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

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.