Vector ducted propeller aerodynamic modeling and verification testing

The tail-sitter aircraft has the advantages of both high-speed cruise and capability of the take-off and land vertically, high-performance control laws usually rely on the accurate flight mechanics model, while its flight aerodynamic modeling is a complex problem. This paper analytically develops an aerodynamic model of its propelling system based on first principles, including three components of a propeller, a vector duct, and eight slipstream vanes. This model can give the propeller’s induced velocity, power, thrust, and moment, also considers the augmented effect of ducted, and the lift and drag forces of vanes. Moreover, a propelling system test bed is built to measure the forces and moments along the body frame, which is composed of the tail-sitter prototype, power, communication link, upper system, and servo drive device. In addition, experimental results show the effectiveness of the model in estimating the thrust and torque with different angles of control vanes, the measured thrust is consistent with the calculated values, and the steering performance of each vane is further examined. The moments exerted on the two axes are linear with the vanes angles, and the symmetry is also well, which is convenient for flight controller design for this type of tail-sitter aircraft.


Introduction
In recent years, unmanned aircraft vehicles (UAVs) have been developing rapidly due to the capabilities of executing tasks in remote and complex environments.The vertical take-off and landing (VTOL) aircraft has comprehensive performance in cruise and hover among various UAVs, especially in scouting, mapping, and emergency rescue applications.As a type of VTOL, tail-sitter UAV has the benefits of high-speed cruise, which satisfies higher flight envelope requirements.Various tail-sitter prototypes thus are developed to explore the integrated design and control technology.For instance, R. Hugh developed a "T-wing" UAV and conducted tests in different flight modes [1].A "V-bat" prototype UAV with a ducted propeller was designed, and flight verification was also tested [2].The advanced flight control technique is the key point for this type of UAV to achieve high performance, especially in the transition stage.In this stage, system dynamics shows strong nonlinear characteristics, and the aerodynamics analysis is a challenging problem.Several controllers have been proposed to cope with this situation.An anti-disturbance control method for tail-sitter UAVs was provided in [3][4][5].Lyu et al. [6] proposed an adaptive controller for a tail-sitter with two rotors.In general, independent control laws are designed to hover and level flight by switching the two modes in a simple logic.In [7], an integrated controller consisting of an observer and an adaptive law was presented which can adapt the full envelope.Highperformance control laws usually rely on the accurate flight mechanics model, therefore, researchers always dedicated to creating a high-fidelity dynamic model of VTOL aircraft.The way of identification is often used on this type of aircraft, for instance, Sun et al. [8] built a digital flight model for the "Caipirinha" tail-sitter based on wind tunnel data, and model accuracy had also been verified.This paper aims to explore the aerodynamic characteristics of the tail-sitter's propelling system, the rest is arranged as follows.In the next section, the aerodynamic model is given based on the first principle including the propeller, duct, and control vanes.The third section introduces the prototype test bed.The truest test is also conducted considering different angles of control vanes.Finally, a brief conclusion is given.

Propeller force
There are many ways to calculate the propeller thrust, the following thrust model is introduced and the duct is taken into account independently [2,9].The velocity at the disk is given as Where  ,  are propeller's angular velocity and radius respectively,  is the twist at the three quarters of blade,    ,  ,  is the body speed measured along three body axis, and    ,  ,  is corresponding wind speed.Therefore, the thrust generated by propeller can be calculated by Where  is air density,  denotes the lift curve slope,  ,  is the number and chord of the blade. denotes the thrust augmentation factor result from the duct, and  ∈ 0 1 .The  represents the induced velocity, and is given by represents the far-field velocity.It is worth noting that there is a lack of explicit expression of thrust and induced velocity, which are usually found by numerical solution iteratively.In general, the convergence of solving process is fast and can enter a lowest error bound within about 10 steps.
After obtaining the thrust, induced velocity and angular velocity, the induced power is given by Therefore, the reaction moment generated by the propeller can be represented as   / .

Ducted fan
The ducted fan can enlarge the thrust of propelling system, and also generate significant drag.The airspeed magnitude at the top of duct is given by Where  is the relative airspeed at the mouth of duct, which is represented as ,  are the body angular rates along the Y axis and Z axis, and  is the length between the mouth of the duct and the gravity center of the whole body.Let us define the ducted fan's attack angle  and sideslip  as follows: The momentum drag generated by duct is given by [10]     cos    sin The  is the inner radius of the duct.Naturally, the moment along the body axis is calculated as

Slipstream vanes
The body frame is described as a right-handed coordinate system.Each slipstream vane deflection will generate the lift and drag forces as Where  is the actual deflection angle of the i-th slipstream vane, which is deemed positive in a clockwise rotation.   is the airflow velocity over the corresponding slipstream vane,  is the area of vane,  and  is the coefficient of vane's drag and lift respectively.Then, the total drag force aligned with the Z-axis is given as The forces aligned with X and Y axis are The moment exerted on the propelling system body is expressed as L vane and d vane are the lengths from the center of duct, and center of gravity of the body to the vane's pressure center respectively.

Thrust test
In this section, the propelling system test bed is built to measure the forces and moments along the body frame, as presented in Figure 1, it is composed of the tail-sitter prototype, power, communication link, upper system, and servo drive device.
Firstly, we consider that there is no slipstream vane deflection, as shown in Figure 2, and it is observed that the measured thrust values are consistent with that calculated by the model, when the propeller's angular velocity reaches about 6000 rpm, the thrust measured is a little over 800 N. Steering performance of each vane is further examined, Figure 3 shows the measured moments generated by the first and second vanes at 50% and 55% throttle rate of engine.
The measured moment values along the X and Y axis are proportionate to the vane's deflection angle roughly, while is not perfectly symmetrical in opposite rotational direction for the Vane 2, it results from the nonlinear and coupling aerodynamic characteristics of vanes.Similarly, force test experiments further validate the handling performance of the 7 th and 8 th control vanes in Figure 4, it can be seen that the moments exerted on the two axes are linear with the vane's angles, and the symmetry is also well,

Conclusion
This paper analytically develops an aerodynamic model for a type of vector ducted propeller, the components of the propeller, ducted fan, and slipstream vanes are considered.Then, a prototype with a test system is constructed to evaluate the force and moment performance, the measured results verify the effectiveness of the model in estimating the thrust, and the handling capability test of the propelling system also shows a satisfactory result.