), 4914–4931 (2014).
[Crossref] [PubMed]

J. Shen, G. Zhang, and X. Wei, “Simulation analysis of dynamic working performance for star trackers,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 27(12), 2638–2647 (2010).
[Crossref] [PubMed]

Zhang, J.

Zhang, S.

S. Wang, S. Zhang, M. Ning, and B. Zhou, “Motion blurred star image restoration based on mems gyroscope aid and blur kernel correction,” Sensors 18(8), 2662 (2018).

Zhang, T.

J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors 11(8), 7341–7363 (2011).
[Crossref] [PubMed]

Zhao, H.

X. Li and H. Zhao, “Analysis of star image centroid accuracy of an aps star sensor in rotation,” Aerosp. Control. Appl. 35(4), 11–16 (2009).

Zhao, J.

C. Zhang, J. Zhao, T. Yu, H. Yuan, and F. Li, “Fast restoration of star image under dynamic conditions via lp regularized intensity prior,” Aerospaceence Technol. 61(1), 29–34 (2017).
[Crossref]

Zheng, J.

Zhou, B.

S. Wang, S. Zhang, M. Ning, and B. Zhou, “Motion blurred star image restoration based on mems gyroscope aid and blur kernel correction,” Sensors 18(8), 2662 (2018).

Acta Astronaut. (2)

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99(1), 231–241 (2014).
[Crossref]

X. Wu and X. Wang, “Multiple blur of star image and the restoration under dynamic conditions,” Acta Astronaut. 68(11), 1903–1913 (2011).
[Crossref]

Aerosp. Control. Appl. (1)

X. Li and H. Zhao, “Analysis of star image centroid accuracy of an aps star sensor in rotation,” Aerosp. Control. Appl. 35(4), 11–16 (2009).

Aerospaceence Technol. (1)

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Figures (15)

Fig. 1
Fig. 1 Imaging process of HDST, (a) structure of the star tracker, (b) motion of a star spot.
Fig. 2
Fig. 2 GFOV imaging trajectory model of HDST, (a) modeling process, (b) velocity vectors.
Fig. 3
Fig. 3 Trajectory centroiding error εtraj vs. parameters |φ|, ωT and γ in GFOV, (a) ωT = 0.2°, β = −30°, γ = 135°, (b) |φ| = 43°, β = −30°, γ = 135°, and (c) ωT = 0.2°, |φ| = 43°.
Fig. 4
Fig. 4 Discretization centroiding error δdisc vs. L and χ when ρ = 0.3 pixels.
Fig. 5
Fig. 5 Discretization centroiding error δdisc vs. L and ρ for imaging direction (a) 0°, (b) 45°.
Fig. 6
Fig. 6 Random centroiding error vs. imaging direction angle χ and imaging length L.
Fig. 7
Fig. 7 Total centroiding error σtotal vs. imaging direction for (a) VL = 0.1 pixels/ms, (b) VL = 1.1 pixels/ms.
Fig. 8
Fig. 8 Total centroiding error σtotal vs. Gaussian radius ρ when χ = 0° and χ = 45° for (a) 20°/s and 40°/s, (b)1°/s and 2°/s, and (c) ρmin 1 and ρmin 2.
Fig. 9
Fig. 9 Centroiding errors vs. imaging length and angular velocity, (1) apartments of centroiding errors, (2) under different rotational angular velocities, (3) optimal exposure time .
Fig. 10
Fig. 10 Optimal Gaussian radius ρ̄, centroiding error σtotal, and optimal exposure time curves influenced by different incident stellar magnitudes and temperatures.
Fig. 11
Fig. 11 Comparison of trajectory centroiding errors of different trajectory models in GFOV.
Fig. 12
Fig. 12 Centroiding error vs. imaging length under different Gaussian radii for imaging directions (a) 0°, (b) 45°.
Fig. 13
Fig. 13 Centroiding error vs. imaging length for imaging direction of 0°, (a) under different angular velocities, and (b) under different incident stellar magnitudes.
Fig. 14
Fig. 14 Setup of the night sky experiment.
Fig. 15
Fig. 15 Star location error vs. imaging length for (a) 5°/s, (b) 25°/s.

Tables (1)

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Table 1 Design parameters of the star tracker

Equations (43)

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{ x i t + Δ t = x i t + ( y i t ω z t + f ω y t ) Δ t y i t + Δ t = y i t ( x i t ω z t + f ω x t ) Δ t .
V x y ( Δ t ) = f ω x y 1 cos 2 ( θ t + ω x y Δ t ) f ω x y 1 cos 2 θ t ,
{ x i t + Δ t = x i t + ( y i t ω z t + f ω y t cos 2 θ t ) Δ t y i t + Δ t = y i t ( x i t ω z t + f ω x t cos 2 θ t ) Δ t .
{ x T = x 0 + f ω cos φ cos 2 θ 0 sin β T y T = y 0 f ω cos φ cos 2 θ 0 cos β T .
{ x T = x T cos ( ω z T ) + y T sin ( ω z T ) y T = y T cos ( ω z T ) x T sin ( ω z T ) .
V L , x = x T x 0 T f ω sin φ tan θ 0 sin ( 2 γ ω sin φ T 2 ) + f ω cos φ cos 2 θ 0 sin ( β ω sin φ T ) ,
V L , y f ω sin φ tan θ 0 cos ( 2 γ ω sin φ T 2 ) f ω cos φ cos 2 θ 0 cos ( β ω sin φ T ) .
V L = V L , x 2 + V L , y 2 = f ω sin 2 φ tan 2 θ 0 + cos 2 φ cos 4 θ 0 + sin 2 φ tan θ cos ( γ β + ω sin φ T / 2 ) cos 2 θ 0 .
I i , j = Φ K η QE G INS 2 π ρ 2 x i 0.5 x i + 0.5 y j 0.5 y j + 0.5 0 T exp { [ x x 0 V L , x t ] 2 + [ y y 0 V L , y t ] 2 2 ρ 2 } d t d y d x ,
I i = j I i , j Φ K η QE G INS 2 π ρ 2 0 T x i 0.5 x i + 0.5 exp { [ x x 0 V L , x t ] 2 2 ρ 2 } d x d t .
I tot = i I i = Φ K η QE G INS T .
L C , X = 1 N i = 0 N 1 x i , L C , Y = 1 N i = 0 N 1 y i N = T Δ t .
ε traj , x = 1 N i = 0 N 1 x i 1 2 V L , x T x 0 , ε traj , y = 1 N i = 0 N 1 y i 1 2 V L , y T y 0 .
ε traj = ε traj , x 2 + ε traj , y 2 .
x ¯ = i x i I i i I i , y ¯ = j y j I j j I j .
δ disc , x = x ¯ x C = i x i I i i I i ( V L , x T 2 + x 0 ) , δ disc , y = y ¯ y C = j y j I j j I j ( V L , y T 2 + y 0 ) .
δ disc , x = { 1 2 i x i [ erf ( x i x 0 + 0.5 2 ρ ) erf ( x i x 0 + 0.5 2 ρ ) ] x 0 V L , x T = 0 1 2 V L , x T i x i 0.5 + 0.5 [ erf ( x i + Δ x x 0 2 ρ ) erf ( x i + Δ x V L , x T 2 ρ ) ] d Δ x x C others ,
i x i 0.5 + 0.5 [ erf ( x i + Δ x x 0 2 ρ ) erf ( x i + Δ x V L , x T 2 ρ ) ] d Δ x = i = 1 N x i { ( x i x 0 + 0.5 ) erf [ ( x i x 0 + 0.5 ) / ( 2 ρ ) ] + 2 ρ 2 f PSF ( x i x 0 + 0.5 ) ( x i x 0 + 0.5 V L , x T ) erf [ ( x i x 0 + 0.5 V L , x T ) / ( 2 ρ ) ] 2 ρ 2 f PSF ( x i x 0 + 0.5 V L , x T ) ( x i x 0 0.5 ) erf [ ( x i x 0 + 0.5 ) / ( 2 ρ ) ] 2 ρ 2 f PSF ( x i x 0 + 0.5 ) + ( x i x 0 0.5 V L , x T ) erf [ ( x i x 0 + 0.5 V L , x T ) / ( 2 ρ ) ] + 2 ρ 2 f PSF ( x i x 0 + 0.5 V L , x T ) } x N x N x 0 + 0.5 V L , x T x N x 0 + 0.5 erf [ ( x / 2 ρ ) ] d x i = 1 N x i x 0 0.5 V L , x T x i x 0 0.5 erf [ x / ( 2 ρ ) ] d x
δ disc , x = { 1 2 i x i [ erf ( x i x 0 + 0.5 2 ρ ) erf ( x i x 0 + 0.5 2 ρ ) ] x 0 V L , x T = 0 0 V L , x T = 1 , 2 , N 1 2 V L , x T [ x N V L , x T i = 1 N x 1 x 0 0.5 V L , x T x i x 0 0.5 erf [ x / ( 2 ρ ) ] d x ] x C others .
δ disc , y = { 1 2 j y j [ erf ( y j y 0 + 0.5 2 ρ ) erf ( y j y 0 0.5 2 ρ ) ] y 0 V L , y T = 0 0 V L , x T = 1 , 2 , M 1 2 V L , y T [ y M V L , y T j = 1 M y j y 0 0.5 V L , y T y j y 0 0.5 erf [ y / ( 2 ρ ) ] d y ] y C others .
δ disc = δ disc , x 2 + δ disc , y 2 .
δ disc ( ρ , V L , x T , V L , y T ) = max ( δ disc ( ρ , V L , x T , V L , y T , x 0 , y 0 , Δ x 0 , Δ y 0 ) Δ x 0 , Δ y 0 [ 0.5 , 0.5 ) .
W × H = [ 0.5 + 3 ρ , 0.5 + V L , x T + 3 ρ ] × [ 0.5 + 3 ρ , 0.5 + V L , y T + 3 ρ ] ,
δ disc ( ρ , L , χ ) = max ( δ disc ( ρ , L cos χ , L sin χ , x 0 , y 0 , Δ x 0 , Δ y 0 ) Δ x 0 , Δ y 0 [ 0.5 , 0.5 ) .
δ disc ( ρ , L , χ ) = { δ disc , y ( ρ , L , 0 ° ) χ = 0 ° , 90 ° , 180 ° , 270 ° 0 χ = 45 ° , 135 ° , 225 ° , 315 ° L 1 [ 0 , δ disc , y ( ρ , L , 0 ° ) ] others .
n i , j 2 = K 2 [ n shot 2 + ( n dark 2 + n read 2 + n adc 2 ) ] = I i , j K + K 2 n add 2 ,
σ rand , x 2 = i ( x ¯ I i ) n i 2 + 2 i j ρ i , j ( x ¯ I i ) ( x ¯ I j ) n i n j ,
σ rand , x 2 = K I tot 2 i ( x i x ¯ ) 2 I i + K 2 n add 2 H I tot 2 i ( x i x ¯ ) 2 .
σ shot , x 2 = 1 λ 2 T 2 i ( x i x ¯ ) 2 λ 2 π ρ 2 0 T x i 0.5 x i + 0.5 exp { [ x x 0 V L , x t ] 2 2 ρ 2 } 1 λ T 2 0 T [ ρ 2 + ( x 0 + V L , x t ) 2 2 x ¯ ( x 0 + V L , x t ) + x ¯ 2 ] d t = 1 λ T [ ρ 2 + 1 12 ( V L , x T ) 2 ] .
σ add , x 2 = H n add 2 λ 2 T 2 x 0 0.5 + 3 ρ x 0 + 0.5 + V L , x T + 3 ρ ( x x 0 V L , x T 2 ) 2 d x 1 12 H n add 2 λ 2 T 2 W 3 .
σ shot , y 2 = 1 λ T [ ρ 2 + 1 12 ( V L , y T ) 2 ] , σ add , y 2 = 1 12 W n add 2 λ 2 T 2 H 3 .
σ rand 2 = ( σ shot , x 2 + σ shot , y 2 ) + ( σ add , x 2 + σ add , y 2 ) = V L λ L ( 2 ρ 2 + 1 12 L 2 ) + 1 12 n add 2 V L 2 λ 2 L 2 W H ( W 2 + H 2 ) .
σ total 2 = δ disc 2 ( ρ , L , χ ) + σ rand 2 ( ρ , L , χ , λ , V L , n add ) .
( ρ ¯ , T ¯ ) = arg min ρ , T { σ total ( ρ , T , χ , v ω ) } s . t . ω [ 0 ° / s , ω max ] , χ [ 0 ° , 360 ° ] ,
v ω = f ω { cos [ arctan ( 1 2 sin ( 2 θ max ) ) ] / cos 2 θ max + sin [ arctan ( 1 2 sin ( 2 θ max ) ) ] tan θ max } ,
Ω ρ , T = [ 0 , σ total ( ρ , L , χ = 0 ° , v ω ) ] [ 0 , σ total ( ρ , L , χ = 45 ° , v ω ) ] ω = ω max .
{ σ shot 2 = V L λ L ( 2 ρ 2 + 1 12 L 2 ) χ = 0 ° or 45 ° σ add 1 2 = 1 2 n add 2 V L 2 λ 2 L 2 ( 1 + 6 ρ ) ( 1 + 6 ρ + L ) [ ( 1 + 6 ρ ) 2 + ( 1 + 6 ρ + L ) 2 ] χ = 0 ° σ add 2 2 = 1 6 n add 2 V L 2 λ 2 L 2 ( 1 2 L + 1 + 6 ρ ) 4 χ = 45 ° ,
L shot = 2 6 ρ , L add 1 = 2.383 ( 1 + 6 ρ ) , L add 2 = 2 ( 1 + 6 ρ ) .
{ σ total 2 ( ρ , χ = 0 ° ) = δ disc , y 2 ( ρ , L , 0 ° ) + σ shot 2 ( ρ , L shot ) + σ add 1 2 ( ρ , L add 1 ) σ total 2 ( ρ , χ = 45 ° ) = σ shot 2 ( ρ , L shot ) + σ add 2 2 ( ρ , L add 2 ) .
{ ρ min 1 = arg min ρ { δ disc , y 2 ( ρ , L , 0 ° ) + σ shot 2 ( ρ , L shot ) + σ add 1 2 ( ρ , L add 1 ) } ρ min 2 = solve ρ { δ disc , y 2 ( ρ , L , 0 ° ) + σ add 2 2 ( ρ , L add 2 ) σ add 1 2 ( ρ , L add 1 ) } ρ ¯ = min [ ρ min 1 , ρ min 2 ] .
T ¯ = L v ω ω [ 0 ° / s , ω max ] .
L 0 [ L shot , L add 1 ] [ L shot , L add 2 ] .
L 0 = min { arg min L { σ total ( ρ ¯ , L , χ = 0 ° , v ω ) } , L add 2 } ω [ 0 ° / s , ω max ] , L [ L shot , L add 2 ] .

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