Direct transformation of coordinates for GPS positioning using the techniques of genetic programming and symbolic regression

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Abstract

Transformation of coordinates usually invokes level-wised processes wherein several sets of complicated equations are calculated. Unfortunately, the accuracy may be corrupted due to the accumulation of inevitable errors between the transformation processes. This paper presents a genetic-based method for generating regressive models for direct transformation from global positioning system (GPS) signals to 2-D coordinates. Since target coordinates for a GPS application can be obtained by using simpler transformation formulas, the computational costs and inaccuracy can be reduced. The proposed method, though does not exclude systematic errors due to the imperfection on defining the reference ellipsoid and the reliability of GPS receivers, effectively reduces the statistical errors when the accurate Cartesian coordinates are known from the independent sources. From the experimental results where the target datums TWD67 is investigated, it seems that the proposed method can serve as a direct and feasible solution to the transformation of GPS coordinates.

Introduction

The global positioning system (GPS) (EI-Rabbany, 2002) is a satellite-based navigation system made up of a network of 24 satellites which broadcasts precise timing signals by radio to GPS receivers, allowing users to determine their locations on Earth. With the popularity of general purposed GPS receivers becoming consumer electronics, GPS has been emerging as a convenient tool for positioning and navigation. Positioning by GPS is carried out in three-dimensional (3-D) geocentric Cartesian coordinates, X, Y, and Z, which is calculated and transmitted to the receivers as coded information. Signals from a general purposed GPS receiver are usually encoded according to the NMEA-0183 standard, carrying positioning information (longitude, latitude, altitude, etc.) in WGS84 (DMA, 1987). As in practical applications of positioning or navigation, it is frequently required to convert the coordinates derived in one geographic coordinate system to the values expressed in another. Coordinates obtained from a GPS receiver also need to be converted if the target coordinates are described in a different way.

Transformations between various coordinate systems involve not only complex, usually non-linear, algebraic formulas, but also some very specific numerical parameters which have been established officially as national and continental geodetic datums. If necessary, the transformation process performs level-wise conversions, starting from the source coordinates to the target coordinates, where each level of conversion takes care of a mapping from one coordinate system to another. Unfortunately, some transformation formulas for specific areas are based on estimation and inherently of errors; and therefore, positioning accuracy is degraded as such errors are accumulated during the transformation process. The more steps for transformations, the more inaccuracies. From the aspect of computational costs, the more steps for transformations usually result in more power consumption; which is a serious problem when GPS is applied to mobile applications.

This study tries to eliminate the levels needed for the transformation of coordinates and to produce a condensed process for GPS positioning. Conceptually, the level-wise transformation process can be viewed as the composition of all transformation functions. Thus, there seemingly exists a set of transformation equations between the signals received from GPS and the target coordinates used in an application. This intuitively forms a problem of regression. If such formulas that perform fewer levels of transformation can be derived, the inaccuracies and computational costs may be reduced. In this paper, a genetic-based method is proposed for finding simpler formulas for direct transformation of coordinates from WGS84 to specific coordinate systems.

Notably, GPS-based positioning needs to consider at least two types of inaccuracy. Firstly, systematic errors exist due to the imperfection on defining the reference ellipsoid (Ashkenazi, 1986), though relative positioning with respect to standard reference points defined in the local datums is accurate and acceptable. Secondly, GPS has several sources of errors, such as the radio signal corruption caused by ionospheric delay, tropospheric delay, satellite clock and receiver clock offsets, receiver noise, receiver calibration and multipath, and the synchronization of positioning data from different tracking stations. Approaches that can partially reduce such errors have been proposed (Liou et al., 2001, Lundberg, 2001, Jwo et al., 2004, Yeh et al., 2006, Zhang et al., 2006). However, eliminating such errors mentioned above is not the scope of this paper.

The proposed method has been successfully applied, but not restrictedly, to the transformation of TWD67 (Tseng and Chang, 1999; MOI, 2006a, MOI, 2006b). The experimental results show that the proposed method can serve as a direct and feasible solution to this problem. This study is not to find a single transformation method for reference systems. The same procedure can be applied to the transformation of other target coordinate systems, provided that accurate coordinates are known from independent sources.

Organization of the paper is as follows. In Section 3, the process of transformation from GPS signals to two-dimensional (2-D) coordinates is briefly introduced. In Section 4, we formulate the level-wise transformation process as a regression problem and define the objective functions. For producing target functional expressions, the proposed methods are presented in Section 5. The results of experiments are given in Section 6. Section 2 presents a brief review on the related works followed by the conclusions of this study in Section 7.

Section snippets

Related work

Finding regressive models is essential in many applications, such as Liu and Wang (2006), Yang et al. (2006), Wu et al. (2007), and Seghouane and Amari (2007). Among these applications, black-box methods, such as artificial neural networks (Tresp, 2000, Aksyonova et al., 2003) and support vector regression (Drucker et al., 1997, Collobert and Bengio, 2001), are widely employed. As mentioned before, black-box methods are not dedicated to describing the relationships between the inputs and

Transformation of coordinates

Basically, to convert coordinates from 3-D space to 2-D space in the same reference system is to flatten the data points by projection. If data points are defined across different reference systems, e.g., between the Polar system and the Cartesian system, they have to be further converted. In this paper, source coordinates received from GPS are in WGS84 which are defined by latitude (φ84), longitude (λ84), and the height (h84). The target coordinates are TWD67 in TM2 (Tseng and Chang, 1999)

Problem formulation

A regression problem is to find descriptive relationships between inputs and outputs so that the expected output corresponding to a new input is predictable. Formally, it can be described as follows. Suppose that D is a set of values appearing at the interval we are interested in and I={(xi,f(xi))|0in,xiD}is a set of paired data. The purpose of regression is to find a function g(x) so that f(xi)g(xi) for all xiD. Usually, the error, εi, between f(xi) and g(xi) is defined as least square

Genetic algorithms, genetic programming, and symbolic regression

Genetic algorithms (Koza, 1992) simulate the natural selection rule, the-best-can-fit, and have been successfully applied to many optimization problems. A genetic algorithm iteratively performs operations of crossover and mutation on the expressions of the target problem and evaluates each new solution with respect to the error function for all input data; until the one with lowest error is accepted. Genetic-based methods are usually applied for solving complex problems where no effective

Data collection and parameters

In order to test our method, sufficient amount of sampling points of inputs g=φ84,λ84,h84 and their corresponding outputs TN(g) and TE(g) are needed from difference sources. For this purpose, two sources of data are collected. One is from the official databases of standard reference points which are maintained by the government (MOI, 2006b). The standard reference points are established by years of surveys and verified periodically. There are 2748 such standard reference points distributed on

Discussion and conclusion

This paper presents a genetic-based method for finding regression functions for transforming coordinates in GPS applications. From the experimental results, it seems that symbolic regression can serve as a feasible solution to the transformation of coordinates. The proposed method, though does not exclude systematic errors due to the imperfection on defining the reference ellipsoid and the reliability of GPS receivers, may reduce inaccuracy statistically in relative positioning on a specific

Acknowledgment

This work was partially supported by National Science Council (NSC), Taiwan, under Grants NSC 94-2218-E-390-001 and NSC 95-2218-E-309-004. The authors would like to thank the comments of anonymous reviewers.

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