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Unbiased Nonlinear Least Squares Estimations of Short-distance Equations

Published online by Cambridge University Press:  23 February 2017

Shuqiang Xue  and
Yuanxi Yang
Shuqiang Xue
Affiliation:
(School of Geological and Surveying Engineering, Chang'an University, Yanta Road, Xi'an 710054, China) (Chinese Academy of Surveying and Mapping, Beijing, 100830, China)
Yuanxi Yang*
Affiliation:
(National Key Laboratory for Geo-information Engineering, Xi'an Research Institute of Surveying and Mapping, Xi'an, 710054, China)
*
E-mail: yuanxi_yang@163.com
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Abstract

Nonlinear least squares estimations have been widely applied in positioning. However, nonlinear least squares estimations are generally biased. As the Gauss-Newton method has been widely applied to obtain a nonlinear least squares solution, we propose an iterative procedure for obtaining unbiased estimations with this method. The characteristics of the linearization error are discussed and a systematic error source of the linearization error needs to be removed to guarantee the unbiasedness. Both the geometrical condition and the statistical condition for unbiased nonlinear least squares estimations are revealed. It is shown that for long-distance observations of high precision, or for a positioning configuration with the lowest Geometric Dilution Of Precision (GDOP), the nonlinear least squares estimations tend to be unbiased; but for short-distance cases, the bias in the nonlinear least squares solution should be estimated to obtain unbiased values by removing the bias from the nonlinear least squares solution. The proposed results are verified by the Monte Carlo method and this shows that the bias in nonlinear least squares solution of short-distance distances cannot be ignored.

Keywords

Distance equationBias Nonlinear least squaresNonlinear error propagation
Type
Research Article
Information
The Journal of Navigation , Volume 70 , Issue 4 , July 2017 , pp. 810 - 828
Copyright
Copyright © The Royal Institute of Navigation 2017 

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