Influence of Wall Effect on Comprehensive Controllability for Ducted Fan Aircraft

Tianfu Ai， Changle Xiang， Bin Xu,N， Wei Fan and Yibo Zhang

（1. Vehicle Research Center, School of Mechanical Engineering, Beijing Institute of Technology,Beijing 100081, China；2. Beijing Institute of Technology Chongqing Innovation Center, Chongqing 401120, China）

Abstract: A novel coaxial ducted fan structure aircraft is proposed to enable the aircraft near ver?tical walls at high altitudes. The state space equation of the system can be obtained by correlation deduction and identification of the whole prototype model. Based on the duct test bench experi?ment and computational fluid dynamics (CFD) simulation analysis, the expressions between the different distances d WE from the rotor center of the prototype to the wall and the thrust, reaction torque, and tilting moment of the system under hovering conditions are obtained. The influence of the wall effect of the prototype is incorporated into the system model to analyze the relationship between distance d WE and the comprehensive controllability of the system. The results show that the system comprehensive controllability vector of other channels changes little with the decrease of the distance d WE, and only the controllability vector of the rolling channel increases significantly. At the same time, the tilting moment also increases significantly, which strengthens the tendency of the prototype to tilt towards the wall.

Key words: coaxial ducted fan aircraft；hovering condition；wall effect；tilting moment；compre?hensive controllability analysis

Research on unmanned aerial vehicle (UAV)is a hotspot in today’s research. The rapid devel?opment of autonomous flight control technology makes UAVs easier and more reliable to operate;furthermore, the superior performance of UAVs makes them suitable to support some advanced research. Currently, the mainstream form of most UAVs structures is open?rotor, which has some advantages and practicalities. However, with the continuous improvement of performance require?ments and the increasingly demanding working conditions and environments, UAVs cannot meet the requirements of various challenges. For ex?ample, most open rotor structure UAVs gener?ally work in an open environment, so they are not competent for high?altitude operations near vertical walls.

With the development of artificial intelli?gence and UAV technology, it is the develop?mental trend in the future to use UAVs and re?lated equipment, instead of humans, to perform air operations tasks. Due to its characteristic ad?vantages, the ducted structure UAV has attrac?ted extensive attention in the academic and in?dustrial fields. Ducted structure UAVs can work in more dangerous, complex, and unknown envir?onments. The existence of the duct can prevent direct contact between the rotor blade and the outside. In case of an accident, the blade will not directly eject toward the human, which also makes its safety performance better. In addition,because of the air flow at the lip, a duct can gen?erate more thrust than an open?rotor aircraft with the same blade size[1?2]. At the same time,with the same blade size, ducted structure UAVs can carry more loads than open?rotor UAVs,making it possible to carry different tools for dif?ferent operations. Therefore, a ducted structure UAV is more suitable than an open structure UAV in a complex environment. All the advant?ages of the ducted structure make the study of ducted fan UAVs have great value[3?4].

Most of the previous academic research on ducted and open UAVs were conducted in open and unconstrained conditions. For example, Fan studied the autonomous control of UAVs in an open environment and proposed an autonomous control method for engineering[5]. Du et al. stud?ied the aerodynamic performance of ducted struc?ture UAVs in an open environment and ana?lyzed the influence of different ducted layout parameters on the aerodynamic characteristics of the system[6]. Fan et al. synthesized previous re?search and analyzed several key technical issues in the formation of flight control in an open en?vironment[7].

The aerodynamic performance of UAVs hov?ering in an unconstrained environment has been widely studied[8?10], but less research has been conducted in a constrained environment[11?12]. Ac?cording to some related papers, there are a few studies on the proximity of UAVs to wall sur?faces or obstacles[11,13], most of which generalize the aerodynamic effects introduced by the dis?tance between the UAVs and the wall as an ex?ternal disturbance. For example, Ref. [14] used an adaptive neural network algorithm to estim?ate the disturbance caused by approaching the wall and compensate for it. Ref. [15] modeled the single?ducted UAV in its entirety and studied the UAV’s proximity to the wall by using a correlated feed?back algorithm to obtain a good tracking per?formance of the system. These methods can off?set the adverse effects of the wall effect caused by the UAVs approaching the wall to a certain extent. However, they do not reveal that the sys?tem controllability is significantly affected by dis?tance differences when the prototype is close to the wall. Therefore, this paper adopts a new re?search method. In this paper, we consider the prototype and the wall as a whole system, and study the controllability of the different whole systems at different distances between the proto?type and the wall. Through the test bench exper?iment and CFD simulation, this paper studies the influence of the distance change on aerody?namic parameters such as the resultant force and resultant moment of the system when the UAV is near the wall at different distances and gener?ates the wall effect. Otherwise, the variation of aerodynamic parameters is taken into account in the model to analyze the relationship between the distance of UAVs to the wall and the integ?rated controllability of the system. At the same time, by analyzing the comprehensive controllab?ility of the system, we can find out some short?comings of the prototype, so as to further optim?ize the structure of the prototype for future re?search, and provide some reference information for a controller design with better performance.Moreover, the study of this paper can provide new ideas for the related study of hovering opera?tion of the prototype near the object at high alti?tudes in the future.

1 System Model Description

In previous studies, we have done a series of related research on ducted fan aerial robots[16?19].With the continued progress of the research, the structure of the prototype is changing constantly.Going through continuous adjustments to its structure, the latest prototype is shown in Fig.1.

Fig. 1 State and input variables of the prototype

Similar to multi?rotor UAVs, the pitch and yaw moments of the aerial robot are generated by the speed difference between the front and rear ducted fan rotors. The rolling moment of the prototype is generated by the control surface at the bottom. In addition, the ducted fan UAVs presented in this paper can carry manipulators and other tools for different tasks to perform hovering observations and operations close to an object in a confined environment.

According to the mechanical characteristics of the duct, from the blade element and mo?mentum theory, the relevant expressions of the resultant force Fband the resultant moment Mbof the front and rear ducts can be obtained, and then the Newton?Euler equation can be used to obtain the rigid dynamic equation of the system in the body coordinate system. The derivation process of rigid body dynamics is similar to Ref. [17].Due to the limited space, the derivation process is not listed in this paper.

Ignoring external disturbances such as gusts,the nonlinear dynamic equations of the proto?type is

Fig. 2 is the prototype model structure. ucol,ulat, ulon, and upedare the normalized control in?puts for altitude, roll, pitch, and yaw channels.After inputting these values in the actuator, it will output the rotor speed and rudder angle of the duct module. The resultant force and torque of the duct module are further output to the kin?ematics equation of the whole machine, and the final output vector of the whole machine is ob?tained.

Fig. 2 Model structure of the prototype

To analyze the controllability of the system,we need to know the state space equation of the system. For Eq.(1), based on the small disturb?ance assumption and the nonlinear dynamic sys?tem linearization theory, the nonlinear system can be simplified into a linear system at the hov?ering equilibrium point of the system. According to the prototype identification method in Refs. [20?22], we can get the state space equa?tion of the nominal system.

2 Analysis of the Influence of Wall Effect on Aerodynamics

Just as Fig.3 shown, a wall effect will occur when the UAV is near the vertical wall. The re?search in this section is mainly about the impact of the aerodynamic effect of the prototype close to the wall. When the prototype is flying close to the wall surface, its operation tool is in the middle of the body, so the Y?axis direction or the negative Y?axis direction of the duct is facing the wall surface.

Fig. 3 Schematic diagram of prototype wall effect

When the wall effect occurs, the wall pre?vents the free movement of gas, slowing down air flow at the duct lip close to the wall, resulting in the increase of pressure in the boosting area and the decrease of negative pressure, as shown in Fig.4. The static pressure increases, resulting in the imbalance of the total lift force of the duct,and the force at FAis greater than that at FB, as shown in Fig.5. At the same time, the unbal?anced force will produce a tilting moment CMy,which makes the system incline to one side of the wall, called the tilting moment.

Fig. 4 Pressure cloud diagram when the duct is close to the wall

Fig. 5 shows that the velocity of the duct flow field does not change too much, nor does turbulence occur when it is close to the wall.Therefore, it is assumed that the inlet of the duct is still uniform. In this section, through the test bench experiment and the CFD numerical simu?lation, we analyze the variation diagrams of aero?dynamic coefficients such as the total lift coeffi?cient, torque coefficient, and tilting moment with the ratio change of the distance dWEfrom the ro?tor center to the wall and the rotor radius R un?der hovering conditions.

Fig. 5 Velocity cloud diagram when the duct is close to the wall

In Fig.6, it is observed that the test bench is made of aluminum alloy and the total weight is 30 kg. As a result of the weight, when the bench is in normal operation, it will be almost free from external interference, which can effectively en?sure the accuracy of the data obtained by the test. The force sensor in Fig.6 is used to measure the lift of the ducted fan, and the two?compon?ent balance is used to measure the lift and torque of the rotors. Because the experimental bench lacks moment equipment for measuring the X and Y directions of the duct, the tilting moment cannot be calculated by the bench, so CFD simu?lation data is needed. Figs. 7?9 show the proces?sed data, in which the dWEratios are 3.3, 3.0, 2.6,2.3, 2.1, 1.9, 1.7, 1.5, and 1.4, and these data connections are plotted as linkage diagrams. The simulation also shows that the wall effect has little effect on the X and Y directions compared with lift and torque in the Z?direction, so the change of the force and torque in the X and Y directions are ignored.

Fig. 6 Test bench for the aerodynamic performance of wall ef?fect

Fig. 7 Connection diagram of total lift coefficient at different distance ratios

Fig. 8 Connection diagram of reaction torque coefficient at dif?ferent distance ratios

The dimensionless parameters such as lift coefficient, torque coefficient, and tilting mo?ment coefficient[23?24]can be presented as

Fig. 9 Connection diagram of tilting moment coefficient at dif?ferent distance ratios

In Eq.(4), T is the lift, and Q is the torque.The air density is ρ=1.225 kg/m3, A=πR2is the rotor area, and ω=wπ/30 is the rotor rotation speed, w is the angular velocity. In Fig. 7?Fig. 9,it can be found that when the ducted fan has the wall effect, its lift and reaction torque have a cer?tain relationship with the distance ratio. By fit?ting the data with the Matlab fitting toolbox, the fitting expressions of these relationships can be obtained.

In an unconstrained environment, the duc?ted fan does not produce wall effects. It is the same as the system modeling; since the proto?type is far away enough from the wall, the wall effect has little influence on it, which can be ig?nored and not taken into account in the model.In a constrained environment, the influence of wall effects should be taken into account in the model. This requires correction of lift and reac?tion torque in an unconstrained environment;that is, lift and torque need to be multiplied by a correction factor for force and torque. Meanwhile,the tilting moment, My, can also be expressed by the distance ratio and added separately in the model. By incorporating the factors, such as total system lift, torque variation value, and tilting moment brought by the wall effect into the ini?tial system model, the dynamic model structure with wall effect can be obtained in Fig. 10. In Fig. 10

α is the correction factor of the total lift of the system with the wall effect. It is related to the distance ratio and is derived from the total lift ratio of the system under the conditions of different distances and the unconstrained envir?onment.

Fig. 10 Model structure of prototype with wall effect

Similar to α, β is the correction factor of the reaction torque, derived from the reaction torque ratio of the system under the conditions of differ?ent distances and the unconstrained environ?ment. it is derived from the ratio of the system reaction torque with different distances and the unconstrained environment. Myis also related to the distance ratio, and H is the fitting expres?sion of the distance ratio and the tilting moment coefficient.

Similar to the analysis process without the wall effect, through linearization and related identification, the system state space equations with the wall effects can also be obtained. Com?pared with the nominal system, the parameters in the state space equations, matrices A and B,have changed with different distance ratios.

3 Analysis of System Comprehensive Controllability

Controllability is an inherent characteristic of the system; it has nothing to do with the con?troller, but also reflects the difficulty of control[25]. The input?output controllability of the system is independent with the controller design,only related to the physical structure of the sys?tem, and can reflect the inherent characteristics of the system. The typical method is obtained by calculating the controllability matrix.

rank[B,AB,··· ,An?1B]=n, where n is di?mension.

Through verification, we find that our nom?inal system is controllable. However, this meth?od does not show how the system controllability will change as the prototype?to?wall distance changes. By singular value decomposition(SVD), the frequency domain response of the system is quantified, which is known as compre?hensive controllability analysis.

The transfer function of the open?loop sys?tem can be written as

The control input vector u of the open?loop system can be represented by an orthogonal basis composed by various control channels.

By introducing the nominal model of the system into Eq.(17), a comprehensive control?lable vector diagram of the nominal model can be made, as shown in Fig. 11.

Fig. 11 Vector graph of comprehensive controllability of nominal model

As we can see from Fig. 11, when there is no wall effect, most of the comprehensive controllab?ility vector components in the low?frequency state are greater than 1, which shows that the corresponding system state has better controllab?ility[26?28].

When dWE/R=2.2, 1.7, 1.4, and other differ?ent ratios, the model of the open?loop system changes differently, which is similar to the ana?lysis without the wall effect. The comprehensive controllability vector diagrams of the nominal model at different distance ratios can be ob?tained.

By analyzing the system comprehensive con?trollability vector diagram at different distance ratios, it can be seen that the comprehensive con?trollability vector components of Fig. 12–Fig. 14 are greater than 1 in most low frequency states,indicating that the system state controllability is better.

Fig. 12 Controllability vector diagram of the system at distance ratio d WE/R =2.2

Fig. 13 Controllability vector diagram of the system at distance ratio d WE/R =1.7

Fig. 14 Controllability vector diagram of the system at distance ratio d WE /R =1.4

For the attitude vector ( p,q,r,?,θ,φ), the roll angular velocity p and roll controllability vector of the roll channel are very small, and it is obviously smaller than the pitch and yaw chan?nels at low frequencies, indicating that the rud?der surface should be optimized to increase the steering torque in the roll direction, so as to im?prove the controllability of the rolling channel.Compared with the nominal system, when dWE/R =2.2, the controllability of the lateral speed is in?creased by about 2%; when dWE/R =1.7, it is in?creased by about 4%; when dWE/R = 1.4, it is in?creased by about 9%.

At low frequencies, the controllability of the yaw channel is better, while at high frequencies,it drops sharply, which is caused by the large yaw direction inertia and insufficient control torque. The Z?axis rotational inertia can be ap?propriately reduced by optimizing the body structure. The influence of the wall effect with distance is mainly reflected in the rolling chan?nel of the system, with little influence in other channels. The reason for this phenomenon is that there are few changes in the force and torque of other channels, and the rolling channel adds an additional tilting moment.

4 Conclusions

This paper focuses on a coaxial ducted fan UAV that can operate close to the wall at high altitudes. Through the test bench experiment and CFD simulation analysis, this paper mainly stud?ies the aerodynamic performance changes of the prototype when the aircraft is affected by the wall effect at different distances from the wall,and then analyzes the comprehensive controllab?ility changes of the system. The conclusions are as follows.

① When the prototype produces a wall ef?fect, its flow field changes little; therefore, it can still be considered that the inflow is uniform. The influence of the wall effect is mainly reflected in the change of the total lift, reaction torque, and tilting moment of the system.

② Compared with the nominal system, the system with the wall effect needs to consider the force and moment changes caused by the wall ef?fect. The lift and reaction torque in an unres?trained environment are corrected by using a cor?rection factor, and adding a tilting moment around the X?axis that inclines to the wall. The changes of these factors can be fitted into the ex?pression between the distance from the rotor cen?ter to the wall and the rotor radius.

③ When the distance ratio dWE/R becomes smaller, the controllability vector of other chan?nels of the system changes little, the controllabil?ity vector of the rolling channel increases to some extent, and the wall effect has a limited influ?ence on the controllability of the prototype. At the same time, the tilting moment caused by the wall effect will also increase greatly. When the distance ratio dWE/R =1.4, the tilting moment is about 1/3 of the reaction torque, which will make the prototype have a strong wall tilting tend?ency and affect the safety performance of the prototype during flight. To avoid UAVs flying as close to the wall as possible, or when the aircraft is close to the wall, relevant controls should be used to eliminate the wall effect.