Ground Risk Buffer Estimation for Unmanned Aerial Vehicle Test Flights Based on Dynamics Analysis

Highlights

What are the main findings?

• The force characteristics of rotor UAVs and fixed-wing UAVs are analyzed and the falling trajectories are calculated based on the dynamics model analysis.
• Both airspace uncertainty and operational risk are considered in test flight and a 3D contour map is generated for quantitative estimation of the ground risk buffer

What is the implication of the main finding?

• The greater safety margin provided by the proposed method is verified using actual test flight certification cases.

Abstract

Unmanned aerial vehicles (UAVs) are regarded as a novel mode for urban air mobility, earning increasing attention on many commercial and civil applications. The risk of UAVs to people on the ground is heightened by airspace range and operational risks, and the quantitative ground risk buffer estimation are highly required to protect the people on the ground. In this work, a ground risk buffer estimation method based on the analysis of the UAVs dynamics is proposed. It is a 3D contour map, incorporated with flight test parameters, to determine the ground risk buffer for both, rotorcraft UAVs and fixed-wing UAVs. The contour map is generated through UAVs dynamics analysis and combines several parameter layers, including altitude and speed at moment of failure occurence, environment conditions and the lift-to-drag ratio. Each location of the map has associated a value that quantifies the area of the ground risk buffer for a specific test flight condition. The ground risk buffer determined by the current Specific Operations Risk Assessment framework using the 1-to-1 principle is provided for comparison. The proposed method exhibits greater safety margin and further proves the potential of the new estimation method in the perspective of risk quantification and practical engineering applications.

1. Introduction

The rapid growth of the applicability of Unmanned Aerial Vehicles (UAVs) has made them a focus on interest for many practical applications. Currently, UAVs have been widely applied in various fields, including agriculture [1,2,3], forestry [4,5,6], environmental monitoring [7], power line inspection [8,9], emergency rescue [10,11], logistics [12], and infrastructure surveying [13]. Test flights are required before a UAV is officially put into service. Operating UAVs for test flights requires that flight should be confined within a pre-defined test mission volume. Once there is in-flight failure, the UAVs will lose power, and the uncontrolled descent of the UAVs will be performed, resulting in a potential impact risk to any third parties at ground. Consequently, in order to enhance operational safety in test flights, the ground risk buffer can be determined from the side of the test mission volume to prevent the impacts between the descending UAVs and the people on the ground. In most case, to ensure the safety of UAVs in test flight, quantitatively analyzing ground risk buffers must be required, which makes the preparation of test mission considerably challenging. Therefore, the ground risk buffers should be cautiously estimated since the buffer size underestimation will compromise safety and expose people to risk, whereas its overestimation will lead to inefficient usage of airspace resources. Apparently, an appropriate estimation of ground risk buffer is necessary for balancing airspace utilization against optional safety of UAVs.

“Interim Regulation on Civil Unmanned Aircraft Flight Management” [14] and the CCAR-92 “Regulations on Civil Unmanned Aircraft Operations Safety Management” [15] set the requirements: in order to ensure the operational safety of UAVs, it is necessary to carry out the airworthiness certification procedure. To improve the operational safety of UAVs during test flights, the safe margin introduced by ground risk buffers is required. However, in practical implementation, the determination of ground risk buffers is often compromised by various elements originating from UAV configuration, environment condition, and mission requirement [16]. The current international risk assessment framework, the Specific Operations Risk Assessment (SORA) proposed by the Joint Authorities for Rulemaking on Unmanned Systems (JARUS), deconstructs the UAV operational risk into the dual dimensions of ground and air risk, and the core of its ground risk mitigation measures involves establishing ground risk buffer based on the 1-to-1 principle. In SORA, the standard approach to controlling ground risk is the “1-to-1 principle”, which refers to applying a ground risk buffer that is as wide as the maximum height of the operational volume [17]. However, the 1-to-1 principle has a speed-adaptation drawback through dynamics simulation, that is, the overestimation of the buffer requirement at low speeds results in a waste of airspace resources, while the underestimation of the buffer size at high speeds results in risks to human safety [18,19].

In recent years, although some efforts have been devoted to develop risk assessment models for analyzing ground risks during UAV operations, the estimation of the ground risk buffer is rarely addressed. Based on probabilistic model, Kim and Bae [20] developed a risk assessment methodology through probabilistic calculations of impact zones. Jiang et al. [21] proposed a novel method for joint assessment of impact area and human fatality of ground crash by a UAV. Yang et al. [22] quantified the separation probability using the Weibull distribution, and established a risk probability model for the separation state to estimate ground risks. However, these methods still have complex procedures for test flight missions. Consequently, based on AI (Artificial Intelligence) new technology, Han et al. [23] used over 20,000 flight hours of operational data to train a Bayesian network-based risk assessment model to identify the failure causes of accidents and reduced the UAV risks. Jiao et al. [24] proposed a dynamic model combining a deep learning and a kinetic model to predict ground risks, thereby extracting the spatial–temporal characteristics of ground risks and accurately predict risk areas. Gigante et al. [25] explored the research in ground risk models for different operation planning stages, which provided practical steps for national authorities to streamline the UAV authorization process. As mentioned in the above method, incorporating UAV dynamics analysis into the assessment framework allows for the quantification of the ground risk buffers, thereby facilitating the estimation of an appropriate ground risk buffer.

Referring to the UAV dynamics analysis, la Cour-Harbo [26] pioneered the use of ballistic trajectory decoupling methods, decomposing coupled differential equations into independent equations to be solved in the horizontal and vertical directions, but this model is complex and does not include safety margin design. Lin and shao [27] used uniform acceleration motion and Simulink simulation to calculate collision probability density radius, thereby achieving risk-constrained path planning, whose physical nature is similar to the mechanism of ground risk buffer setting. It is worth noting that Meng et al. [28] combined ballistic falling dynamics models and Monte Carlo simulations, with the 95% confidence interval envelope at each trajectory point serving as the UAV’s entire trajectory’s crash buffer protection zone, without using the ground risk buffer definition method in SORA. Moreover, Che Man et al. [29] investigated the effects of different multirotor drones’ failure modes on its crash trajectory and crash area compared to the ballistic model using Automated Dynamic Analysis of Mechanical Systems (ADAMS) and MATLAB (v2024a) cosimulation methods. However, such probability sampling methods suffer from computational intensity flaws, with single simulation being time-consuming, severely limiting engineering practicality.

In summary, current methodologies for determining ground risk buffer sizes predominantly model the UAV’s crash dynamics as a simplified mass point descent process. However, existing research exhibits notable limitations, the scope of studies remains narrow, model construction is overly complex, and sensitivity to input parameters is pronounced, such as environmental temperature, air pressure, and drag coefficient. Furthermore, the real-time computation capability is insufficient, and the existing model relies on offline simulation calculation, which is difficult to meet the timeliness requirements of dynamic risk mitigation in condition like urban logistics. To address these concerns, this work proposes a estimation method for ground risk buffer based on UAV dynamics analysis, where a contour map is generated to directly determine the ground risk buffers in test flight missions, and the proposed method is verified through a typical test flight certification case, so as to provide basic theoretical support for the risk calculation and reference for engineering application. The main contributions of this work are (1) Innovative Analytical Framework. This work is the first to generate a contour map to quantitatively estimate the ground risk buffers, which addresses the trade-off between airspace utilization and operation risks. (2) Rapid Response of Model. The analytical framework demonstrates rapid computational requirement, thereby enhancing its applicability for ground risk buffer estimation in test flight missions and providing reliable theoretical guidance.

The structure of this paper is organized as follows: Section 2 proposes ground risk buffer size establishment methods for rotorcraft UAVs and fixed-wing UAVs, respectively, with detailed derivation of the computational process. Section 3 elaborates the buffer sizes calculation procedure using open-source data, conducts systematic comparison of different computational methods, thoroughly analyzes parameter influence patterns, and finally demonstrates practical applications through test flight certification cases. Section 4 concludes the research findings of this study.

2. Method

The ground risk buffer is the area on the ground surrounding the footprint of operational volume. If a UAV loses control and flies out of the test flight mission volume, it should be contained to end its flight within the ground risk buffer. The length of the buffer zone is the upper limit of the horizontal flight distance between the location of the UAV’s failure and its impact with the ground. In actual operations, the buffer size is based on the maximum possible fall distance; thus, the buffer size is determined by the maximum altitude and maximum speed that might be reached during test flights. Based on the force characteristics of UAVs after failure, the main discussion will focus on rotorcraft UAVs and fixed-wing UAVs, which will be introduced separately below, and the flowchart of the proposed method is shown in Figure 1.

2.1. Rotorcraft UAV

To calculate the ground risk buffer size of a rotorcraft UAV, this section considers the scenario where the UAV is operating at maximum speed at the highest point on the edge of the operational volume and suddenly fails. At this time, the altitude of the UAV is h, the horizontal speed is $v_F$, and the vertical speed is $v_0$. It is assumed that when the rotorcraft UAV fails, all motors will stop running and will not provide any lateral or rotational motion to the UAV. The UAV is subjected solely to gravity and air resistance, so its trajectory can be considered a damped projectile trajectory, also known as ballistic trajectory.

A damped projectile trajectory represents the motion of an object where the only major force acting on the object is gravity $F_g$, as shown in Figure 2. This type of motion can be easily described using Newtonian mechanics, and the resulting differential equations can be solved analytically. However, any object moving in the atmosphere is subject to aerodynamic forces, which are usually divided into lift $F_l$ and drag $F_d$. For a damped projectile trajectory, lift is very small or nonexistent. Adding a drag term to the differential equation makes its solution more complex.

To simplify the calculation, it is assumed that there is only air drag in the downward direction of motion, and no air drag in the forward direction. This means that the horizontal distance value calculated by the method will be greater than the actual horizontal distance, which also means that the ground risk buffer identified here is an upper estimate.

For the vertical direction, the UAV is subjected to upward air resistance $F_d$ and downward gravity $F_g$. According to Newton’s law, the equation holds as follows:

$$F_g-F_d=mg-{\displaystyle \frac{1}{2}}\rho A{C}_d{v}^2=m{\displaystyle \frac{dv}{dt}}$$

(1)

where m is the mass of the UAV, and g is the gravitational coefficient, taken as 9.81 m/s². A is the frontal area of the UAV during descent; $C_d$ is the drag coefficient, which depends on the shape and material of the UAV, that remains constant in most test flights; $\rho$ is the air density, which is related to the environmental parameters of the UAV’s operating site, and v is the velocity of the UAV, which depends on the time t.

Suppose the operation site of the UAV has a temperature of T in degrees Celsius and an air pressure of P in kPa, then the air density $\rho$ can be estimated by the standard atmospheric model [30]:

$$\rho =\rho \left(P,T\right)={\displaystyle \frac{P}{0.2869\left(T+273.15\right)}}$$

(2)

Keeping on considering the force analysis equation, since ${\displaystyle \frac{dv}{dt}}={\displaystyle \frac{dv}{dy}}{\displaystyle \frac{dy}{dt}}=v{\displaystyle \frac{dv}{dy}}$, Equation (1) above can be rewritten as

$$mv{\displaystyle \frac{dv}{dy}}=mg-{\displaystyle \frac{1}{2}}\rho A{C}_d{v}^2$$

(3)

Define $v_T=\sqrt{{\displaystyle \frac{2mg}{C_{d}A\rho }}}=\sqrt{{\displaystyle \frac{g}{\beta }}}$, which is the terminal velocity of the falling UAV. Here, $\beta ={\displaystyle \frac{C_{d}A\rho }{2m}}$, which is related to the UAV parameters (frontal area, mass, and drag coefficient) and environmental parameters (temperature, and atmospheric pressure). The differential equation can then be derived as

$$vdv=[g-{\displaystyle \frac{\rho A{C}_d{v}^2}{2m}}]dy$$

(4)

By substituting the values of $v_T$ and $\beta$ into Equation (4), a more simplified form can be obtained, as follows:

$${\displaystyle \frac{v}{v_T^2-v^2}}dv=\beta dy$$

(5)

Integrating it and substituting the initial condition $v\left(y=0\right)=0$, the velocity as a function of altitude is obtained, as follows:

$$v^2=v_T^2\left(1-e^{-2\beta y}\right)$$

(6)

The time of descent t can be obtained by integrating

$$t={\int }_0^h{\displaystyle \frac{1}{v\left(y\right)}}dy=-{\displaystyle \frac{1}{2\sqrt{\beta g}}}ln\left({\displaystyle \frac{e^{\beta h}-\sqrt{e^{2\beta h}-1}}{e^{\beta h}+\sqrt{e^{2\beta h}-1}}}\right)$$

(7)

Since the air resistance in the forward direction of the UAV is ignored, the maximum horizontal distance the UAV traveled is the product of the horizontal speed at failure $v_F$ and the descent time t:

$$x_m=v_F·t$$

(8)

2.2. Fixed-Wing UAV

For a fixed-wing UAV, consider the same scenario where the UAV loses power at altitude h with a horizontal speed $v_F$. In order to calculate the ground risk buffer for fixed-wing UAVs, we assume that there is neither yaw nor roll during UAV gliding and that the problem is reduced to a two-dimensional problem considering only the lateral and vertical directions. Therefore, the estimated maximum horizontal distance traveled by the fixed-wing UAV is the length of the buffer zone.

However, unlike a rotorcraft UAV, the trajectory of a fixed-wing UAV after losing power can be considered as a gliding trajectory. Gliding trajectories are distinguished from damped projectile trajectories in that the lift for a gliding trajectory is not provided by the rotors. When the fixed-wing UAV fails and loses thrust, the lift does not disappear. Therefore, for the gliding trajectory, lift cannot be ignored, which further complicates the dynamics equations.

For a failed fixed-wing UAV, consider the lift $F_l={\displaystyle \frac{1}{2}}\rho A{C}_l{v}^2$, drag $F_d={\displaystyle \frac{1}{2}}\rho A{C}_d{v}^2$ and gravity $F_g=mg$. From Newton’s law, the UAV satisfies the equation:

$$m{\displaystyle \frac{d\overrightarrow{v}}{dt}}={\overrightarrow{F}}_d+{\overrightarrow{F}}_l+{\overrightarrow{F}}_g$$

(9)

where A, unlike the Equation (1), is the wing reference area, m is the mass of the UAV, $\rho$ is the air density, which can be estimated using the standard atmospheric model, i.e., Equation (2). $C_l$ and $C_d$ are the lift and drag coefficients, respectively, and their ratio is denoted as $\eta ={\displaystyle \frac{C_l}{C_d}}$. Note that here, $\overrightarrow{v}$, ${\overrightarrow{F}}_d$, ${\overrightarrow{F}}_l$, and ${\overrightarrow{F}}_g$ are vectors in comparison to Equation (1).

This Equation (9) can be iteratively solved accurately using the Runge–Kutta method. However, performing high-fidelity calculations for the entire gliding process requires a large number of iterations. This would be cumbersome and inflexible if coupled partial differential equations were solved finely for each task. It is necessary to quickly approximate the buffer size by some simplification.

As shown in Figure 3, at the point of power loss, the induced drag will cause the UAV’s speed to decrease, and the induced lift will also slowly decrease. This stage is called the transition stage. After a period of time, the UAV enters the gliding stage, in which the UAV descends to the ground at a fixed angle.

During the gliding stage, where the angle between the flight path and the ground is $\gamma$, the fixed-wing UAV reaches a force balance state, that is

$$\left\{\begin{array}{c}mgcos\gamma ={\displaystyle \frac{1}{2}}\rho A{C}_l{v}^2\hfill \\ mgsin\gamma ={\displaystyle \frac{1}{2}}\rho A{C}_d{v}^2\hfill \end{array}\right.$$

(10)

Thus, the angle $\gamma$ of the UAV’s flight path is $tan\gamma ={\displaystyle \frac{C_d}{C_l}}={\displaystyle \frac{1}{\eta }}$, and the horizontal distance traveled by the UAV can be expressed as

$$x=x_0+x_1=x_0+{\displaystyle \frac{h_1}{tan\gamma }}\approx x_0+{\displaystyle \frac{h}{tan\gamma }}$$

(11)

To simplify the calculation of the horizontal distance during the transition stage, we consider abstracting the transition phase as a process in which the altitude remains h and the velocity is reduced from $v_F$ to v. In the process, the deceleration is mainly due to air resistance, which is given by the law of conservation of energy that

$${\displaystyle \frac{1}{2}}m{v}_F^2-{\displaystyle \frac{1}{2}}m{v}^2\approx {\int }_{x_0}{F}_{d}ds={\displaystyle \frac{1}{2}}\rho A{C}_l{v}^2{x}_0$$

(12)

Since $v=\sqrt{{\displaystyle \frac{mgcos\gamma }{\frac{1}{2}\rho S{C}_l}}}=\sqrt{{\displaystyle \frac{2mg}{\rho S{C}_l}}}\sqrt{cos\gamma }=v_F\sqrt{cos\gamma }$, the horizontal distance $x_0$ during the transition stage can be simply estimated as

$$x_0={\displaystyle \frac{m\left(v_F^2-v_1^2\right)}{\rho A{C}_l{v}_1^2}}={\displaystyle \frac{v_F^2\left(1-cos\gamma \right)}{2gcos\gamma }}$$

(13)

In summary, the total horizontal distance $x_m$ traveled by the fixed-wing UAV can be approximated as follows:

$$x_m=x_0+x_1\approx {\displaystyle \frac{v_F^2\left(1-cos\gamma \right)}{2gcos\gamma }}+{\displaystyle \frac{h}{tan\gamma }}$$

(14)

where $\gamma$ is the angle between the gliding trajectory and the horizontal direction, which can be calculated by $\gamma =arctan({\displaystyle \frac{C_d}{C_l}})=arctan({\eta }^{-1})$.

It is worth noting that the value of ${\displaystyle \frac{1-cos\gamma }{cos\gamma }}$ in Equation (14) above is usually quite small. When ${\displaystyle \frac{1-cos\gamma }{cos\gamma }}<0.01$, the difference exceeds 2 orders of magnitude, i.e., $\eta ={\displaystyle \frac{1}{tan\gamma }}>{\left(tan\left(arccos\left({\displaystyle \frac{1}{1.01}}\right)\right)\right)}^{-1}\approx 7.05$, and the first term $x_0$ in Equation (14) can be ignored, and this formula can be further simplified as

$$x_m=\left\{\begin{array}{cc}h\eta ,\hfill & \hfill \eta >7.05\\ {\displaystyle \frac{v_F^2\left(1-cos\gamma \right)}{2gcos\gamma }}+h\eta ,\hfill & \hfill \eta \le 7.05\end{array}\right.$$

(15)

For a fixed-wing UAV, the maximum horizontal distance traveled after a failure can be considered as the distance covered during the gliding stage. This trajectory is mainly determined by the lift-to-drag ratio during the glide stage and the failure altitude. It should be noted, however, that the actual post-failure path may not strictly follow a steady glide trajectory. The estimated distance presented here represents the upper bound of possible flight paths under the given conditions.

3. Results

This section initially calculates the buffer sizes for both rotorcraft UAVs and fixed-wing UAVs through simulation computations combined with publicly available data. It further discusses the impact of environmental parameters and UAV-specific parameters on these buffer sizes.

3.1. Rotorcraft UAV Analysis

3.1.1. Simulation Results of Rotorcraft UAV

For the environmental parameters, a temperature of 10 °C and an atmospheric pressure of 103 kPa are chosen as default settings in this paper. Using the atmospheric model in Equation (2), the calculated air density is 1.2679 kg/m³.

Ultimately, the area and mass parameters for each rotorcraft UAV are summarized in

Table 1. Parameters for modeled rotorcraft UAVs.

Name of UAV	Mass (kg)	Area (m2)	Parameter $\mathit{\beta }$
DJI Air 3S [31]	0.724	0.0215	0.016943
ARK40 [32]	46	1.1272	0.013981
XAG P30 [33]	38.5	0.6453	0.009563
DJI T30 [34]	78	0.9193	0.006725

, taken from [31,32,33,34]. After obtaining the environmental and UAV parameters, the formula $\beta ={\displaystyle \frac{C_{d}A\rho }{2m}}$ is used for the calculation of the ground risk buffer. Regarding the drag coefficient among the UAV parameters, this section references the findings of [35], which evaluated the drag coefficient of the DJI Phantom 3 using computational fluid dynamics (CFD). This modeling and simulation work indicates that the DJI Phantom 3, without blade guards, has an average drag coefficient of 0.93 in the vertical direction and 1.08 in the horizontal direction. Therefore, in the calculations presented in this section, the drag coefficient is approximately chosen as 0.9. The calculation of the frontal area, i.e., the cross-sectional area, is obtained by subtracting the air area that does not belong to the UAV from the UAV’s length multiplied by its width. The mass parameters are obtained from publicly available data, with the maximum take-off weight selected for calculations.

Figure 4 provides the above UAV buffer size versus failure altitude and speed. Comparing the four types of rotorcraft UAVs, it can be observed that parameter $\beta$ is positively correlated with the buffer zone length, the larger the parameter $\beta$, the larger the calculated buffer size under the same altitude and speed conditions. In practical applications, it is challenging to obtain real-time information on the altitude and speed at the failure point, as we cannot predict when the UAV might fail.

3.1.2. Parameter Sensitivity Analysis

To explore the impact of environmental parameters on the buffer size calculated by this method, taking ARK40 as an example, considering a flight speed of 10 m/s and a flight altitude of 100 m at the failure point, the buffer sizes calculated under different air pressures and environmental temperatures are compared with those under reference condition (with temperature set at 10 °C and air pressure at 103 kPa). The resulting ratios are shown in Figure 5a below, where considering the ground risk buffer under reference conditions, the ground risk buffer under any temperature and pressure condition can be calculated, thereby simplifying the complexity of ground risk buffer estimation in test flight missions. In this figure, compared with the reference condition, the greatest change in the buffer size occurs at temperature of 40 °C and air pressure of 80 kPa, with a maximum ratio of approximately 5.84%. This indicates that environmental parameters have a relatively small effect on the UAV’s buffer size. Therefore, if it is not feasible to specify the environmental parameters, the air density under standard conditions (i.e., air density of 1.225 kg/m³ at a temperature of 15 °C and an air pressure of 101.325 kPa) can be selected for the calculation.

Similarly, fixing the environmental parameters, the buffer sizes obtained under different masses and drag coefficients are compared with those under reference condition (with a drag coefficient of 0.9 and a mass of 46 kg). The resulting ratios are illustrated in Figure 5b above. As before, the greatest change in the buffer size is located at the lower right corner of the image, indicating that the smaller the mass, the larger the buffer size; the larger the drag coefficient, the larger the buffer size. Moreover, compared to mass, the drag coefficient has a significant impact on the size of the ground risk buffer. When the UAV is unloaded (the mass is about 25 kg) and the drag coefficient is set to 1.3, the variation reaches 30.29%. This implies that in practical applications, an overestimation of the drag coefficient will result in an enlarged buffer size. If the available airspace is ample, the drag coefficient can be roughly estimated as a larger value to ensure the buffer size is sufficiently large to guarantee the safety of test flights.

3.1.3. Comparison with Other Methods

In this section, ARK40 is selected as an example to analyze the buffer size computed by different methods in comparison with the 1-to-1 principle proposed in SORA.

In SORA, the standard approach to controlling ground risk is the “1-to-1 principle”, which refers to applying a ground risk buffer that is as wide as the maximum height of the operational volume [17]. In Figure 6, the downward black dashed line denotes the 1-to-1 principle in SORA, and the vertical black dashed line denotes the length of the ground risk buffer required for this principle.

The methods used for the comparison includes both parabolic and damped projectile trajectories. The damped projectile trajectory, where the horizontal and vertical drag are both considered, is obtained using the Runge–Kutta method [36]. The parabolic trajectory, on the other hand, completely ignores the air drag, and its trajectory and horizontal distance are related only to the horizontal speed and altitude at the time of failure, making it an oversimplified computational method.

For the ARK40 UAV, the flight altitude is set to 25 m, with horizontal flight speeds set to 10 m/s, 12 m/s, and 15 m/s, respectively. It can be seen from Figure 6a, that the distances calculated by the three methods are all less than the buffer size required by the 1-to-1 principle. However, as the speed increases, the 1-to-1 rule begins to be inapplicable, as shown in Figure 6b. Particularly, when the speed exceeds 15 m/s, as depicted in Figure 6c, the distances calculated by all three methods surpass that implied by the 1-to-1 principle.

The results indicate that the damped projectile trajectory, due to the involvement of horizontal drag, yields the shortest calculated horizontal distance traveled. This is followed by the parabolic trajectory. The method presented in this paper, which accounts for vertical air drag, results in an extended falling time, thereby increasing the horizontal distance. In practical applications, the actual crash positions follow a Gaussian distribution [37], with its center being the impact point determined by the damped projectile trajectory. Due to inaccuracies in estimating UAV parameters and environmental parameters, strictly adhering to the damped projectile trajectory for buffer size calculations may result in the UAV exceeding the ground risk buffer. However, the method presented in this paper, by considering vertical drag with neglecting horizontal drag, provides a buffer size with sufficient safety margin.

Under the proposed simulation framework, the dimensions of the operational volume are defined according to the test mission profile, with a length of 100 m and a width of 200 m. The test UAV remains the ARK40, with parameter $\beta$ is set to 0.013981. The maximum flight altitude and speed for the mission are 25 m and 15 m/s, respectively. The objective is to rapidly determine the corresponding ground risk buffers within the designated test airspace.

Figure 7 provides a schematic comparison of the ground risk buffers estimated by four different methods. The central blue area represents the operational volume, i.e., the airspace in which the UAV operates normally during flight tests. Personnel involved in the test flight mission must remain outside the red dashed boundary to ensure safety. According to the 1-to-1 principle, the ground risk buffer is defined as equal to the flight altitude of 25 m, resulting in the smallest estimate among all methods. In contrast, force equilibrium analysis and numerical simulations yield a glide distance of 32.38 m. The severe underestimation of the ground risk buffer using the 1-to-1 principle would expose personnel to potential risks. The Monte Carlo-based method further incorporates stochastic process. Simulations are conducted for conditions where the UAV exits the operational volume boundary in various directions, with a total of 100,000 computational trials performed. The ground risk buffer size is estimated within a 95% confidence interval, yielding a result of 32.35 m. This value is consistent with the trajectory simulation but requires significantly greater computational time, amounting to 17.35 s. In contrast, the ground risk buffer size estimated using the contour map method is 35.87 m, representing an upper bound for the flight distance without considering horizontal drag. All calculations are conducted on a desktop computer with an Intel Core i5-14400F CPU. This value can be rapidly obtained from Figure 4b without incurring additional computational cost.

3.1.4. Case Analysis of Rotorcraft UAV

A recent airworthiness safety certification flight test of a certain quadcopter UAV as an example is used to introduce the flight test procedures and the steps for calculating the ground risk buffer. It also provides a schematic diagram of the operational volume and ground risk buffer during the flight test.

On 20 December 2024, the UAV underwent a speed test flight. The flight test verified that the UAV possesses the capability for normal flight operations at its proposed maximum altitude above ground level, flying at its intended maximum speed with both maximum and minimum take-off weights. The parameters used in the case study of rotorcraft UAV is shown in

Table 2. Parameters used in case study of rotorcraft UAV.

Parameter	Value	Parameter	Value
Maximum take-off weight	113 kg	Frontal area	1.0516 m2
Minimum take-off weight	53 kg	Ambient temperature	15 °C
Flight altitude	40 m	Barometric pressure	101 kPa
Maximum speed	13.8 m/s	Drag coefficient	0.96

.

The UAV’s test flight adopts the following flight procedures: (1) Confirm that the UAV is configured to execute the flight mission with Mass 1 (fully loaded, 113 kg) and that the ballast is properly attached. (2) Check the flight route and confirm that the flight altitude is the altitude of the test flight site +40 m. (3) Start up the UAV, power it on, connect it to the ground control station, and ensure that all personnel are outside the zone of the ground risk buffer. (4) After the UAV takes off, fly in a straight line at a maximum speed of 13.8 m/s, confirm that the UAV completes the entire flight process according to the vertical take-off, cruise, and landing procedures, and observe if there are any abnormal warnings from the ground station during the flight. (5) Once again, with Mass 2 (unloaded, 53 kg), repeat steps (2) to (4) to complete another sortie of the flight mission.

To conduct the UAV test flight, a dedicated airspace has been applied for; the airspace is centered at the latitude and longitude coordinates N23°20′15.50″ E113°29′52″. It covers a circular area with a radius of 1 km, and the flight altitude is permitted to be less than 200 m above ground level.

Based on the above test flight mission description, as the UAV’s flight trajectory follows a straight line, the operational volume for the test flight has been designated as a cuboid with 1.5 km in length, 200 m in width, and 40 m in height. On the day of the test flight, the ambient temperature was 15 degrees Celsius, with a barometric pressure of 101 kPa. The test UAV has a frontal area of 1.0516 m², with an unloaded mass of 53 kg and fully loaded mass of 113 kg, respectively. Since the drag coefficient was not determined for the test UAV, it was set to 0.96 based on referenced simulation results.

According to the above parameters and the formula $\beta ={\displaystyle \frac{C_{d}A\rho }{2m}}$, the parameter $\beta$ is calculated to be 0.011635 and 0.005457 for the UAV when it is fully loaded and unloaded, respectively. Taking into account the possible maximum altitude of 40 m and the maximum speed of 13.8 m/s for this test flight, the required buffer sizes for fully loaded and unloaded conditions are determined to be 42.52 m and 40.86 m, respectively. The designated flight area and corresponding ground risk buffer are illustrated in Figure 8. Since the ground risk buffer size is very close in the two cases, only the unloaded case is plotted in Figure 8.

3.2. Fixed-Wing UAV Analysis

3.2.1. Simulation Results of Fixed-Wing UAV

For the section of fixed-wing UAVs, the lift-to-drag ratio $\eta$ and the altitude h are the key parameters for calculating the buffer size, and the lift-to-drag ratio $\eta$ is crucial in determining the buffer size of a fixed-wing UAV. There are many factors affecting the lift-to-drag ratio, and for a certain shape-determined UAV, the lift-to-drag ratio is mainly a function of the flight speed and the angle of attack. In addition to this, in actual operation, the change in environment will also inevitably lead to small variation in the lift-to- drag ratio.

The lift-to-drag ratio $\eta$ can be calculated using, i.e., CFD, or obtained by experimental measurement in a wind tunnel. However, it is very expensive to accurately calculate the variation of the lift-to-drag ratio with each influence factor in order to calculate the length of the ground risk buffer. In practice, we use the maximum lift-to-drag ratio of the UAV to calculate the ground risk buffer. The maximum lift-to-drag ratio ${\eta }_{max}$ of the UAV can be estimated by the following equation [38]:

$${\eta }_{max}={\displaystyle \frac{1}{2\sqrt{C_{D_0}K}}}={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\pi Ae}{C_{D_0}}}}$$

(16)

where K is drag-due-to-lift factor, which is presented by $K={\displaystyle \frac{1}{\pi Ae}}$. $C_{D_0}$ is zero-lift drag coefficient, A is the aspect ratio of the wing, and e is the span efficiency factor.

Therefore, this section calculates the buffer size for flight altitudes from 0 to 1000 m and lift-to-drag ratios from 0 to 30. Once the maximum flight altitude and the maximum lift-to-drag ratio of the test UAV are known, the required buffer size can be quickly obtained from Figure 9.

It is noteworthy that the demarcation of the ground risk buffer is predicated on the maximum distance under the ideal scenario of the UAV gliding in a straight line without any yaw or roll. However, in actual flight, once the UAV fails, it is difficult to maintain the balance of the UAV and the possibility of rolling is extremely high, so the actual forward distance after failure is often smaller than the buffer size calculated by the introduced method.

3.2.2. Comparison to Other Methods

In this section, the buffer sizes obtained from fixed-wing UAVs with different lift-to-drag ratios are analyzed under different computational methods. The simulated fixed-wing UAV is set to a speed of 20 m/s and a cruising altitude of 50 m. Figure 10 shows the schematic diagram of the trajectories obtained using the theoretical calculation method, the method in this paper, and the 1-to-1 principle with lift-to-drag ratios of 5, 10, and 15, respectively. In Figure 10, the black dashed line represents the 1-to-1 principle of SORA. The theoretical calculation method employs the Runge–Kutta method [36] to solve a system of partial differential equations, resulting in the gliding trajectory.

The results show that the traveled distance of the fixed-wing UAV exceeds the expectation of the 1-to-1 principle in all three cases presented above. The 1-to-1 principle tends to underestimate the traveled distance in the application of fixed-wing UAVs, and therefore, other methods for determining the buffer size need to be explored. The buffer size calculated by the proposed method is similar to the result obtained by numerically solving the system of partial differential equations. This indicates that the proposed method is able to estimate the buffer length quickly and avoids the complex iterative computation process.

It is essential to clarify that in Appendix E of SORA v2.5 [17], it is stated that the 1-to-1 principle may not be applicable to certain UAV configurations, including fixed-wing UAVs. The comparisons of methods in this section also corroborate this statement. SORA further mentions that for cases where the 1-to-1 principle is not suitable, alternative methods such as damped projectile or gliding trajectory calculations may be necessary to define the ground risk buffer. The method proposed in this paper represents a feasible and rapid approach to calculating buffer sizes based on gliding trajectories.

For the fixed-wing UAV, the dimensions of the operational volume remain 100 m in length and 200 m in width. The maximum flight altitude for this test profile is 30 m, with the lift-to-drag ratio during the glide phase is set to 10. Figure 11 illustrates a schematic comparison of the ground risk buffers estimated by four different methods. According to the 1-to-1 principle, the ground risk buffer is defined as 30 m. In contrast, force equilibrium analysis and numerical simulation yield a maximum glide distance of 311.74 m. Following the same simulation framework as applied to rotorcraft UAV, 100,000 Monte Carlo-based trials are conducted on a local server, requiring a total computation time of 40.68 s and resulting in an estimated ground risk buffer size of 311.58 m. The ground risk buffer size estimated using our contour map method is 300 m, which corresponds to the glide distance under force equilibrium conditions. All calculations are conducted on a desktop computer with an Intel Core i5-14400F CPU.

3.2.3. Case Analysis of Fixed-Wing UAV

Similarly, in this section, the test flight of a certain type of vertical take-off fixed-wing UAV, that recently performed airworthiness safety certification, is taken as an example. The test flight procedure with the buffer size calculation step is presented, and the schematic diagram of the ground risk buffer is given.

On 9 May 2024, the UAV underwent a test flight to evaluate its maximum range. Under standard operational conditions and within the parameters specified in the flight manual, the UAV was equipped with a fully charged battery system and fueled to its maximum capacity. It flew at the maximum take-off weight of 145 kg to verify the actual range achievable with a full fuel load.

The UAV’s test flight adhered to the following procedures: (1) establish a rectangular flight path at an altitude of 50 m; (2) execute a normal take-off and maintain flight along the rectangular path for no less than 4 h; (3) monitor the UAV’s responses to ensure correct operation, and verify that its attitude, altitude, and speed remain normal; and (4) record the measured maximum range.

The airspace is defined as a rectangular area below a true altitude of 300 m, enclosed by the following coordinates: Coordinate 1: 29°23′1″ N 104°37’33″ E; Coordinate 2: 29°21′20″ N 104°36′22″ E; Coordinate 3: 29°20′41″ N 104°37′43″ E; and Coordinate 4: 29°22′20″ N 104°38′51″ E.

During the test flight, the operational volume was designated as the part of the rectangular airspace excluding the ground risk buffer. The UAV was permitted to fly only within this operational volume during the maximum range test flight. Given that the vertical accuracy of the UAV during fixed-wing flight is 4 m, the buffer size was calculated with an altitude of 54 m. According to data provided by the UAV manufacturer, the lift-to-drag ratio is about 15.78. Based on these parameters, using Equation (15), the required buffer size for this test flight was calculated to be 852.12 m. The resulting test flight area and the ground risk buffer are illustrated in Figure 12. The test flight was conducted within the designated operational volume, with personnel positioned outside the ground risk buffer.

4. Conclusions

In order to address the estimation of ground risk buffer under UAVs test flight, this work proposes a framework of ground risk buffer estimation based on the analysis of the UAVs dynamics. By integrating UAVs dynamics analysis with several parameter layers such as UAV type and failure conditions, a 3D contour map is constructed in this framework for estimating the ground risk buffer sizes for both rotor and fixed-wing UAVs in test flight condition. Analyzing the motion characteristics of different UAVs after failure, the calculations for corresponding damped projectile trajectories and gliding trajectories are simplified, offering a novel estimation method for ground risk buffer using a contour map. Based on the method presented in this paper, contour maps are provided that allow for the retrieval of the necessary ground risk buffer sizes once the key parameters of the test flight subjects are determined. Finally, according to recent examples from UAV airworthiness safety department, specific buffer calculation examples for designated test flight subjects are presented and applied in UAV test flights.

Overall, this paper investigates the estimation of ground risk buffer sizes in UAV test flight missions and introduces a novel estimation method based on dynamic analysis. In this work, a contour map is used to determine the ground risk buffer sizes, thereby addressing the trade-off between airspace utilization and operation risks. However, in practical applications, the consideration of dynamic variables remains insufficient, and the types of test flight missions are not taken into account. Despite not fully capturing the precise trajectory of in-flight failures, the proposed method offers valuable reference estimation for ground risk buffer during test flights, demonstrating potential for guiding practical UAV test flights. Future work will focus on expanding the method to more diverse test flights and developing the dynamic model for ground risk buffer by incorporating actual flight trajectory.