Tracking the maneuvering spacecraft propelled by swing propulsion of constant magnitude

doi:10.23919/JSEE.2020.000014

Journal of Systems Engineering and Electronics ›› 2020, Vol. 31 ›› Issue (2): 370-382.doi: 10.23919/JSEE.2020.000014

收稿日期:2019-04-25 出版日期:2020-04-30 发布日期:2020-04-30

Tracking the maneuvering spacecraft propelled by swing propulsion of constant magnitude

Guang ZHAI¹(), Yanxin WANG¹(), Qi ZHAO^2,*()

¹ School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
² Systems Engineering Research Institute, Beijing 100094, China

Received:2019-04-25 Online:2020-04-30 Published:2020-04-30
Contact: Qi ZHAO E-mail:gzhai@bit.edu.cn;18811690075@163.com;zhaoqibit@163.com
About author:ZHAI Guang was born in 1979. He received his Ph.D. degree in control science and engineering from Harbin University of technologyin 2009. He is an associated professor in the School of Aerospace Engineering from Beijing Institute ofTechnology. His research interests are spacecraft navigation, guidance and control technology. E-mail: gzhai@bit.edu.cn|WANG Yanxin was born in 1996. She received her M.S. degree in the School of Aerospace Engineering from Beijing Institution of Technology in 2019. Her research interests are unscented Kalman filter, reentry tracking, and distributed multisensor data fusion. E-mail: 18811690075@163.com|ZHAO Qi was born in 1991. Hereceived her M.S. degreein the School of Aerospace Engineering from Beijing Institution of Technology in 2017.He is an engineer in Systems Engineering Research Institute. His research interests are target location, Kalman filter, and data fusion. E-mail: zhaoqibit@163.com
Supported by:
the National Natural Science Foundation of China(11872109);the National Key R & D Program of China(2019YFA0706500);This work was supported by the National Natural Science Foundation of China (11872109) and the National Key R & D Program of China (2019YFA0706500)

摘要/Abstract

Abstract:

This paper proposes partially norm-preserving filtering for a class of spacecraft in the presence of time-varying, but constant-magnitude maneuver. The augmented state Kalman filter (ASKF) is commonly used to track the maneuvering spacecraft with unknown constant propulsion; however, if the maneuver varies via time, the estimation performance will be degraded. To promote the tracking performance of the ASKF in case of time-invariant, constant-magnitude disturbance, the partially norm-preserving ASKF is developed by applying the norm constraint on the unknown maneuver. The proposed estimator, which is decomposed into two partial estimators and iteratively propagated in turns, projects the unconstrained maneuver estimation onto the Euclidian surface spanned by the norm constraint. The illustrative numerical example is provided to show the efficiency of the proposed method.

Key words: maneuvering spacecraft, tracking, augmented state, norm-preserving, Kalman filter

. [J]. Journal of Systems Engineering and Electronics, 2020, 31(2): 370-382.

Guang ZHAI, Yanxin WANG, Qi ZHAO. Tracking the maneuvering spacecraft propelled by swing propulsion of constant magnitude[J]. Journal of Systems Engineering and Electronics, 2020, 31(2): 370-382.

图/表 9

Initialization	${\boldsymbol{x}}_{0 / 0} = {\rm{E}}\{{\boldsymbol{x}}_0 \}$ ${\boldsymbol{d}}_{0 / 0} = {\rm{E}}\{{\boldsymbol{d}}_0 \}$ ${\rm{E}}\{({\boldsymbol{x}}_0 - \widehat {{\boldsymbol{x}}}_0)({\boldsymbol{x}}_0 - \widehat {{\boldsymbol{x}}}_0)^{\rm{T}} \} ={\boldsymbol{P}}_{0 / 0}^{{\boldsymbol{x}}{\boldsymbol{x}}}$ ${\rm{E}}\{({\boldsymbol{x}}_0 - \widehat {{\boldsymbol{x}}}_0)({\boldsymbol{b}}_0 - \widehat {{\boldsymbol{b}}}_0)^{\rm{T}} \} ={\boldsymbol{P}}_0^{{\boldsymbol{x}}{\boldsymbol{b}}}$${\rm{E}}\{({\boldsymbol{d}}_0 - \widehat {{\boldsymbol{d}}}_0)({\boldsymbol{d}}_0 - \widehat {{\boldsymbol{d}}}_0)^{\rm{T}} \}= {\boldsymbol{P}}_{0 / 0}^{dd}$
$\widehat {{\boldsymbol{x}}}_{k + 1}$ estimation	$\widehat {{\boldsymbol{x}}}_{k + 1 / k} = {\boldsymbol{A}}_k \widehat {{\boldsymbol{x}}}_k + {\boldsymbol{G}}_k \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{d}}}} _k$ ${\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{x}}} = ({\boldsymbol{A}}_k {\boldsymbol{P}}_k^{{\boldsymbol{x}}{\boldsymbol{x}}} +{\boldsymbol{G}}_k \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{P}}}}_k^{{\boldsymbol{d}}{\boldsymbol{x}}}){\boldsymbol{A}}_k^{\rm{T}}+$$({\boldsymbol{A}}_k \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{P}}}}_k^{{\boldsymbol{x}}{\boldsymbol{d}}} + {\boldsymbol{G}}_k \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{P}}}}_k^{{\boldsymbol{d}}{\boldsymbol{d}}}){\boldsymbol{G}}_k^{\rm{T}}+ {\boldsymbol{m}}{\mathit{\Gamma}}_k {\boldsymbol{Q}}{\boldsymbol{m}}{\mathit{\Gamma}}_k^{\rm{T}}$${\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{d}}} = ({\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{d}}{\boldsymbol{x}}})^{\rm{T}} = {\boldsymbol{A}}_k \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{P}}}}_k^{{\boldsymbol{d}}{\boldsymbol{x}}} + {\boldsymbol{G}}_k \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{P}}}}_k^{{\boldsymbol{d}}{\boldsymbol{d}}}$ ${\boldsymbol{P}}_{k + 1}^{{\boldsymbol{x}}{\boldsymbol{x}}} = {\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{x}}} - {\boldsymbol{K}}_{k + 1}^{\boldsymbol{x}} {\boldsymbol{C}}_{k + 1} ({\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{x}}})^{\rm{T}}-$ ${\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{x}}} {\boldsymbol{C}}_{k + 1}^{\rm{T}} ({\boldsymbol{K}}_{k + 1}^{\boldsymbol{x}})^{\rm{T}} + {\boldsymbol{K}}_k^{\boldsymbol{x}} {\boldsymbol{m}}{\mathit{\Theta}} ({\boldsymbol{K}}_{k + 1}^{\boldsymbol{x}})^{\rm{T}}$ ${\boldsymbol{K}}_{k + 1}^{\boldsymbol{x}} = {\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{x}}} {\boldsymbol{C}}_k^{\rm{T}} {\boldsymbol{m}}{\mathit{\Theta}} ^{ - 1}$$\widehat {{\boldsymbol{x}}}_{k + 1} = \widehat {{\boldsymbol{x}}}_{k + 1 / k} + {\boldsymbol{K}}_{k + 1}^{\boldsymbol{x}}{\boldsymbol{m}}\eta_{k + 1}$
$\widehat {{\boldsymbol{d}}}_{k + 1}$ estimation	$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{d}}}}_{k+1/k}=\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{P}}}}_k$${\boldsymbol{P}}^{{\boldsymbol{d}}{\boldsymbol{d}}}_{k+1/k}=\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{P}}}}^{{\boldsymbol{d}}{\boldsymbol{d}}}_k$${\boldsymbol{K}}^{\boldsymbol{d}}_{k+1}={\boldsymbol{P}}^{{\boldsymbol{x}}{\boldsymbol{d}}}_{k+1/k}{\boldsymbol{C}}^{\rm{T}}_{k+1}{\boldsymbol{m}}{\mathit{\Theta}}^{-1}$$\widehat {{\boldsymbol{d}}}_{k + 1} = \widehat {{\boldsymbol{d}}}_{k + 1 / k} + {\boldsymbol{K}}_{k + 1}^{\boldsymbol{d}} {\boldsymbol{m}}\eta_{k + 1} $
$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{d}}}}_{k+1}$ projection $\kappa _k > 1$ or$\kappa _k < \kappa _T $	$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{K}}}}^{\boldsymbol{d}}_{k+1}={\boldsymbol{K}}^{\boldsymbol{d}}_{k+1}+\left(\frac{\rho-\\|\widehat{{\boldsymbol{d}}}_{k+1}\\|}{\\|\widehat{{\boldsymbol{d}}}_{k+1}\\|}\right)\widehat{{\boldsymbol{d}}}_{k+1}\dfrac{{\boldsymbol{m}}\eta^T_{k+1}{\boldsymbol{m}}{\mathit{\Theta}}^{-1}_{k+1}}{\overline{\eta}_{k+1}}$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{P}}}}_{k + 1}^{{\boldsymbol{d}}{\boldsymbol{d}}} = {\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{d}}{\boldsymbol{d}}} -\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{K}}}} _{k + 1}^{\boldsymbol{d}} {\boldsymbol{C}}_{k + 1} ({\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{d}}{\boldsymbol{x}}})^{\rm{T}} -$ ${\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{d}}{\boldsymbol{x}}} {\boldsymbol{C}}_{k + 1}^{\rm{T}}(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{K}}}} _{k + 1}^{\boldsymbol{d}})^{\rm{T}} +\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{K}}}} _{k + 1}^{\boldsymbol{d}}{\boldsymbol{m}}{\mathit{\Theta}} (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{K}}}} _{k + 1}^{\boldsymbol{d}})^{\rm{T}} $ $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{d}}}}_{k+1}=\dfrac{\rho}{\\|\widehat{{\boldsymbol{d}}}_{k+1}\\|}\widehat{{\boldsymbol{d}}}_{k+1}$
Coupling update	$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{P}}}}_{k + 1}^{{\boldsymbol{x}}{\boldsymbol{d}}} =(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{P}}}}_{k + 1}^{{\boldsymbol{d}}{\boldsymbol{x}}})^{\rm{T}} =$ ${\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{d}}} - {\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{x}}} {\boldsymbol{C}}_{k + 1}^{\rm{T}}(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{K}}}} _{k+1}^{\boldsymbol{d}})^{\rm{T}}-$${\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{d}}} {\boldsymbol{C}}_{k + 1}^{\rm{T}} ({\boldsymbol{K}}_{k + 1}^{\boldsymbol{x}})^{\rm{T}} +\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\boldsymbol{K}}}} _{k + 1}^{\boldsymbol{d}} {\boldsymbol{m}}{\mathit{\Theta}} ({\boldsymbol{K}}_{k + 1}^{\boldsymbol{x}})^{\rm{T}}$ Constrained estimation ${\boldsymbol{P}}_{k + 1}^{{\boldsymbol{x}}{\boldsymbol{d}}} =({\boldsymbol{P}}_{k + 1}^{{\boldsymbol{d}}{\boldsymbol{x}}})^{\rm{T}} =$ ${\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{d}}} - {\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{x}}} {\boldsymbol{C}}_{k + 1}^{\rm{T}}({\boldsymbol{K}} _{k+1}^{\boldsymbol{d}})^{\rm{T}}-$${\boldsymbol{P}}_{k + 1 / k}^{{\boldsymbol{x}}{\boldsymbol{d}}} {\boldsymbol{C}}_{k + 1}^{\rm{T}} ({\boldsymbol{K}}_{k + 1}^{\boldsymbol{x}})^{\rm{T}} +{\boldsymbol{K}} _{k + 1}^{\boldsymbol{d}} {\boldsymbol{m}}{\mathit{\Theta}} ({\boldsymbol{K}}_{k + 1}^{\boldsymbol{x}})^{\rm{T}}$ Unconstrained estimation

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Tracking the maneuvering spacecraft propelled by swing propulsion of constant magnitude

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