Study On Some Theoretical Problems Of The Earth Gravity Field In Ballisitcs

Space trajectory is a curve, and it is need the precise geophysical environment to send the missle into the designed position which generally is the tarket. Earth's gravity field is the most important geophysical environment content. Due to the fast movement characteristics of the missile, how quickly and accurately obtain the gravity field information is the key to achieve a success launch of the missile and the task of precision strike. Therefore, we studied the earth's gravity field theory problems for the ballistics.The disturbance gravity vector outside the earth can be calculated by spherical harmonics exapnsion. Depending on the dunctional of the harmonic series, associated lengendre functions must be computed numerically which in general is based on recurrence relations. The calculation speed and the stabiliy of a variety of associated Legendre functions are analyzed in different environments. The experiments and results provide a reference for engineer application. Another important issue of the earth's gravity potential model is the computation of trigonometric functions of chich the calculation speed should be fast. The running speed of trigonometric recursion is compared with the call of trigonometric runctions through numerical simulations. The simulations results have shown that recursive calculation speed has obvious advantages.Coordination translation of spherical functions between new and old system in sptial coordination has been deduced, and the translation of geo-coordination between the two systems is introduced using spherical triangle functions. There are two methods that select the new pole point for the new coordination system. Simulation on the accurancy and the computed speed of the new spherical functions is conducted and the results show that the two methods can improve the calculation speed of gravity unit. In addition, using Clenshaw summation algorithm, the method that the missle is along the new equator can improve the calculation speed greatly. Spherical harmonics expansion is singularity when the position is in the area near the pole. Therefore, the errors of gravity units calculated by the new geopotential model are large by the method that the missle is along the new maridial.The geometry impact of vertical deviation on trajectory is analysed through theory, and the analysis of the effect of vertical deviation error on impact point error is gaven in the article. The relationship between impact point geometric error and deflection of vertical is deduced based on the analysis of the results. In addition, the formula of the calculation of missile impact point geometric errors by vertical deviation is analysised in launch coordination system and the errors are in three vectors. An example is simulated to determ the impacts of vertical deviation on various distance trajectories. We assume that the vectors of vertical deviation in two directions are both 20" and the astronomical geodetica azimuth is 90°. The results of simulation show that impact point error of geometric error caused by vertical deviation is closely related with the distance of trajectory. The errors caused by vertical deviation must be eliminated when the distance of trajecy is over 1000 km.The advantages of the virtual point masses model are analysed and the key issues of the construction for the model are discussed. Furthermore, the method of the structure of the virtual sphere model is formatted in the paper. The analysis of the model error transformation is gaven and the analysis is taken as a base condition to build the virtual sphere radius. Point masses model sometimes should extand the area of gravity field, thus a method of the extanding is proposed. And the efficient of the method is verified by the simulation data. Based on geopotental model with the low degree and order, the point masses model is formatted through the gravity anomaly calculated by EGM2008. Then, the radius direction vectors of gravity disturbance computed by peopotatial model and new point masses model are used to analyse the precision of the virtual model. In addition, the compution speed of the gravity units calculated by point masses model is comparied with the speed by geopotential. Turthermore, the point masses model is analysed with corresponding degree and order of geopotential.Polynomial fitting and spline interpolation are used to compute the gravity disturbance rapidly. In order to assign the gravity disturbance correltly and fast, the optimal number of the section of the trajectory is designed and the criteria of the optimal order of the polynomial are proposed. In addition, the actualization of the optimal number and the criteria are discussed in the paper. Various experiments are designed for three sections of the trajectory and different order polynomial. The results of experiments show that:the trajected should be divided into many sections, the advantages of the polynomial fitting are that the calculation speed is fast and the memory requirements are less. Using a Uniform B spline gravitational perturbations can also be achieved fast assignment algorithm requires less memory, while using B-spline interpolation can be more flexible on the trajectory piecewise interpolation, to reduce unnecessary interpolation nodes. The best approximation standards of the uniform and un-uniform B-spline are proposed for the assignment of gravity disturbance. However, the standards are in the case the the approximation error must approve enough accurancy. Additionally, the implementation of the method is probed.Spherical cap harmonic model is studied to calculate the gravity disturbance vector fast. The common and difference between spherical cap harmonic method and spherical harmonic method are compared and analysed. The approximation of the two methods are also compared and discussed. The non-integral degree associated legendre functions are introduced. However, according to the analysis of the non-integral degree associated lengedre function, the approximation degree of the spherical cap harmonic model is limited, thus the advantage of this method is not obviously. Tests prove that sphcrical cap harmonic model can replace goopetential model with corresponding degree and order. What is more, the application space of the spherical cap harmonic model for trajectory is gaven through model truncating error formula.The relationship between the distance of trajectory and the motion terminal parameter of the missle is studied and based on the relationship, the first derivative of the trajectory distance error repect to the three parameters is analysed. At the same tim, the relationshop tetween the the trajectory distance error and the three parameters error is researched. In addition, we assume two cases with hight and anayse the trajectory error with different cases. The analysis offers the reference for optimization of trajectory design and mathematical modeling. Simulation is on the asumation that the numerical differential equations are simplied and the missle speed angle is replaced by a simulated function. The results of the tests on ballistic motion phase show that: rajectory deviation of the speed is very sensitive to the long-range missile, lOmGal errors caused by gravity field error rate can reach 2 km trajectory deviation.