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].

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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 co-ordinate system (Figure 2), Cartesian 3D (metric) co- (m), geographical ({), 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 ’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 GNSS receivers, in- the Earth's ellipsoid description, as defined in Table 1 cluding those embedded in smartphones, utilize the and outlined in Equations 2 and 3. geographical reference frame, the position estimates intended to serve as the input data set for position- Table 1 – Earth’s ellipsoid description as defined by the WGS-84 ing error calculation should be first transformed to the WGS84-based 3D Cartesian reference frame using Parameter Name WGS-84 value Equations 7-9, following the application of Equation 1. Semi-major axis a 6378137 m This requires knowledge of the Earth's dynamics pa- Flattening f 1/298.257223563 rameter values (Table 1 and Equations 1 and 2) [1,5,8]. Alternative methods may be applied using the coordi- nates of real and estimated positions only, as present- ef2 =-$ ()2 f (2) ed in Section 3.3 [8]. a y = 22 1 - e $ sin { (3) 3.3 Where: Reference [8] provides a simple means for the e – eccentricity; calculation of the so-called geographical distance be- f – flattening; tween two points on the ellipsoid. Approximating the a – equatorial radius (semi-major axis); shape of the Earth as an ellipsoid, the algorithm con- { – geographical latitude; siders the geographical coordinates of positions and y – Earth’s radius of curvature in the prime vertical. the radius of the sphere R only. The haversine formula Equations 4-6 outline the procedure of co-ordinates is defined as inEquation 10: transformation from the WGS84-based 3D Cartesian 2 HH1 - cos haversine H ==sin (10) reference frame to the WGS84-based spatial geo- ^ h b 22l graphical reference frame. By defining the intermediary parameter a as in -1 Z 2 y Equation 11: { = arctg $ 1 - e (4) 22 y + h XY+ a k 2 {{21- 2 mm2 - 1 ^ h a = sinc+ os {{12$$cossin (11) Y b 22l b l m = arctg (5) X the great circle angular distance (in rad) can be calcu- lated using Equation 12: XY22+ ) h = - y (6) cos { ca=-tana21, a (12) ^h The transformation process from the WGS84- from which the geographical distance between the two based spatial geographical reference frame to the points is given by Equation 13: WGS84-based 3D Cartesian reference frame is out- lined in Equations 7-9. dR= $ c (13)

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The algorithm comprising Equations 10-13 can be rewritten as a single Equation 14: 20 [ m ]

2 {{21- 2 mm21- dR= 2 $$arcsin sinc+ os {{12$$cossin b 22l bbll (14) 15

10 4. ASSESSMENT OF THE HAVERSINE 5 FORMULA METHOD PERFORMANCE Haversine

Calculated distance Calculated Cartesian We assessed the performance of the haversine 0 formula approach by comparing the positioning error 0 50 100 150 200 data with the data obtained using the WGS84 meth- Case No. od. The results are presented in this section with the Figure 3 – Calculated distance using Cartesian and aim to propose a standard operation procedure for haversine algorithms positioning error estimation in the field of navigation applications in location-based services and intelligent transport systems. -1 [ m ] Two methods for positioning error calculations (Car- -2 tesian, Section 3.2, and haversine, Section 3.3) were used for algorithm development and were deployed in -3 the open-source statistical programming environment -4 Absolute error error Absolute for statistical computing R [9]. A series of GNSS-based position observables was -5 artificially created in R, using the algorithm defined in 0 5 10 15 20 25 Equation 15. Cartesian distance [m]

--6 6 Figure 4 – Absolute distance calculation error, with {{10=+ii$$10 ,,mmi =+0 10 i = 12,,f 00 (15) _ i Cartesian values used as the reference The series of position estimates with coordinates in the spherical reference frame 15 was translated into -20 a series of position estimates with Cartesian coordi- -30 nates using 7-9. The coordinates of the reference point [ m ] -40 ({0,m0) were translated into Cartesian coordinates -50 (x0,y0). The algorithm 12-14 was applied to the series of -60 observables in spherical coordinates 15 to determine error Relative -70 the distance from the reference point ({0,m0). This pro- -80 cedure generated a series of haversine distance esti- 0 5 10 15 20 25 mates, dh, expressing position errors as distances be- Cartesian distance [m] tween two points on a sphere representing the Earth. Figure 5 – Relative distance calculation error, with The method 1 was applied to the series of observ- Cartesian values used as the reference ables in Cartesian framework to determine the dis- tance from the reference point (x0,y0). This procedure generated a series of Cartesian distance estimates, 5. DISCUSSION d C, expressing position errors as distances between Two most commonly used methodologies for de- two points in the 3D space, without reference to any termination of GNSS positioning accuracy based on object or shape. raw GNSS observables were presented. The results of Two series of distances were compared for their ab- a simulation (Figure 3) show different behavior of the Cartesian and the haversine approaches. With the Car- solute values (Figure 3), absolute distance calculation tesian approach used as a reference, the performance errors with d C series as a reference (Figure 4), and of the haversine approach was evaluated in both abso- relative distance calculation errors with d C series as lute and relative terms. As the Cartesian values were a reference (Figure 5). Absolute distance is defined as considered true, the absolute error caused by the the Euclidian distance in 3D space. Relative distance implementation of the haversine approach is mostly is determined in relative terms, as the ratio of absolute proportional to the actual distance to be presented distance error over true distance. (Figure 4). Considering the relative terms (Figure 5), the

308 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 haversine approach shows poor performance at small 1 Sveučilište u Zagrebu Fakultet elektrotehnike distances, while stabilizing to the value of approxi- i računarstva, Unska 3, 10000 Zagreb, Hrvatska mately 20% for larger actual distances. 2 Sveučilište u Rijeci, Tehnički fakultet Reducing the domain to navigation applications Vukovarska 58, 51000 Rijeka, Hrvatska 3 (which includes location-based services and intelli- Sveučilište u Zagrebu, Fakultet prometnih znanosti gent transport systems) used by non-stationary ob- Vukelićeva 4, 10000 Zagreb, Hrvatska jects, the haversine approach still performs relatively poorly. Finally, the initial calculation of a GNSS position IZRAŽAVANJE POZICIJSKIH POGREŠAKA U GNSS estimate is conducted in a GNSS receiver using the SUSTAVU POMOĆU UDALJENOSTI Cartesian reference system, making it a more natural SAŽETAK solution for determination of GNSS position accuracy. The haversine formula method reduces the inaccu- U radu se analiziraju dvije metode određivanja pozici- racy in positioning error estimation caused by incorrect jskih pogrešaka u globalnom navigacijskom satelitskom sus- simplifications in the process. However, the mitigation tavu (eng. Global Navigation Satellite System – GNSS). An- effects of the haversine formula are of limited scope. alizirane metode služe za procjenu uspješnosti određivanja This finding led to a recommendation for utilization of položaja, a zasnivaju se na izračunu razlika između mjerenog WGS84-expressed coordinates of position estimates, i stvarnog položaja. Jedna metoda koristi Kartezijev 3D taken from dedicated APIs of GNSS SDR receivers, for koordinatni sustav, a druga sferni koordinatni sustav. Za studying GNSS positioning performance in LBS and određivanje udaljenosti koristi se metoda udaljenosti u Kar- tezijevom referentnom sustavu te „haversine“ formule (for- ITS application scenarios. mula polovine sinus versusa). Istraživanje ukazuje na neri- ješene probleme u procjeni performansi GNSS određivanja 6. CONCLUSION položaja pri uporabi procjene položaja u geografskom ref- Frequent encounters with miscalculated values erentnom sustavu. Zbog spomenutih problema za procjenu of GNSS positioning errors derived from true and ob- uspješnosti određivanja položaja preporuča se korištenje served positions motivated this study. We identified a postupaka procjena pogreške položaja zasnovanih na orig- problem in false assumption of identity between po- inalnom WGS84 standardu, primijenjenim u programski sition coordinates in geographical and 3D Cartesian određenim radio prijamnicima (SDR). reference systems that is further utilized in position error estimation based on positions expressed in the KLJUČNE RIJEČI geographical reference system, a common practice pogreške određivanja položaja GNSS-om; procjena učinka; in traditional low-cost receiver design. We challenged izračun udaljenosti; programski određen radio prijemnik the problem by introducing the haversine formula, a (SDR); method for transformation from the geographical to the Cartesian 3D reference system (WGS84 in case of REFERENCES GNSS utilization). We then examined the performance [1] Petrovski I, Tsujii T. 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