1. Introduction
Urban traffic congestion has worsened due to increasing urbanization. This has led to the development of electric vertical take-off and landing (evtol) aircraft concepts [
1,
2]. The tiltrotor evtol configuration, which combines features from traditional fixed-wing and rotor aircraft, shows potential for sustainable transport over medium to long distances, particularly in areas with complex routes and severe congestion [
3].
Aerodynamic modeling is critical in determining the dynamic characteristics and simulation fidelity of maneuvering stability for evtol. Furthermore, aerodynamic modeling can offer an affordable and replicable simulation platform for designing flight control systems, assessing maneuvering stability, modifying designs, and other relevant aspects of evtol operations. Concurrently, aerodynamic modeling is the primary method used to describe evtol aerodynamic effects [
4]. It reflects the variation characteristics of these effects with factors such as airspeed, configuration, tilt angle, and control surface deflection angle.
The aerodynamic characteristics of tiltrotor evtol aircraft vary significantly between different flight modes, particularly during the transition phase caused by nacelle tilting [
5,
6]. Due to the influence of flight conditions and rotor tilting motions, tiltrotor evtol aircraft exhibit high complexity, strong coupling, and nonlinearity in their aerodynamic characteristics. These variations in rotor-induced velocities, as well as the effects of rotor downwash and wake, make aerodynamic modeling extremely challenging.
Some aerodynamic propulsion modeling methods for lift + cruise evtol aircraft are proposed in references [
7,
8,
9,
10]. System identification techniques and computational fluid dynamics (CFD) simulations are used to develop flight dynamic models and validate the predictive capabilities of aerodynamic propulsion models. However, aerodynamic modeling methods for tiltrotor evtol still rely on traditional physical mechanisms for aerodynamic modeling. References [
11,
12] suggest that traditional methods are not effective in simulating certain aerodynamic characteristics, such as rotor wake and downwash. Concurrently, there is a certain contradiction between the realism of models established by traditional methods and the real-time performance of simulations. Therefore, most aerodynamic modeling methods used in real-time simulation environments are based on simplified physical mechanisms, which limits their ability to perform high-fidelity simulations.
The use of algorithms and computational capabilities has led to the widespread adoption of data-driven aerodynamic modeling methods based on neural networks, commonly used neural network methods in aerodynamic prediction include MLP, RBFNNs, CNNs, RNNs, and GANs [
13,
14,
15]. These methods do not require the establishment of complex mathematical formulae based on physical mechanisms [
16,
17]. Instead, they learn the hidden dynamic characteristics of the system through sample data. Data-driven methods can address high-dimensional, multiscale, and nonlinear problems that are difficult to solve with traditional methods.
The aim of this study is to validate the use of data-driven modeling methods in aerodynamic modeling during the transition phase of tiltrotor evtol aircraft. This study analyses the complex relationship between dimensionless aerodynamic force coefficients of tiltrotor aircraft and their influencing factors. A multilayer perceptron (MLP) neural network is selected for multivariate nonlinear regression for data-driven aerodynamic modeling. The data-driven model was constructed from the wind tunnel test data of XV-15 and trained using an MLP neural network. A comparison was conducted between the predictive performance of a data-driven model and a mathematical model based on physical mechanisms. The results demonstrate that the data-driven model can capture aerodynamic characteristics that are challenging to express in mathematical models.
This paper is structured as follows:
Section 2 introduces the relationship between dimensionless aerodynamic coefficients of tiltrotor evtol and their influencing factors.
Section 3 describes the advantages of data-driven methods based on MLP neural networks and the construction of the dataset.
Section 4 presents the evaluation criteria and training methods for the neural network. The predictive performance of the data-driven method is compared with that of mathematical models to validate the accuracy of the established neural network model. Conclusions are drawn in
Section 5.
2. Analysis of Aerodynamic Characteristics of Tiltrotor evtol
Tiltrotor evtol aircraft have unique aerodynamic characteristics, including interactions between numerous control surfaces, rotor–fuselage interactions, rotor–rotor interactions, and rapid changes in aerodynamics during the transition phase as the rotor pitch angle varies. Therefore, a suitable aerodynamic modeling strategy for tiltrotor evtol must combine aspects of traditional aircraft modeling strategies.
Figure 1 shows the tiltrotor evtol Vahana that Airbus conducted research and testing on several years ago.
Aerodynamic modeling for fixed-wing aircraft usually entails generating aerodynamic interpolation tables or representing dimensionless aerodynamic force and moment coefficients as functions of aircraft states and controls [
8]. The aerodynamic force and moment coefficients, which are dimensionless, are commonly expressed as functions of various parameters, including angle of attack (
), sideslip angle (
), angular rate (
), and control surface deflections (elevator deflection angle
, aileron deflection angle
, rudder deflection angle
). These coefficients are known as response variables, while explanatory variables include factors such as airflow angle. The dimensionless aerodynamic forces and moment coefficients of a fixed-wing aircraft, such as lift coefficient
, drag coefficient
, and pitch moment coefficient
, can be expressed as follows:
Rotorcraft aerodynamic modeling relies heavily on computational or flight test data due to the difficulty in scaling rotorcraft proportionally and equipment limitations for wind tunnel testing [
8]. It is important to note that unlike fixed-wing aircraft, rotorcraft aerodynamic modeling requires a different approach due to their unique characteristics [
18]. Since stability axes and wind axes become undefined in hover, modeling is generally only performed in the body axes for rotorcraft. The formulation in terms of body-axis velocity components, as opposed to airflow angles
and
, allows the state variables to be defined in hover and reflects the fact that fuselage angle of attack and angle of sideslip are less physically meaningful for describing rotorcraft aerodynamics [
8]. Explanatory variables for rotorcraft modeling often include body-axis velocity components (
), angular rates (
), pilot control inputs, and rotor states such as flapping and inflow.
Tiltrotor evtols currently have smaller rotor diameters and do not use cyclic pitch control, which makes rotor section modeling less critical, except for specific rotor states [
19]. Therefore, the aerodynamic modeling of tiltrotor evtol includes most of the characteristics of fixed-wing aircraft. However, the aerodynamic characteristics of the rotor section are more evident in the fuselage–rotor interaction.
Figure 2 shows the relationship between the dimensionless aerodynamic coefficients and moments of tiltrotor evtol and the control and state variables.
Variations in the state and control variables of tiltrotor evtol can cause changes in airflow speed and direction, as well as induced velocity characteristics, across different parts of the fuselage. These changes can result in complex aerodynamic features, such as rotor wake and rotor–fuselage interactions, which ultimately affect the aerodynamic forces of different components. It is important to note that these aerodynamic characteristics are interdependent, which means characteristics located in the second layer of
Figure 1 can impact each other.
However, most aerodynamic modeling work for evtol to date has used analytical or semiempirical models for research or application purposes. These traditional methods cannot fully express certain special aerodynamic characteristics, such as the impact of rotor wakes on different parts of the fuselage, which greatly simplifies highly complex aerodynamics. The discrepancy between mathematical models and wind tunnel data is evident in the limitation of accuracy.
Figure 3 shows this for the gravity-to-lift ratio of the XV15 tiltrotor aircraft versus airspeed [
20], highlighting the inability of mathematical models to fully capture the effects of rotor wakes.
3. Data-Driven Modeling Methods
The analysis of the mechanism and influencing factors of the transition phase of the tiltrotor evtol in the
Section 2 reveals that its complex aerodynamic characteristics result from the interaction of multidimensional factors. This interaction determines a nonlinear relationship between the input and output of the aerodynamic characteristic model that needs to be established. This type of relationship is reflected not only between factors but also between layers. Therefore, we selected an MLP neural network for aerodynamic modeling. This type of network is capable of solving complex nonlinear relationships.
3.1. Data-Driven Network Model Design
The multilayer perceptron (MLP) is a type of feedforward neural network with a simple connectivity pattern. It comprises an input layer, one or more hidden layers, and an output layer. Each hidden layer contains multiple neurons [
21]. The MLP operates by connecting each neuron in a layer to every neuron in the next layer with appropriate weights and biases. Input data propagate from the input layer to the output layer through forward propagation.
The input data are received by the network through the input layer neurons. Each neuron in a hidden layer receives a weighted sum of signals from all neurons in the previous layer. This weighted sum is then passed through an activation function, such as sigmoid or ReLU, which introduces nonlinearity and enables the network to learn complex patterns. The output of the activation function becomes the output of that neuron and the input for neurons in the next layer. MLP calculates its error by comparing its predictions with the expected outputs using a loss function at the output layer. The error is then propagated backward through the network using backpropagation, which employs the chain rule to calculate the contribution of each weight to the error. The weights in each layer are then adjusted based on the error signal and a learning rate to minimize the overall error. The iterative process of forward pass, error calculation, and backpropagation with weight updates continues during training until the network achieves an acceptable level of performance [
22].
MLP networks can be used for multifunctional learning, including classification and regression tasks. MLP demonstrates proficient performance in handling multivariable nonlinear regression problems with its classic architecture [
21]. The relationship between the dimensionless aerodynamic coefficients introduced in
Section 2 is similar in structure to the MLP neural network and essentially belongs to multiple nonlinear regression. Therefore, this article introduces an MLP neural network based on the input–output and influence factors described in the
Section 2. The hidden layer neurons represent the unique aerodynamic characteristics and state variables of the tiltrotor evtol. However, in these complex coupling relationships, some factors do not influence each other. Therefore, these unrelated neurons should not be connected. The MLP topology diagram for the unique aerodynamic characteristics of the tiltrotor evtol is obtained as
Figure 4.
During network training, the structural parameters of the MLP are determined based on the coupling characteristics of aerodynamic features and their training effect. To achieve non-interconnected neurons, the weight of the corresponding connections in the weight matrix can be set to zero. This reflects the unique aerodynamic characteristics of the tiltrotor evtol.
This article employs a network comprising multiple layers of neurons, each comprising a LeakyReLU activation function. The network is trained using the Adam optimizer to calculate the weights of the MLP neural network model. All of these components are implemented in Python code based on the PyTorch framework.
3.2. Sample Data Requirements and Generation
The aerodynamic characteristics and rotor layout of the XV15 tiltrotor aircraft are similar to some tiltrotor evtols, resulting in largely consistent aerodynamic characteristics. This article conducts data-driven aerodynamic modeling on the tilting rotor evtol, based on wind tunnel test data of the transition maneuver of XV15. Due to space limitations, this article focuses on data-driven model construction for longitudinal aerodynamics, the generation of sample data, model training, and validation.
According to reference [
20,
23], for the longitudinal transition maneuver of the XV15 tiltrotor aircraft, aerodynamic forces and moments (
) are defined as functions of a set of flight state values (
). Among them,
is the Mach number,
is the angle of attack of the fuselage,
is the tilt angle of the nacelle,
is the flap/aileron angle mode selection, and
is the elevator deflection angle. During the transition phase, the value ranges of variables mentioned above for XV15 are as
Table 1.
By considering the range constraints of explanatory variables during the transition phase of the tiltrotor evtol and the inclined transition corridor determined in reference [
20,
23], a multidimensional sample space can be established. This sample space allows for a one-to-one correspondence between the aerodynamic coefficients and the explanatory variables. Additionally, there are constraints between variables, in addition to the range constraints of the variables themselves.
Figure 5 shows the transition corridor resulting from the constraint between the tilt angle
and the horizontal velocity
. When the nacelle inclination angle is 0°, XV15 is in helicopter mode, and when the nacelle inclination angle is 90°, it is in aircraft mode. The dataset needed for the neural network is constructed by selecting points from a vector space consisting of flight envelopes and constraints in multiple dimensions. It is important to note that the aerodynamic coefficient profile has both linear and nonlinear regions that correspond to a single explanatory variable. The nonlinear region has a higher density of points compared with the sparser points in the linear region. The constructed sample dataset comprises 2000 sample points, with each point represented in the form of
.
5. Conclusions
A neural network structure that conforms to the aerodynamic characteristics and mechanism of the tiltrotor evtol was constructed using an MLP neural network. To create a dataset for neural network training and validation, wind tunnel test data from the XV15 tiltrotor with similar aerodynamic characteristics and structure were selected. The dataset was constructed by analyzing the sample data structure and constructing a sample vector space with constraints. Finally, the optimal hyperparameter combination for the neural network was selected by designing evaluation indicators. Predictions were then made based on the test set under different tilt angle conditions. The simulation results show the following:
Neural-network-based data-driven methods can accurately predict the aerodynamic characteristics of tiltrotor evtol aircraft during the transition phase, with an error rate of less than 2% compared with wind tunnel test data.
Traditional physical modeling methods exhibit larger errors in representing the complex aerodynamic characteristics of tiltrotor aircraft due to the limitations of mathematical formulae, typically up to nearly 10%.
The MLP neural network structure constructed based on the aerodynamic characteristic mechanism of the tiltrotor evtol is effective.
In summary, this study shows that data-driven methods using neural networks can predict the aerodynamic characteristics of tiltrotor evtol aircraft during the transition phase. Due to the high fidelity fitting and relatively simple modeling process of MLP for nonlinear and strongly coupled aerodynamic characteristics, this method can be extended and applied to the full degree of freedom and full flight envelope aerodynamic modeling of tiltrotor evtol aircraft.