Assessment of the Steering Precision of a UAV along the Flight Profiles Using a GNSS RTK Receiver

Abstract

Photogrammetric surveys are increasingly being carried out using Unmanned Aerial Vehicles (UAV). Steering drones along the flight profiles is one of the main factors that determines the quality of the compiled photogrammetric products. The aim of this article is to present a methodology for performing and processing measurements, which are used in order to determine the accuracy of steering any drone along flight profiles. The study used a drone equipped with a Global Navigation Satellite System (GNSS) Real Time Kinematic (RTK) receiver. The measurements were performed on two routes which comprised parallel profiles distant from each other by 10 m and 20 m. The study was conducted under favourable meteorological conditions (windless and sunny weather) at three speeds (10 km/h, 20 km/h and 30 km/h). The cross track error (XTE), which is the distance between a UAV’s position and the flight profile, calculated transversely to the course, was adopted as the accuracy measure of steering a UAV along the flight profiles. Based on the results obtained, it must be concluded that the values of XTE measures for two representative routes are very similar and are not determined by the flight speed. The XTE68 measure (p = 0.68) ranged from 0.39 m to 1.00 m, while the XTE95 measure (p = 0.95) ranged from 0.60 m to 1.22 m. Moreover, analyses demonstrated that the statistical distribution of the XTE measure was most similar to the gamma and Weibull (3P) distributions.

1. Introduction

An Unmanned Aerial Vehicle (UAV) is an aircraft capable of performing a flight with no pilot on board. Therefore, the aircraft’s flight must be performed autonomously, in pre-programmed mode, or using remote control. Another commonly used term for a UAV is a drone [1].

Originally, UAVs were used for military applications such as land mapping, surveillance zone, performing reconnaissance and as long-range weapons. Nowadays, drones are used in many different areas, including civilian applications. Compared to classical solutions, i.e., manned aerial vehicles, UAVs are characterised by significantly lower operating costs. The fact that drones offer a cheaper alternative has helped UAVs revolutionise the aeronautics industry [1,2,3].

The most important factors that affect the development of drones are the continuous advances being made in the field of microelectronics, radio technology and miniaturisation. UAVs are characterised by their high manoeuvrability, small size and high availability, as well as they enable the performance of complex tasks. Due to the continued development of drones for civilian applications, the number of sectors using UAVs is increasing, which contributes to their universality and the continued expansion of their capabilities. The main areas of drone application include [3,4]: archaeology and architecture [5,6,7,8,9], agriculture [10,11,12,13,14], crisis management [15,16,17,18,19], environment [20,21,22,23,24,25], forestry [26,27,28,29,30], as well as traffic monitoring [31,32,33,34,35]. In order to use a UAV in the above-mentioned applications, it is necessary to know its accuracy of steering along the flight profiles. This parameter is very important from the point of view of ensuring navigational safety [36]. Presented below are the results of several research during which the accuracy of steering a drone was determined.

Bhattacherjee et al. [37] used Radio Frequency (RF) sensors to detect and locate UAVs through passive monitoring of signals emitted by drones. Moreover, the errors of steering a UAV along the flight profiles were filtered using the Extended Kalman Filter (EKF). The study used Keysight N6854A RF sensors and DJI Inspire 2 drone. The measurements were performed in an open area in Dorothea Dix Park (Raleigh, NC, USA). The study demonstrated that depending on the measurement section, the accuracy of steering a UAV along the flight profiles amounted to 4.76 m (standard deviation) when using the EKF filter and 6.6 m (standard deviation) when using RF sensors.

Brunet et al. [38] developed a method for detecting, tracking and controlling the UAV’s flight parameters based on a stereo image. The study used a multi-engine drone equipped with oCamS-1CGN-U stereo camera and RotorS simulator proposed by Furrer et al. [39], as well as Micro Aerial Vehicle (MAV) virtual environment. Simulation tests demonstrated that the accuracy of steering a UAV along a 3D spiral route amounted, in the best case, to 0.09 m (Root Mean Square (RMS)).

Connor et al. [40] used a UAV to compile 3D maps of radiation in the Fukushima region (Ōkuma, Japan). Their study used a lightweight (3 kg) quadrocopter equipped with Pixhawk autopilot and Go Pro Hero 5 Black camera. The experimental tests demonstrated that the accuracy of steering a UAV along the parallel flight profiles (distant from each other by 1.5 m) was, on average, 0.47–0.68 m, while the max deviation of the drone from the planned route ranged from 0.91 m to 1.02 m.

Goel et al. [41] developed a new cooperative location prototype which uses the exchange of information between a UAV and static anchor nodes in order to precise positioning of the drone. The study used an octocopter equipped with Pixhawk autopilot, GoPro camera, Raspberry Pi 3 computer, GPS u-blox receiver, UWB P410 radio and a modem for communication and data transmission. The experimental tests demonstrated that the designed system allows the UAV’s positioning accuracy to be obtained at a level of 2–4 m in the absence of a GNSS signal, provided that continuous communication is ensured in the cooperative network.

Nemra and Aouf [42] presented a new Global Positioning System (GPS)/Inertial Navigation System (INS) data fusion scheme based on the nonlinear State-Dependent Riccati Equation (SDRE) filter used to locate UAVs. The simulation tests demonstrated that the SDRE algorithm enables more accurate steering of a drone along the planned route as compared to the Kalman Filter (KF), EKF and Unscented Kalman Filter (UKF) algorithms. The accuracy of steering a UAV along the flight profiles amounted to: 4.05 m (RMS_x), 1.26 m (RMS_y) and 3.07 m (RMS_z) for the SDRE algorithm.

Paraforos et al. [43] presented a methodology for determining the positioning accuracy of images taken by a UAV. For this purpose, the authors used a prismatic mirror installed on the drone, tracked by a total station. The study used DJI Matrice 100 drone, Parrot Sequoia + multi-spectral camera, Trimble 360 Prism prismatic mirror, GNSS receiver receiving corrections from EGNOS system and Trimble SPS930 total station. The measurements were conducted at University of Hohenheim (Hohenheim, Germany). Experimental studies demonstrated that the accuracy of steering a UAV along the flight profiles (cross track error (XTE)) in the 2D plane amounted to 0.65 m (standard deviation) and 2.33 m (p = 0.95).

Shah et al. [44] presented a sliding mode based lateral (cross-track) control scheme along straight and circular sections for UAVs. The system is designed to minimise the XTE error, i.e., the distance between a UAV’s position and the planned route. Simulation tests demonstrated that the XTE error of a UAV on 400-metre straight sections did not exceed 5 m, while on curved sections, it was approx. 2 m.

Vílez et al. [45] developed an algorithm for trajectory generation and flight tracking using the Robot Operating System (ROS) for AR.Drone 2.0 quadrocopter. To design the flight route, Bézier curves and third-degree polynomials were used. The study performed 40 simulations using the Gazebo simulator and 46 real measurements for different flight trajectories. The tests demonstrated that the greatest mean error of steering a UAV along the flight profiles with a length of 40 m was 2.8 m (simulation) and 5.06 m (real measurements) if the Bézier curves were used for flight trajectory modelling. However, the greatest mean error of steering a UAV along the flight profiles with a length of 60 m was 3.73 m (simulation) and 5.13 m (real measurements) if third degree polynomials were used for flight trajectory modelling.

Yang et al. [46] used UAVs for the inspection of transmission lines. The authors noted wind gusts which have an adverse effect on the steering of a drone along the planned route. The article proposed a sliding mode robust adaptive control algorithm for an aerial vehicle to solve this problem. Simulation tests used a UAV equipped with PX4 autopilot and GPS u-blox NEO-M8N receiver. Tests demonstrated that the accuracy of steering a UAV along the flight profiles was: 0.16–0.21 m (for the northing), 0.31–0.58 m (for the easting) and 0.63–1.21 m (for the height).

Yang et al. [47] proposed a nonlinear double model for multisensor-integrated navigation using the Federated Double-Model EKF (FDMEKF) for small UAVs. The proposed algorithm was compared to other existing filters (Multi-Sensor Complementary Filtering (MSCF) and Multi-Sensor EKF (MSEKF)) using the data derived from different instruments. Experimental tests demonstrated that in terms of the accuracy of steering a UAV along the planned route, the proposed filtering method (FDMEKF) was more accurate than the MSCF and MSEKF filters. The accuracy of steering a UAV along the flight profiles amounted to: 0.658 m (standard deviation) and 0.223 m (RMS) (for the northing), 0.503 m (standard deviation) and 0.523 m (RMS) (for the easting), as well as 0.691 m (standard deviation) and 0.512 m (RMS) (for the height) for the FDMEKF algorithm.

Based on the review of the literature, it should be stated that the accuracy of steering a UAV depends on many factors, such as the type of drone, positioning system, weather conditions, etc. Therefore, the aim of this article is to present a methodology for performing and processing measurements, which are used to calculate the accuracy of steering any UAV along flight profiles. It was also decided to determine the impact of the speed and shape of the route on the accuracy of steering a drone supported by the GNSS RTK receiver.

This article is structured as follows: Section 2 describes the measurement equipment (UAV and GNSS RTK receiver) used when performing the photogrammetric surveys. Moreover, this section presents the photogrammetric survey planning and performance, as well as specifies how the data recorded during the measurements were compiled. Section 3 calculates the accuracy measures of steering a UAV along the flight profiles (XTE68 and XTE95) and determines the statistical distributions that are best fitted to the XTE distributions. The paper concludes with final (general and detailed) conclusions that summarise its content.

2. Materials and Methods

2.1. Measurement Equipment

DJI Phantom 4 RTK is a quadrocopter from a popular Phantom series and its first representative from the DJI Enterprise segment. This is all thanks to the RTK module, which is directly integrated with a drone and provides real-time positioning data with one-centimetre accuracy to improve the accuracy of image metadata. Omnidirectional obstacle detection was achieved thanks to intelligent obstacle detection sensors during both outdoor and indoor flights. The UAV has a 24 mm wide-angle lens (35 mm equivalent focal length). Photos and films are recorded on a 1” Complementary Metal-Oxide-Semiconductor (CMOS) sensor with a resolution of 20 Mpx. The camera has a mechanical shutter, which allows a series of images to be taken efficiently without the risk of blurring. The technical data of DJI Phantom 4 RTK drone, along with DJI camera pre-installed on it, are presented in

Table 1. The technical data of DJI Phantom 4 RTK drone.

Technical Data	DJI Phantom 4 RTK Drone	Photo
Takeoff weight	1391 g
Max service ceiling ASL	6000 m
Max ascent speed	5–6 m/s
Max descent speed	3 m/s
Max speed	50–58 km/h
Max flight time	30 min
Mapping accuracy	Mapping accuracy meets the requirements of the ASPRS Accuracy Standards for Digital Orthophotos Class Ⅲ
GSD	(H/36.5) (cm/px), H—aircraft altitude relative to shooting scene (m)
Data acquisition efficiency	Max operating area of approx. 1 km2 for a single flight

and

Table 2. Technical data of DJI camera.

Technical Data	DJI Camera	Photo
Sensor	1″ CMOS, 20 Mpx
Lens	FOV: 84° Focal length: 8.8 mm/24 mm Aperture width: f/2.8–f/11 Sharpness: 1 m–∞
ISO range	100–12,800
Electronic shutter speed	8–1/8000 s
Max image size	4864 × 3648 (4:3) or 5472 × 3648 (3:2)
Photo format	JPEG
Data recording	MicroSD 128 GB

.

As regards the GNSS RTK receiver, DJI company provided no detailed information on the device model. The manufacturer’s website only mentions that the receiver has the ability to track signals transmitted by satellites of the following systems: GPS (L1/L2), GLObal Navigation Satellite System (GLONASS) (L1/L2), BeiDou Navigation Satellite System (BDS) (B1/B2) and Galileo (E1/E5a). However, the positioning accuracy is 1 cm + 1 ppm (RMS) in the horizontal plane and 1.5 cm + 1 ppm (RMS) in the vertical plane.

2.2. Photogrammetric Survey Planning and Performance

Before starting the study, it was necessary to specify the longitudinal (o_forward) and transverse (o_side) coverage of images, distance between flight profiles (d_side), camera height above the ground surface (H), flight speed (V) and Ground Sample Distance (GSD). As a first step, it was decided to determine the longitudinal and transverse coverage of images (Figure 1), which can be calculated using the following formulas [48,49]:

$$o_{forward}=\left(1-\frac{d_{forward}\cdot f}{H\cdot w}\right)\cdot 100\%$$

(1)

$$o_{side}=\left(1-\frac{d_{side}\cdot f}{H\cdot w}\right)\cdot 100\%$$

(2)

where:

• o_forward—longitudinal coverage of images (%),
• o_side—transverse coverage of images (%),
• d_forward—distance between successive images (m),
• d_side—distance between flight profiles (m),
• f—camera focal length (mm),
• H—camera height above the ground surface (m),
• w—camera sensor width (mm) [50].

Figure 1. The graphic interpretation of the longitudinal and transverse coverage of images.

Figure 1. The graphic interpretation of the longitudinal and transverse coverage of images.

The longitudinal coverage of images should be at least 70–90%, while the transverse coverage of images must not be less than 60–80% [51,52]. For the purposes of the photogrammetric surveys, both the longitudinal and transverse coverage of images was assumed to be 70%.

Another parameter to be set was the flight speed. It follows from the literature review [53,54,55,56] that the UAV’s photogrammetric flights can be performed at speeds as high as 50–70 km/h. Since the operational speed of DJI Phantom 4 RTK drone is approx. 25 km/h, it was decided to carry out the study at three speeds of: 10 km/h (2.8 m/s), 20 km/h (5.6 m/s) and 30 km/h (8.3 m/s).

The camera height above the ground surface (H) could then be determined. In order to perform the flight along the planned routes at three speeds: 10 km/h, 20 km/h and 30 km/h, it was necessary to set the height H at 105 m. UAV photogrammetric flights are commonly performed at heights of approx. 100 m, as they allow the GSD at a level of 2–3 cm to be obtained [57,58]. The ground sample distance can be calculated using the following formula:

$$\mathrm{G}SD=\frac{H\cdot w}{f\cdot i_w}$$

(3)

where:

• GSD—field pixel size (cm/px),
• i_W—image width (px).

When the longitudinal coverage of images (o_forward), camera height above the ground surface (H) and selected technical parameters of the camera used (w = 25.4 mm and f = 24 mm) were known, the min. distance between the flight profiles (d_forward) could be calculated using the following relationship:

$$d_{forward}=\frac{\left(100\%-o_{forward}\right)\cdot \left(H\cdot w\right)}{f}$$

(4)

The routes were designed in such a manner that the flight profiles were parallel to each other. Based on the relationship (4), it was assumed that the mutual distance between the flight profiles would be 10 m (Figure 2a–c) and 20 m (Figure 2d–f).

The routes were designed using the Trimble Business Center (TBC) software. The coordinates of the route turning points were then exported to files with the extension *.kml. These points were recorded as geodetic coordinates referenced to the World Geodetic System 1984 (WGS-84) ellipsoid with an accuracy of eight decimal places. The planned UAV routes set in this manner was uploaded to the dedicated DJI Go software. The program is used for planning routes on which UAVs can move in automatic/autonomous mode. In this way, the prepared data are sent via Wi-Fi and Bluetooth modules from the drone’s control unit, on which the DJI Go software is installed, to the unmanned aerial vehicle. After setting all the necessary default parameters, the flight along the set points in automatic mode could be started.

On 9 August 2022, the photogrammetric surveys were performed to assess the accuracy of steering a UAV along the flight profiles. Moreover, an attempt was made to determine the optimal trajectory of the drone’s movement, with the parallel profile method subjected to verification. It was decided to carry out the study at an airport dedicated to unmanned aircraft in Gdynia (Poland). Since the meteorological conditions can significantly affect the obtained results, the measurements were performed under suitable weather conditions, i.e., no precipitation, windless weather (wind speed not exceeding 6–7 m/s) and sunny day. Such conditions allowed clear, evenly lit images to be taken. In addition, the receiver used the nearest reference station (located in Gdańsk) of the VRSNet.pl GNSS geodetic network to generate RTK corrections. Thanks to them, it is possible to determine position coordinates of a UAV with an accuracy of 1–1.5 cm (RMS) in the horizontal and vertical planes. Moreover, in order to assess the accuracy of steering a UAV along the flight profiles, it is necessary to determine the position data sampling frequency [59]. The value of this parameter is dependent on the GNSS receiver that is installed on the drone. Similar to other studies conducted by other authors [60,61], it was decided to record position data at a frequency of 1 Hz.

2.3. Data Processing

During the photogrammetric surveys, the GNSS RTK receiver recorded its position coordinates in relation to the WGS-84 ellipsoid. In order to calculate the distances between the designed routes and the routes covered by the UAV, it was decided to convert the measured coordinates into a uniform reference system to enable the determination of the XTE value in metres. Therefore, it was decided to transform coordinates to the PL-2000 system, which, in accordance with the ordinance of the Council of Ministers on the national spatial reference system, has been the basic geodetic coordinate system in force in Poland since 2012 [62]. The angular coordinates were converted into Cartesian coordinates in the PL-2000 system based on the following mathematical relationships [61,63]:

$$\begin{array}{l}x=m_0\cdot N\cdot \left[\frac{S(B)}{N}+\frac{{(\Delta L)}^2}{2}\cdot \sin (B)\cdot \cos (B)+\right.\frac{{(\Delta L)}^4}{24}\cdot \sin (B)\cdot {\cos }^3(B)\cdot (5-t^2+9\cdot {\eta }^2+4\cdot {\eta }^4)+\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left.\frac{{(\Delta L)}^6}{720}\cdot \sin (B)\cdot {\cos }^5(B)\cdot (61-58\cdot t^2+t^4+270\cdot {\eta }^2-330\cdot {\eta }^2\cdot t^2+445\cdot {\eta }^4\right]\end{array}$$

(5)

$$\begin{array}{l}y=m_0\cdot N\cdot \left[\Delta L\cdot \cos (B)+\frac{{(\Delta L)}^3}{6}\cdot {\cos }^3(B)\cdot (1-t^2+{\eta }^2)\right.+\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{{(\Delta L)}^5}{120}\cdot {\cos }^5(B)\cdot \left.(5-18\cdot t^2+t^4+14\cdot {\eta }^2-58\cdot {\eta }^2\cdot t^2+13\cdot \eta )\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathrm{500,000}+\frac{L_0}{3}\cdot \mathrm{1,000,000}\end{array}$$

(6)

where:

• m₀—scale factor (–),
• N—ellipsoid normal (radius of curvature perpendicular to the meridian) (m),
• S(B)—meridian arc length from the equator to the arbitrary latitude (B) (m),
• ΔL—distance between the point and the central meridian (rad),
• B,L—ellipsoidal coordinates of the point (°),
• L₀—longitude of the central meridian (°),

The other parameters of projection to the two-dimensional (2D) Cartesian coordinates in the PL-2000 system include:

$$t=\tan \left(B\right)$$

(7)

$$\eta =\sqrt{\frac{e^2\cdot {\cos }^2(B)}{1-e^2}}$$

(8)

where:

• η—ellipse distortion orientation angle (−),
• e—first eccentricity (−).

The distance between the designed routes and the routes covered by the UAV then had to be calculated. The article defined it using the XTE measure, i.e., the distance that is determined between a UAV’s position and the flight profile, which is measured transversely to the course. The concept of the XTE measure can be found in [61,64]. In order to determine this variable, it was necessary to describe each route mathematically. As they only comprised straight sections, each segment of the route could be represented using the linear function expressed with the following formula:

$$x_{i,j}=a_{i,j}\cdot y_{i,j}+b_{i,j}$$

(9)

where:

• i—route number (−),
• j—section number of the i-th route (−),
• x_i,j, y_i,j—flat coordinates of the point j recorded by a GNSS RTK receiver on the i-th route in the PL-2000 system (m),
• a_i,j—slope of the straight line j for the i-th route, defined as follows (−):

$$a_{i,j}=\frac{x_{i,j+1}-x_{i,j}}{y_{i,j+1}-y_{i,j}}$$

(10)

• b_i,j—y-intercept of the straight line j for the i-th route, defined as follows (m):

$$b_{i,j}=x_{i,j}-a_{i,j}\cdot y_{i,j}$$

(11)

Once the course of each segment of the designed route had been mathematically determined, it was possible to calculate the distances between the designed routes and the routes covered by the UAV. However, two variants had to be considered when performing the calculations. In the first of them, a straight line perpendicular to the designed route segment could be drawn through the measurement point (Figure 3a), which was not possible in the second variant (Figure 3b). Variant I determines the distance between a measurement point and the designed route segment using the following formula:

$$XT{E}_{i,k}=\frac{\left|-a_{i,j}\cdot y_p{}_{{}_{i,k}}+x_p{}_{{}_{i,k}}-b_{i,j}\right|}{\sqrt{{\left(-a_{i,j}\right)}^2+1}}$$

(12)

where:

• k—point number recorded by a GNSS RTK receiver (−),
• $x_p{}_{{}_{i,k}},y_p{}_{{}_{i,k}}$—flat coordinates of the point k recorded by a GNSS RTK receiver on the i-th route in the PL-2000 system (m).

However, variant II calculates the distance to the nearest turning point using the following formula:

$$XT{E}_{i,k}=\sqrt{{\left(x_p{}_{{}_{i,k}}-x_{i,j}\right)}^2+{\left(y_p{}_{{}_{i,k}}-y_{i,j}\right)}^2}$$

(13)

Figure 3. The graphical method for determining the distance between the measurement point and the designed route for variants I (a) and II (b).

Figure 3. The graphical method for determining the distance between the measurement point and the designed route for variants I (a) and II (b).

3. Results

After calculating the distances between the designed routes and the routes covered by the UAV, a statistical analysis of this variable was conducted. To this end, two accuracy measures were determined: XTE68 and XTE95. They describe which value is not exceeded by 68% (XTE68) and 95% (XTE95) of the distances between the designed routes and the routes covered by a UAV [61]. The values of the XTE68 and XTE95 measures for the routes comprising parallel profiles distant from each other by 10 m (

Table 3. The accuracy measures of steering a UAV for the route comprising parallel profiles distant from each other by 10 m.

Accuracy Measure	Parallel Profiles Distant from Each Other by 10 m
	V = 10 km/h	V = 20 km/h	V = 30 km/h
Time period	11:12:46–11:19:55	13:53:49–13:57:49	10:56:31–10:59:37
Number of measurements	430	241	187
XTE68	0.44 m	3.94 m	0.70 m
XTE95	0.60 m	4.17 m	1.09 m

) and 20 m (

Table 4. The accuracy measures of steering a UAV for the route comprising parallel profiles distant from each other by 20 m.

Accuracy Measure	Parallel Profiles Distant from Each Other by 20 m
	V = 10 km/h	V = 20 km/h	V = 30 km/h
Time period	10:28:32–10:32:41	10:46:29–10:48:46	10:18:28–10:20:17
Number of measurements	250	138	110
XTE68	1.00 m	0.39 m	0.93 m
XTE95	1.22 m	0.65 m	1.18 m

) are presented below.

The study also analysed the distribution of a random variable, i.e., the distance between the designed route and the route recorded by a GNSS RTK receiver. To this end, the free software Easy Fit was used, which checks, by means of statistical tests such as the Anderson-Darling test, chi-square test and Kolmogorov-Smirnov test [65,66,67,68], whether the distribution of a random variable in the population differs from the assumed theoretical distribution. The program then classifies the probability distributions according to the obtained statistics for the particular statistical tests [61]. For the purposes of the study, the most commonly used theoretical distributions in navigation were used: Beta, Cauchy, chi-square, exponential, gamma, Laplace, logistic, lognormal, normal, Pareto, Rayleigh, Student’s and Weibull [69,70,71,72]. Moreover, the statistical analyses ignored the results obtained on the route comprising parallel profiles distant from each other by 10 m at a speed of 20 km/h (Figure 2b) due to the lack of sample representativeness. It follows from the conducted analyses that the empirical distribution of the distances between the designed routes and the routes covered by the UAV is the most similar to the gamma distribution (Kolmogorov-Smirnov test) and Weibull (3P) distribution (Anderson-Darling and chi-square tests). Similar results of the studies into the empirical distributions of navigation positioning system errors were obtained in [69,70,71,72].

The Cumulative Distribution Function (CDF) of the two-parameter gamma distribution of the XTE variable is expressed by the following formula [73,74]:

$$F\left(\mathrm{XTE}\right)=\frac{{\Gamma }_{\mathrm{XTE}}\left({\alpha }_g\right)}{\Gamma \left({\alpha }_g\right)}$$

(14)

where:

• α_g—gamma shape parameter (α_g > 0) (−),
• β_g—gamma scale parameter (β_g > 0) (−),
• Γ(α_g)—gamma function defined as follows (−):

$$\Gamma \left({\alpha }_g\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{t}^{{\alpha }_g-1}\cdot e^{-t}dt}$$

(15)

• Γ_XTE(α_g)—incomplete gamma function defined as follows (−):

$${\Gamma }_{\mathrm{XTE}}\left({\alpha }_g\right)={\displaystyle \underset{0}{\overset{\mathrm{XTE}}{\int }}{t}^{{\alpha }_g-1}\cdot e^{-t}dt}$$

(16)

The CDF of the three-parameter Weibull distribution of the XTE variable is expressed by the following formula [75,76]:

$$\mathrm{G}\left(\mathrm{XTE}\right)=1-e^{-{\left(\frac{\mathrm{XTE}-\gamma }{{\beta }_w}\right)}^{{\alpha }_w}}$$

(17)

where:

• α_w—Weibull shape parameter (α_w > 0) (−),
• β_w—Weibull scale parameter (β_w > 0) (−),
• γ—location parameter (XTE ≥ γ) (−).

CDFs of the gamma and Weibull (3P) distributions of the XTE variable are presented on the graphs generated by the Easy Fit software (Figure 4).

In statistics and probability calculus, the value of the XTE_p is referred to as the quantile of the order p, where 0 ≤ p ≤ 1, in the empirical distribution P_XTE of the XTE variable, for which the following inequalities are met:

$$P_{\mathrm{XTE}}\left(\left(-\infty ,{\mathrm{XTE}}_p\right]\right)\ge p$$

(18)

$$P_{\mathrm{XTE}}\left(\left[{\mathrm{XTE}}_p,+\infty \right)\right)\ge 1-p$$

(19)

where:

• p—probability of the XTE variable occurring in the population under study (−).

In the next stage of the work, it was decided to determine how much the empirical CDF differs from the theoretical one for the particular distributions. To this end, the differences between the probabilities of the XTE variable occurrence for the empirical and theoretical gamma (Δp_g(XTE)) and Weibull (3P) (Δp_w(XTE)) distributions were calculated using the following formulas:

$$\Delta p_g\left(\mathrm{XTE}\right)=F_n\left(\mathrm{XTE}\right)-F\left(\mathrm{XTE}\right)$$

(20)

$$\Delta p_w\left(\mathrm{XTE}\right)={\mathrm{G}}_n\left(\mathrm{XTE}\right)-\mathrm{G}\left(\mathrm{XTE}\right)$$

(21)

where:

• F_n(XTE)—empirical CDF of the gamma distribution of the XTE variable (−),
• F(XTE)—theoretical CDF of the gamma distribution of the XTE variable (−),
• G_n(XTE)—empirical CDF of the Weibull (3P) distribution of the XTE variable (−),
• G(XTE)—theoretical CDF of the Weibull (3P) distribution of the XTE variable (−).

The graph of the differences between the probabilities of the XTE variable occurrence for the empirical and theoretical gamma, Weibull (3P) and normal distributions is provided in Figure 5.

4. Discussion

Based on the results presented in Table 3 and Table 4, it must be concluded that the values of accuracy measures of steering a UAV for two representative routes are very similar to each other and are not determined by the flight speed. The differences between the particular XTE68 (0.39–1.00 m) and XTE95 (0.60–1.22 m) errors were approx. 60 cm for a GNSS RTK receiver. Moreover, the statistical analyses ignored the results obtained on the route comprising parallel profiles distant from each other by 10 m at a speed of 20 km/h (Figure 2b) due to the lack of sample representativeness. This resulted from the fact that the photogrammetric flight was performed 3 or 4 h later than the other flights due to the discharge of all available batteries powering the drone. Despite the almost identical measurement conditions (the only difference was the wind direction and force), the values of accuracy measures XTE68 (3.94 m) and XTE95 (4.17 m) were four times greater than the metrics obtained on the other routes.

On the basis of Figure 4, it can be concluded that the quantiles of the order of 0.68 and 0.95 (percentiles) of the XTE variable for the gamma distribution amount to 0.81 m and 1.70 m, respectively. Moreover, the quantiles of the order of 0.68 and 0.95 of the XTE variable for the Weibull (3P) distribution amount to 0.83 m and 1.68 m, respectively. This should be interpreted to mean that 68% of the distances between the designed routes and the routes covered by the UAV is not greater than 0.81 m or 0.83 m and, similarly, 95% of the elements of this population do not exceed the value of 1.68 m or 1.70 m.

Based on Figure 5, it can be observed that the empirical and theoretical distributions for the gamma and Weibull (3P) distributions are characterised by a high degree of fit. The max difference between the probabilities of the XTE variable occurrence for the empirical and theoretical gamma and Weibull (3P) distributions ranges from −0.02 to 0.04. However, for normal (Gaussian) distribution, which is the most important theoretical probability distribution used in statistics, the analysed measure is several times larger and amounts to approx. ±0.08.

5. Conclusions

The accuracy of steering a UAV along the flight profiles is very important from the point of view of ensuring navigational safety [36]. Due to the low accuracy of steering a UAV (keeping it on the profile), dangerous situations may occur, such as collision with people, resulting in severe damages and/or injuries [77,78,79], collision with static and other flying objects, e.g., buildings and powerlines, other UAVs, as well as manned aircraft [77,78], or flights in restricted zones [80].

Research results have shown that GNSS RTK receivers can be successfully used in UAVs for applications related to, among others, archaeology and architecture, crisis management, forestry, photogrammetric surveys, precision agriculture, or in traffic monitoring [80,81].

The study proved that the use of a GNSS RTK receiver would allow high accuracy of steering a UAV along the flight profiles to be achieved despite many factors that affect it. These include, e.g., initial state errors, intent update errors, dynamic intent updates, vehicle performance errors, weather forecast or trajectory modelling errors [59,81,82,83]. As part of future research, it is planned to determine the effect of the above-mentioned factors on the accuracy of steering a UAV along the flight profiles.