Parameter estimation in mathematical models of the ESRO 1A attitude dynamics (including atmospheric effects) using numerical differentiation of measured data with smoothing splines
Abstract
Measured data of the attitude dynamics and the orbit of the ESRO 1A spacecraft were used for the development of a mathematical model for the attitude behavior of the spacecraft, including, in particular, the aerodynamic interaction with the atmosphere of the earth. The attitude behavior was determined by two strong permanent magnets aligning the spacecraft along the magnetic field lines of the earth, by a damping system with magnetic hysteresis rods, by the gravity gradient, and by the atmospheric interaction. A mathematical model for the attitude dynamics is presented. The model of the damping by the magnetic hysteresis rods, which was developed and used in previous studies was improved. A model for the atmospheric interaction is proposed. The mathematical model contains unknown parameters, which were estimated by fitting the Euler equations for the attitude behavior to processed measured attitude and orbit data. These processed data were obtained by data reduction algorithm, in which numerical differentiation of measured data with smoothing splines plays a critical role. The estimation of the parameters in the model, which determine the model aerodynamic torques and the model torques of the hysteresis rods, turned out to be unsatisfactory. It is shown that this can be explained by bias in the processed data, caused by numerical differentiation with smoothing splines. Due to this bias, the torques computed from the data contain errors of the same order of magnitude as the relatively small aerodynamic and hysteresis torques. It is evident, that the validity of the modeling fo the aerodynamic torques and of the torques generated by the hysteresis rods cannot be established with the present data reduction algorithms. It is concluded that, given the low level of torques to be fitted, alternative ways of numerical differentiation should be considered in order to improve the results. These are briefly indicated.
 Publication:

NASA STI/Recon Technical Report N
 Pub Date:
 September 1976
 Bibcode:
 1976STIN...7718186B
 Keywords:

 Differential Calculus;
 Esro 1 Satellite;
 Mathematical Models;
 Satellite Attitude Control;
 Spline Functions;
 Aerodynamic Drag;
 Damping;
 Data Reduction;
 Hysteresis;
 Magnets;
 Numerical Analysis;
 Satellite Drag;
 Torque;
 Launch Vehicles and Space Vehicles