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The inertia matrix typically varies during the satellite lifetime due to fuel consumption, possible collisions and damages. The importance of an accurate estimation arises especially for agile satellites, where control laws directly proportional to the inertia matrix are typically adopted to respond the stringent performance constraints on the on-board attitude control system. During the CNES mission MICROCARB test campaign, it has been highlighted that when performing the FIXED TARGET pointing maneuver, even though the control loop is robust enough to deal with an inertia relative error of up to 15%, ensuring it to be below 5% can significantly reduce the stabilization period and the oscillation frequencies. The purpose of the study is twofold. Firstly, to compare two inertia estimation methods and secondly, to study the observability of the input data to use for the estimation. The first estimation method, the Instrumental Variable (IV) [1], is a variation of the Least Square (LS) method that remains consistent for a class of noise signals which makes the LS estimates biased. Exactly as the LS, the IV method is based on a prediction model structure, which is linearly parameterized. However, in the IV cost function, a matrix Z (called the instrumental variable) is added in order to achieve unbiased estimates. The second method is based on an Unscented Kalman Filter [2], a derivative-free alternative to the Extended Kalman Filter (EKF), which uses a statistical linearization technique of a nonlinear function by linear regression. To do that, a set of chosen sample points called sigma-points are propagated through the true nonlinear system and the posterior mean and covariance are captured accurately to the second order for any nonlinearity. The performances of IV and UKF methods have been compared in terms of precision and robustness and validated using data from the MICROCARB AOCS Simulator. For missions that do not integrate maneuvers particularly designed to estimate the satellite inertia, a telemetry data analysis is crucial to find the most dynamically interesting moment of the satellite life to launch the estimation. For this purpose, four observability metrics (namely the A-Optimality, the D-Optimality, the E-Optimality, and the Condition Number criteria [3] [4]), integrated with qualitative data analysis, are tested on a typical MICROCARB day. Once the best range of the dataset is identified, the estimation is launched and it results that for both the IV and the UKF methods the relative error remains within 2% on the three principal axes parameters ????????????, ???????????? and ????????????. Finally, with the intent of trying to get the most from the available data, the observability is computed on each guidance mode of the MICROCARB day and the ten best ones are combined in a fictitious maneuver to be tested as input for the estimation algorithms. As a result of the whole research, a MATLAB tool named CALIBRI to study the input data observability and consequently estimate the satellite inertia has been added to the CNES heritage. References [1] In-Orbit Data-Driven Parameter Estimation for Attitude Control of Satellites, PhD Thesis, Universite de Lorraine, 2020, C. Nainer [2] Kornienko, A., Dhole, P., Geshnizjani, R., Jamparueang, P., and Fichter, W., Determining Spacecraft Moment of Inertia Using In-Orbit Data, GNC 2017: 10th international ESA conference on Guidance, Navigation and Control Systems, June 2017, Salzburg, Austria. [3] Franceschini, G. and Macchietto, S. Model-based design of experiments for parameter precision: State of the art. In: Chemical Engineering Science 63.19 (2008), pp. 48464872. [4] Walter, E. and Pronzato, L. Qualitative and quantitative experiment design for phenomenological models - a survey. In: Automatica 26.2 (1990), pp. 195213. |
