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Expression of GNSS Positioning Error in Terms of Distance Filić M, Filjar R, Ševrović M. Expression of GNSS Positioning Error in Terms of Distance MIA FILIĆ, mag. inf. et math.1 Transport Engineering E-mail: [email protected] Preliminary Communication RENATO FILJAR, Ph.D.2 Submitted: 4 Nov. 2016 E-mail: [email protected] Accepted: 11 Apr. 2018 MARKO ŠEVROVIĆ, Ph.D.3 (Corresponding author) E-mail: [email protected] 1 University of Zagreb Faculty of Electrical Engineering and Computing Unska 3, 10000 Zagreb 2 University of Rijeka, Faculty of Engineering Vukovarska 58, 51000 Rijeka, Croatia 3 University of Zagreb Faculty of Transport and Traffic Sciences Vukelićeva 4, 10000 Zagreb, Croatia EXPRESSION OF GNSS POSITIONING ERROR IN TERMS OF DISTANCE ABSTRACT methodology development, and simplifications used, This manuscript analyzes two methods for Global Nav- thus undermining the value of research and providing igation Satellite System positioning error determination for meaningless and non-comparable results [2]. positioning performance assessment by calculation of the In our research, we address the problem by assess- distance between the observed and the true positions: one ing the possible approaches to express GNSS position- using the Cartesian 3D rectangular coordinate system, and ing accuracy for navigation (i.e., used for applications the other using the spherical coordinate system, the Carte- to non-stationary objects, such as those belonging to sian reference frame distance method, and haversine for- location-based services (LBS) and intelligent trans- mula for distance calculation. The study shows unresolved issues in the utilization of position estimates in geographical port systems (ITS) [3]), and propose a solution for a reference frame for GNSS positioning performance assess- common methodology that will allow for comparison ment. Those lead to a recommendation for GNSS positioning of independently conducted GNSS positioning assess- performance assessment based on original WGS84-based ments. GNSS position estimates taken from recently introduced The manuscript is structured as follows. Section 2 data access from GNSS software-defined radio (SDR) receiv- discusses the problem. Section 3 outlines two com- ers. mon methodologies for the determination of GNSS positioning accuracy. Section 4 presents the results KEY WORDS of performance comparison of the methodologies pre- GNSS positioning error; performance assessment; distance sented in Section 3. In Section 5, the results of Section calculation; software-defined radio (SDR); 4 are discussed with the aim to propose the common practice in determination of GNSS positioning accu- 1. INTRODUCTION racy performance for independent studies. Section 6 An assessment of a positioning method's accura- concludes the manuscript and outlines the plan for cy is essential in the estimation of the performance near-term future research activities. of various position-based navigation, communication, and information systems and services [1]. Inexperi- 2. PROBLEM DESCRIPTION AND RESEARCH enced use of different methodologies for experimental BACKGROUND assessments frequently leads to miscalculations and inability to compare different experimental evaluations Continuous monitoring of GNSS service perfor- of positioning services. The problem is emphasized mance usually comprises the task of determining in the evaluation of the performance of positioning GNSS positioning error observables in navigation services (such as those based on the Global Naviga- applications for further analysis [3,4,5]. A common tion Satellite System, GNSS), where different accu- cost-effective approach calls for direct collection of po- racy assessment methodologies are frequently ap- sition error estimates from GNSS receivers, if such a plied without a deeper understanding of their nature, feature is in operation [1, 4]. Promet – Traffic & Transportation, Vol. 30, 2018, No. 3, 305-310 305 Filić M, Filjar R, Ševrović M. Expression of GNSS Positioning Error in Terms of Distance Most commercially available single-frequency Equation 1 and depicted in Figure 1 [1,4]. Since the GNSS receivers present the results of the position es- GNSS position estimation process usually takes place timation process based on reception of satellite rang- in the so-called navigation application domain and ing signals [4] in geographical coordinates, describing with presumed utilization of the 3D Cartesian WGS84 the estimated position with geographical longitude co-ordinate system (Figure 2), Cartesian 3D (metric) co- (m), geographical latitude ({), and height (h) above the ordinates (x, y, z) are used, rather than angular ones. mean sea level [1, 2]. Such an approach in expressing 2 2 2 dT(,12Tx) =+21--xy21yz+-21z (1) measurement results complies with the requirements ^^hh^ h of numerous GNSS-based applications [3]. However, In respect of location-based telecommunication expression of position estimates in that way causes a services (LBS) and intelligent transport system (ITS) significant increase in miscalculation and misinterpre- services, operators and users are more concerned with planar (horizontal) positioning errors, rather than tation of GNSS positioning error estimates. general and height-related ones [3]. This results in the In general, the GNSS positioning error is estimat- nowadays increasingly common positioning error mis- ed by calculation of the Euclidian distance between calculation, where researchers presume flat Earth’s GNSS-estimated and true positions, as expressed in surface and equalize the geographic coordinates of a position (geographical latitude and geographical lon- z gitude) with the planar (horizontal) components of a position in the WGS84 Cartesian three-dimensional z 2 system (x,y) in the process of horizontal GNSS error calculation. Such a practice leads inevitably to incor- rect and misleading results in an increasingly frequent number of studies. The traditional design of the GNSS receiver only worsens the situation, since the GNSS t 2 position estimation process is neither transparent nor d accessible by third-party applications. In recent de- z1 y1, y2 y velopments, raw pseudo-range measurements from software-defined radio (SDR) GNSS receivers [6] have x1 t1 become accessible through dedicated application protocol interfaces (APIs) [7], thus allowing access to partial results of GNSS-based position estimation x 2 performed in the WGS84 3D Cartesian reference co- x ordinate system [6]. Still, many researchers and engi- neers take the least-resistance approach in analyzing Figure 1 – Problem description: point 1 as true position, the final, rather than partial or original, results of the and point 2 as an estimated one position estimation process. z 3. HAVERSINE-BASED GNSS POSITIONING ERROR DETERMINATION Assessment of GNSS position estimation error for x,y,z point on surface navigation applications (including location-based ser- vices, road traffic information systems, and road-use charging) does not require the appropriate level of con- fidentiality [3]. The accuracy of GNSS positioning error determination derived from GNSS position estimates { y is to be assessed to advise the choice of a position Prime meridian m error estimation method that will result in a significant- Equator ly smaller additional estimation error than the GNSS positioning error itself [1]. We aimed at the assessment of feasibility to utilize x the geographical coordinates in a GNSS positioning performance analysis, providing a suitable method was applied for correct transformation of the WGS84 Figure 2 – Definition of position in the WGS84 reference position estimates into geographical ones. The hav- frame ersine formula [8] was selected for the coordinate 306 Promet – Traffic & Transportation, Vol. 30, 2018, No. 3, 305-310 Filić M, Filjar R, Ševrović M. Expression of GNSS Positioning Error in Terms of Distance system transformation. In the assessment of the hav- Xh=+()y{$ cosc$ os m (7) ersine formula approach to solving the problem, we selected the WGS84 data [2] as control. Yh=+()y{$$cossin m (8) 3.1 Transformation formulae Ze=+[(y{1 - 2)]h $ sin (9) A GNSS-based position estimate is usually given Depending on the choice of programming environ- in the 3D Cartesian (rectangular) reference frame. A ment, the angular values should be expressed in [rad] common GNSS position estimation algorithm deals in practical deployment of the system of Equations 2-9. with the observables and positions in the WGS84 3D Cartesian reference frame [1,2]. 3.2 3D Cartesian reference frame distance Commercial-grade GNSS receivers usually provide method position estimates in the spatial geographical refer- ence frame, outlining the position's latitude, longitude, Providing the real and estimated positions (x1,y1,z1) and height above the Earth's mean sea level. Provision and (x2,y2,z2), respectively, are given in the 3D Carte- of position estimates in the spatial geographical ref- sian reference frame, the position estimation error erence frame allows for user-oriented presentation of can be defined as the shortest distance between two the results of the position estimation process [1]. points in space, as outlined by Equation 1. Transformations between the two reference frames Numerous authors consider Equation 1 to be the can be conducted using transformation formulae, as determination of the position estimation error per defi- described in the rest of this section. nition. However, as the series of position estimates The transformation formulae use the parameters of provided by commercial-grade
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