Orbit Determination By Fitting Probability Distribution On Space Object Admissible Region Using Gaussian Mixed Model

In recent years,the number of spacecrafts and space debris orbiting the Earth keeps increasing,which could lead to substantial increase in the collision risk of space objects.Therefore,the capability of space object surveillance and orbit cataloging must be strengthened to ensure space services such as collision warning and avoidance more efficiently.Orbit cataloging includes the catalog maintenance of cataloged objects and incorporation of newly discovered objects,with the latter involving three orbit determination phases for each new object.The first is the initial orbit determination(IOD),the second the improvement of orbit accuracy using relatively more observation data,and finally,when the orbit determination accuracy is high enough,the new object is put into the catalog and it becomes a cataloged object.The constantly expanded capability of space object detection allows collection of a huge amount of observation data of space objects,many of which are un-cataloged small objects whose observed orbit arcs could be very short.As many of the too short arcs(TSAs)are only optically observed,classical IOD methods,including the Gauss method and Laplace method,could be unsuitable to the TSA observation scenario,mainly because the essence of all the classical methods is to determine an object’s orbital geometry from angles over a TSA.Unfortunately,it is difficult to confine the shape of the orbit track using TSA data having observation uncertainty.The applicability problem of the classic methods makes the TSA angles-only IOD a research focus in Astrodynamics in the past two decades.In order to solve the problem of the TSA angles-only IOD,this thesis proposes an IOD method which uses sparse TSA angles data of a space object to fit a probability density function(PDF)of the object’s admissible region,where the PDF is expressed by the Gaussian Mixed Model(GMM).This method firstly constrains the solution of the orbit parameters through a variety of orbital mechanics conditions,so that the solution parameters can be contained within a specific solution domain referred to as the Admissible Region(AR).Then the stochastic characteristic of the solutions within the AR is described by a given PDF,thus forming the Probabilistic Admissible Region(PAR)which makes the IOD algorithm focus on a specifically constrained solution domain at the very beginning.Considering that the probability distribution of the orbit parameters is unknown at the initial moment,this thesis assumes the initial probability distribution to be a two-dimensional uniform distribution for two unknown parameters on the PAR.Then,the GMM is fit using angles data,and an epoch-by-epoch filtering approach will eventually result in a PDF which contains orbit parameters.To Combine multiple TSA angles data together,the Gaussian Mixed Unscented Kalman Filter(GMUKF)estimation algorithm under Bayes’ theorem is applied to estimate unknown orbit parameters epoch by epoch,therefore,the final solution is a maximum a posteriori estimation.In addition,the Square-Root Unscented Kalman Filter(SR-UKF)is used instead of the conventional UKF in the implementation of algorithm to overcome the problem of filter divergence caused by the computer numerical truncation error.At the same time,the application of the Open MP parallel computing technique in the orbital propagations,which are computationally very cost,has significantly improved the efficiency of the algorithm.Experimental results show that,when the observation accuracy is 0.5″,more than50% of the estimated distances between the observing station and space objects have errors less than 7 km,and more than 98% less than 12 km,and the error of the distance change rate is under 10 m/s.When the accuracy of observation data increases to 0.1″,more than 98% of the estimated distances have errors less than 1 km,while the error of the distance change rate is under 2 m/s.The experimental results also show that,when the uncertainty of observation is small,the GMUKF algorithm outperforms other IOD methods in terms of the accuracy.In addition,this method has good convergence and stability.All these properties make the proposed IOD method a valuable solution option to the TSA sparse angles IOD problem,and the GMM approach may find applications in Geodesy and Geomatics.