Tracking the maneuvering spacecraft propelled by swing propulsion of constant magnitude

doi:10.23919/JSEE.2020.000014

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

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.

Figures/Tables 9

Table 1

Summary of the constrained estimator"

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

Table 1

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References 30

1	KALMAN R E, BUCY R S. New results in linear filtering and predition. Transactions of the ASME, Series D: Journal of Basic Engineering, 1961, 83 (1): 95- 108. doi: 10.1115/1.3658902
2	LUENBERGER D. An introduction to observers. IEEE Trans. on Automatic Control, 1971, 16 (6): 596- 602.
3	DAHMANI M, MECHE A, KECHE M, et al. An improved fuzzy alpha-beta filter for tracking a highly maneuvering target. Aerospace Science and Technology, 2016, 58, 298- 305. doi: 10.1016/j.ast.2016.08.029
4	CHEN J, PATTON R J. Optimal filtering and robust fault diagnosis of stochastic systems with unknown disturbances. IEE Proceedings——Control Theory and Applications, 1996, 143 (1): 31- 36.
5	HSIEH C S. Robust two-stage Kalman filters for systems with unknown inputs. IEEE Trans. on Automatic Control, 2000, 45 (12): 2374- 2378.
6	DAROUACH M, ZASADZINSKI M. Unbiased minimum variance estimation for systems with unknown exogenous inputs. Automatica, 1997, 33 (4): 717- 719. doi: 10.1016/S0005-1098(96)00217-8
7	GILLIJNS S, DE MOOR B. Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica, 1997, 43 (5): 934- 937.
8	CHENG Y, YE H, WANG Y Q, et al. Unbiased minimum-variance state estimation for linear systems with unknown input. Automatica, 2009, 45 (2): 485- 491. doi: 10.1016/j.automatica.2008.08.009
9	BORISOV A V, PANKOV A R. Optimal filtering in stochastic discrete-time systems with unknown inputs. IEEE Trans. on Automatic Control, 1994, 39 (12): 2461- 2464.
10	SUNDARAM S, HADJICOSTIS C N. Optimal state estimators for linear systems with unknown inputs. Proc. of the IEEE Conference on Decision and Control, 2006, 4763- 4768.
11	GILLIJNS S, DE MOOR B. Unbiased minimum-variance input and state estimation for linear discrete-time systems. Automatica, 2007, 43 (1): 111- 116.
12	ALOUANI A T, RICE T R, BLAIR W D. A two-stage filter for the state estimation in the presence of dynamical stochastic bias. Proc. of the American Control Conference, 1992, 1784- 1788.
13	KELLER J Y, DAROUACH M. Optimal two-stage Kalman filter in the presence of random bias. Automatica, 1997, 33 (9): 1745- 1748. doi: 10.1016/S0005-1098(97)00088-5
14	FRIEDLAND B. Treatment of bias in recursive filtering. IEEE Trans. on Automatic Control, 1969, 14 (4): 359- 367. doi: 10.1109/TAC.1969.1099223
15	LEE S, LEE J, HWANG I. Maneuvering spacecraft tracking via state-dependent adaptive estimation. Journal of Guidance, Control, and Dynamics, 2016, 39 (9): 2034- 2043. doi: 10.2514/1.G001567
16	KO H C, SCHEERES D J. Tracking maneuvering satellite using thrust-fourier-coefficient event representation. Journal of Guidance Control Dynamics, 2016, 39 (11): 1- 9.
17	GOFF G M, BLACK J T, BECK J A. Orbit estimation of a continuously thrusting spacecraft using variable dimension filters. Journal of Guidance, Control, and Dynamics, 2015, 38 (12): 2407- 2420. doi: 10.2514/1.G001091
18	HOLZINGER M J, SCHEERES D J, ALFRIEND K T. Object correlation, maneuver detection, and characterization using control-distance metrics. Journal of Guidance, Control, and Dynamics, 2012, 35 (4): 1312- 1325.
19	LEE S, HWANG I. Interacting multiple model estimation for spacecraft maneuver detection and characterization. https://doi.org/10.2514/6.2015-1333.
20	LEE H, TAHK M J. Generalized input-estimation technique for tracking maneuvering targets. IEEE Trans. on Aerospace and Electronics System, 1999, 35 (4): 1388- 1402. doi: 10.1109/7.805455
21	BAR-SHALOM Y, BIRMIWAL K. Variable dimension filter for maneuvering target tracking. IEEE Trans. on Aerospace and Electronics System, 1982, 18 (5): 621- 629.
22	CLOUTIER J R, LIN C F, YANG C. Enhanced variable dimension filter for maneuvering target tracking. IEEE Trans. on Aerospace and Electronics System, 1993, 29 (3): 786- 797. doi: 10.1109/7.220930
23	HU X, HU X, HUANG Y. A nonlinear variable dimension estimator for maneuvering spacecraft tracking via the unscented Kalman filter. Proc. of the IEEE International Conference on Information and Automation, 2008, 226- 231.
24	JULIER S J, JR LAVIOLA J J. On Kalman filtering with nonlinear state equality constraints. IEEE Trans. on Signal Processing, 2007, 55 (6): 2774- 2784. doi: 10.1109/TSP.2007.893949
25	KO S, BITMEAD R R. State estimation for linear systems with state equality constraints. Automatica, 2007, 43 (8): 1363- 1368. doi: 10.1016/j.automatica.2007.01.017
26	WOODBURY D, JUNKINS J. On the consider Kalman filter. https://doi.org/10.2514/6.2010-7752.
27	CHOUKROUN D, BAR-ITZHACK I Y, OSHMAN Y. A novel quaternion Kalman filter. IEEE Trans. on Aerospace and Electronic Systems, 2006, 42 (1): 174- 190. doi: 10.1109/TAES.2006.1603413
28	CRASSIDIS J L, MARKLEY F L, CHENG Y. Survey of nonlinear attitude estimation methods. Journal of Guidance, Control, and Dynamics, 2007, 30 (1): 12- 28.
29	ZANETTI R, MAJJI M, BISHOP R H, et al. Norm-constrained Kalman filtering. Journal of Guidance Control & Dynamics, 2009, 32 (5): 1458- 1465.
30	FORBES J R, DE RUITER A H J, ZLOTNIK D E. Continuous-time norm-constrained Kalman filtering. Automatica, 2104, 50 (10): 2546- 2554.

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