Computational Fluid Dynamics (CFD)-Enhanced Dynamic Derivative Engineering Calculation Method of Tandem-Wing Unmanned Aerial Vehicles (UAVs)

Abstract

Dynamic derivatives are critical for evaluating an aircraft’s aerodynamic characteristics, dynamic modeling, and control system design during the design phase. However, due to the multiple iterations of the design phase, a method for calculating dynamic derivatives that balances computational efficiency and accuracy is required. This work presents a CFD-enhanced engineering calculation method (CEHM) for calculating tandem-wing UAVs’ dynamic derivatives. A coupling-effect-driven estimation strategy is proposed to incorporate the contribution of the rear wing to the longitudinal dynamic derivatives, and it accounts for the aerodynamic coupling effects between the front and rear wings. To enhance the accuracy of the dynamic derivative calculations, we put forward a dynamic derivative-correction mechanism based on the CFD method. It achieves three types of parameters from the static derivative CFD simulations to enhance accuracy, including parameters for aerodynamic force coefficient fitting, the dynamic pressure ratio, and the upwash and downwash gradients. The CEHM method is applied to compute the dynamic derivatives of the SULA90 tandem-wing UAV, with results compared to those obtained from the traditional engineering estimation tools (XFLR5 and OpenVSP). The simulation experiment results show that the proposed method not only calculates the acceleration derivatives but also provides higher calculation accuracy. To further validate the method’s effectiveness, open-loop model verifications were conducted using field flight test data of the SULA90. The field flight test results show that the CEHM method’s predicted results align closely with the measured flight data. The proposed method calculates dynamic derivatives in seconds, balancing accuracy and computational cost, making it highly suitable for tandem-wing aircraft during the design phase. Furthermore, this approach is generalizable and can be applied to other aircraft configurations.

1. Introduction

In recent years, tandem-wing configurations have re-emerged as a critical design option for UAVs and eVTOL vehicles due to their potential for high lift-to-drag ratios and a compact footprint [1,2,3,4,5,6]. Aligned with Munk’s stagger theorem [7], the tandem-wing configuration can potentially improve aircraft performance under optimized vertical stagger and lift distribution, leading to reduced induced drag and enhanced lift compared to conventional single-wing designs. Otherwise, this configuration combines the benefits of a morphing wing structure and wings with a relatively high aspect ratio and is widely used in the tube-launched loitering munition [8]. However, due to the more pronounced coupling effects between wings with a tandem-wing configuration compared to a traditional configuration [9], the downwash and upwash make the dynamic modeling of the tandem-wing aircraft more intricate. When modeling aircraft dynamics, three types of aerodynamic derivatives are crucial: static, control, and dynamic. Remarkably, dynamic derivatives determine an aircraft’s flying qualities. Predicting them reliably, quickly, and sufficiently early in the design stage is increasingly essential.

The aerodynamic investigation of tandem-wing aircraft primarily encompasses two key aspects. First, aerodynamic configuration optimization is investigated through parametric studies to achieve maximum aerodynamic efficiency. Researchers have conducted CFD simulations and wind tunnel experiments to analyze the effects of stagger distance, wingspan difference, and dihedral angles on flow field characteristics [1,10]. Broering et al. [11] identified the optimal chordwise spacing for lift enhancement in low-Reynolds-number regimes. Subsequent parametric studies have systematically revealed the aerodynamic coupling mechanisms of geometric parameters under such conditions [12,13,14,15]. The second aspect involves a dynamic stability investigation with the explicit consideration of coupling effects. Marcin et al. [16] discovered nonlinear relationships between span factors and dynamic stability characteristics. Tomasz et al. [17] identified directional instability and spiral–roll mode coupling in a tailless tandem-wing configuration, with a Dutch roll divergence persisting across speed ranges. The determination of dynamic derivatives predominantly employs four methodologies: wind tunnel experimentation [18], aircraft system identification [19,20,21], empirical engineering estimation [22], and computational fluid dynamics (CFD) simulations [23]. Among these, empirical estimation and CFD approaches prevail in aircraft design workflows due to their adaptability during early-stage development. The empirical estimation framework, rooted in inviscid flow theory and semi-empirical correlations, achieves high computational efficiency, including slender-body theory (SBT) [24,25], lifting-line theory (LLT) [26], lifting-surface theory (LST) [27], and the vortex lattice method (VLM) [28]. Verified for low-to-moderate angle of attack (AOA) conditions, empirical engineering estimation methods have been proven effective for calculating dynamic derivatives [29,30]. Commonly used tools include XFLR5 [31], Tornado VLM [32], AVL [33], OpenVSP [34], and DATCOM [35]. Within CFD-based approaches, dynamic derivative computation are bifurcated into time-domain and frequency-domain strategies. The time-domain methods include harmonic perturbation methods [36,37], conical motion methods [38,39,40], quasi-steady methods [41,42,43], and forced oscillation methods [44,45,46]. Frequency-domain methods primarily rely on the harmonic balance method (HBM) and its variants [47,48,49,50,51].

While contemporary CFD techniques have become the predominant approach for dynamic derivative analysis due to their high fidelity, the substantial computational overhead imposes critical constraints on design iteration efficiency, which is particularly acute in the concept design phase, where rapid responsiveness is essential. There remains an urgent need for tandem-wing aircraft aerodynamic analysis to develop dynamic derivative-prediction methodologies that balance moderate accuracy with computational efficiency. Grigorios et al. [52] developed a framework for the direct calculation of stability derivatives based on the unsteady subsonic 3D Source and Doublet Panel Method (SDPM), integrating empirical engineering estimation methods. They applied this framework to a Blended Wing Body (BWB) configuration and demonstrated that the static aerodynamic derivatives closely matched those obtained from CFD methods. However, the study did not compare the dynamic aerodynamic derivative results. Gao et al. [53] developed a dynamic modeling approach for a tandem-wing aircraft, where static and control derivatives were obtained via CFD simulations, and dynamic derivatives were calculated using the Tornado Vortex Lattice Method (VLM) tool. However, the calculation of dynamic derivatives did not utilize the results from static derivative CFD simulations, and the results lacked proper validation. The study conducted by Wang et al. [54] involved a tandem-wing aircraft weighing 4.3 kg and utilized lift curve slopes obtained from static derivative CFD simulations in conjunction with computational engineering methods to determine the longitudinal dynamic derivatives. However, it did not extend to the computation of acceleration and lateral-directional dynamic derivatives, and the interaction effects between the front and rear wings were not taken into account. Lua et al. [55] investigated the influence of installation angles and the distance between the two wings, deriving an approximate mathematical model to describe the relationship between these parameters. Zhang and Yu [9] investigated the coupling effects between the front and rear wings during the unsteady deployment process of a foldable tandem-wing UAV under low-Reynolds-number conditions. Their study revealed that the horizontal and vertical distances between the wings, as well as the wing deflection angle, significantly impact the coupling effects. However, these studies primarily focused on the factors influencing the coupling effect without examining its impact on dynamic derivatives or deriving formulas for such derivatives.

Since static derivative CFD simulations do not involve extensive unsteady calculations, their computational time is significantly less than that of dynamic derivative CFD simulations. Therefore, combining static derivative CFD simulations with empirical engineering estimation methods for dynamic derivative calculation has emerged as a viable approach. We propose a CFD-enhanced dynamic derivative engineering calculation method (CEHM) of tandem-wing UAVs. This method corrects the empirical calculation formulas for dynamic derivatives by incorporating the aerodynamic characteristics of tandem-wing aircraft. Three types of aerodynamic parameters are extracted from high-fidelity static derivative CFD simulation results to enhance the accuracy of the calculations. Three types of aerodynamic parameters are extracted from high-fidelity static derivative CFD simulation results to enhance the accuracy of the calculations. The proposed method not only calculates acceleration derivatives, which traditional empirical engineering estimation tools cannot, but also provides higher calculation accuracy. The rest of the paper is organized as follows: Section 2 introduces the aerodynamic configuration of a typical tandem-wing UAV and the model structure for aerodynamic force and moment coefficients. Section 3 demonstrates the coupling-effect-driven estimation strategy for dynamic derivative calculation. We also propose the dynamic derivative correction mechanism based on the CFD method for improving the accuracy. Section 4 presents the technical parameters of the research object. The results calculated using the CEHM method and two aerodynamic estimation tools (XFLR5 and OpenVSP ) are compared with the dynamic derivative CFD simulation results. Finally, field flight tests are conducted for open-loop model validation. Section 5 concludes the paper and provides an outlook on future work.

2. Tandem-Wing UAV Model Structure

2.1. Tandem-Wing UAV Configuration

The tandem-wing UAV discussed in this paper features a configuration with a high aspect ratio of both the front and rear wing, as shown in Figure 1a. The center of gravity (CG) of the UAV is located between the front and rear wings, which makes the contribution of the front wing to the pitching moment significant impossible to ignore. Here, $b_{*}$ represents the wingspan, $c_{*}$ represents the wing chord, and $l_{*}$ represents the distance from the aerodynamic center AC to the CG, measured along the body axis. The subscript (*) denotes different aerodynamic components, $fw$ represents the front wing, $rw$ represents the rear wing, and $vt$ represents the vertical tail. It is assumed that the AC of each aerodynamic component is located at one-quarter of its chord length. As shown in Figure 1b, z is the rolling moment arm of the vertical tail, and $z_{vt}$ is the vertical distance from the AC of the vertical tail to the aircraft reference line, measured normally to the body axis. The relationships between z, $z_{vt}$ and $l_{vt}$ are

$$z=z_{vt}\mathit{cos}\alpha -l_{vt}\mathit{sin}\alpha ,$$

(1)

where $\alpha$ denotes the angle of attack (AOA).

The reference wing area S, reference wing span b, and reference wing chord c are defined as

$$S=b_{fw}{c}_{fw}+b_{rw}{c}_{rw}$$

(2)

$$b=b_{fw}$$

(3)

$$c=c_{fw}$$

(4)

2.2. Aerodynamic Model Structure

For small UAVs, the effects of Froude, Mach, Strouhal, and Reynolds numbers are generally neglected, and the mass and inertia of the aircraft dominate those of the surrounding air [19,56]. A general model of the aerodynamic forces and moments acting on a fixed-wing UAV is given in Equation (5) [57].

$$\begin{gathered} \begin{array}{c}\left[\begin{array}{c}D\\ Y\\ L\end{array}\right]={\displaystyle \frac{1}{2}}\rho V_a^2\left[\begin{array}{c}{C}_D\left(\alpha ,\dot{\alpha },\beta ,p,q,r,{\delta }_a,{\delta }_e,{\delta }_r\right)\\ C_Y\left(\alpha ,\beta ,\dot{\beta },p,q,r,{\delta }_a,{\delta }_e,{\delta }_r\right)\\ C_L\left(\alpha ,\dot{\alpha },\beta ,p,q,r,{\delta }_a,{\delta }_e,{\delta }_r\right)\end{array}\right]\hfill \end{array} \\ \left[\begin{array}{c}{L}_A\\ M_A\\ N_A\end{array}\right]={\displaystyle \frac{1}{2}}\rho V_a^2\left[\begin{array}{c}b{C}_l\left(\alpha ,\beta ,\dot{\beta },p,q,r,{\delta }_a,{\delta }_e,{\delta }_r\right)\\ c{C}_m\left(\alpha ,\dot{\alpha },\beta ,p,q,r,{\delta }_a,{\delta }_e,{\delta }_r\right)\\ b{C}_n\left(\alpha ,\beta ,\dot{\beta },p,q,r,{\delta }_a,{\delta }_e,{\delta }_r\right)\end{array}\right], \end{gathered}$$

(5)

where $\rho$ denotes air density, $V_a$ represents airspeed, and dynamic pressure is defined as $Q={\displaystyle \frac{1}{2}}\rho V_a^2$. $\beta$ is the angle of sideslip (AOS), while $\dot{\alpha }$ and $\dot{\beta }$ signify the rates of change of AOA and AOS, respectively. Angular rates p, q, and r correspond to rotational velocities about the $X_b$, $Y_b$, and $Z_b$ axis of the body frame ${\mathcal{F}}^b$. ${\delta }_a$, ${\delta }_e$, and ${\delta }_r$ are the controls of the aileron, elevator, and rudder. D, Y, and L are the aerodynamic drag, side, and lift forces, acting along the negative $X_w$, positive $Y_w$, and negative $Z_w$ axis of the wind frame ${\mathcal{F}}^w$. The dimensionless aerodynamic coefficients for these forces are $C_D$, $C_Y$, and $C_L$, respectively. $L_A$, $M_A$, and $N_A$ are the aerodynamic rolling, pitching, yawing moments, acting along the $X_b$, $Y_b$ and $Z_b$ axes. The dimensionless aerodynamic coefficients of these moments are $C_l$, $C_m$, and $C_n$.

A commonly adopted simplification assumes decoupled longitudinal and lateral dynamics for UAVs operating at a small AOA and AOS. Under this paradigm, the lift force coefficient $C_L$ and pitch moment coefficient $C_m$ depend on $\alpha$, $\dot{\alpha }$, q and ${\delta }_e$ while remaining independent of lateral-directional variables $\beta$, p, r, ${\delta }_a$ and ${\delta }_r$. Conversely, the roll $C_l$ and yaw $C_n$ coefficients, on the other hand, are independent of $\alpha$, q, and ${\delta }_e$ and are related to $\beta$, p, r, ${\delta }_r$ and ${\delta }_a$. Notably, a residual coupling persists through the AOS-dependent drag force, which retains bidirectional sensitivity due to airframe symmetry about the $X_w{Z}_w$-plane, rendering the drag force coefficient $C_D$ an even function of $\beta$.

In summary, the aerodynamic force and moment coefficients are expressed in Equation (6), and each aerodynamic coefficient comprises three parts: static, control, and dynamic derivatives.

$$\begin{array}{c}\left[\begin{array}{c}{C}_D\\ C_Y\\ C_L\end{array}\right]=\left[\begin{array}{c}{C}_D\left(\alpha \right)+\Delta C_D\left(\beta \right)+\Delta C_D\left({\delta }_e\right)+C_{Dq}\overline{q}\\ C_Y\left(\beta \right)+\Delta C_Y\left({\delta }_r\right)+C_{Yr}\overline{r}\\ C_L\left(\alpha \right)+\Delta C_L\left({\delta }_e\right)+C_{Lq}\overline{q}+C_{L\dot{\alpha }}\overline{\dot{\alpha }}\end{array}\right]\hfill \\ \left[\begin{array}{c}{C}_l\\ C_m\\ C_n\end{array}\right]=\left[\begin{array}{c}{C}_l\left(\beta \right)+\Delta C_l\left({\delta }_a\right)+\Delta C_l\left({\delta }_r\right)+C_{lp}\overline{p}+C_{lr}\overline{r}\\ C_m\left(\alpha \right)+\Delta C_m\left({\delta }_e\right)+C_{mq}\overline{q}+C_{m\dot{\alpha }}\overline{\dot{\alpha }}\\ C_n\left(\beta \right)+\Delta C_n\left({\delta }_a\right)+\Delta C_n\left({\delta }_r\right)+C_{np}\overline{p}+C_{nr}\overline{r}\end{array}\right],\hfill \end{array}$$

(6)

where $\overline{q}={\displaystyle \frac{qc}{2V_a}}$, $\overline{\dot{\alpha }}={\displaystyle \frac{\dot{\alpha }c}{2V_a}}$, $\overline{p}={\displaystyle \frac{pb}{2V_a}}$, $\overline{r}={\displaystyle \frac{rb}{2V_a}}$.

Static derivatives are aerodynamic derivatives related to the airflow angle, including $\alpha$ and $\beta$. A lookup table method is used for modeling, including $C_D\left(\alpha \right)$, $\Delta C_D\left(\beta \right)$, $C_Y\left(\beta \right)$, $C_L\left(\alpha \right)$, $C_l\left(\beta \right)$, $C_m\left(\alpha \right)$, and $C_n\left(\beta \right)$. Taking the non-deflected state as the baseline, the effects of control surface deflections are expressed incrementally and also modeled using lookup tables, including $\Delta C_D\left({\delta }_e\right)$, $\Delta C_Y\left({\delta }_r\right)$, $\Delta C_L\left({\delta }_e\right)$, $\Delta C_l\left({\delta }_a\right)$, $\Delta C_l\left({\delta }_r\right)$, $\Delta C_m\left({\delta }_e\right)$, $\Delta C_n\left({\delta }_a\right)$ and $\Delta C_n\left({\delta }_r\right)$. Dynamic derivatives mainly consist of angular rate derivatives associated with three-axis angular rates and acceleration derivatives related to airflow angular rates. The angular rate derivative is a key component of the dynamic derivatives and plays a major role in aircraft dynamics and control. Due to the use of a decoupling modeling approach, the angular rate derivatives considered in this paper include $C_{Dq}$, $C_{Yr}$, $C_{Lq}$, $C_{lp}$, $C_{lr}$, $C_{mq}$, $C_{np}$, and $C_{nr}$. Although the effect of the acceleration derivatives is relatively small, this paper considers the AOA acceleration derivatives $C_{L\dot{\alpha }}$ and $C_{m\dot{\alpha }}$ for modeling the aerodynamic characteristics of the highly maneuverable flight. The AOS rate $\dot{\beta }$ has a minor influence than the roll angular rate p and the yaw angular rate r, so the AOS acceleration derivatives are neglected [58]. Dynamic derivatives are modeled using constant parameters. This paper assumes metric units in the aerodynamic model, with angles, including control surface deflections, and angular rates provided in radians and radians per second.

3. CFD-Enhanced Dynamic Derivative Engineering Estimation Method (CEHM)

3.1. Coupling the Effect-Driven Estimation Strategy

3.1.1. Longitudinal Dynamic Derivatives

For conventional configurations, when the CG resides near the AC, the horizontal tail demonstrates a more substantial effect on longitudinal stability derivatives compared to the wing [22]. However, in tandem-wing UAV architectures, the CG is typically located between the front and rear wings, resulting in both lifting surfaces contributing dominantly to stability derivative characteristics. As shown in Figure 2, the flight AOA of UAV is $\alpha$. By considering the downwash effect from the front wing on the rear wing (inducing a downwash angle of attack, $\epsilon$) and the upwash effect from the rear wing on the front wing (inducing an upwash angle of attack, $\vartheta$), the actual effective AOAs for the front wing and rear wing become ${\alpha }_f=\alpha +\epsilon$ and ${\alpha }_r=\alpha -\vartheta$, respectively. Static aerodynamic derivatives obtained through Reynolds-Averaged Navier–Stokes (RANS) simulations inherently capture these bidirectional interference effects. To align with the CFD framework, all aerodynamic coefficients (e.g., lift, drag, and moments) are expressed as functions of the UAV’s flight AOA $\alpha$, rather than the effective angles (${\alpha }_f$ or ${\alpha }_r$). For example, the front wing’s lift coefficient $C_L^{fw}\left(\alpha \right)$ derived from CFD simulations represents the actual lift force when the AOA is ${\alpha }_f=\alpha +\epsilon$, even though it is parametrized in terms of $\alpha$. Similar reasoning applies to the rear wing’s coefficients. Detailed methodologies for processing CFD results are provided in Section 3.2.

We assume that the UAV undergoes a positive pitch rate q for the front and rear wings, the AOA variation of the front wing is $\Delta {\alpha }_{fw}=-{\displaystyle \frac{q{l}_{fw}}{V_a}}$, the AOA variation of the rear wing is $\Delta {\alpha }_{rw}={\displaystyle \frac{q{l}_{rw}}{V_a}}$, and the detailed derivations are given in Appendix A.1. We define the downwash gradient of the rear wing as ${\displaystyle \frac{d\epsilon }{d\alpha }}$ and the upwash gradient of the forewing as ${\displaystyle \frac{d\vartheta }{d\alpha }}$. The upwash angle variation $\Delta \vartheta$ on the front wing induced by the rear wing’s AOA variation $\Delta {\alpha }_{rw}$ is quantified as $\Delta \vartheta ={\displaystyle \frac{d\vartheta }{d\alpha }}\Delta {\alpha }_{rw}$. Correspondingly, the downwash angle variation $\Delta \epsilon$ on the rear wing induced by the front wing’s AOA variation $\Delta {\alpha }_{fw}$ is quantified as $\Delta \epsilon ={\displaystyle \frac{d\epsilon }{d\alpha }}\Delta {\alpha }_{fw}$. Therefore, the change in the lift coefficient induced by the front and rear wing for the entire UAV can be expressed as

$$\begin{array}{c}\hfill \Delta C_L^{fw}=C_L^{fw}\left(\alpha +\Delta {\alpha }_{fw}+\Delta \vartheta \right)-C_L^{fw}\left(\alpha \right)=-C_{L_{\alpha }}^{fw}{\eta }_{fw}\left({\displaystyle \frac{S_{fw}}{S}}\right)\left({\displaystyle \frac{q}{V_a}}\right)\left(l_{fw}-{\displaystyle \frac{d\vartheta }{d\alpha }}{l}_{rw}\right)\end{array}$$

(7)

$$\begin{array}{c}\hfill \Delta C_L^{rw}=C_L^{rw}\left(\alpha +\Delta {\alpha }_{rw}-\Delta \epsilon \right)-C_L^{rw}\left(\alpha \right)=C_{L_{\alpha }}^{rw}{\eta }_{rw}\left({\displaystyle \frac{S_{rw}}{S}}\right)\left({\displaystyle \frac{q}{V_a}}\right)\left(l_{rw}+{\displaystyle \frac{d\epsilon }{d\alpha }}{l}_{fw}\right),\end{array}$$

(8)

where $\Delta C_L^{*}$ denotes the variation in the entire aircraft lift coefficient induced by the wing, $C_L^{*}\left(\alpha \right)$ represents the lift coefficient at AOA $\alpha$ obtained from CFD simulations, $C_{L_{\alpha }}^{*}$ represents the lift curve slope of the wing, ${\eta }_{*}={\displaystyle \frac{Q^{*}}{Q}}$ represents dynamic pressure ratio, $S_{*}$ donates the wing area, and the subscript (*) can be $fw$ and $rw$. The detailed derivation processes for Equations (7) and (8) are provided in Appendix A.1.

For brevity, we denote ${\displaystyle \frac{d\vartheta }{d\alpha }}$ as ${\vartheta }^{\prime }$ and ${\displaystyle \frac{d\epsilon }{d\alpha }}$ as ${\epsilon }^{\prime }$. Due to the minor influence of the fuselage and vertical tail on the longitudinal dynamic derivatives, the $C_{Lq}$ is given by

$$C_{Lq}=2·\left({\displaystyle \frac{C_{L_{\alpha }}^{rw}{S}_{rw}{\eta }_{rw}\left(l_{rw}+{\epsilon }^{\prime }{l}_{fw}\right)-C_{L_{\alpha }}^{fw}{S}_{fw}{\eta }_{fw}\left(l_{fw}-{\vartheta }^{\prime }{l}_{rw}\right)}{S·c}}\right)$$

(9)

By using a similar method, $C_{Dq}$ and $C_{mq}$ are given by

$$C_{Dq}=2{\displaystyle \frac{{\left(C_D^{fw}\right)}^i·S_{fw}+{\left(C_D^{rw}\right)}^i{S}_{rw}}{Sc}}$$

(10)

$$C_{mq}=-2·\left({\displaystyle \frac{C_{L_{\alpha }}^{rw}{S}_{rw}{\eta }_{rw}\left(l_{rw}+{\epsilon }^{\prime }{l}_{fw}\right)l_{rw}+C_{L_{\alpha }}^{fw}{S}_{fw}{\eta }_{fw}\left(l_{fw}-{\vartheta }^{\prime }{l}_{rw}\right)l_{fw}}{S·c^2}}\right)$$

(11)

In Equation (10),

$${\left(C_D^{fw}\right)}^i=C_{D\alpha }^{fw}\left(-l_{fw}+{\vartheta }^{\prime }{l}_{rw}\right)+2C_{D{\alpha }^2}^{fw}\alpha \left(-l_{fw}+{\vartheta }^{\prime }{l}_{rw}\right)$$

(12)

$${\left(C_D^{rw}\right)}^i=C_{D\alpha }^{rw}\left(l_{rw}+{\epsilon }^{\prime }{l}_{fw}\right)+2C_{D{\alpha }^2}^{rw}\alpha \left(l_{rw}+{\epsilon }^{\prime }{l}_{fw}\right)$$

(13)

where $C_{D_{\alpha }}^{*}$ and $C_{D_{{\alpha }^2}}^{*}$ are the first- and second-order coefficients of the quadratic fitting curve of $C_D\left(\alpha \right)$, and the subscript (*) can be $fw$ and $rw$. The acceleration derivatives $C_{L\dot{\alpha }}$ and $C_{m\dot{\alpha }}$ are measures of the time-lag effects on the lift force and pitching moment when the aircraft experiences a change in the AOA with respect to time. The calculation formulas are given by [22]

$$C_{L\dot{\alpha }}=2C_{L_{\alpha }}^{rw}{\displaystyle \frac{S_{rw}\left(l_{rw}+l_{fw}\right)}{Sc}}{\eta }_{rw}\left({\displaystyle \frac{\mathrm{d}\epsilon }{\mathrm{d}\alpha }}\right)$$

(14)

$$C_{m\dot{\alpha }}=-C_{L\dot{\alpha }}\left({\displaystyle \frac{l_{rw}}{c}}\right),$$

(15)

where ${\displaystyle \frac{\mathrm{d}\epsilon }{\mathrm{d}\alpha }}$ is the downwash gradient at the rear wing that can be estimated using the static derivatives CFD simulation results discussed in Section 3.2.

3.1.2. Lateral-Directional Dynamic Derivatives

The lateral-directional dynamic derivatives’ calculation formulas are the same as those of traditional configuration aircraft and are given by [22]

$$C_{Y_r}=-{\displaystyle \frac{2}{b}}A{C}_{Y_{\beta }}^{vt}$$

(16)

$$C_{lp}=\left|2\left({\displaystyle \frac{z}{b}}\right)\left({\displaystyle \frac{z-z_{vt}}{b}}\right)\right|C_{Y_{\beta }}^{vt}-\sum _i^{fw,rw}{\displaystyle \frac{\left(C_{L_{\alpha }}^i+C_D^i\left(\alpha \right)\right)c_i{b}_i^3}{6S{b}^2}}$$

(17)

$$C_{lr}={\displaystyle \frac{-2}{b^2}}AB·C_{Y_{\beta }}^{vt}+\sum _{*}^{fw,rw}{\displaystyle \frac{c_{*}{b}_{*}^3{C}_L^{*}\left(\alpha \right)}{3S{b}^2}}$$

(18)

$$C_{np}=-2A\left({\displaystyle \frac{z-z_{vt}}{b^2}}\right)C_{Y\beta }^{vt}-\sum _{*}^{fw,rw}{\displaystyle \frac{\left(C_L^{*}\left(\alpha \right)-C_{D_{\alpha }}^{*}-2C_{D_{{\alpha }^2}}^{*}\alpha \right)c_{*}{b}_{*}^3}{6S{b}^2}}$$

(19)

$$C_{nr}={\displaystyle \frac{2}{b}}{A}^2{C}_{Y\beta }^{vt}-\sum _{*}^{fw,rw}{\displaystyle \frac{C_D^{*}\left(\alpha \right)c_{*}{b}_{*}^3}{3S{b}^2}},$$

(20)

where $C_{Y\beta }^{vt}$ represents the rate of change of the vertical tail side force with respect to the AOS. $C_D^{*}\left(\alpha \right)$ is the drag force coefficient of the wing, $C_L^{*}\left(\alpha \right)$ is the lift force coefficient of the wing, and the subscript (*) can be $fw$ and $rw$. In the above equations, A and B are given by

$$\begin{array}{c}A=\left(l_{vt}\mathit{cos}\alpha +z_{vt}\mathit{sin}\alpha \right)\hfill \\ B=\left(z_{vt}\mathit{cos}\alpha -l_{vt}\mathit{sin}\alpha \right)\hfill \end{array}$$

(21)

Thus far, we have established the estimation formulae for the dynamic derivatives of the tandem-wing UAV. The UAV’s geometric parameters, such as S, b, and c, are derived from aerodynamic design specifications. However, the aerodynamic parameters in these formulas, such as $C_{L_{\alpha }}^{*}$, $C_{D_{\alpha }}^{*}$, ${\eta }_{*}$ and ${\displaystyle \frac{\mathrm{d}\epsilon }{\mathrm{d}\alpha }}$, conventionally rely on inviscid flow theory and semi-empirical correlations, fundamentally limiting prediction accuracy in traditional methodologies. To mitigate this constraint, the present study integrates static-derivative CFD simulation data to refine dynamic derivative computation accuracy.

3.2. Dynamic Derivative Correction Mechanism Based on the CFD Method

The method for enhancing the accuracy of dynamic derivatives calculations based on static derivative CFD simulation results can be broadly divided into two key aspects, as illustrated in Figure 3. The first component emphasizes the aerodynamic analysis of individual configuration elements. This involves analyzing the CFD simulation data for the front and rear wing to extract fitting parameters related to the lift and drag force coefficients as a function of the AOA, as well as for the vertical tail, to obtain the fitting parameters related to the side force coefficient as a function of the AOS. The second component investigates aerodynamic coupling between the front and rear wings. This includes measuring the dynamic pressure on both wings to compute the dynamic pressure ratio, calculating the washout gradient of the rear wing caused by the downwash effect from the front wing and the upwash gradient of the front wing caused by the rear wing.

Through this analysis, three types of aerodynamic parameters are derived.

(1)

The aerodynamic-force-related parameters. Within the linear range of the lift coefficient variation, the lift force coefficient data for both the front and rear wings, as functions of AOA, are typically modeled using a linear function, as shown in Equation (22). The drag coefficient data for both the front and rear wings, as a function of AOA, are generally approximated with a quadratic polynomial, as depicted in Equation (23). Similarly, the side force coefficient data for the vertical tail as a function of AOS are fitted with a linear function, as indicated in Equation (24).

$$C_L^{*}\left(\alpha \right)=C_{L_0}^{*}+C_{L_{\alpha }}^{*}\alpha$$

(22)

$$C_D^{*}\left(\alpha \right)=C_{D_0}^{*}+C_{D_{\alpha }}^{*}\alpha +C_{D_{{\alpha }^2}}^{*}{\alpha }^2$$

(23)

$$C_Y^{vt}\left(\beta \right)=C_{Y_{\beta }}^{vt}·\beta ,$$

(24)

where the subscript (*) can be $fw$ and $rw$.

(2)

The dynamic pressure ratios. The dynamic pressure measured one body length from the nose is the reference dynamic pressure Q. The dynamic pressures measured at a quarter chord length from the leading edge of the front and rear wings are referred to as the front wing dynamic pressure $Q^{fw}$ and the rear wing dynamic pressure $Q^{rw}$, respectively. Therefore, the formulas for calculating the front and rear wing dynamic pressure ratios are as follows:

$${\eta }_{fw}={\displaystyle \frac{Q^{fw}}{Q}}$$

(25)

$${\eta }_{rw}={\displaystyle \frac{Q^{rw}}{Q}}$$

(26)

(3)

The rear wing washout and front upwash gradient. To investigate the aerodynamic interactions between the front and rear wings, static CFD simulations were conducted, excluding the standalone wings for comparison. The rear wing lift coefficient from the CFD simulation omitting the front wing is denoted as $C_L^{rwsi}\left(\alpha \right)$, and the front wing lift coefficient from the simulation omitting the rear wing is $C_L^{fwsi}\left(\alpha \right)$. In the full-aircraft static derivative CFD simulation, the rear wing lift coefficient is $C_L^{rw}\left(\alpha \right)$ and the front wing lift coefficient is $C_L^{fw}\left(\alpha \right)$. Using the rear wing lift curve slope derived from Equation (22), the downwash gradient ${\displaystyle \frac{d\epsilon }{d\alpha }}$ on the rear wing at this AOA is calculated using Equation (27). Similarly, the upwash gradient ${\displaystyle \frac{d\vartheta }{d\alpha }}$ on the front wing is calculated using Equation (28).

$${\displaystyle \frac{d\epsilon }{d\alpha }}=d\left({\displaystyle \frac{\left(C_L^{rwsi}\left(\alpha \right)-C_L^{rw}\left(\alpha \right)\right)}{C_{L_{\alpha }}^{rw}}}\right)/d\alpha$$

(27)

$${\displaystyle \frac{d\vartheta }{d\alpha }}=d\left({\displaystyle \frac{\left(C_L^{fw}\left(\alpha \right)-C_L^{fwsi}\left(\alpha \right)\right)}{C_{L_{\alpha }}^{fw}}}\right)/d\alpha$$

(28)

4. Simulation and Field Experimental Results

4.1. SULA90 UAV

The research object in this paper is a 10 kg class UAV, named SULA90 [58,59], which is designed by the Beijing Institute of Technology, as shown in Figure 4. This small, tandem-wing UAV utilizes a conventional control setup, featuring ailerons on the front wings, elevators on the rear wings, and rudders on the vertical tails. It is powered by a single electric motor positioned at the rear of the fuselage, which drives a custom-designed, carbon-fiber folding propeller. The UAV’s geometric and inertial properties are detailed in

Table 1. The main inertial and geometric properties of SULA90.

Parameter	Value
Mass m	12.8 kg
Body $X_b$-axis moment of inertia $J_{xx}$	0.510 kg/m2
Body $Y_b$-axis moment of inertia $J_{yy}$	1.739 kg/m2
Body $Z_b$-axis moment of inertia $J_{zz}$	2.073 kg/m2
Product of inertia about the $X_b$ and $Z_b$ axis $J_{xz}$	−0.018 kg/m2
Front wing span $b_{fw}$	$1.86\mathrm{m}$
Rear wing span $b_{rw}$	$1.34\mathrm{m}$
Vertical tail span $b_{vt}$	$0.3\mathrm{m}$
Front wing chord $c_{fw}$	$0.113\mathrm{m}$
Rear wing chord $c_{rw}$	$0.133\mathrm{m}$
Vertical tail chord $c_{vt}$	$0.069\mathrm{m}$
Front wing area $S_{fw}$	$0.21{\mathrm{m}}^2$
Rear wing span $S_{rw}$	$0.17{\mathrm{m}}^2$
Vertical tail span $S_{vt}$	$0.0514{\mathrm{m}}^2$
Body length $l_{bd}$	$1\mathrm{m}$
Front wing moment arm $l_{fw}$	$0.271\mathrm{m}$
Rear wing moment arm $l_{rw}$	$0.540\mathrm{m}$
Vertical tail moment arm $l_{vt}$	$0.468\mathrm{m}$
Vertical tail chord $c_{vt}$	$0.069\mathrm{m}$
Vertical distance of vertical tail $z_{vt}$	$0.15\mathrm{m}$
Aileron control input ${\delta }_a$ and deflection angle	$\left[-1,1\right]$, $\left[-30,30\right]deg$
Elevator control input ${\delta }_e$ and deflection angle	$\left[-1,1\right]$, $\left[-30,30\right]deg$
Rudder control input ${\delta }_r$ and deflection angle	$\left[-1,1\right]$, $\left[-30,30\right]deg$

.

4.2. Reference Values Obtained by Dynamic Derivative CFD Simulations

To verify the effectiveness of the proposed method, this paper uses the dynamic derivative CFD simulations as reference values. We calculate the dynamic derivatives in both the longitudinal and lateral directions. A forced oscillation analysis method with a simple harmonic oscillation (SHO) mode is adopted to obtain longitudinal coupled dynamic derivatives $C_{*q}+C_{*\dot{\alpha }}$ [45,60], and the quasi-steady method is used for solving the longitudinal angular rate derivative $C_{*q}$ [43], where the subscript * is L, D and m. Researchers have applied the aforementioned method for dynamic derivative CFD simulations across different flight platforms and have validated its effectiveness using wind tunnel data [46,61]. According to Equation (6), we only need to consider the angular rate derivatives for the roll and yaw directions. Therefore, steady roll and yaw movements are simulated to obtain the $C_{*p}$ and $C_{*r}$, where the subscript * can be Y, l, and n. As the UAV primarily operates in cruise flight, the CFD simulation uses a cruise flight speed of 30 m/s and an altitude of 300 m above sea level. All numerical simulations were conducted in ANSYS FLUENT 2023R1 with tailored methodologies for distinct aerodynamic derivatives. For the longitudinally coupled dynamic derivatives, a transient solver is employed, a transient pressure-based solver incorporating the Sliding Mesh Method (SMM) [62] was used to simulate UAV oscillatory motions, utilizing the realizable $k-\epsilon$ turbulence model with standard wall functions. The first-order implicit transient formulation is utilized, with the Roe-FDS flux scheme and cell-based gradient reconstruction for the least squares applied for numerical accuracy. The second-order upwind scheme is used for spatial discretization. The Courant number is set to 20, and default under-relaxation factor values are employed. A fixed-time-step-size marching scheme with the user-specified method is implemented to ensure stability and precision. In parallel, the angle rate derivatives simulations implemented a steady-state framework through the Multiple Reference Frame (MRF) [63] approach, retaining an identical turbulence model while adopting a coupled pressure–velocity solver for enhanced convergence. Spatial discretization preserved second-order upwind accuracy with identical Courant number conditioning, complemented by standardized relaxation factors.

The entire computational fluid domain is a spherical region with a diameter of 20 m. Using ANSYS Fluent Meshing 2023R1, an unstructured computational grid was generated. The process began with the creation of a triangular surface mesh, followed by the configuration of boundary layer parameters, and concluded with the generation of a poly-hexcore volume mesh. In this context, the first layer’s height and the number of boundary layers are directly linked to the $Y-plus$ value, which is determined by the turbulence model employed in the simulation. For RANS-based CFD simulations of external aerodynamic flows, three turbulence models are typically used: Spalart–Allmaras (SA), $k-\epsilon$, and $k-\omega$. The SA model is a single-equation approach and performs well for simple geometries but shows limitations in predicting shear and separated flows. The $k-\omega$ model is a two-equation approach; while offering high accuracy for boundary layer resolution, it imposes stringent grid requirements ($Yplus\le 1$ and at least 15 boundary layers). For example, achieving $Yplus=1$ on the SULA90 would require a first-layer thickness of $0.01$ mm, significantly increasing mesh density and compromising grid quality at wing–body junctions due to extreme aspect ratios. To balance accuracy and computational efficiency, the realizable $k-\epsilon$ model was selected. This model permits a relaxed $Yplus$ threshold of 30 and a reduced boundary layer count of 10 while still capturing essential boundary layer physics. The first layer’s height was set to $0.3$ mm using the last ratio offset method with a transition ratio of $0.272$. The UAV surface mesh and boundary layers on the UAV wall are depicted in Figure 5a. The far field was assigned a pressure–far-field boundary condition, while the UAV wall was defined as the no-slip wall.

The Sliding Mesh Method (SMM) is a dynamic mesh technique for or longitudinal coupled dynamic derivative simulations. The fluid domain is partitioned into static and dynamic zones, as shown in Figure 5b. The dynamic zone encompasses the mesh within a 5 m diameter inner sphere and the UAV wall, actuated by longitudinal oscillatory motions governed by a User-Defined Function (UDF) file, while the static zone extends to the far-field boundary. In contrast, angular rate derivatives simulations are steady-state simulations, only requiring a single fluid domain, though configured with corresponding rotational velocities to replicate the UAV’s uniform rotation. The longitudinal coupled dynamic derivative case utilized a volumetric mesh comprising $11.867$ million elements, whereas the angular rate derivatives case employed $9.192$ million elements. The above simulations were run on a workstation AMD EPYC 9554 series 128-core CPU, finishing in 64 h and 20 min. The detailed procedure is as follows.

(1) Longitudinal coupled dynamic derivative analysis: The AOA vibration equation is given by

$$\alpha ={\alpha }_0+{\alpha }_m\mathit{sin}\left(\omega t\right),$$

(29)

where ${\alpha }_0$ is the initial flight AOA and is set as zero, and the oscillation amplitude ${\alpha }_m$ is set as 2 deg. The equation of the oscillation angular rate is $\omega ={\displaystyle \frac{2V_{a}k}{c}}$. In the equation, the reduced frequency $k=0.02$, $V_a=30\mathrm{m}/\mathrm{s}$, so $\omega =9.23077\mathrm{rad}/\mathrm{s}$. The numerical simulations adopt a fixed time step ($\Delta t=0.02$ s) with a convergence threshold limiting iterations to 50 per step. A total of 144 time steps are calculated, thereby yielding a complete harmonic oscillation period of $2.88$ s. The results from $2.1$ to $2.88$ s were fitted using the Least Squares Method (LSM). The $C_L$ and $C_m$ fitting formula is shown in Equation (30), and the $C_D$ fitting formula is shown in Equation (31).

$$C_{*}=C_{*0}+{\displaystyle \frac{C_{*\alpha }}{{\alpha }_m}}\mathit{sin}\left(\omega t\right)+{\displaystyle \frac{C_{*q}+C_{*\dot{\alpha }}}{k{\alpha }_m}}\mathit{cos}\left(\omega t\right)(*=L,M)$$

(30)

$$C_D=C_{D0}+C_{D\alpha }\Delta \alpha +C_{D\dot{\alpha }}{\displaystyle \frac{\dot{\alpha }c}{2V}}+C_{Dq}{\displaystyle \frac{qc}{2V}}+{\displaystyle \frac{1}{2}}{C}_{D{\alpha }^2}{\left(\Delta \alpha \right)}^2$$

(31)

The fitting curves of $C_L$, $C_m$ and $C_D$ for the whole SULA90 are shown in Figure 6a–c. The fitting curves of $C_L^{fus}$, $C_m^{fus}$ and $C_D^{fus}$ for the fuselage are shown in Figure 7a–c. The detailed computational method for longitudinal coupling derivatives is presented in Appendix A.2.

(2) Longitudinal pitch angular rate derivative analysis: The AOA is also set to zero, and the pitch rate q is taken as $2.4\mathrm{rad}/\mathrm{s}$, $5\mathrm{rad}/\mathrm{s}$, $10\mathrm{rad}/\mathrm{s}$, and $15\mathrm{rad}/\mathrm{s}$. A linear fit is used to analyze how $C_L$, $C_m$ and $C_D$ change with respect to $\overline{q}$ ($\overline{q}={\displaystyle \frac{qc}{2V_a}}$), as shown in Figure 8a. The slope of the fitting line represents the pitch angular rate derivative $C_{*q}$, where the subscript * can be L, D and m.

Table 2. The CFD simulation longitudinal dynamic derivatives.

Parameter	Value	Parameter	Value	Parameter	Value
$C_{L\dot{\alpha }}+C_{Lq}$	$14.2575$	$C_{Lq}$	$12.8540$	$C_{L\dot{\alpha }}$	$1.4035$
$C_{m\dot{\alpha }}+C_{mq}$	$-136.5106$	$C_{mq}$	$-134.1540$	$C_{m\dot{\alpha }}$	$-2.3566$
$C_{D\dot{\alpha }}+C_{Dq}$	$1.5176$	$C_{Dq}$	$1.3421$	$C_{D\dot{\alpha }}$	$0.1755$
$C_{L\dot{\alpha }}^{fus}+C_{Lq}^{fus}$	$2.0445$	$C_{Lq}^{fus}$	$1.8395$	$C_{L\dot{\alpha }}$	$0.2050$
$C_{m\dot{\alpha }}^{fus}+C_{mq}^{fus}$	$-3.8531$	$C_{mq}^{fus}$	$-3.634$	$C_{m\dot{\alpha }}$	$-0.2191$
$C_{D\dot{\alpha }}^{fus}+C_{Dq}^{fus}$	$0.4619$	$C_{Dq}^{fus}$	$0.4228$	$C_{D\dot{\alpha }}$	$0.0391$

shows the detailed longitudinal dynamic derivative results for the whole UAV and the fuselage. It can be observed that the fuselage primarily affects the drag dynamic derivatives (e.g., $C_{Dq}^{fus}$ and $C_{D\dot{\alpha }}^{fus}$) while having minimal impact on the lift dynamic derivatives (e.g., $C_{Lq}^{fus}$ and $C_{L\dot{\alpha }}^{fus}$) and pitching moment dynamic derivatives (e.g., $C_{mq}^{fus}$ and $C_{m\dot{\alpha }}^{fus}$).

(3) Lateral-directional angular rate derivative analysis: The AOA is also set to zero, and the roll rate p is taken as $0.1\mathrm{rad}/\mathrm{s}$, $0.4\mathrm{rad}/\mathrm{s}$, $0.6\mathrm{rad}/\mathrm{s}$, and $1\mathrm{rad}/\mathrm{s}$ for a total of four simulations. A linear fit is used to analyze how $C_l$ and $C_n$ change with respect to $\overline{p}$ ( $\overline{p}={\displaystyle \frac{pb}{2V_a}}$ ), as shown in Figure 8b. The slope of the fitting line represents the roll angular rate derivatives, so we obtain $C_{lp}=-0.7513$ and $C_{np}=-0.11066$. The AOA and yaw rate r are the same as roll angular rate derivative analysis. A linear fit is used to analyze how $C_Y$, $C_l$ and $C_n$ change with respect to $\overline{r}$ ( $\overline{r}={\displaystyle \frac{rb}{2V_a}}$ ), as shown in Figure 8c. The slope of the fitting line represents the yaw angular rate derivatives, so we obtain $C_{lr}=0.5491$ and $C_{nr}=-0.1106$ and $C_{Yr}=0.6888$.

4.3. Comparison Results

Research indicates that the accuracy of dynamic derivatives depends on the chosen control method. The aircraft primarily employs two control methods: open-loop and closed-loop control. Compared to closed-loop control, open-loop control demands higher precision [22,64]. W. B. Blake [30] summarized the accuracy criteria for both methods in

Table 3. Dynamic derivative accuracy criteria [30].

Parameter	Open Loop	Closed Loop
$C_{*q}\left(*=L,D,m\right)$	$25\%$	$50\%$
$C_{*\dot{\alpha }}\left(*=L,D,m\right)$	$25\%$	$50\%$
$C_{lp}$	$0.5$/rad	$0.2$/rad
$C_{np}$	$0.05$/rad	$0.2$/rad
$C_{lr}$	$0.1$/rad	$0.2$/rad
$C_{nr}$	$0.1$/rad	$0.2$/rad

. Since the modern UAV primarily utilizes closed-loop control, the proposed method meets the precision requirements for closed-loop control, which means it is suitable for engineering applications. So, the relative difference of the longitudinal dynamic derivative should not exceed $50\%$, and the absolute difference of the lateral-directional derivative should not exceed $0.2$/rad.

The dynamic derivative calculations in the CEHM method require three types of aerodynamic parameters, which are derived from static derivatives CFD simulation results. The CFD simulations used in the CEHM are presented in Appendix A.3, while the calculation process and results of the three types of parameters are provided in Appendix A.4 and the detailed results of the static derivatives CFD simulations are provided in Appendix A.5. To validate the effectiveness of the CEHM method, two widely used aerodynamic estimation software tools were employed for comparison: XFLR5 (version 6.61) and OpenVSP (version 3.41.2). Since the CEHM method omits the influence of the fuselage when estimating dynamic derivatives, XFLR5 and OpenVSP also neglect fuselage effects to ensure consistency in the calculations. The simulation models for these tools are shown in Figure 9. The computation times for CEHM and XFLR5 are on the order of seconds, while OpenVSP requires minutes. The CFD simulations employed as input for CEHM required 24 h and 46 min, equating to a quarter of the computational duration entailed by traditional full-CFD methods. The dynamic derivative results calculated at zero AOA are summarized in

Table 4. The dynamic derivatives of different methods.

Param.	CFD	CEHM	AD	RD %	XFLR5	AD	RD %	OpenVSP	AD	RD %
$C_{Lq}$	$11.0145$	$12.157$	$1.14$	$10.37$	$14.245$	$3.231$	$29.33$	$14.058$	$3.044$	$27.63$
$C_{mq}$	$-130.52$	$-136.541$	$6.021$	$4.61$	$-148.24$	$17.720$	$13.58$	$-146.050$	$15.530$	$11.90$
$C_{Dq}$	$0.9193$	$0.7321$	$0.187$	$20.36$	NAN	NAN	NAN	$0.4771$	$0.442$	$48.10$
$C_{L\dot{\alpha }}$	$1.1985$	$1.5819$	$0.383$	$31.99$	NAN	NAN	NAN	NAN	NAN	NAN
$C_{m\dot{\alpha }}$	$-2.1375$	$-3.7354$	$1.598$	$74.76$	NAN	NAN	NAN	NAN	NAN	NAN
$C_{Y_r}$	$0.6888$	$0.448$	$0.241$	$34.96$	$0.1211$	$0.568$	$82.42$	$0.1170$	$0.572$	$83.01$
$C_{lp}$	$-0.7513$	$-0.6371$	$0.114$	$15.20$	$-0.49469$	$0.257$	$34.16$	$-0.4997$	$0.252$	$33.49$
$C_{lr}$	$0.5491$	$0.3668$	$0.182$	$33.20$	$0.12706$	$0.422$	$76.86$	$0.1128$	$0.436$	$79.46$
$C_{np}$	$-0.1106$	$-0.0891$	$0.022$	$19.44$	$-0.06131$	$0.049$	$44.57$	$-0.05025$	$0.060$	$54.57$
$C_{nr}$	$-0.22131$	$-0.116$	$0.105$	$47.58$	$-0.02977$	$0.192$	$86.55$	$-0.03737$	$0.184$	$83.11$

. Using the CFD simulation results as the reference values, we evaluate the effectiveness of the CEHM method by calculating the relative differences for longitudinal dynamic derivatives and the absolute difference for lateral-directional dynamic derivatives.

The relative differences (RDs) for the dynamic derivatives are summarized in Table 4. To facilitate a more meaningful comparison, the dynamic derivatives in the CFD calculation results provided in the table are adjusted by subtracting the fuselage effects according to Table 2. Given the relatively minor influence of the fuselage on lateral-directional dynamic derivatives, the CFD-derived lateral-directional dynamic derivatives presented in the table are based on full-aircraft computational results. For $C_{Lq}$, the relative differences for the CEHM method remain below $25\%$, satisfying the stringent criteria for open-loop control. The relative differences for XFLR5 and OpenVSP closely approach $25\%$. Regarding $C_{mq}$, all three methods exhibit relative differences within $15\%$, with the CEHM method achieving the highest precision at a relative difference of $4.61\%$. While XFLR5 is incapable of computing $C_{Dq}$, OpenVSP demonstrates a substantial relative discrepancy of $48.1\%$, which aligns with closed-loop control prerequisites. In contrast, the CEHM method yields a relative difference of $20.36\%$ for $C_{Dq}$, which meets the open-loop control requirement. Furthermore, XFLR5 and OpenVSP lack the capability to compute acceleration derivatives such as $C_{L\dot{\alpha }}$ and $C_{m\dot{\alpha }}$, which are critical for simulating high-maneuverability flight dynamics. The CEHM method, conversely, not only calculates these acceleration derivatives but also attains an acceptable accuracy. For $C_{Yr}$, both XFLR5 and OpenVSP exhibit relative errors exceeding $80\%$, whereas the CEHM method demonstrates the smallest relative difference at $34.96\%$, accompanied by an absolute difference of $0.2408$/rad, as illustrated in Table 4; this value approaches the permissible tolerance range. For $C_{lr}$ and $C_{lp}$, only the CEHM method produces results within the required accuracy. For $C_{nr}$ and $C_{np}$, although all three methods meet the requirements, the CEHM method achieves the highest accuracy, with an absolute difference of about half of the results from XFLR5 and OpenVSP.

4.4. Open-Loop Validation of the Models Based on Field Flight

4.4.1. Validation Process and Metrics

To further validate the effectiveness of the method proposed in this paper, we employed an open-loop model verification approach based on field flight data [54], with the framework shown in Figure 10. We use the open-source Flight Dynamic Model software (FDM) code JSBSim [65,66] to develop the CEHM method FDM named the CEHM plant, which uses CFD simulation results for static and control derivatives; the detailed derivatives are in Appendix A.4. Still, its dynamic derivatives are obtained using the CEHM method. Two pitch and roll maneuver data segments from the flight test data are selected for individual validation. The control surface deflections from these data segments are used as inputs to the FDM. The inputs are fed into the CEHM plant, and the predicted angular rates are compared with the measured data. The pitch control surface output in the flight test data is denoted as ${\delta }_e^{re}$, the roll control surface output is ${\delta }_a^{re}$, the recorded pitch angular rate is $q_{re}$, and the recorded roll angular rate is $p_{re}$. In the predicted outputs from the FDM, the predicted pitch angular rate is denoted as $q_{pr}$ and the predicted roll angular rate as $p_{pr}$. The control surface deflections for roll and yaw should be 0 or near 0 in the pitch maneuver flight data segment to minimize interference from different control inputs. Similarly, the control surface deflections for pitch and yaw should also be 0 or near 0 for the roll maneuver flight data segment.

The goodness of fit between the model’s predicted output and the measured values is evaluated using the coefficient of determination ($R^2$) and the root mean square error ($RMSE$) [67], as shown in Equations (31) and (32), respectively. A value of $R^2$ closer to 1 indicates a better fit, while values closer to 0 suggest a poor fit. Similarly, an $RMSE$ value closer to zero indicates that the model’s predicted values are closer to the measured data, reflecting a better model fit.

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(X_i-Y_i\right)}^2}{{\sum }_{i=1}^n{\left(\overline{Y}-Y_i\right)}^2}}$$

(32)

$$\mathit{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(X_i-Y_i\right)}^2},$$

(33)

where $X_i$ is the predicted ith value and $Y_i$ is the actual ith value. $\overline{Y}$ is the mean of the actual values, and $\overline{Y}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{Y}_i$. n is the total number of the values.

4.4.2. Field Flight Data Collection

The electrical components of SULA90 are shown in Figure 11. It is equipped with an onboard Pixhawk autopilot named CUAV X7+ Pro. The autopilot is running PX4 firmware and records the flight data in ulog form. The sensor package includes a Global Positioning System (GPS) receiver and an airspeed sensor. Pulse width modulation (PWM) control signals applied to the control surface actuators are recorded and mapped using a servo-actuator model to control surface deflection angles. The sample rate for the translational acceleration and angular rate measurements is 200 Hz. The sample rate for the EKF-derived data is 100 Hz. The control effector PWM commands and the propeller rotational speed are sampled at 50 Hz.

The field flight used for this paper is only conducted in negligible wind conditions. The test adopts a taxi takeoff method, and the total test flight time is 6 min 42 s. We set a rectangular flight route to collect flight data for model validation, as shown in Figure 12, and it should be noted that the entire flight process is not shown in the figure. The pitch maneuver data segment is selected from the steady-level flight phase, where the roll and yaw deflections are small. The roll maneuver data segment is selected from the turn flight phase, with small pitch and yaw deflections.

4.4.3. The Model Validation Results

A 20 s flight data segment, from 163 s to 183 s, was extracted for roll maneuver analysis. The aileron control input ${\delta }_a^{re}$ is shown in Figure 13a, and two lateral maneuvers with turns can be observed. As illustrated in Figure 13b, the results of the CEHM plant are acceptable. The RMSE between $p_{pr}$ and $p_{re}$ is $0.102\mathrm{rad}/\mathrm{s}$, and the $R^2$ is $0.7829$. A segment data with relatively constant throttle output is used to minimize the influence of throttle input variations, as shown in Figure 14a. The input is a total of 30 s of steady straight flight data, from 225 s to 255 s of total flight time, with an average flight speed of $30\mathrm{m}/\mathrm{s}$, as well as the elevator control amount ${\delta }_e^{re}$ from the flight test data, as shown in Figure 14b. The model prediction value $q_{pr}$ matches well with the flight record value $q_{re}$, as shown in Figure 14c. The RMSE between $q_{pr}$ and $q_{re}$ is $0.0175\mathrm{rad}/\mathrm{s}$, and the $R^2$ is $0.7550$. The results indicate that the CEHM method is effective.

5. Conclusions

This paper presents a method named CEHM for calculating the dynamic derivatives of small tandem-wing UAVs, optimizing both computational efficiency and accuracy. This method derives engineering estimation formulas for the longitudinal and lateral-directional dynamic derivatives, tailored to the unique configuration of the tandem-wing aircraft. Furthermore, three types aerodynamic parameters were extracted from static CFD simulations to enhance the accuracy. In contrast to the lengthy computational times (tens of hours) required by CFD methods, the proposed method delivers results in seconds. Compared to the traditional aerodynamic estimation tools XFLR5 and OpenVSP, the CEHM method calculates the acceleration derivatives and provides higher accuracy. The proposed method yields results strongly in agreement with CFD simulations, particularly for the longitudinal and lateral angular rate derivatives. Although the accuracy for acceleration and yaw angular rate derivatives is slightly lower, their effect on aerodynamic coefficients is minimal, thus ensuring that the method meets the modeling requirements in the design phase. To comprehensively validate the method’s effectiveness, we use a 10 kg class tandem-wing UAV as the research object, and open-loop verification of the FDM developed by the CEHM method was performed using flight test data. Steady-level flight data with pitch control surface deflection were extracted to validate the longitudinal model, while turning flight data with roll control surface deflection were used to validate the lateral model. The angular rates’ values predicted with the CEHM method align well with the measured data. The results demonstrate that the FDM has significant model prediction capability and further validate the effectiveness of the CEHM method.

The method described in this paper also applies to the dynamic derivative calculations of traditional aircraft configurations. Future work will focus on incorporating the effects of the airframe on dynamic derivatives to enhance the accuracy of acceleration and yaw moment derivatives. Additionally, the influence of the propeller on dynamic derivatives will be integrated into the estimation formulas.