Kinematic Studies in the Field of Accuracy and Position Determination of Aircrafts Using the PPP Technique by the GPS System

This article discusses the results of the measurements of position accuracy in 3D space for a UAV object using real-time kinematic measurements of the GPS system. The aim of the research presented in the work was to determine all three components (x, y, z) describing the location of the UAV object during the flight and making comparative characteristics using the PPS navigation service. Comparison of the differences between the two methods showed that the obtained results are characterized by a 95% of certainty. The results obtained in this experiment showed the accuracy of height measurement using the PPS GNSS system at a level of 10-80 centimeters. The results of this study also indicate that kinematic measurements can be very useful in the work of the air traffic controller and engineers of other specialties involved in position determination using satellite techniques. The article also presents a mathematical model defining the process of determining the position of a UAV object during the flight, taking into account kinematic measurements. In the final part of the work, practical conclusions were presented based on the analysis and selected simulation tests.


Introduction
In the introductory part of this work, the authors focused on making a critical analysis of the subject of research, i.e. the problems, determining the accuracy of kinematic objects (aircraft, UAV), based on the precise positioning of PPP (Precise Point Positioning). The article also describes various work parameters in the conducted simulation tests.
In 2002 Shen and Gao published an article, in which they described the kinematic solution of PPP technique for 6 Canadian active control stations. In this aspect, the observational data were tested using two-phase receivers with a sampling frequency of 1 [s] and another convergence time of 6-8 [h].
For this purpose, precise IGS (International GNSS Service) orbits were used with a sampling rate of 15 [min] and IGS satellite clocks with an interval of 30 [s] were used for estimation, where the term GNSS means global navigation satellite system (Global Navigation Satellite System) [1]. Estimated mean square RMS (Root Mean Square) for static PPP information solutions, inside the preamble with corrections provided by geostationary systems, where these stations are shown 10 [cm] in the horizontal direction and 12 [cm] in the height direction [2].
In turn, Gao and Wojciechowski processed two air kinematic trajectories with the angle of reaching the signal to the receiver antenna of 10 [°] to obtain a solution to the PPP concept. The P3 software package, which was implemented at the University of Calgary in Canada, was used primarily to obtain a PPP solution for this purpose. The  [min] respectively. The obtained solution was compared twice, the first with a differential system solution of DGPS (Differential Global Positioning System) for multi-reference data stations, the second with GIPSY-OASIS II software. The first comparison with the DGPS system showed 8 [cm] in the area of latitude and longitude and 0.5 [m] in height. In the second comparison with the GIPSY-OASIS II software, it provided 5 cm horizontally and up to several meters [3].
In turn, Abdel-salam compared the accuracy of the kinematic PPP solution by a differential solution with  [4]. To calibrate the on-board sensors, Ovstedal and other authors tested the accuracy of the TerraPos software compared to the DGPS system. Three hours of continuous flight observation data were processed using this software package, where observation data had an interval of 1 [s]. The estimation using the PPP technique for data with two frequencies was based on a linear combination free from the ionosphere.
In addition, precise orbits, satellite clocks and ground orientation parameters were provided from the IGS service. Estimated horizontal accuracy using the TerraPos software compared to the DGPS system was up to 10 [cm] with an average of 5 [cm].
In turn, Schwieger together with other authors compared the online services GIPSY-OASIS and CSRS-PPP in relation to the DGPS system. They achieved an RMS height of better than 0.5 [m] from PPP solutions with the speed of availability of observational data around 60 [%]. In contrast, Anquela together with other authors analyzed the kinematic trajectory with an observation interval of 5 seconds.
This kind of trajectory was tested using a car with a two-frequency GNSS system receiver, where the tests were carried out in good weather conditions, the sky was clean enough to get a good view of the satellite [5], [6].
The kinematic coordinates of the PPP solution were estimated using the MagicGNSS software, and the solution using the DGNSS system was considered as a reference solution for accuracy calculations. The estimated SD for absolute errors of the GNSS-PPP system showed 8 [cm] in the east and north. What is more, it has achieved a double accuracy dm in the upward direction [7], [8].
Further analyzing the literature on the subject of the study, it should be noted that in 2014 Abdallah and Schwieger explored the impact of the static initialization time for a kinematic solution using PPP. Two kinematic trajectories were tracked using a measuring vehicle in the open area with a dual frequency receiver, where the observation data had an interval of 1 [s].
Observation processes began after about 1 [h] as socalled time of initialization and then 30 [min] of kinematic measurements (Fig. 1). Kinematic coordinates by the PPP technique were obtained twice. First using the GIPSY-OASIS II V 6.3 software package that uses JPL satellite products. The second of the CSRS-PPP online service that uses IGS satellite products.
The estimated coordinates of the PPP solution were compared with the solution using the DGNSS system as a reference solution. The GIPSY-OASIS software showed after the initialization time 20 minutes less than 5 [cm] accuracy in the east and north and less than 10 [cm] for height.
Online service CSRS-PPP technique achieved with initialization time from 10 [min] with the same accuracy [9], [10]. A recent Ceylan study together with other authors assessed the kinematic estimation using the PPP method for urban areas.
In Turkey, three kinematic trajectories were observed using a dual-frequency GNSS receiver with an angle of signal reaching the receiver antenna of 10 [°] and the observation interval of 1 [s]. The DGNSS system solution from the Leica Geo-Office 5.0 software has been recognized as a reference solution. The PPP estimation was performed using the CSRS-PPP online service.
It should be noted that the disadvantage of this kind of research is that the signals of the GNSS system have been lost due to the high buildings surrounding this area.
The three trajectories had different lengths, different numbers of observational data, and different views of satellites, and various estimates of accuracy were provided in this study according to observation conditions [11], 12].
The estimated standard deviation of differences between the DGNSS system solutions and the PPP technique showed that the first trajectory provided accuracy of 5 [cm] for east, north and 10 [cm] for height.
The second trajectory has reached less than 5 [cm] in all directions. The third displayed 10 [cm] for east and north and over 15 [cm] for height. In addition, the third trajectory provided the worst accuracy due to the low visibility of the satellite [13], [14].
In the next subsection, the authors of this work presented an estimated model for accuracy purposes, based on the PPP solution.

An estimated model for the solution of PPP technique
The key aspect of estimating the PPP solution depends on the application of the linear free-ionosphere combination. The next part is explained in detail with the traditional implementation of the PPP technique using the smallest quadratic estimation method [15], [16], [17]. Because accurate satellite orbit information and clock are known, satellite clock error and satellite position , , are precisely known. The estimation equations can be reformed as seen in subsequent equations marked as (1) and (2): Because the geometrical range is equal to the distance between the co-ordinate satellites , , and coordinates of the receiver , , the estimated equations are updated as shown in equations (3) and (4). Thus, known observations are satellite coordinates, pseudo-transmitters 1 , 2 and carrier phases Φ 1 , Φ 2 from the RINEX file.
The co-ordinates of the receiver, receiver clock error , troposphere zenith delay and variable infinities floating are not known. The traditional estimation of the PPP solution is carried out by the smallest square fit. This type of estimation methodology is used in the Bernese GNSS system software [18], [19], [20].
According to the estimate using the GNSS Bernese system software, receiver clock errors are initially eliminated and inserted into the final estimate as known values.
Two vectors of unknowns are defined as ̂1 and ̂2. ̂1 containing the coordinates of the troposphere receiver , , troposfery and integers of ambiguity [equation (6)], and the unknown ̂2 refers to the receiver clock error, which is first estimated, as shown in the following equation (5): The matrix design 1 for the main estimation was formed in the equation (7): ( , ) The first line refers to phase observations for one satellite and the second one also mentions about code observation. In addition, the entire designed A1 matrix is linearized, as can be seen in equation (8), where n refers to the total number of satellites observed: The second designed matrix A2 refers to the clock error of the receiver, as shown in the following equation (9).

Methodology of conducted research
The aim of the article was to determine the possibility of designating the spatial location of objects (aircrafts, unmanned aerial vehicles) with GPS/GNSS system methods using different measurement modes [21], [22], [23]. Depending on the nature of GNSS system receivers and their place of mounting, the following modes of determining the position of moving objects have been distinguished:  static (e.g. for the measurement of x, y, z coordinates of an aircraft on the surface of the airport),  dynamic (kinematic) of moving receiver (e.g. when defining the course of an aircraft in 3D space). It should be noted that the accuracy of determining the position of a given object in three-dimensional space depends on many factors, but mainly on the quality of the receiver itself and the number of radio channels operating the reception and processing of navigation signals of the GPS/GNSS system, antenna (external, internal) and the conditions of the measurement itself (availability of the horizon, no factors that affect the quality decrease of signal reception).
It is also important to acquire attributes (features of the measured objects), in the case of aircraft, the most decisive factor directly influencing the accuracy of the measurement is its altitude.
Extremely important in conducting measurements in difficult airspace conditions is proper planing of the measurement mission using the current file "almanac" and knowledge of obscuring the horizon of the measurement site (both vertically and horizontally). This avoids many failures and thus saves a lot of time.
In situations requiring greater precision of measurements, a differential GPS (differential GPS) method is applied with the use of differential corrections to data from GPS system satellites [24], [25].
The process of making measurements in mode of the so-called post-processing requires the use of a correction file registered by GNSS system base stations, i.e. reference stations (base station) at the same time as observations made in the field by a mobile receiver (rover station) or movable receiver.
This has a significant impact on increasing the accuracy and certainty of determining the position of the observed point in space. The tests were carried out on different days, months and under different atmospheric conditions.

Results and analysis of satellite clock research
This chapter examines the impact of the satellite clock interval. Satellite clocks of pseudo-random code signals with the interval of 5 [s] and 30 [s] were tested. To understand how the satellite clock interval affects the observation data at different sampling rates, two categories of observation data were processed.
The first category refers to observation data with a sampling rate of 5 [s]. The second category exchanges data on fast sampling with a sampling frequency of 1 [s].
To obtain an optimal solution from the kinematic solution of the PPP technique, observational data from different sets of data was processed with an observation interval of 5 [s]. This data sampling interval is suitable for the high satellite clock rates provided by the CODE center, operating at the Astronomy Institute of the University of Bern. This means that there is no interpolation process for the needed satellite clocks. The following chapter shows the kinematic results of a PPP technique with various satellite clocks for a set of static data and real hydrographic kinematic measurements.

Static measurement data
Due to the huge amount of information contained within the preamble, only the observation time of 6 hours was considered for the GPS CORS (Continuously Operating Reference Stations). To assess the accuracy of the estimation, a kinematic solution of PPP with reference coordinates for these stations was compared. The error in the east, north and height directions was calculated.
The RMS values for all plans are summarized in the table below (Table 1), where two loops were carried out to estimate the PPP solution. The first loop aims to check the estimated residual values of each satellite for each information contained within the preamble. The default solution using satellite clocks 5 [s] shows some eliminated observation data. On the other hand, due to the interpolation process for 30 [s] satellite clocks, observational data that have satellite debris larger than limits are marked as bad observations. The estimated accuracy in this case is significantly reduced. Therefore, the default settings for this estimate are adapted to deactivate this screening filter, which means that for the second loop all observations were considered without elimination. This solution is called a modified solution in this case [26], [27], [28]    for all solutions. The left part shows the error graph over time, the vertical axis refers to the error [m], and the horizontal axis shows the weekly time of the GPS system [29], [30], [31], [32]. The right side shows the east-north and east-height error graphs. The purpose of these plots is to show the density of errors for the estimated information contained within the preamble. A more fluent solution was obtained from satellite clocks 5 [s], as shown in Figure 1, compared to the clock obtained from satellite clocks 30 [s]. This means that in this case a smaller range of errors is obtained. In detail, figure 2 shows the kinematic errors of PPP station 0384 default solution for pseudorandom code signals [duration 30s].
On the basis of Fig. 2 it can be concluded that the largest coordinates have the coordinates of the height. Two leaps in height are shown in dashed lines. In turn, figure 3 shows errors for the modified solution. It can be observed that a significant improvement was achieved compared to the default solution.

Summary and final conclusions
In order to determine the kinematic accuracy of the PPP concept solution, various aspects were explored: (1) The influence of the satellite clock interval, (2) The influence of the tropospheric zenith delay model TZD (Troposphere Zenith Delay) and (3) Implementation of the height limit for measurements.
Satellite clock intervals play an important role in estimating the accuracy of the kinematic solution of the PPP concept. Two satellite clocks provided by the CODE center were used in the kinematic processing of the PPP concept. These clocks have a range of 30