# Identification of Lateral-Directional Aerodynamic Parameters for Aircraft Based on a Wind Tunnel Virtual Flight Test

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{a}, δ

_{r}) meet the identifiable requirements, to determine the frequency and amplitude range of the signal [17,18]; and the accuracy of the identification results of various aerodynamic derivatives can be guaranteed by designing complex orthogonal multi-sine optimization excitation signals [19,20]. These two methods require a large amount of calculation and complicated steps, and they are not suitable for the excitation signal design of aircraft with closed-loop flight control law, because the flight control law will have a feedback effect on the initial signal input, resulting in a large difference between the final control surface deflection and the initial input signal.

## 2. Flight Dynamics Model for Wind Tunnel Virtual Flight Test

_{e}during pitch control, and the reverse deflection is used as the rudder δ

_{r}during yaw control, thus decoupling the operation between longitudinal and directional. Define the V-tail control surface downwards deviation as positive, upwards deviation as negative, and V-tail surface angles of the left and right sides as δ

_{V−L}and δ

_{V−R,}respectively. The equivalent elevator and rudder angles are defined as δ

_{e}= 0.5 × (δ

_{V−L}+ δ

_{V−R}) and δ

_{r}= 0.5 × (δ

_{V−L}− δ

_{V−R}).

#### 2.1. Flight Dynamics Modelling

_{b}z

_{b}plane under the body coordinate system, i.e., I

_{xy}= I

_{yz}= 0. The lateral-directional rotation dynamics and kinematic equations of the test model around the centre of mass are the same as free flight, as shown in Equation (1) [23].

_{x}, I

_{y}, and I

_{z}are the inertia of the three axes; I

_{xz}is the cross inertia; p, q, and r are the roll rate, pitch rate and yaw rate, respectively; and L and N are the rolling and yawing moments of the aircraft, respectively.

_{y−b}is the combined external force on the y-axis of the aircraft under the body coordinate system. Since the model has no translational motion, the combined external force F

_{y−b}= 0.

_{bg}is the rotation matrix from the ground coordinate system to the body coordinate system; F

_{x}, F

_{y}, and F

_{z}are the forces of the support rod on the model in the x-, y-, and z-axis directions, respectively; and X, Y, and Z are the aerodynamic forces on the model in the body coordinate system, respectively.

_{y−b}of the model in the body coordinate system derived from Equation (3) into Equation (2), the centre-of-mass dynamics equation of the model is obtained as shown in Equation (4).

_{g}at the centre of mass of the model, with Ox

_{g}pointing in the direction of wind velocity, Oz

_{g}vertically downwards, and Oy

_{g}perpendicular to the Ox

_{g}z

_{g}plane to the right. In the wind tunnel test, assuming that the direction of the incoming flow is constant, the x-axis of the air coordinate system, ground coordinate system, and flight path coordinate system all coincide. Thus, u, v, and w can be solved using the attitude angle and the coordinate system transformation matrix, which, in turn, directly solves for the airflow angles α and β, as shown in Equation (6).

#### 2.2. Linearization and Decoupling for Equations of Motion

_{y−b}of the model in Equation (3) be 0, the expression of the force acting on the model by the support device F

_{y}can be obtained:

_{*}= 0°, and the result of the linearization is shown in Equation (9).

_{y}can be expressed in the form of a linear superposition of aerodynamic forces and gravity. The linear expression for the force ΔF

_{y}acting on the model by the support device can be obtained from Equation (9) as:

_{y−β}, F

_{y−p}, F

_{y−r}, F

_{y−ϕ}, and F

_{y−δr}denote the derivatives of the side force generated by the support device in the body coordinate system, with respect to the sideslip angle, roll rate, yaw rate, roll angle, and rudder deflection, respectively; the specific expressions are shown in Equation (11).

_{y}, shown in Equations (7) and (10), are substituted into these linearised equations of motion to obtain the linearization results for the test model at the base state of V

_{*}and α

_{*}, as shown in Equation (12):

_{1}and B

_{1}represent the free flight lateral-directional stability matrix and the control matrix of the control surface, respectively, and A

_{2}represents the matrix of the additional motion of the model caused by the support force. In addition, the deflection of the V-tail control surface generates side force and the constraints of the support device have an impact on the control matrix, as shown in B

_{2}. The expressions for each lateral-directional dynamic derivative in Equation (12) are shown in Table 2.

_{2}and B

_{2}matrices in Equation (12) need to be eliminated. For ease of description, Equation (12) is rewritten into the form shown in Equation (13). The mathematical expressions in the box are the differences in the lateral-directional equations of motion between the wind tunnel virtual flight and the free flight.

## 3. Design Method for Excitation Signals Based on Frequency Domain Analysis

#### 3.1. Selection of Excitation Signal Type

#### 3.2. Design of Excitation Signal Parameters

_{l}, the high frequency limit ω

_{h}, and the signal amplitude |A|. In this paper, a method of excitation signal parameter design based on frequency domain analysis is proposed to design the frequency band and amplitude magnitude of the excitation signal. Bode diagram analysis of the equations of motion of the aircraft is carried out to observe the amplitude response of each parameter to be identified with the change in excitation signal frequency. The frequency range of the excitation signal is determined by ensuring that the amplitude response of the aerodynamic forces or moments caused by each motion variable (β, p, r, ϕ), as well as the manipulation variables (δ

_{a}, δ

_{r}), is sufficiently large. Finally, the amplitude of the excitation signal is adjusted, so that the signal has high energy in the designed frequency band.

_{β}Δβ(ω)/δ

_{r}|, |Y

_{r}Δr(ω)/δ

_{r}|, and |Y

_{δr}Δδ

_{r}(ω)/δ

_{r}| of Y

_{β}Δβ, Y

_{r}Δr, and Y

_{δr}Δδ

_{r}with the variation of rudder frequency according to the side force equation.

_{r}should not differ significantly. The time domain curves of each yawing moment component, with different frequencies of rudder signal inputs, are given in Figure 10.

_{β}Δβ, Y

_{r}Δr, and Y

_{δr}Δδ

_{r}in the side force observation equation should not differ significantly.

_{i}; ${\left|A\right|}_{{\stackrel{-}{L}}_{\beta}\Delta \beta}$, ${\left|A\right|}_{{\stackrel{-}{L}}_{p}\Delta p}$, ${\left|A\right|}_{{\stackrel{-}{L}}_{r}\Delta r}$, and ${\left|A\right|}_{{\stackrel{-}{L}}_{\delta r}\Delta \delta r}$ are the response amplitudes of each component of the rolling moment; and ${\left|A\right|}_{{Y}_{\beta}\Delta \beta}$, ${\left|A\right|}_{{Y}_{\delta r}\Delta \delta r}$, and ${\left|A\right|}_{{Y}_{r}\Delta r}$ are the response amplitudes of each side force component. ${\left|A\right|}_{Cn-{\omega}_{i}}$, ${\left|A\right|}_{Cl-{\omega}_{i}}$, and ${\left|A\right|}_{CY-{\omega}_{i}}$ are 1/10 of the total response amplitudes of the yawing moment, rolling moment, and side force at a rudder signal frequency of ω

_{i}, respectively.

_{δr}

_{·Cn}$\in $[ω

_{1}~ω

_{2}] in Figure 9, i.e., the amplitude responses of all yawing moment components are above the identifiable boundary, all yawing moment derivatives are identifiable. Therefore, the frequency band of the rudder excitation signal determined by the yawing moment derivatives is in the range of ω

_{δr}

_{·Cn}. The same method can be used to obtain the frequency band range of the rudder excitation signal determined by rolling moment, as well as side force derivatives, such as ω

_{δr}

_{·Cl}and ω

_{δr}

_{·CY}. To ensure that all lateral-directional aerodynamic derivatives are identifiable, it is necessary to take the intersection of ω

_{δr}

_{·Cn}, ω

_{δr}

_{·Cl}, and ω

_{δr}

_{·CY}to obtain the frequency band of the rudder excitation signal ω

_{δr}.

_{δa}can be obtained.

_{δa}

_{·Cn}, ω

_{δa}

_{·Cl}, and ω

_{δa}

_{·CY}are the frequency band ranges of the aileron signals determined by the yawing moment, rolling moment, and side force derivatives, respectively.

## 4. Step-by-Step Identification Method for the Aerodynamic Parameters

#### 4.1. Identification Method of Side Force Derivatives

_{m}can be used as an observed variable to identify side force derivatives. The identification model of side force derivatives is shown in Equation (18).

_{m}is the side force of the whole test model directly measured by the force balance; Δβ, Δp, Δr, and Δδr are the amount of change in sideslip angle, roll rate, yaw rate, and V-tail control surface, which can be measured directly by the sensors. The parameter to be identified is Θ = [Y

_{β}, Y

_{p}, Y

_{r}, Y

_{δr}].

_{a}, δ

_{r}) of the aircraft; and v is the measurement noise matrix. In the wind tunnel test, the measurement accuracy of the strain balance is high, and v can be regarded as white noise with a mean value of 0, as shown in Equation (20):

_{β}, Y

_{p}, Y

_{r}, and Y

_{δr}are identified by the least square method and substituted into Table 2 to solve the aerodynamic derivatives C

_{yβ}, C

_{yp}, C

_{yr}, and C

_{yδr}.

#### 4.2. Identification Method of Rolling and Yawing Moment Derivatives

_{1}, c

_{2}, c

_{3}, c

_{4}, c

_{5}, and c

_{6}are shown in Equation (23). The lateral-directional moment derivatives to be identified are Θ = [L

_{β}, L

_{p}, L

_{r}, L

_{δa}, L

_{δr}, N

_{β}, N

_{p}, N

_{r}, N

_{δr}].

_{m}, r

_{m}, β

_{m}, and ϕ

_{m}measured by the experiment are chosen as the observation variables, so the observation equation is shown in Equation (24).

_{p}, v

_{r}, v

_{β}, and v

_{ϕ}are measurement noise. Aerodynamic derivatives of the test model can be obtained by identifying aerodynamic parameters of the nonlinear motion model shown in Equations (22)–(24). The most widely used method is the output error method based on maximum likelihood estimation [29]. General nonlinear dynamic equations can be expressed in the form shown in Equation (25):

_{a}, δ

_{a}); y is the output variable (p, r, β, ϕ); z is the observed variable (p

_{m}, r

_{m}, β

_{m}, ϕ

_{m}); f and g are the general nonlinear functions; and Θ is the parameter to be identified. Assume that the measurement noise v is Gaussian white noise with zero mean and a covariance matrix of R.

_{0}, the Gauss–Newton solution algorithm shown in Equation (27) is used to iterate to find the optimal parameters to be identified [29].

## 5. Verification of Identification Results Based on the Wind Tunnel Virtual Flight Test

#### 5.1. Facility for Wind Tunnel Virtual Flight Tests

_{cmd}, and the direction of the yaw axis is controlled by the sideslip angle command β

_{cmd}.

_{p}= 0.25, K

_{P−ϕ}= 2.18, K

_{I−ϕ}= 0.14, K

_{β}= 0.63, K

_{r}= 0.37, K

_{P−β}= 0.10, K

_{I−β}= 0.50, and K

_{ari}= 0.33.

#### 5.2. Design of the Excitation Signal

#### 5.2.1. Determination of the Rudder and Aileron Frequency Range

_{δr}

_{·Cn}= [1.1, 5.3] rad/s, ω

_{δr}

_{·Cl}= [0.7, 8.7] rad/s, and ω

_{δr}

_{·CY}= [0.8, 12.0] rad/s, respectively. Taking the intersection of ω

_{δr}

_{·Cn}, ω

_{δr}

_{·Cl}, and ω

_{δr}

_{·CY}, the band range of the rudder excitation signal is obtained as ω

_{δr}= ω

_{δr}

_{·Cn}$\cap $ω

_{δr}

_{·CY}$\cap $ω

_{δr}

_{·Cl}= [1.1, 5.3] rad/s. In the same way, the band range of the aileron excitation signal can be obtained as ω

_{δa}= ω

_{δa}

_{·Cn}$\cap $ω

_{δa}

_{·CY}$\cap $ω

_{δa}

_{·Cl}= [1.2, 5.2] rad/s.

_{δr}= [1.1, 5.3] rad/s and ω

_{δa}= [1.2, 5.2] rad/s, approximately between 0.5 and 2 times the Dutch roll modal frequency.

#### 5.2.2. Determination of the Signal Amplitude

_{δa}= ω

_{δr}= 1.5~5 rad/s in this experiment. To make the sideslip of the test model change within the range of ±2°, the amplitude of the aileron excitation signal is set to 6°, and that of the rudder excitation signal is set to 4°. The lateral-directional excitation signal is shown in Figure 17.

#### 5.3. Identification Results of Aerodynamic Derivatives

#### 5.3.1. Identification of Side Force Derivatives

_{e}= −5.6°. The outer ring of the flight control law is the instruction of 0° sideslip angle and 0° roll angle. The excitation signal is directly input on the V-tail control surface and aileron on both sides, and the side force Y

_{m}of the test model is measured in real time through the force balance.

_{m}, sideslip angle Δβ

_{m}, yaw rate Δr

_{m}, roll rate Δp

_{m}, and V-tail control surface deflection angle Δδ

_{r}of the model measured in the test are substituted into Equation (19), and the side force derivatives of the model are identified by the least squares method, shown in Equation (21). The identification results are shown in Table 4, with the first column of the table showing the aerodynamic derivatives measured in the traditional wind tunnel test. Comparing the identification results with the conventional wind tunnel measurements, it can be seen that the identification results of side force derivatives are close to the reference values, and the deviations are within 10%.

#### 5.3.2. Identification of Rolling and Yawing Moment Derivatives

_{i}is the experimental measurement, $\stackrel{-}{z}$ is the mean value of the observed data, and y

_{i}is the theoretical output; a GOF value close to 1 indicates that the two sets of data fit well [33].

#### 5.4. Verification of the Modified Identification Model

_{1}and B

_{1}matrices in Equation (12) and eliminating the A

_{2}and B

_{2}matrices generated by the influence of the support device, a modified lateral-directional motion model for wind tunnel virtual flight can be constructed, as shown in Equation (29).

## 6. Conclusions

- (1)
- The lateral-directional flight dynamics equations of the wind tunnel virtual flight test model are established. By linearizing the equations of motion to describe the wind tunnel virtual flight test as a decoupled form of the effects of free flight aerodynamic forces and support forces on the model motion, the differences between the lateral-directional dynamics of wind tunnel virtual flight and free flight can be analysed more intuitively, thus establishing a model for the identification of aerodynamic parameters.
- (2)
- Based on the identifiable requirements of aerodynamic derivatives, the amplitude–frequency characteristics of the equations of motion are analysed to establish the type selection and parameter design method of the lateral-directional excitation signal. The frequency of the lateral-directional excitation signal needs to be between 0.5–2 times the frequency of the Dutch roll mode. Therefore, a suitable actuator needs to be selected for different flight states, so that the deflection rate of the control surface is fast enough. In addition, to identify the aerodynamic model at high angle of attack or high sideslip, it is necessary to design appropriate flight control law to ensure the stability of the test model.
- (3)
- In this paper, a step-by-step method for the identification of side force, rolling, and yawing moment derivatives is established. The identification of the side force derivatives can be achieved by measuring the aerodynamic force of the test model with a force balance. The differences between the identification results of the aerodynamic derivatives and the conventional wind tunnel measurements are within 10%. The lateral-directional motion response of the identified model is basically consistent with the wind tunnel virtual flight test data, and the GOF of all motion variables are greater than 0.95.
- (4)
- The modified identification model can well-simulate the lateral-directional motion of the conventional wind tunnel test model, and the goodness-of-fit is higher than 0.95, which verifies the correctness of the proposed method.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Traditional wind tunnel test. (

**a**) Static force test; (

**b**) Dynamic derivatives and large oscillation test.

**Figure 7.**Shape changes of two signal types in the time domain. (

**a**) The 3211-multipole square wave; (

**b**) Frequency sweep.

**Figure 8.**Energy distribution of two types of signals in the frequency domain. (

**a**) The 3211-multipole square wave; (

**b**) Frequency sweep.

**Figure 9.**Amplitude–frequency characteristic curve of the yawing moment component with the change in rudder frequency.

**Figure 10.**The time domain response of each yawing moment component with different rudder frequency inputs. (

**a**) ω = 0.5 rad/s; (

**b**) ω = 3 rad/s; (

**c**) ω = 10 rad/s.

**Figure 15.**Amplitude–frequency characteristics of lateral-directional aerodynamic parameters varying with rudder frequency. (

**a**) Side force; (

**b**) Yawing moment; (

**c**) Rolling moment.

**Figure 16.**Amplitude–frequency characteristics of lateral-directional aerodynamic parameters varying with aileron frequency. (

**a**) Side force; (

**b**) Yawing moment; (

**c**) Rolling moment.

**Figure 18.**Comparison of the simulation results for the side force observation model with experimental data.

**Figure 20.**Comparison of the identification model for wind tunnel virtual flight before and after correction with the free flight model.

Parameters | Description | Instruments |
---|---|---|

ϕ, θ, ψ | Roll angle, pitch angle, yaw angle | Inertial measurement unit |

p, q, r | Roll rate, pitch rate, yaw rate | Inertial measurement unit |

α, β | Angle of attack and sideslip | Numerical solution |

δ_{a}, δ_{r} | Aileron, rudder deflection | Rotary encoder |

F_{x}, F_{y}, F_{z} | Force of support device in the x-, y-, and z-axis directions (body coordinate system) | Strain gauge balance |

Side Force Derivatives | Yawing Moment Derivatives | Rolling Moment Derivatives |
---|---|---|

$\stackrel{-}{Y}$_{β} = $\frac{{Y}_{\beta}}{m{V}_{*}}$ = $\frac{{C}_{y\beta}{\stackrel{-}{q}}_{*}S}{m{V}_{*}}$ | ${N}_{\beta}$ = ${C}_{n\beta}{\stackrel{-}{q}}_{*}S$b | ${L}_{\beta}$ = ${C}_{l\beta}{\stackrel{-}{q}}_{*}S$b |

$\stackrel{-}{Y}$_{p} = $\frac{{Y}_{p}}{m{V}_{*}}$ = $\frac{{C}_{yp}{\stackrel{-}{q}}_{*}S}{m{V}_{*}}\frac{b}{2{V}_{*}}$ | ${N}_{p}$ = ${C}_{np}{\stackrel{-}{q}}_{*}S$b$\frac{b}{2{V}_{*}}$ | ${L}_{p}$ = ${C}_{lp}{\stackrel{-}{q}}_{*}S$b$\frac{b}{2{V}_{*}}$ |

$\stackrel{-}{Y}$_{r} = $\frac{{Y}_{r}}{m{V}_{*}}$ = $\frac{{C}_{yr}{\stackrel{-}{q}}_{*}S}{m{V}_{*}}\frac{b}{2{V}_{*}}$ | ${N}_{r}$ = ${C}_{nr}{\stackrel{-}{q}}_{*}S$b$\frac{b}{2{V}_{*}}$ | ${L}_{r}$ = ${C}_{lr}{\stackrel{-}{q}}_{*}S$b$\frac{b}{2{V}_{*}}$ |

$\stackrel{-}{Y}$_{δa} = $\frac{Y\delta a}{m{V}_{*}}$ = $\frac{{{C}_{y\delta a}\stackrel{-}{q}}_{*}S}{m{V}_{*}}$ | ${N}_{\delta a}$ = ${C}_{n\delta a}{\stackrel{-}{q}}_{*}S$b | ${L}_{\delta a}$ = ${C}_{l\delta a}{\stackrel{-}{q}}_{*}S$b |

$\stackrel{-}{Y}$_{δr} = $\frac{Y\delta r}{m{V}_{*}}$ = $\frac{{{C}_{y\delta r}\stackrel{-}{q}}_{*}S}{m{V}_{*}}$ | ${N}_{\delta r}$ = ${C}_{n\delta r}{\stackrel{-}{q}}_{*}S$b | ${L}_{\delta r}$ = ${C}_{l\delta r}{\stackrel{-}{q}}_{*}S$b |

$\stackrel{-}{L}$_{I} = $\frac{{L}_{i}+({I}_{xz}/{I}_{z}){N}_{i}}{{I}_{x}-{I}_{xz}^{2}/{I}_{z}}$, $\stackrel{-}{N}$_{i} = $\frac{{N}_{i}+({I}_{xz}/{I}_{z}){L}_{i}}{{I}_{z}-{I}_{xz}^{2}/{I}_{x}}$, i∈(β, p, r, δ_{a}, δ_{r}) |

Parameters | Proportions | Full-Size Aircraft | Test Model |
---|---|---|---|

Wing span b (m) | 1/9 | 36 | 4.00 |

Mean aerodynamic chord c (m) | 1/9 | 10.41 | 1.16 |

Wing area S (m^{2}) | (1/9)^{2} | 241 | 2.98 |

Mass m (kg) | (1/9)^{3} | 49,149 | 67.42 |

Pitch moment of inertia I_{y} (kg·m^{2}) | (1/9)^{5} | 4,044,856 | 68.50 |

Yaw moment of inertia I_{z} (kg·m^{2}) | (1/9)^{5} | 5,166,787 | 87.50 |

Roll moment of inertia I_{x} (kg·m^{2}) | (1/9)^{5} | 1,210,504 | 20.50 |

Product of inertia I_{xz} (kg·m^{2}) | (1/9)^{5} | 171,242 | 2.90 |

Parameters | Traditional Wind Tunnel Measurements | $\mathbf{Identification}\mathbf{Results}\widehat{\mathsf{\Theta}}$ | Deviation (%) | Standard Deviation $\frac{\mathbf{s}\left(\widehat{\mathsf{\Theta}}\right)}{\left|\widehat{\mathsf{\Theta}}\right|}\%$ | |
---|---|---|---|---|---|

Side force derivatives | C_{y}_{β} | −0.400 | −0.391 | −2.25 | 2.60 |

C_{y}_{δr} | 0.121 | 0.111 | −8.26 | 6.55 | |

C_{y}_{r} | 0.719 | 0.689 | −4.17 | 5.89 | |

Rolling moment derivatives | C_{l}_{β} | −0.083 | −0.090 | 8.43 | 3.55 |

C_{l}_{p} | −0.085 | −0.080 | −5.88 | 4.36 | |

C_{l}_{r} | 0.229 | 0.237 | 3.49 | 1.59 | |

C_{l}_{δa} | −0.031 | −0.029 | −6.45 | 6.47 | |

C_{l}_{δr} | 0.020 | 0.021 | 5.00 | 7.23 | |

Yawing moment derivatives | C_{n}_{β} | 0.073 | 0.070 | −4.11 | 1.23 |

C_{n}_{p} | −0.0102 | −0.010 | −1.96 | 8.25 | |

C_{n}_{r} | −0.066 | −0.063 | −4.55 | 4.05 | |

C_{n}_{δr} | −0.062 | −0.060 | −3.23 | 3.55 |

GOF | p | r | β | ϕ |
---|---|---|---|---|

Identification model and wind tunnel virtual flight test | 0.98 | 0.99 | 0.97 | 0.98 |

GOF | p | r | β | ϕ |
---|---|---|---|---|

Flight dynamics model before correction | 0.84 | 0.82 | 0.77 | 0.83 |

Flight dynamics model after correction | 0.97 | 0.99 | 0.98 | 0.97 |

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## Share and Cite

**MDPI and ACS Style**

Tai, S.; Wang, L.; Wang, Y.; Lu, S.; Bu, C.; Yue, T.
Identification of Lateral-Directional Aerodynamic Parameters for Aircraft Based on a Wind Tunnel Virtual Flight Test. *Aerospace* **2023**, *10*, 350.
https://doi.org/10.3390/aerospace10040350

**AMA Style**

Tai S, Wang L, Wang Y, Lu S, Bu C, Yue T.
Identification of Lateral-Directional Aerodynamic Parameters for Aircraft Based on a Wind Tunnel Virtual Flight Test. *Aerospace*. 2023; 10(4):350.
https://doi.org/10.3390/aerospace10040350

**Chicago/Turabian Style**

Tai, Shang, Lixin Wang, Yanling Wang, Shiguang Lu, Chen Bu, and Ting Yue.
2023. "Identification of Lateral-Directional Aerodynamic Parameters for Aircraft Based on a Wind Tunnel Virtual Flight Test" *Aerospace* 10, no. 4: 350.
https://doi.org/10.3390/aerospace10040350