Article citation information:
Krasuski, K., Ćwiklak, J. Application of the DGPS method for the precise positioning of an aircraft in air transport. Scientific Journal of Silesian University of Technology. Series Transport. 2018, 98, 65-79. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2018.98.7.
APPLICATION OF THE DGPS METHOD FOR THE PRECISE POSITIONING OF AN AIRCRAFT IN AIR TRANSPORT
Summary. This article presents research results concerning the determination of the position of a Cessna 172 aircraft by means of the DGPS positioning method. The position of the aircraft was recovered on the basis of P1/P2 code observations in the GPS navigation system. The coordinates of the aircraft were designated due to the application of the Kalman forward-filtering method. The numerical calculations were conducted using RTKLIB software in the RTKPOST module. In the scientific experiment, the authors used research materials from the test flight conducted by a Cessna 172 aircraft in the area of Dęblin in the Lublin Voivodeship in south-eastern Poland. The research experiment exploited navigation data and GPS observation data recorded by the geodetic Topcon Hiper Pro receiver mounted in the cockpit of the Cessna 172 and installed on the REF1 reference station. The typical accuracy for recovering the position of the Cessna 172 with the DGPS method exceeds in the region of 2 m. In addition, the authors specify the parameters of availability, integrity and continuity of GNSS satellite positioning in air navigation. The obtained findings of the scientific experiment were compared with the International Civil Aviation Organization’s (ICAO’s) technical standards.
Keywords: air navigation, air transport, DGPS method, accuracy, ICAO, Kalman filter, integrity
The DGPS positioning technique refers to a differential measurement approach, which can be executed both in near real time and in postprocessing. The DGPS measurement method requires a rover receiver and a base reference station from the user. In near real time, the coordinates of the rover receiver’s antenna are determined on the basis of differential corrections sent via the NTRIP protocol in the RTCM format from the service of the reference station’s network . In the case of calculations during postprocessing, the coordinates of a rover receiver are determined on the basis of registered raw GNSS satellite observations by the rover receiver and the base reference station. In the DGPS measurement technique, mostly single-frequency (or dual-frequency) code observations are used from one or more GNSS navigation systems .
In practice, the DGPS positioning technique allows for reducing or eliminating a number of systematic errors in GNSS satellite measurements. Systematic errors that are related to the satellite clock and the receiver are completely eliminated in the DGPS method. In this way, it is possible to remove the satellite clock error correction, the receiver clock error correction, the relativistic satellite clock correction, TGD hardware delay for the satellite and the RDCB instrumental bias for the receiver. On the other hand, the impact of the ionosphere correction and the troposphere correction is reduced at the differentiation stage for the observation equations of the mathematical model . It is worth mentioning that, in the DGPS measurements, it is crucial to determine the characteristics of the antenna of the rover receiver and the base reference station.
The DGPS measurement method is used for positioning in static and kinematic modes. In the kinematic mode, the method of DGPS positioning provides, for example, the designation of the precise position of the aircraft in air navigation . The recovery of a reliable aircraft position affects the improvement in the safety of air operations in airspace. In addition, the technique of DGPS positioning is important in the development of aircraft approach procedures for landing with the use of the GNSS system in air transport .
The aim of this investigation is to recover the possibility of aircraft coordinates using the DGPS positioning method in air navigation. In the test research, we recovered the position of a Cessna 172 aircraft by executing a test flight around the airfield in Dęblin. The position of the aircraft was recovered using RTKLIB software in the RTPOST module. Satellite data were used for the numeric calculations, which were obtained from a Topcon Hiper Pro received mounted on board the Cessna 172 and installed as a physical reference station at the military airfield in Dęblin. The calculations were made in the postprocessing mode for the GPS code observations.
2. RESEARCH METHODOLOGY
The basic observation equations in the DGPS positioning method rely on the use of the operation of double difference of GPS code observations, as follows [4,8]:
is the operator of the double difference for code measurements, which allows for comparing the difference in code measurements for two satellites tracked by two receivers
is the operator of a single difference for code measurements, which allows for determining the difference in code measurements for two satellites tracked by one receiver
is the vector in the space between the base station () and the rover receiver () mounted on board the aircraft
is the value of the double code difference (expressed in metres) on the vector between the satellites and on the L1 frequency in the GPS system
is the value of the double code difference (expressed in metres) on the vector between the satellites and on the L2 frequency in the GPS system
is the geometric distance of the vector for the double code difference (expressed in geocentric coordinates XYZ)
is the value of the ionosphere delay on the L1 frequency for the double code difference
is value of the ionosphere delay on the L2 frequency for the double code difference
is the relationship of the ionosphere delay on the L1 and L2 frequency
is the scaling coefficient
is the L1 frequency in the GPS system
is the L2 frequency in the GPS system
is the value of the troposphere delay for the double code difference
is the multipath effect and noise measurement at the L1 frequency for the code measurements
is the multipath effect and noise measurement at L2 frequency for the code measurements
The observation equations (1) were recorded for the code observations P1/P2 for the carrier frequencies L1/L2 in the GPS navigation system. In Equation (1), the unknown parameters are the coordinates of the aircraft involved in the geometrical distance factor. The parameters of the ionosphere and troposphere delays are expressed by deterministic models. The values of the multipath effect are expressed on the basis of empirical models. The observation model from Equation (1) is usually solved in two stages, using Kalman filtering; see below :
a) Process of “prediction”:
is the matrix of coefficients
is the estimation of the values of the designated parameters a priori from the previous step
is the estimation of the values of covariance a priori from the previous step
is the prediction of the state value
refers to the predicted covariance values
is the variance matrix of the noise of the measurement process
b) Process of “correction”:
is the covariance matrix of measurements
is the matrix of partial derivatives
is the Kalman gain matrix
is the vector of measured values
is the unit matrix
refers to the parameters determined a posteriori
is the covariance matrix of parameters determined a posteriori
The Kalman filtering process is performed sequentially for all measured epochs registered by the GNSS receiver mounted on board the aircraft. Additionally, in the stochastic process of developing the GPS observations, the accuracy of positioning the aircraft is also determined. It should be emphasized that the designated coordinates of the aircraft and their accuracies are expressed in the geocentric coordinates XYZ.
3. RESEARCH EXPERIMENT
The verification of applying the DGPS technique in air navigation was carried out in an air experiment using a Cessna 172 aircraft. The air experiment was conducted on a military airfield in Dęblin and in the surrounding area. The test flight on the Cessna 172 was made in the morning, from 09:39:03 to 10:35:03, according to the time of the GPS navigation.
shows the trajectory of the Cessna 172 on the horizontal plane. The coordinates
of the aircraft were expressed using BLh ellipsoidal coordinates (B: latitude,
L: longitude, h: ellipsoidal height). In order to transform the coordinates of
the aircraft from the XYZ geocentric coordinate frame into the BLh ellipsoidal
frame, the Helmert transformation was used [15, 18]. Figure 1 shows the
location of the reference station REF1, which was used to recover the precise
trajectory of the flight of the Cessna
– Latitude: 51° 33’ 19.92606” N
– Longitude: 21° 52’ 08.72275” E
– Ellipsoidal height:
Fig. 1. The horizontal trajectory of the flight of the Cessna 172
During the flight test, the geodetic receiver Hiper Pro was installed on board the Cessna 172 (see Figure 2). The aim of the Topcon Hiper Pro rover receiver was to collect raw GNSS observations in order to recover the coordinates of the aircraft in postprocessing. The frequency of data registration in the rover receiver was also equal to 1 s. Furthermore, the SAMSET system, which monitored the position of the aircraft in near real time, was installed on board the Cessna 172.
Fig. 2. The GNSS receiver in the pilot’s cabin of the Cessna 172
synchronization of GNSS observations from the Topcon Hiper Pro rover receiver
and the receiver of the REF1reference station allowed for the designation of
the Cessna 172’s position, as well as the determination of the positioning
accuracy. The coordinates of the aircraft were designated on the basis of
a single baseline (spatial vector ), i.e., baseline (vector) REF1-Cessna
- GNSS system: GPS system
- GNSS observations: code observations P1/P2 in the GPS system
- Construction of the observation equations: double difference for code observations in the GPS system
- Data source of the ephemeris GPS satellites: GPS navigation data message
- Source of the GPS observation: RINEX 2.11 file
- Method for determining the coordinates of the GPS satellites: based on the parameters of the Kepler orbit
- Correction of the pseudorange from the satellite to the receiver antenna: applied
- Effect of the earth’s rotation: applied
- Sagnac effect: applied
- Correction of the satellite clock: eliminated
- Relativistic effects: eliminated
- TGD hardware delay: eliminated
- Receiver hardware delay: eliminated
- Troposphere model: Saastamoinen
- Ionosphere correction model: Klobuchar
- Source of ionosphere correction: GPS navigation data message
- Receiver antenna phase centre: based on the ANTEX IGS08 file
- Elevation angle: 10°
- Observation weighting: applied
A priori standard
deviation of code observations:
- Initial values of aircraft coordinates: based on the RINEX file header
- Frame of coordinates: geocentric XYZ and ellipsoidal BLh (ultimately ETRF ‘89)
- Method of calculations: Kalman forward-filtering
- Positioning method: DGPS/DGNSS
- Positioning mode: kinematic
- Computational mode: postprocessing
- Interval of calculations: 1 s
- Blunder error detection in GPS measurements: RAIM module algorithm
- Number of iterations in the measurement epoch: five
- Maximum value of the DOP coefficient: 30
- Final recording of coordinates: coordinates in the XYZ geocentric frame and the BLh ellipsoidal frame
- Correction of the receiver clock: eliminated
- Geodynamic and tidal effects: applied
- Rover receiver: Topcon Hiper Pro mounted in a Cessna 172 aircraft
- Base receiver: Topcon Hiper Pro fixed at the REF1 reference station
4. RESEARCH RESULTS
The examination of the use of GNSS satellite technology in air navigation is focused on determining four basic positioning parameters: availability, accuracy, continuity and integrity. The availability parameter is determined based on the visibility of the GNSS constellation during the measurement session. In addition, when tracking the GNSS satellite constellation, no break must appear in the satellite positioning due to the lack of navigation data and observation data. Therefore, monitoring the available satellites of a given constellation of the GNSS system (e.g., the GPS system) is of crucial importance. In accordance with Annex 10 to the Convention on International Civil Aviation, entitled “Radio Communication”, Volume I “Radio Navigation Aids”, a typical parameter value of the availability of the GPS system is 0.99 (99%) . This means, de facto, that, during the executed air test, the continuity of tracking a GPS constellation equals at least 0.99 of the duration of the whole flight. Thus, the lack of data or GPS system failure may occur only in the case of 1% of the duration of the flight test. Figure 3 shows the number of available GPS satellites during the executed test flight in Dęblin on 1 June 2010.
Fig. 3. Number of satellites in the GPS constellation
Based on Figure 3, it can be concluded that the number of available GPS satellites during the test flight ranged from five to nine. Therefore, when executing the test flight, the tracking of the GPS constellation was still available. Likewise, navigation data were not missing. Therefore, the availability parameter of the constellation of GPS satellites was above 0.99 (99%), which complies with the ICAO requirements. It should be added that the number of available GPS satellites in Figure 3 de facto expresses the total number of GPS satellites tracked jointly by the rover receiver mounted on board the Cessna 172 and the REF1 reference station.
An important parameter in determining the quality of satellite positioning is the accuracy of the set position. The accuracy parameter in the GNSS measurements is represented by the values of the standard deviation for the designated coordinates of the aircraft. In this case, the accuracy of the set position of the Cessna 172 aircraft can be referred to geocentric XYZ coordinates, as below:
is the accuracy of the aircraft position along the X-axis
is the accuracy of the aircraft position along the Y-axis
is the accuracy of the aircraft position along the Z-axis
or adequately expressed in the coordinates of the ellipsoidal BLh, as follows :
is the covariance matrix in the geodetic frame (BLh),
is the transition matrix from the geocentric (XYZ) to the geodetic frame (BLh)
is the standard deviation in latitude
is the standard deviation in longitude
is the standard deviation in ellipsoidal height
Fig. 4. The accuracy of the Cessna 172 aircraft in the XYZ geocentric frame
shows the values of positioning accuracy of the Cessna 172 in the XYZ
geocentric frame; see Equation (4). The average positioning accuracy along the
shows the values of positioning accuracy in the BLh ellipsoidal frame; see
Equation (5). The average positioning accuracy of geodetic latitude B is equal
Fig. 5. The accuracy of the Cessna 172 aircraft in the BLh geodetic frame
to the Convention on International Civil Aviation, entitled “Air
Communication”, Volume I “Radio Navigation Aids”, specifies the technical
standards for the parameter of accuracy of satellite positioning using the GPS
navigation system in civil aviation . The ICAO imposed the framework for
commissioning the GPS system on civilian users in aviation. The ICAO’s accuracy
standards are matched with air operations for a specific flight plane of an
aircraft in civil aviation. For navigation on the horizontal plane, the
accuracy of flight navigation LNAV ranges from
shows the positioning accuracy of the Cessna
average value of the MRSE parameter is
Fig. 6. The values of the MRSE parameter
Fig. 7. The values of the HPL/VPL parameters
The parameter of integrity for the DGPS satellite positioning in civil aviation is of the utmost importance during the execution of air operations in airspace. In civil aviation, the integrity parameter is specified by means of safety levels. In practice, the levels of safety are referenced to navigation on both the horizontal and the vertical planes. On the horizontal plane, the safety level is determined by the HPL parameter, and by means of the VPL parameter on the vertical plane. The approximate values of the HPL and VPL safety parameters are determined on the basis of Equation (7) :
for the horizontal plane
for the vertical plane 
shows the obtained values for the HPL and VPL parameters. The average value of
the HPL parameter is
Annex 10 to the Convention on International Civil Aviation, entitled “Air Communication”, Volume I “Radio Navigation Aids”, specifies the technical standards for the parameter of integrity for satellite positioning using the GNSS navigation system in civil aviation . The integrity values for GNSS satellite positioning in civil aviation were specified for the selected type of aircraft approach for landing. Within the framework of the ICAO’s technical standards, there are three types of aircraft approach to landing with the GNSS sensor:
– Non-precision approach (NPA)
– Approach procedures with vertical guidance (APV)
- Precision approach (PA)
the largest civilian passenger and transport airports have implemented
technical regulations for the NPA with the GNSS sensor. The framework for the
operation and application of the GNSS sensor for this approach was introduced
by the Polish Air Navigation Services Agency. It must be underlined that, with
regard to the NPA, the accuracy of determining the position of the aircraft on
the horizontal plane is equal to
The obtained values of the HPL and VPL parameters can be used directly or indirectly to determine the continuity of GNSS satellite positioning in civil aviation. The parameter of continuity specifies and defines the gaps in tracking down a moving object with the use of GNSS satellite techniques. Furthermore, the parameter of continuity indicates possible failure and a lack of data from the GNSS positioning system. The mathematical formula to determine the parameter of continuity is as follows :
is the maximum alert value on the horizontal plane
is the maximum alert value on the vertical plane
parameter values of HAL and VAL specify the maximum alert levels for the
integrity of GNSS positioning during the selected type of aircraft approach for
landing. The continuity parameter is exceeded when the HPL is larger than HAL,
or when VPL is larger than VAL. Within the framework of the NPA using GNSS, the
HAL value is
In this section, the obtained trajectory of an aircraft from a DGPS application was verified and compared with results from the DGLONASS solution. The aircraft position was estimated using the DGLONASS method in the RTKPOST library within the RTKLIB software package. The Kalman filter solution was applied as a stochastic scheme of designation for the aircraft coordinates in the RTKLIB program. The coordinates of the aircraft were recovered with an interval of 1 s using the DGLONASS method. The aircraft coordinates from the DGLONASS solution are referenced to the ETRF’89 frame, similar to the DGPS method.
Fig. 8. The difference in the XYZ geocentric coordinates of the aircraft between the DGPS and DGLONASS solutions
The difference in the aircraft coordinates in the geocentric XYZ frame between the DGPS and DGLONASS solutions was calculated as follows:
is the x coordinate of the aircraft based on the DGPS solution – see Equation (1)
is the x coordinate of the aircraft based on the DGLONASS solution
is the y coordinate of the aircraft based on the DGPS solution – see Equation (1)
is the y coordinate of the aircraft based on the DGLONASS solution
is the z coordinate of the aircraft based on the DGPS solution – see Equation (1)
is the z coordinate of the aircraft based on the DGLONASS solution
presents the values of the coordinates based on a comparison of the DGPS and DGLONASS
solutions. The mean difference for the x coordinate of the aircraft equals
This article analysed the
applicability of the DGPS positioning method in determining aircraft
coordinates in air navigation. To this end, we recovered the position of a
Cessna 172 aircraft in postprocessing. The calculations were carried out using
RTKLIB software in the RTKPOST module by exploiting GPS code observations. The
mathematical model for designating the aircraft position was based on the use
of observation equations for dual code differences. In the research experiment,
we used the P1/P2 code observations recorded by the geodetic receivers mounted
on board the Cessna 172 and the Topcon Hiper Pro receiver installed at the
reference station REF1. The materials for research came from a test flight,
which was conducted at the military airfield in Dęblin. Within the framework of
the conducted research, we recovered the trajectory of the Cessna 172 using the
Kalman forward-filtering forward base solution. The article also analysed the
GNSS satellite positioning for determining the parameters of availability,
accuracy, integrity and continuity in civil aviation. The parameter of
availability of the GPS satellite constellation was 100%, which facilitated a
continuous navigation solution for the position of the Cessna 172 aircraft. The
accuracy of the designated coordinates of the Cessna 172 was higher than
The authors would like to thank Tomoji Takasu for making available the RTKLIB software package on the website www.rtklib.com.
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Received 07.12.2017; accepted in revised form 25.02.2018
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