An Iterative Robust Regularization Method for GPS Positioning

Abstract

In shaded areas, Global Positioning System (GPS) signals may suffer significant degradation, making the measurements to be contaminated by different kinds of errors. These errors such as outliers arising due to multipath and line-of-sight signals can degrade the accuracy of GPS positioning. Moreover, the poor geometric constellation of satellites that the receiver is tracking may cause inverse problems, resulting in unstable and unreliable GPS positioning solutions. To tackle these problems, an iterative robust regularization method (IRRM) for the GPS positioning is proposed in this article. The IRRM employs the Cholesky factorization method to preclude computing the inverse of matrices, making it numerically stable and be easily implemented in engineering. Also included in this work is the derivation of the sufficient condition for the IRRM. Performance comparisons of the proposed algorithm with the least squares, the Cholesky factorization method for iterative least squares (CM), and the robust estimation are demonstrated via two static positioning problems. The experimental results, presented herein, exhibit the improved performance of the IRRM over conventional algorithms.

Appendix

Numerical stability often refers to the accuracy of an algorithm in the presence of rounding errors. An algorithm is unstable if rounding errors cause large errors in the result. In the appendix, we are going to have a brief discussion on the numerical stability.

Several applications in engineering, physics, and applied mathematics can be modeled with the help of the generalized eigenvalue problem (GEP). Most of them concern the largest eigenvalue and the associated eigenvector. However, for a singular system, the smallest eigenvalue is equally of importance to the largest one. It is numerically unstable to solve them by simply solving for the generalized inverse of ${\varvec{A}}$.

Consider the following two systems: (39) and (40),

$$\begin{aligned} {\varvec{A}}^T{\bar{\varvec{P}}\varvec{A}}\delta {\varvec{X}}={\varvec{A}}^T{\bar{\varvec{P}}}\delta {\varvec{L}} \end{aligned}$$

(39)

and

$$\begin{aligned} ({\varvec{A}}^T{\bar{\varvec{P}}\varvec{A}}+\alpha I)\delta {\varvec{X}}={\varvec{A}}^T{\bar{\varvec{P}}}\delta {\varvec{L}} \end{aligned}$$

(40)

where (39) is derived from (4), and (40) is the associated regularization system.

With regard to system (39), ${\tilde{\varvec{N}}}={\varvec{A}}^T{\bar{\varvec{P}}\varvec{A}}$ is almost singular for ill-conditioned problems. If $\tilde{\lambda }_{\max }$ and $\tilde{\lambda }_{\min }$ denote the maximum and the minimum eigenvalues of ${\tilde{\varvec{N}}}$, respectively, then

$$\begin{aligned} {\tilde{\varvec{N}}}\sim diag(\tilde{\lambda }_{\max } ,\;\bullet \;,\;\bullet \;,\;\tilde{\lambda }_{\min } ) \end{aligned}$$

(41)

where the symbol “$\sim$” denotes the matrix similarity; $\bullet$ stands for the other eigenvalues, such that $\bullet \in \left[ {\tilde{\lambda }_{\min } ,\;\tilde{\lambda }_{\max }} \right]$. We do not consider the specific values of $\bullet$ since they do not have any effect on the following discussions.

Corresponding to the analysis of Sect. 3, we use the 2-norm condition number to describe the stability of the proposed method. It is well known that if the condition number is close to 1, the matrix is well-conditioned which means its inverse can be computed with high accuracy. If the condition number is much larger than 1, then such a matrix is almost singular, and the computation of its inverse, or solution of the system, is prone to large numerical errors.

The 2-norm condition number of (39) is given by

$$\begin{aligned} \kappa ( {\tilde{\varvec{N}}})_2 ={\tilde{\lambda }_{\max }} /{\tilde{\lambda }_{\min }} \end{aligned}$$

(42)

By introducing the stable function, we have

$$\begin{aligned} ( {{\tilde{\varvec{N}}}+\alpha I})\sim diag(\tilde{\lambda }_{\max } +\alpha ,\bullet ,\bullet ,\tilde{\lambda }_{\min } +\alpha ) \end{aligned}$$

(43)

The associated condition number of (40) is

$$\begin{aligned} \kappa ( {( {{\tilde{\varvec{N}}}+\alpha I})})_2&= {( {\tilde{\lambda }_{\max } +\alpha })}/ {( {\tilde{\lambda }_{\min } +\alpha })} \nonumber \\&= \frac{1}{\tilde{\lambda }_{\min }} \cdot \frac{\tilde{\lambda }_{\min } ( {\tilde{\lambda }_{\max } +\alpha })}{( {\tilde{\lambda }_{\min } +\alpha })}\nonumber \\&\le \frac{1}{\tilde{\lambda }_{\min }} \cdot \frac{\tilde{\lambda }_{\max } ( {\tilde{\lambda }_{\min } +\alpha })}{( {\tilde{\lambda }_{\min } +\alpha })}\text{= }\kappa ( {\tilde{\varvec{N}}})_2 \end{aligned}$$

(44)

Clearly, the ill-conditioning has been weakened. Hence, the solution of the regularization system is more numerically stable.