a. Uncrewed aerial system

The system used for this study is a rotary-wing UAS, specifically the DJI Spreading Wings S900 hexacopter (Table 1). Using open-source ArduCopter firmware (V4.0.5), the PixHawk 2.1 Cube Orange autopilot controls the system. The flight control unit utilizes a Here3 GPS antenna as a GNSS receiver. During each flight mission, the autopilot logs the flight parameters of the system, computed with its extended Kalman filter, at a frequency of 10 Hz. The only component retained from the original DJI electronics are the six built-in arms with electronic speed controllers (ESC). The servos responsible for retracting the landing gears have been removed to reduce weight.

Table 1.

Relevant specifications of the UAS used in this study.

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The hexacopter’s body is enclosed in a Styrofoam sphere (Fig. 1 and Fig. A1 in appendix A). The structure is composed of two parts with a thickness of 2 cm each. Using 3D printed acrylonitrile butadiene styrene (ABS) parts and two sets of Velcro straps for each arm, both halves are connected to the multicopter frame. The lower part of the sphere has been manually carved to accommodate the landing gears. As a result of this shape, symmetry is increased with respect to horizontal wind while a partial shelter is provided for electronics from external elements. Detailed discussion of this encasing will be included in section 5.

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Modified DJI S900 hexacopter.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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This system is powered by a pair of lithium polymer six cells batteries each rated at 12 Ah. The weight of the whole system is 7.3 kg, including the batteries: the maximum flight time is 24 min. In this configuration the hexacopter proved to fly up to a peak ground speed (GS) of 19 m s⁻¹.

Mission Planner is the software used on the ground station. This open-source program allows users to create missions with specific commands, simulate them, and then store them on the system. Several other parameters including the altitude, battery level, and orientation are monitored through the same interface during flight.

b. Wind estimation theory

2) Tilt angle Γ

The tilt angle is defined as the angle between the vehicle reference frame vertical vector (i₃) and the body reference frame vertical vector (b₃) (Neumann and Bartholmai 2015; Palomaki et al. 2017). Figure 2 shows a simplified representation of the airborne system and tilt angle. Making use of the rotation matrix $\mathsf{R}$_B→V it is possible to obtain an expression for the tilt angle as a function of the pitch and roll angle only. Indeed, by expressing the b₃ vector in the vehicle reference frame,

${\mathbf{b}}_3^V={\mathsf{R}}_{B\to V}\cdot {\mathbf{b}}_3^B,$(5)

${\mathbf{b}}_3^V=\left[\begin{array}{c}{c}_{\varphi }{s}_{\theta }{c}_{\psi }+s_{\varphi }{s}_{\psi }\\ c_{\varphi }{s}_{\theta }{s}_{\psi }-s_{\varphi }{c}_{\psi }\\ c_{\varphi }{c}_{\theta }\end{array}\right].$(6)

Thus, the tilt angle can be computed using the three components of the ${\mathbf{b}}_3^V$ vector:

$\Gamma =\arctan \left[\frac{\sqrt{{(c_{\varphi }{s}_{\theta }{c}_{\psi }+s_{\varphi }{s}_{\psi })}^2+{(c_{\varphi }{s}_{\theta }{s}_{\psi }-s_{\varphi }{c}_{\psi })}^2}}{c_{\varphi }{c}_{\theta }}\right].$(7)

By developing the calculation, it is possible to notice that the tilt angle does not depend on the yaw angle ψ:

$\Gamma =\arctan \left[\frac{\sqrt{c_{\varphi }^2{s}_{\theta }^2+s_{\varphi }^2}}{c_{\varphi }{c}_{\theta }}\right].$(8)

However, the actual value of ψ cannot be ignored when determining the wind direction.

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Attitude of a multicopter while flying at constant speed or while hovering under the influence of horizontal atmospheric wind. The multicopter tilts (Γ) toward the wind direction so that it can balance the aerodynamic force (F_aero).

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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a. Flight description and meteorological conditions

For a broader understanding of the conditions in Poltringen on the measurement day, we use ERA5 (Hersbach et al. 2018) data from Baden-Württenberg. There was low wind speed, low cloud cover, and warm temperatures for 23 February, with maximum temperatures of 17°C toward the afternoon and lowest temperatures of 10°C toward the evening. A low surface wind speed around 1 m s⁻¹ and an easterly wind direction were experienced during the first two flights (Table 2 and Fig. 3, 1400–1500 UTC).

Table 2.

Missions sequence. In Fig. 4, the E–W direction is represented by the blue dots and N–S by the orange dots. The surface wind speed (SWS) and direction (SWD) are obtained using ERA5.

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Maps describing the atmospheric conditions (surface wind speed, direction, temperature, and pressure) at Poltringen airfield (48.545°N, 8.947°E; red dot) on the day of the calibration flights, 23 Feb 2021. The four plots describe the evolution of these parameters from (top left) 1400 UTC to (bottom right) 1700 UTC. The plots were generated using ERA5 with a 9 km grid resolution. Below the maps, a timeline shows when the four flights have been performed with respect to the four weather maps provided.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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The wind conditions changed between 1500 and 1600 UTC, and therefore between flight 2 and flight 3. The wind direction changed from east to southeast at 1600 and finally to south at 1700. At the same time, the surface wind speed increased to around 2 m s⁻¹ at 1600 UTC and to 2–3 m s⁻¹ at 1700 UTC (Fig. 3, 1600–1700 UTC).

During the mission, the UAS flew at an altitude of 50 m on a straight line in the direction of a waypoint and finally back to the starting point. This basic flight pattern was carried out for different GS from 1 to 14 m s⁻¹ (safety limit). To gather more data for model calibration, the procedure was repeated for a direction perpendicular to the first flight. For each GS, the missions were planned so that equal numbers of data points would be collected, resulting in shorter distances for lower speeds and greater distances for higher speeds (Fig. 4).

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Example of the calibration flight missions (8–1 m s⁻¹). Blue and orange points indicate the limits of each section at a specific ground speed. Since the multicopter reaches a steady condition faster at lower velocities, less distance needs to be covered.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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Due to the battery endurance, the total number of flights has been divided into four missions. The mission sequence can be found in Table 2. An illustration of the GS behavior during the first mission is presented in Fig. 5. It is evident that the autopilot effectively maintains the desired GS for both the forward and backward segments. For higher GS values, the acceleration phase is extended, resulting in longer sections of the flight path being planned accordingly. The air density was determined by measuring atmospheric parameters at ground level.

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An example of the ground speed data recorded during mission 2 (N–S direction; see Table 2). Data filtered for calibration are represented by the orange part. For each ground speed, two datasets were obtained from flying first toward the waypoint and then back.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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The key to detecting any nonzero atmospheric wind is to fly forward and backward on a straight line. If a wind is present during these flights, it will interfere with the UAS with the same magnitude (assuming it is constant), but in the opposite direction. Therefore, this influence can be detected and corrected after recording, resulting in final calibration using the TAS rather than the GS. The next sections (sections 3b and 3c) deal with the data analysis and the data correction in order to tackle this issue.

b. Data analysis

First, the autopilot data have been filtered to be within ±0.05 m s⁻¹ around the desired GS. The band is uniform for all the tested velocities (orange segments in Fig. 5). Based on the Euler angles, the tilt angle and the extended drag coefficient have been computed by applying Eq. (8) and the inverse of Eq. (24):

$C_A=\frac{\tan (\Gamma )}{K{V}^2}.$(27)

At this point the only way to solve Eq. (27) is to assume undisturbed wind conditions (TAS = GS) for the moment and use the multicopter V_GS instead of the TAS(V):

$C_A=\frac{\tan (\Gamma )}{K{V}_{\mathrm{G}\mathrm{S}}^2}.$(28)

Figure 6 shows the Γ(V_GS) data and the C_A|_GS(Γ) data obtained for all the tested GSs for one flight direction.

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(a) Tilt angles values recorded along the E–W direction (missions 1 and 3) for the ground speeds from 1 to 8 m s⁻¹ and from 10 to 14 m s⁻¹, respectively. (b) Extended drag coefficient values obtained using Eq. (28) along the same direction plotted against the respective tilt angle values. The color represents the different ground speeds.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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During the first mission, which started at 1403 UTC, the data (mapping GSs from 8 to 1) show no discernible difference between the forward and backward part, and they are generally very well overlapped. In contrast, data gathered during the mission No. 3 mapping higher GSs, started at 1533 UTC reveal two very well-separated clouds of points: this is clear evidence of the presence of wind influencing the calibration process. Due to the wind, the multicopter has to tilt more when it is facing it (headwind); on its way back, it must tilt less since it is being pushed from behind (tailwind). There is also a similar pattern for the second flight direction: it shows that the wind velocity increased between the second and third mission. This can be seen as well in Fig. 6b where 8 lines are present for the velocities from 1 to 8 m s⁻¹ while for the flights from 10 to 14 m s⁻¹ we have two well-separated segments for each velocity, due to the presence of wind.

c. Data correction

Assume two unknown wind components u and υ are present during calibration flights. Thus, the situation would be similar to that shown in Fig. 7. Due to the relatively short flight time, we can assume that the wind is constant for every GS along the same direction. At this point, Eq. (27) is used to model the UAS behavior in both forward (F) and backward (B) sections using the true airspeed (V). The tilt angles for the forward and backward calibration flights, Γ_F and Γ_B, are described by

${\Gamma }_F=\arctan [K{C}_{A,F}{V}_F^2],$(29)

${\Gamma }_B=\arctan [K{C}_{A,B}{V}_B^2].$(30)

The TASs for the forward and backward part are modeled by simple trigonometric relations as in Fig. 7:

${\Gamma }_F=\arctan \{K{C}_{A,F}[{(V_{\mathrm{G}\mathrm{S}}-u)}^2+{\upsilon }^2]\},$(31)

${\Gamma }_B=\arctan \{K{C}_{A,B}[{(V_{\mathrm{G}\mathrm{S}}+u)}^2+{\upsilon }^2]\}.$(32)

Note that C_A,F and C_A,B are not the same values shown in Fig. 6b since those values were calculated using Eq. (28). Using C_A,F|_GS and C_A,B|_GS to solve Eqs. (31) and (32) would result in the trivial solution (u, υ) = (0, 0) since those value were calculated under the hypothesis of zero wind disturbance.

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Effect of a nonnegligible horizontal wind (u, υ) during the calibration flights. (a) When the multicopter flies toward the waypoint, the wind disturbance increases the ground speed (V_GS), leading to a higher true airspeed (V). (b) As the multicopter returns to the starting position, the same wind disturbance causes a decrease in the true airspeed. The red lines on the multicopter represent its front part—as can be seen in Fig. 1—as well as the direction of motion.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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Thus, the system [Eqs. (31) and (32)] has four unknowns (C_A,F, C_A,B, u, υ) and therefore cannot be solved as it is since it theoretically has infinite solutions. Therefore, it becomes imperative to employ an approximation for two of the unknowns. Considering that the function C_A is defined as the arctangent of the tilt angle [Eq. (28)], and given that the tilt angle typically falls within a range of less than 25°, it is reasonable to adopt a linear approximation. For a particular V_GS, the two sets of data points representing C_A,F and C_A,B can be visualized as lying on a straight line that passes through the origin of the coordinate axes of Fig. 6. The slope of this line is determined by the V_GS value itself. An average between forward and backward data clouds is then used to calculate the corrected extended drag coefficient C_A|_GS, shown in Fig. 8b, and solve the system. It is essential to note that the selection of this method is purely heuristic, and there might exist alternative solutions that could prove effective as well. For instance, one could consider a scenario where two different estimates for the two unknown drag coefficients align more closely with the actual values during forward and backward flight. However, a deliberate decision was made to opt for the average as the chosen approach. By using these new extended drag coefficient values, updated tilt angle values ($\overline{\Gamma }$) are calculated by using Eq. (28):

${\overline{C}}_A={\text{mean}(C_{A,F}|{}_{\mathrm{G}\mathrm{S}},C_{A,B}|{}_{\mathrm{G}\mathrm{S}})|}_{\forall \hspace{0.17em}\mathrm{G}\mathrm{S}},$(33)

$\overline{\Gamma }={\text{arctan}(K{\overline{C}}_A{V}_{\mathrm{G}\mathrm{S}}^2)|}_{{}_{\forall \hspace{0.17em}\mathrm{G}\mathrm{S}}}.$(34)

As a result, ${\overline{C}}_A$ values can be used to solve the system of Eqs. (31) and (32) by approximating $C_{A,F}=C_{A,B}={\overline{C}}_A$. Despite the fact that equivalent values for the extended drag coefficient remain a heuristic approximation, u and υ—the only unknowns—are consistent with ERA5 of the mean wind (Table 3 and Fig. 3).

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(a) The red points mark the tilt angle data of Eq. (34) (after the postprocessing wind disturbance correction where GS = TAS). The blue line represents a third-order polynomial fit from (35) between the tilt angle and horizontal wind velocity. (b) The points mark the extended drag coefficient data of Eq. (33) (after the postprocessing wind disturbance correction where GS = TAS). The color map represents the true airspeed. The blue line represents the exponential decay of the C_A model in (36).

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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Table 3.

ERA5 surface wind speed and the wind speed obtained by solving Eqs. (31) and (32) (UAS WS: $\sqrt{u^2+{\upsilon }^2}$) for the four calibration flights. The UAS WS is the average of the atmospheric wind speeds obtained for each GS during one mission.

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d. Calibration models

After removing any atmospheric wind disturbance, it can be seen from Fig. 8 that the $\overline{\Gamma }$ and ${\overline{C}}_A$ values have a more regular trend. At this point, they can be fitted to define our wind estimation models.

For the first model, the corrected tilt angle and the tested GS are directly related for the estimation of the horizontal wind velocity. On the other hand, similar to the one used by Wetz et al. (2021), the second model describes the extended drag coefficient C_A as a function of tilt angle.

a. Validation flights

In this section, the two models obtained from the calibration flight data are assessed for their quality. A Metek USA-1 ultrasonic anemometer mounted on a 99 m mast was used to compare the horizontal wind estimation. The comparison flights have been performed at the German Meteorological Service Boundary Layer Field Site of Falkenberg, Brandenburg, close to the MOL-RAO observatory site, in the framework of the VALUAS project, during the FESSTVaL field campaign in June 2021.

Two sonic anemometers of the same type are mounted at 50 and 90 m altitude on the tower. These sensors provide a fast sampling of the three wind-vector components at 20 Hz, with a measurement range from 0 to 60 m s⁻¹ and a declared accuracy of 0.01 m s⁻¹ at 5 m s⁻¹. Due to their ability to resolve turbulence eddies at 10 Hz, these sensors are considered reliable references.

On 17 and 18 June 2021, several flights were conducted under variable atmospheric conditions, covering a range of 0.3–12.2 m s⁻¹. In these missions, the UAS hovered alongside the tower at the same altitude as the anemometers. A safety distance of approximately 10 m was maintained between the tower and the aircraft. During the wind validation, eight flights were conducted, with a total hovering time of more than 1.5 h (Table 4).

Table 4.

Validation flights summary. The wind speed and direction range refers to the ultrasonic anemometer data.

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A 10 Hz resampling has been performed on the sonic anemometer data so that the multicopter data can be compared. To identify a potential time lag due to the safe distance between the UAS and tower, a cross correlation has been performed between the UAS and tower data. After synchronizing the two data series, the lag was removed.

1) Noise level analysis

For understanding the minimum time scales of atmospheric eddies that the multicopter can resolve and where noise levels corrupt the measurements, the power spectral density (PSD) of horizontal wind can be useful. The average spectrum of all our flights (orange line in Fig. 9) was calculated using raw 10 Hz UAS data. Anemometer data are used to compute the blue spectrum. Kolmogorov (1941) −5/3 decay in quasi-isotropic turbulence is represented by the black line. In Fig. 9, there is a significant agreement between the multicopter and the reference up to 0.1 Hz (once every 10 s). No significant difference is present between the two different wind estimation models.

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Comparison between the power spectral density of the horizontal wind speed of the sonic anemometer (blue) and the UAS (orange). The black line is reported as a reference of the Kolmogorov −5/3 turbulence decay law.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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2) Wind speed and direction retrieval analysis

For comparison of the time series, data have been resampled to 0.2 Hz (once every 5 s to prevent aliasing) based on the PSD plots: by doing so, unwanted oscillations are removed, and the analysis becomes more meaningful.

The plot of the flight No. 7 at 90 m is presented as an example in Fig. 10. During this mission the wind velocity varied between 5 and 11 m s⁻¹. The black line in Fig. 10a represents the ultrasonic anemometer (reference), while the green and red lines represent the result of the two wind estimation models together with their uncertainty bands. In Fig. 10b the wind direction estimation is plotted against the reference direction. Overall agreement is satisfactory. There is only a systematic discrepancy between UAS and sonics in the wind direction (usually around 15°). The complete time-series plot can be found in appendix B.

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Comparison between the horizontal wind vector detected by the ultrasonic anemometer (black) and by the multicopter. The plots show around 300 s of hovering at 90 m altitude for flight 7. All the time series have been resampled to 0.2 Hz for the comparison. (a),(b) Wind magnitude obtained with the two models using the UAS tilt angle (green and red). (c) Horizontal wind direction.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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Figure 11 shows the comparison of the complete dataset collected during the 8 validation flights. It is possible to notice how low winds cause the accuracy of both models to decrease. In general, it was noticed a higher deviation of the two model with respect to the sonic when the wind speed was lower than 3 m s⁻¹. Since uniform band filtering of calibration data results in a higher percentage error at low speeds, both models are susceptible to errors in these conditions. For the C_A model, the drag coefficient uncertainty is higher at low calibration speeds. Last, Fig. 8 show that the average of velocity 3 m s⁻¹ is not as close to the fit as the others, which indicates that the fitting could introduce errors depending on the wind speed.

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Comparison between the horizontal wind vector detected by the ultrasonic anemometer and by the multicopter throughout all the eight validation flights. (a) Wind magnitude obtained with the direct model. (b) Wind magnitude obtained with the C_A model.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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Figure 11 shows also a tendency of the C_A model to deviate from the linear behavior for the highest wind speeds recorded. This effect could be possibly corrected by either selecting a different fit function by extending the calibration GS range so that the final value of the extended drag coefficient could be better identified.

Table 5 presents the mean bias error (MBE) and RMSE for the two types of models. From the results, the accuracy of the two models is comparable and falls below 0.7 m s⁻¹ after subtracting the MBE from both signals. Even though the DJI S900 multicopter modified for this study is several kilograms heavier than the other systems, the RMSE is close to previous studies.

Table 5.

Statistics of the wind-vector estimation. The RMSE is computed after removing the MBE between the multicopter and ultrasonic anemometer signals.

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MBE and the RMSE for the direction estimation are also reported in Table 5. The main issue in the direction estimation is the presence of a constant offset between the multicopter and the reference sensor of around 15° also reported by Wetz et al. (2021). The computation of the direction involved the yaw angle [Eq. (26)]: there is a possibility that the magnetometer has drifted from its original calibration or is simply slightly moved with respect to its original position, resulting in the constant offset. Wind direction variation is well captured by the multicopter, and the corrected RMSE is below 8°.

b. Parallel flights

To determine if the dome effectively increases the drag of the multicopter, two identical DJI S900s were used in parallel missions where one was equipped with the dome and the other was not. The frames of the multicopters were the same, and the flight controller parameters were also set to be identical for both vehicles. The S900 without the dome was given additional weight to simulate the weight of the dome. The two UAS were commanded to hover at 90 m with a 20 m distance between them, in a way that the wake of one would not affect the other. A total of 3 h of data were recorded between 16 and 22 November 2022 (Table 6) at the German Meteorological Service Boundary Layer Field Site of Falkenberg. During these days the copter with the dome recorded tilt angles ranging from 3° to 19.5°. This provided sufficient data for comparison across a large portion of the recorded tilt angles during the calibration flights. A cross correlation was performed on the raw time series of the two multicopters to correct for any potential delays due to the different hovering positions. The data were then resampled to a frequency of 0.2 Hz.

Table 6.

Parallel flights summary.

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Figure 12 illustrates the comparison of tilt angles between the two systems. It is evident that the multicopter with the Styrofoam sphere consistently measures higher values for the tilt angle. The linear fit (red line) indicates a 16% increase in tilt angle resulting from the larger cross-sectional area. This means that for the same input range (wind speed), the output (tilt angle) will be spread over a wider range, increasing the sensitivity of the copter as a wind sensor.

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Comparison of the tilt angle of two DJI S900, one with the sphere and the other one without. In total, 3 h of hovering are plotted with the tilt angle of the UAS with the dome ranging from 3° to 19.5°. The red line represent the linear fit of the cloud of points.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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a. Advantages of the multicopter shell

The spherical Styrofoam shell that encloses the multicopter offers several advantages. First, it protects the exposed electronics from precipitation and damage. Second, its symmetric shape ensures that the same cross-sectional area is always exposed to the wind, regardless of the wind direction. This is similar to the approach taken by Neumann and Bartholmai (2015), who used a quadcopter with a nearly symmetric, cylindrical fuselage for wind measurement and found that the radial orientation of the system to the wind direction was negligible. The same can be said for the spherical shell and hexacopter configuration used in this study, as it offers an even more symmetric shape than a quadcopter with four rotors.

The modified DJI S900 utilized in this study has a significantly larger fuselage cross section compared to the unmodified DJI S900, leading to an increase in air resistance but only a moderate increase in weight. This causes the multicopter to adopt a larger tilt angle in order to compensate for wind speed. Figure 13a shows the relationship between input (wind speed) and output (tilt angle) for both copter configurations. The line corresponding to the no dome configuration was obtained by lowering the direct model fit function by 16%. While the sensitivity is not constant due to the polynomial fit, the configuration with the dome always has a 16% higher local sensitivity than the no dome configuration (Fig. 13b).

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(a) Dome and no dome calibration curves comparison. (b) Dome and no dome sensitivity comparison.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0010.1

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b. Parameters’ influence: Vertical velocity and mass

To a first approximation, the presence of a vertical wind velocity and the change in UAS mass have similar effects on the system despite their substantial differences.

So far, the vertical wind component has been ignored. However, its presence would create a drag force that would add or subtract weight to the system. Furthermore, a change in the UAS mass, such as one resulting from a different battery configuration or the addition of new sensors, affects the vertical force component as well.

c. Parameters’ influence: Air density

d. Uncertainties and limitations

The system can only maintain a constant GS under the conditions of Eq. (47). Nevertheless, we do not take into account gusts when calculating this simple equation. It is precisely gusts that destabilize the system during flight. In light of the autopilot’s ability to stabilize the vehicle without creating a hazardous situation, the maximum atmospheric wind speed will have to be reevaluated.

Using steady equilibrium as a basis, the differential equations describing the motion of the UAS can be simplified advantageously in this study. As a result, it is possible to derive the key equation of the model, Eq. (24) in a matter of steps. However, it is assumed that the copter tilt angle represents the actual wind speed at any given time. This hypothesis ignores any transients that may occur during a flight. It is necessary to build more complex models that also incorporate the dynamics of the system in order to explain this phenomenon. Nevertheless, this would require accurate analysis of the aircraft’s geometric and aerodynamic characteristics (e.g., moments of inertia around the three axes of rotation).