The Mixed Integer Least Square Estimation And Its Application In Gps Positioning

The observation model based on GPS phase observations includes real-valued parameters and integer parameters , so it is called mixed integer linear model. Based on GPS positioning, a survey of parameter estimation in mixed integer linear model is . The mian work and innovation are summarized as follows: investigation1. LAMBDA method is the most widely used ambiguity solution method, Its is used under the transformation from the mixed integer least square estimation into the integer least square estimation. Three methods of transforming are derived in this paper, and which were analysed from the point of theory and practical application. Among them, derivation method is theoretically feasible, but it is difficult to be applied. Least Square method and QR decomposition method are based on the same mixed integer least square criteria, so the solutions of parameters are same, but the efficiency of them is different. Practical test showed that the least square method was more efficient than QR decomposition method.2. The succes rate of LAMBDA method depends on the precision of the float solution and its covariance matrix. In GPS rapid positioning, with a few epochs of observations, the normal equation is ill-posed, they will lead to larger error in ambiguity float solution and incapability of fixing integer ambiguity. Many current methods improve the ill condition of normal equation through an additional prior information or the change of estimation criteria.3. The mixed integer least square estimation was put forward for ambiguity solution, this method do not need to transform the mixed integer square estimation into an integer least square estimation, so it is appliable for any forms of estimation criterias. The current optimal method of the mixed integer programming is branch and bound method, which was used in this paper. Tests showed that when given the prior information of the parameters, the mixed integer least square estimation can obstain more accurate solution than LAMBDA method.4. LAMBDA is more efficient than the mixed integer least squares estimation method. The influential elements of the efficiency of branch and bound method includes: the choice of branch variable, the determination of the upper bound, the sloution of the child problem. Choosing the biggest diagonal entry of weight matrix is most optimal.Traditional method determine the upper bound for the infinite, in this paper the upper bound was determined through bounding the variables and in the whole processing, the upper bound will be updated as long as the new integer solutions are obstained. In this paper several efficient methods were selected, which include the original dual path following method, least square projection, application of Matlab Optimization toolbox. Test shows that least square projection method is most efficient When Branch and bound method is used, the sub-exponential increase in order with the improving of the problem size, based on the carrier phase double difference model, the dimension reduction methods was put forward. Though many methods are presented for improving the efficiency of branch and bound mathod. But test showed that its efficiency still is not as good as LAMBDA method. Finally, the reason was analysed.5. The expression of success probability of integer ambiguity was presented based on Voronoi cell. The Voronoi cell has been limited by an infinite number of hyperplanes, the method of finding all the vertices of the Voronoi cell and active constrained hyperplanes was presented. Due to the complexity of the Voronoi cell, the calculation of the success probability of integer GPS ambiguity is difficult, but it is easy to obstain ambiguity accuracy bounds. In this paper, the tightest bounds of rectangular, ellipsoidal, and spherical types were derived for the voronoi cell.Tests showed the rectangle of maximum area fit the voronoi cell well, the first ellipsoidal type of fitting results was poorly, the sphere type and the second ellipsoidal type of fitting results were more poorly. The corresponding lower and upper probabilistic bounds were calculated. The spherical and ssecond ellipsoidal types of fitting were worse than the first ellipsoidal and the rectangular. To different the weight matrixs, the larger of the condition number of it, the more narrow of the actual shape of Voronoi cell is , the worse lower and upper probabilistic bounds are. At last the probability distribution formula were derived.