A Research About The Modeling And Simulation Of The Binary Passive Positioning Of The Aircrafts’ Active Segment Trajectories

The model of the air crafts’active segment trajectories has a complex structure. It is a typical multivariable system of differential equations that with high order, and high time varying characteristics. The types of the air crafts are changed with the values of the model’s parameters. The binary passive locating method has a higher precision than the single locating methods and a better concealment than the active positioning mode,and needs less observation platforms than the methods of the many star locating.A real-time observation of the air crafts’active segment trajectories is realized by this method.This thesis is based on the topic B of the "Huawei’s cup"2012national graduate mathematical contest in modeling. In this thesis, the derived steps of the point by point intersection method that used to calculate the observed values of the air crafts’active segment trajectories are detailed described.In the second chapter, the conversion between the different coordinate systems using the method of "coordinate system transformation matrix of the Euler angle representation" is realized, and the problem of various coordinate systems are not uniform during the binary passive location is effectively solved. Also in this chapter, an alignment of the observing time points using the cubic spline interpolation method is made to solve the problem of observing time points are inconsistent, and the calculation results of the air crafts’active segment trajectories in the form of3d space diagrams was finally presented.The nonlinear least square methods are developed by the linear least square methods. Using the criterion of minimizing the errors’sum of squares, they can be used as the kinds of methods that are used to estimate the parameters of the nonlinear static models. Levenberg-Marquardt method is the most widely used nonlinear least squares algorithm. The gradient method is used to calculate the maximums and minimums in this algorithm. And the advantages of the gradient method and the newton method are possessed by this algorithm at the same time.The equations’system about the air crafts’active segment trajectories is a multivariate second-order differential system, and has serious nonlinearities. In the third chapter of this thesis, an estimation about the related parameters of the air crafts’active period’s moving equations is made by using the Levenberg-Marquardt method, and some parameters’estimates of some special time points are showed in the form of a table. The final fitting residual error is in the order of magnitude of10m which is showed in the simulation results.