#### Abstract

Vertical takeoff and landing (VTOL) capability has extended the application of unmanned aerial vehicle (UAV) significantly. In this paper, simulation modeling and flight test were employed to investigate the hovering stability of a tail-sitter UAV named Egretta. The hovering stability simulation model was developed based on a simplified rigid body flight dynamic and the time-averaged propeller slipstream flow distribution. Meanwhile, a testing vehicle with PID controllers was built and tested to verify the hovering stability model. It was found that the Egretta UAV can achieve stable hovering in the roll, pitch, and yaw directions. The simulation model has demonstrated accuracy in predicting the hovering stability and dynamic responses with large perturbations in both trend and magnitude. Moreover, the simulation model can be extended to analyze the hovering stability of tail-sitter UAVs with different sizes. The simulation model will be very useful for initial stability sizing and PID optimization investigation.

#### 1. Introduction

The vertical takeoff and hovering capability allow flight vehicles to operate in confined areas and perform special tasks such as emergency rescue and fixed location surveillance. Helicopters play an important role in situations where vertical takeoff and hovering are required. However, the range and speed of helicopters are restricted by the fundamental working principle. The tilted rotor fixed-wing crafts such as Bell V22 and V280 integrate the advantage of the helicopter and fixed-wing to achieve vertical takeoff, long range, and high speed simultaneously. With the fast development of electronic devices, flight vehicles such as a quadcopter, quadcopter with fixed-wing, and tail-sitter configurations raised quickly and have been applied in various applications. Among different configurations which possess vertical takeoff and hovering capability, the tail-sitter configuration has attracted lots of research and practical application attention due to its high aerodynamic efficiency at cruise, structure simplicity, and efficient propulsion system. The Egretta tail-sitter VTOL UAV has demonstrated more than 90 mins endurance and 100 km range in one complete flight mission including takeoff and landing and is currently applied in monitoring inspections, intelligent prewarning, and scientific research data collection.

For tail-sitter configuration starting in the 1950s [1], the hovering stability and control are complex as it involves aerodynamics at a high angle of attack and propeller slip-flow characteristics. Researchers have studied tail-sitter stability from propeller slip-flow simulation, flight controller model development, and experiment test perspectives. Stone studies the propeller slip-flow characteristics by applying the blade element method [2, 3]. In addition, a panel method with 2D experimental airfoil data correction was employed to simulate the aerodynamics and slip-flow wing interactions of a T-wing tail-sitter configuration. These simulation data serve as the important database to build a 6 DOF flight control simulation. Hunsaker and Snyder investigated propeller slip-flow properties numerically based on blade element theory combined with momentum conservation [4]. The isolated propeller model produces results with a good agreement as compared to experiments. However, when combining with wings, only qualitative results are obtained. Beck et al. built a wind tunnel model equipped with an active high-lift system and a powered propeller to study the aerodynamic effects of propeller slipstream [5]. The wind tunnel testing results characterize and quantify the interactions between a propeller and a high-lift wing by using systematic variations of blowing momentum and propeller thrust. Bouquet developed a prediction method to model aerodynamic effects caused by the propeller slipstream on the longitudinal stability and control, but it has no enough validation data [6]. Cui et al. experimentally studied the influence of different parameters on the aerodynamic characteristics of a multielement wing of VTOL under propeller slipstream, and found that the Gurney flaps can enhance the lift and turning angle, further increasing the aerodynamic performance of a multielement wing [7]. Recently, Misiorowski et al. employed Helios with a delayed detached eddy simulation model to study the propeller slip-flow properties in hovering and forward flight of a biplane tail-sitter aircraft [8]. Propeller slip-flow properties are mainly investigated based on numerical simulations. For tail-sitter flight controllers, Knoebel et al. developed a quaternion feedback hover controller and trajectory generation scheme based on Goldstein’s momentum slip-flow theory [9]. Numerical simulation of the flight controller was carried out while no comparison with the actual flight data. Forshaw and Lappas studied a PID and LQR-based controller for tail-sitter driven by two helicopter cyclic pitch propellers [10]. Vortex lattice method and blade element momentum theory are employed for aerodynamics and propeller slip-flow model, respectively. It was found that the LQR controller is more robust and can stabilize the system for a wider matrix variation. Verling et al. studied the full attitude controller where one unified controller allows high-performance flight control in all the flight modes without mode switching [11]. Qassar et al. and He et al. derived optimization algorithms, such as adaptive control method and adaptive fault-tolerant control strategy, and simulated the performance of aircraft [12–14]. Ajel et al. presented a control design of roll motion for VTOL UAV based on the model reference adaptive control (MRAC) scheme in the hovering flight phase, which resulted in better robustness performance of the controller [15]. Najm et al. derived an NLPID-IADRC framework to stabilize the motion of quadrotor while tracking a specified trajectory with disturbances and parameter uncertainties [16]. Model predictive controller (MPC) was also applied in VTOL UAV [17]. Li et al. present a model predictive controller (MPC) for position control of a VTOL tail-sitter UAV in hover flight. The simulation and experimental results show that the proposed MPC position controller has good trajectory tracking performance and robust position holding capability [18]. Wang et al. modeled a distributed propulsion tail-sitter with four electrically driven propellers and analyzed the closed-loop stability of the tail-sitter with thrust-vectoring control and trailing-edge elevators control, proving that the proportional–Derivative controller with thrust vectoring as an effector guaranteed the pitch loop stability [19]. Humaidi et al. derived a sliding mode backstepping controller and disturbance observer-based sliding mode backstepping controller for roll channel and found that the latter gives better transient and steady-state characteristics [20]. Qassar et al. extended the super-twisting sliding mode controller (STSMC) by gray-wolf optimization method for the roll motion of VTOL UAV in hovering flight, which is under nonparametric uncertainty. The simulation results show that STSMC with gray-wolf optimization method can enhance the performance of the controller when referring to tracking error and chattering behavior [21]. Instead of using a slip-flow model, wind tunnel test was employed to obtain the aerodynamic force and moment of the entire tail-sitter UAV over a range of elevon deflections, propeller RPM, and the freestream velocity. Both simulation and flight tests of the controller have been carried out [22]. It was found that the unified controller was capable and effective in controlling tail-sitter configurations. Bapst et al. also investigated the state-of-art unified single controller for tail-sitter control [23]. Flight test and simulation results showed good consistency. Forshaw et al. [24] employed the aerodynamic model called theory by Lustosa et al. [25] to develop a unified model-free controller for tail-sitter. The simulation and flight test showed promising results. Experimental flight tests for tail-sitter provide very useful real data to verify the aerodynamic model and flight controllers, and several researchers like Verling et al., Forshaw et al., Zhou et al., Swarnkar et al., and Li et al. have carried out tail-sitter flight test at various flight state such as hovering, transition, and fixed-wing flight [11, 24, 26–28].

For stability model development, due to the inviscid nature of the blade element method and panel method, the aerodynamic data and propeller slip-flow properties in current research papers have limitations to perform hovering stability and control investigation for tail-sitter configuration. On the other hand, for those employing wind tunnel aerodynamic force and moment data, the drawback is that numerous data has to be collected for a specific tail-sitter model and propeller size. Once the size of the tail-sitter or the propeller is changed, wind tunnel tests have to be carried out again. Hence, it is not suitable for controller design purposes.

In this paper, experimental tests are employed to obtain the propeller slip-flow characteristics, which serve as the database for flight controllers. Then, robust PID controllers based on PX4 architecture and simplified decoupled rigid body flight dynamic models are developed to investigate the pitch, roll, and yaw stability at hovering, respectively. Finally, indoor flight tests are carried out to verify the decoupled stability model in roll, pitch, and yaw motion.

#### 2. Tail-Sitter Hovering Stability Model

Egretta is a vertical takeoff and landing (VTOL) UAV with tail-sitter configuration, developed at the Southern University of Science and Technology (SUSTech) in China. As shown in Figure 1, the two elevons receive mixed control signals from the yaw rotation around the *x*-axis and the pitch rotation around the *y*-axis. The two propellers are independently controlled for the rotation around *z*-axis and the height control. Based on PX4 controller architecture, roll angle is defined as the angle rotating around *z*-axis. Pitch angle is around *y*-axis, and yaw angle is around *x*-axis. Three main parts, namely, the data lookup tables, the decoupled controllers, and the simplified dynamics models, are presented here to develop the hovering stability models.

##### 2.1. The Data Lookup Tables

To develop the stability models, two data lookup tables (see Tables 1–2 and Figure 2) are prepared. Time-averaged propeller slip-flow speeds at various RPM are obtained experimentally. Table 1 shows the sample data at a fixed , where is the propeller diameter, is the radial distance from measurement location to propeller center, and is the streamwise distance from measurement location to propeller center. The detailed experimental procedure for the data in Table 1 can be found in the subsequent Section 3. Figure 2 shows the lift coefficient of NACA0015 at a large range of , which is experimentally tested by Sheldahl and Klimas [29]. The data in Figure 2 are interpolated to obtain the section of lift for the elevons, which is used to solve the simplified pitch and yaw dynamic model. The static propeller thrust is shown in Table 2, which is based on analytical simulations using vortex theory by APC [30]. These data are used for the simplified roll dynamic model. For data interpolation, the linear interpolation function interp1 and interp2 in MATLAB are employed to obtain the required data from the data lookup tables.

##### 2.2. The Simplified Dynamics Models

To explore the insight of the P-PID controller on each stability mode, decoupled roll, pitch, and yaw models are developed. For all three modes, it is assumed that the equilibrium RPM satisfies the hovering condition. Figure 3 shows the Egretta reference frame and free body diagram at hovering state. The simplified rigid body dynamic equations for each model are presented.

###### 2.2.1. Roll Model

where is the roll angle, is the roll rate, is the roll moment due to propeller thrust difference, is the thrust line distance to the *x*-axis, and is the roll moment due to roll rate damping.

###### 2.2.2. Pitch Model

As lift is one order magnitude larger than drag, in addition, the pitch moment for drag is negligible as compared to that for the lift. Hence, the pitch moment contribution due to drag is ignored for the pitch model. The net lift is obtained by integrating the section lift on the elevon. As shown in Figure 4, is the time averaged slip-flow speed at a specific elevon section in the span-wise direction. and are the chord length and span of the elevon, respectively. Hence where is the air density and is the section lift coefficient which is a function of the elevon deflection angle .

The simplified pitch dynamic model is shown in Equations (4) and (5), where is the pitch angle, is the pitch rate, is the pitch moment due to elevon deflection, and are the net lift from the elevon deflection and , respectively, is the elevon distance to the center of gravity, and is the pitch moment due to pitch rate damping.

###### 2.2.3. Yaw Model

The same as for the pitch model, the drag contribution to the yaw moment is assumed to be zero. The net lift is obtained in the same way as that for the pitch model. The simplified dynamic model for the yaw model is given in
where is the yaw angle, is the yaw rate, is the yaw moment due to elevon deflections, and are the net lift from the elevon deflection and , respectively, is the net lift acting location of the elevon to the *x*-axis, and is the yaw moment due to yaw rate damping.

##### 2.3. The Decoupled Stability Simulation Models

Based on PX4 open-source architecture, the decoupled P-PID roll, pitch, and yaw controller diagram are developed, respectively, and the simulation models that contain controllers, aerodynamics lookup table, propeller slip-flow database, thrust lookup table, and rigid body equation of motions are shown in Figure 5. For the decoupled roll simulation model, the reference roll angle is commanded as the input signal, and a P loop converts the difference between the reference roll angle and the real roll angle into the roll rate command input signal. Then, the difference between the command roll rate and the real roll rate single goes through a PID loop, and the output signal from the PID loop serves as the motor RPM difference for propeller 1 and propeller 2. The RPM and thrust lookup tables discussed are used to convert the RPM difference into thrust difference. As a result, the thrust difference between propeller 1 and propeller 2 is used to solve the simplified roll dynamics model for roll angle and roll rate. The complete roll control diagram is shown in Figure 5(a).

**(a) Roll motion simulation model**

**(b) Pitch motion simulation model**

**(c) Yaw motion simulation model**

The decoupled pitch simulation model works similarly to that of the roll model (see Figure 5(b)). The key difference is that the output signal from the pitch rate PID loop goes into the servo motor which results in a corresponding elevator deflection. Based on the aerodynamic lookup table and the propeller slip-flow database, the aerodynamic forces due to the elevator deflection are used to solve the simplified dynamics model. The difference between the decoupled yaw and pitch simulation model is that the yaw rate is the command signal (see Figure 5(c)). The P loop for yaw angle only works under zero yaw command.

#### 3. Experimental Test

##### 3.1. Propeller Slip-Flow Measurement

The testing propeller is a APC nylon version. The testing rig with a 0.8 m height from the ground is located in an indoor open space to avoid external wind perturbation. SMART SENSOR AS856S anemometer with an accuracy of 0.1 m/s is employed to measure the time-averaged slip-flow speed. The time-averaged slip-flow speed measurement location is in the sympatric plane of the rotating disk with the interval of and . Figure 6 shows the propeller slip-flow speed testing rig. The detailed experimental procedure is as follows:

**(a) Experimental schematic**

**(b) Side view**

**(c) Rear view**

Step 1. Select the test point based on the x-r location coordinate, and put the anemometer to the corresponding test point

Step 2. Arm and start the propeller and measure and record the time-averaged propeller slip-flow speed at the test point with an RPM range of [6200 : 200 : 7000]

Step 3. Go back to Step 1, and move the anemometer to the next point in the x-r location, and then, repeat Step 2 until all the positions in the x-r location are covered.

##### 3.2. Indoor Hovering Test

As shown in Figure 7, the testing vehicle is protected by a light string during the test for safety purposes. The detailed specification can be found in Table 3. For data acquisition, the onboard sensors include barometers, gyroscope, GPS, and magnetometers. Table 4 shows the accuracy of these sensors. During the test, the height is auto-controlled by the height P-PID controller. A programmed pulse perturbation with an interval of 0.3 s is initialized for roll angle, pitch angle, and yaw rate, respectively. The oscillation characteristics are recorded onboard for further analysis.

#### 4. Results and Discussion

##### 4.1. Slip-Flow Distribution

Based on experimental tests, the time-averaged slip-flow speed in the and space at a fixed is shown in Figure 8. The maximum average slip-flow speed is larger than 18 m/s, which is close to the cruise speed of Egretta in fixed-wing mode. Large slip-flow speed concentrates in the range between 0 and 0.25 region. At a fixed , the slip-flow speed decreases as further expands, up to ; the slip-flow speed becomes zero at the range of investigated. Besides, as increases, the slip-flow speed remains a relatively constant value and then starts to decrease. As shown in Figure 8(b), the slip-flow speed increases linearly as RPM increases in the range of RPM investigated regardless of the location. Together with the experimental aerodynamic data of NACA0015 by [29], these experimental data of the slip-flow speed distribution are employed to develop the decoupled hovering stability models. As the aerodynamic and slip-flow speed data have taken air viscosity, flow separation, and flow speed in space distribution into consideration, the hovering stability model based on these data is expected to be more realistic.

**(a)**

**(b)**

##### 4.2. Hovering Stability and Dynamic Characteristics

For the pitch mode dynamic response, a pitch angle command with a 0.3-s impulse perturbation is initiated, where the set point refers to the input signal to the UAV system. As shown in Figure 9, the pitch angle command results in a saturated pitch rate command (). Due to the physical pitch control authority limitation, the real pitch angle only reaches 60 deg, and the real pitch rate is saturated at 80 deg/s. Furthermore, the pitch angle lagging time is around 0.3 s. Large aerodynamic damping in pitch mode is one reason for the lagging. When the impulse is finished, without oscillation, the pitch angle returns to the equilibrium state. The simulation results from the decoupled pitch model capture the dynamic responses of the pitch angle and pitch rate in both trend and magnitude, with a goodness of fit larger than 93.44%, which is calculated with the goodness-of-fit function in MATLAB.

As shown in Figure 10, the dynamic responses of the roll mode indicate oscillations with an around 1-second oscillation period. Two reasons will result in the oscillations. One is the smaller aerodynamic damping in roll motion. The other reason is due to the imperfect tuned PID parameters. A large discrepancy can be observed between the command signal and the dynamic response for both roll angle and roll rate. One reason is due to the short impulse command signal (0.3 seconds). Another reason is due to the limitation of the UAV control power in the roll motion. The dynamic responses are not able to follow the command signal exactly even in the situation of actuator situations. Nevertheless, similar to the pitch model simulation results, the decoupled roll model also results in good agreement with the hovering flight test response for both roll angle and roll rate, with a goodness of fit larger than 96.51%. For the decoupled yaw mode, the simulation yaw rate response also agrees well with the real flight test results (see Figure 11). The goodness of fit for yaw rate is smaller as compared to that of pitch and roll rate. The possible reason is that the cross-flow and unsteady aerodynamic delay effects are stronger in the yaw dynamics responses.

Besides the hovering stability models, it is also very important to develop the stability models for other flight phases such as vertical takeoff, front transition, fixed-wing cruise, back transition, and vertical landing. The stability models for the entire flight phase require extensive database including propellers and aerodynamics, which are in progress.

#### 5. Conclusions

Realizing the importance of hovering stability for the practical applications of VTOL flight vehicles, decoupled hovering stability models in roll, pitch, and yaw mode for a novel tail-sitter UAV are developed, respectively. To have more realistic stability models, experimental propeller slip-flow and elevon aerodynamic data are used. Besides, indoor hovering dynamic responses tests of the Egretta UAV are carried out to validate the stability model results. After extensive work, the decoupled hovering stability models demonstrate their capability of capturing the hovering dynamic response of Egretta UAV in both trend and magnitude with the goodness of fit more than 79% (more than 93% for pitch and roll motion). These validated models will be very useful and effective to perform PID optimization. Furthermore, as the experimental slip-flow data are not restricted to a specific UAV model, the hovering stability models are not only capable of analyzing hovering stability, but also be able to perform conceptual stability design.

#### Data Availability

All data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The authors disclose receipt of the following financial support for the research of this paper: This study was supported by the National Natural Science Foundation of China (Grant No. 92152107) and the Science and Technology Innovation Committee of Shenzhen (Grants No. JCYJ20180504165704491).

#### Supplementary Materials

(1) Hovering test video. Two test videos show the stability controller flight test of the Egretta tail-sitter VTOL UAV, which are available at https://youtu.be/rv3LkAySExk and https://youtu.be/xQp5vD8zHQU. (2) Full phase video. There is a short but complete video shows the flight status of the Egretta, including takeoff, transition, cruise flight, wind resistance, landing, and other working conditions, which is available at https://youtu.be/mnnj9KuEMPU.* (Supplementary Materials)*