Estimation of flexible space tether state based on end measurement by finite element Kalman filter state estimator

Abstract This paper proposes a novel finite element Kalman filter to estimate the unmeasurable state of space tether systems based on the measured state at its ends only. The finite element method calculates the unmeasurable internal state as the virtual measurement based on the dynamic model of the system by imposing the input of measured state at the boundary to the model using the Lagrange multiplier method in the spatial space. Combining the real and virtual measurement into a hybrid measurement model of the system, the full state is reconstructed and propagated in the temporal space by the extended Kalman filter. Two state-space system models, the dynamics-based and kinematics-based state models, in the Kalman filter are explored. The observability and stability of the newly proposed finite element Kalman filter are examined and proved. The advantages of the proposed state estimator are (i) the singularity in the virtual measurement of state caused by the number of internal state greater than the number of state measured at the boundary is eliminated in the statistic meaning by the Kalman filter, and (ii) the effects of noises of the observation data and the uncertainties of model discretization are considered and minimized. The correctness and effectiveness of the proposed state estimator is demonstrated by the numerical analysis of a space tether system orbiting around the Earth. The results show the proposed state estimator with only measured state at the ends of the tether successfully provides an accurate time history estimation of geometric configuration and motion of the entire tether. Moreover, the results also show the difference caused by the dynamics-based and kinematics-based system models in the state estimator is negligible. The kinematics-based system model should be used in the state estimator due to its significantly low computational load. Finally, the proposed method can be easily applied for the state estimation process for other space tethered spacecraft systems.

[1]  Robert L. Williams,et al.  Linear State-Space Control Systems , 2007 .

[2]  Mutsuto Kawahara,et al.  Estimation of river current using reduced Kalman filter finite element method , 2009 .

[3]  Zheng H. Zhu,et al.  On libration suppression of partial space elevator with a moving climber , 2019, Nonlinear Dynamics.

[4]  Stevan Dubljevic,et al.  Discrete-Time Kalman Filter Design for Linear Infinite-Dimensional Systems , 2019, Processes.

[5]  M. Kemal Leblebicioğlu,et al.  State estimation of transient flow in gas pipelines by a Kalman filter-based estimator , 2016 .

[6]  Denny Oetomo,et al.  Random Weighting, Strong Tracking, and Unscented Kalman Filter for Soft Tissue Characterization , 2018, Sensors.

[7]  Zheng H. Zhu,et al.  Flight Dynamics and Control Strategy of Electric Solar Wind Sails , 2020 .

[8]  Franz Newland,et al.  Libration Control of Bare Electrodynamic Tethers Considering Elastic–Thermal–Electrical Coupling , 2016 .

[9]  Alessandra Celletti,et al.  A “Spring–mass” model of tethered satellite systems: properties of planar periodic motions , 2010 .

[10]  Yunpeng Wang,et al.  Using consecutive point clouds for pose and motion estimation of tumbling non-cooperative target , 2019, Advances in Space Research.

[11]  Roger B. Sidje,et al.  Expokit: a software package for computing matrix exponentials , 1998, TOMS.

[12]  Konrad Reif,et al.  The extended Kalman filter as an exponential observer for nonlinear systems , 1999, IEEE Trans. Signal Process..

[13]  Wim Desmet,et al.  Virtual microphone sensing through vibro-acoustic modelling and Kalman filtering , 2018 .

[14]  Zheng H. Zhu,et al.  Dynamics and operation optimization of partial space elevator with multiple climbers , 2019, Advances in Space Research.

[15]  Panfeng Huang,et al.  A state estimation scheme with minimal sensor configuration for three-body tethered satellite formations , 2020 .

[16]  Yongmin Zhong,et al.  Kalman Filter Finite Element Method for Real-Time Soft Tissue Modeling , 2020, IEEE Access.

[17]  Panfeng Huang,et al.  A space tethered towing method using tension and platform thrusts , 2017 .

[18]  Shaker A. Meguid,et al.  Characteristics of coupled orbital-attitude dynamics of flexible electric solar wind sail , 2019, Acta Astronautica.

[19]  Zheng H. Zhu,et al.  Three-Dimensional High-Fidelity Dynamic Modeling of Tether Transportation System with Multiple Climbers , 2019, Journal of Guidance, Control, and Dynamics.

[20]  Denis Zanutto,et al.  Analysis of Propellantless Tethered System for the De-Orbiting of Satellites at End of Life , 2013 .

[21]  Shaker A. Meguid,et al.  Nonlinear FE-based investigation of flexural damping of slacking wire cables , 2007 .

[22]  Murat Efe,et al.  Stability of the Extended Kalman Filter When the States are Constrained , 2008, IEEE Transactions on Automatic Control.

[23]  Zheng H. Zhu,et al.  Dynamic modeling of cable towed body using nodal position finite element method , 2011 .

[24]  N. Koizumi,et al.  Kalman filter finite element method applied to dynamic ground motion , 2009 .

[25]  Gangqi Dong,et al.  Autonomous robotic capture of non-cooperative target by adaptive extended Kalman filter based visual servo , 2016 .

[26]  Andrey Polyakov,et al.  Design of interval observers and controls for PDEs using finite-element approximations , 2018, Autom..

[27]  Shaker A. Meguid,et al.  Multiphysics elastodynamic finite element analysis of space debris deorbit stability and efficiency by electrodynamic tethers , 2017 .

[28]  Roberta Veloso Garcia,et al.  Nonlinear filtering for sequential spacecraft attitude estimation with real data: Cubature Kalman Filter, Unscented Kalman Filter and Extended Kalman Filter , 2018, Advances in Space Research.

[29]  Vladimir S. Aslanov,et al.  The motion of tethered tug–debris system with fuel residuals , 2015 .

[30]  Zheng H. Zhu,et al.  Long-term dynamic modeling of tethered spacecraft using nodal position finite element method and symplectic integration , 2015 .

[31]  W. Desmet,et al.  Model based virtual intensity measurements for exterior vibro-acoustic radiation , 2019 .

[32]  Giorgio Battistelli,et al.  Distributed Finite-Element Kalman Filter for Field Estimation , 2017, IEEE Transactions on Automatic Control.