2.1 6-DOF governing equations and physical descriptions

The six degree-of-freedom (6-DOF) dynamics of a UAV are formulated from the fundamental principles of Newtonian mechanics. This framework accounts for translational motion along the three orthogonal axes as well as their rotational behaviours, thereby encompassing the complete rigid-body dynamics of the system (Schmidt [Reference Schmidt44]; Brandt et al. [Reference Brandt, Stiles, Bertin and Whitford45]). The governing equations are expressed as follows:

(1) \begin{align}\left[ {\begin{array}{c}{{F_x}}\\[5pt] {{F_y}}\\[5pt] {{F_z}}\end{array}} \right] = \left[ {\begin{array}{c}{{F_{{x_A}}} + {F_{{x_W}}} + {F_{{x_T}}}}\\[5pt] {{F_{{y_A}}} + {F_{{y_W}}}{\rm{\,\,\,\,\;}}}\\[5pt] {{F_{{z_A}}} + {F_{{z_W}}}{\rm{\,\,\,\,\;}}}\end{array}} \right] = m\left[ {\begin{array}{c}{\dot u}\\[5pt] {\dot v}\\[5pt] {\dot w}\end{array}} \right] + \left[ {\begin{array}{c}{m({qw - rv})}\\[5pt] {m({ru - pw})}\\[5pt] {m({pv - qu})}\end{array}} \right]\!,\end{align}

(2) \begin{align}\left[ {\begin{array}{c}{{M_x}}\\[5pt] {{M_{}}}\\[5pt] {{M_z}}\end{array}} \right] = \left[ {\begin{array}{c}{{M_{{x_A}}}}\\[5pt] {{M_{{y_A}}}}\\[5pt] {{M_{{z_A}}}}\end{array}} \right] = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{{I_{xx}}} & {}{ - {I_{xy}}} {}& { - {I_{xz}}}\\[5pt] { - {I_{xy}}}& {}{{I_{yy}}}& {}{ - {I_{yz}}}\\[5pt] { - {I_{xz}}}& {}{ - {I_{yz}}}& {}{{I_{zz}}}\end{array}} \right]\left[ {\begin{array}{c}{\dot p}\\[5pt] {\dot q}\\[5pt] {\dot r}\end{array}} \right] + \left[ {\begin{array}{c}{r( {{I_{xy}}p - {I_y}q + {I_{yz}}r} ) - q( {{I_{xz}}p + {I_{yz}}q - {I_z}r} )}\\[5pt] {p( {{I_{xz}}p + {I_{yz}}q - {I_z}r} ) - r( {{I_{xy}}q - {I_x}q + {I_{xz}}r} )}\\[5pt] {q( {{I_{xy}}q - {I_x}p{ + _{xz}}r} ) - p( {{I_{xy}}p - {I_y}q + {I_{yz}}r} )}\end{array}} \right]\!.\end{align}

In this formulation, ${F_x}$ , ${F_y}$ and ${F_z}$ correspond to the external forces imposed on the UAV along the body-fixed $x$ , $y$ and $z$ directions, respectively. The velocity components aligned with these axes are denoted by $u$ , $v$ and $w$ , while the overdot notation designates their associated linear accelerations. In parallel, the angular velocity components about the three axes are represented by $p$ , $q$ and $r$ , with their time derivatives describing the corresponding angular accelerations. The aerodynamic and control-induced moments acting about the $x$ , $y$ and $z$ axes are denoted by ${M_x}$ , ${M_y}$ and ${M_z}$ , respectively. The rotational inertia distribution is encapsulated in the inertia tensor, which is a symmetric second-order matrix comprising nine terms that quantify the coupling of angular momentum with respect to the three principal axes. The force components ${F_x}$ , ${F_y}$ and ${F_z}$ can each be decomposed into aerodynamic ( ${F_{{ - _A}}}$ ) and gravitational ( ${F_{{ - _W}}}$ ) contributions, while ${F_x}$ additionally includes a thrust component ( ${F_{{x_T}}}$ ) generated by the propulsion system. In contrast, the moments ${M_x}$ , ${M_y}$ and ${M_z}$ arise solely from aerodynamic effects.

These governing relations provide a unified mathematical structure that captures the full spectrum of the motion of an aircraft, encompassing translational displacements and rotational dynamics in concert. Solving these relations yields three linear velocity components – longitudinal, lateral and vertical components – together with the three angular rates – roll, pitch and yaw rates. Taken together, these quantities specify the instantaneous state of the trajectory and attitude of the aircraft in space.

In this study, a mathematical formulation of 6-DOF aircraft dynamics is established within the body-axis coordinate framework, where the coordinate system is rigidly attached to the aircraft and moves synchronously with it. Unlike the inertial-axis system, which remains stationary relative to the Earth, the body-axis system provides several distinct advantages for modeling flight mechanics. First, the inertia tensor remains constant in this framework, thereby simplifying the calculation of rotational dynamics by avoiding time-dependent mass distribution variations. Second, aerodynamic variables such as the angle-of-attack and sideslip angle can be extracted directly, without the need for additional geometric transformations. Third, thrust forces are naturally aligned with the longitudinal axis of the aircraft, ensuring that propulsion is represented in a physically consistent manner. These benefits collectively increase the computational efficiency and fidelity of the system, making the body-axis framework particularly suitable for flight simulations and control system designs (Schmidt [Reference Schmidt44]).

However, the use of a noninertial coordinate system introduces several additional complexities that must be accounted for. The governing equations must incorporate the inertial and Coriolis terms that arise from the rotating frame of reference. Furthermore, body-axis velocity components, while convenient for aerodynamic analyses, must be transformed into ground-relative speeds expressed in the inertial frame to accurately describe flight trajectories with respect to the Earth. Similarly, gravitational forces, which act in a fixed direction in the inertial frame, must be reoriented according to the instantaneous aircraft attitude in the body frame. The angular velocities describing roll, pitch and yaw motions must also be appropriately converted in the inertial system to preserve the correct coupling with external torques. Moreover, while aerodynamic forces are often computed in the body frame, they require transformations in the stability-axis system to provide precise decompositions into lift, drag and side forces, which are crucial for aerodynamic performance evaluations.

To address these requirements, a set of transformation matrices is employed to link the stability, body and inertial coordinate systems (Schmidt [Reference Schmidt44]). The first transformation matrix projects body-axis quantities, such as forces and velocities, into the Earth-fixed inertial frame by accounting for the instantaneous roll, pitch and yaw angles. The second matrix connects the stability axis and body axis, thereby enabling the aerodynamic forces to be conveniently represented in the body frame through the incorporation of the angle-of-attack. The third matrix converts angular velocities expressed in the body frame to the inertial frame by accounting for roll and pitch orientations. The coordinated use of these transformations ensures that all relevant physical quantities – linear velocities, angular rates, aerodynamic forces and applied moments – are consistently and accurately mapped across the required reference frames. This comprehensive approach provides a robust and elegant mathematical foundation for simulating the full dynamic behaviour of aircraft, thereby supporting both theoretical analyses and practical control system implementations.

In the present study, the baseline UAV engine power is first determined from the maximum rated power, $H{P_{{\rm{max}}}}$ , scaled by the throttle setting, ${\delta _{th}}$ , to represent the commanded power output of the propulsion system. This preliminary power level is subsequently corrected to account for power degradation associated with reduced air density at the operating flight altitude through the density ratio, $\sigma $ , yielding the effective available shaft power $H{P_{{\rm{max}}}}\,{\delta _{th}}\,\sigma $ . The altitude-corrected engine output is then converted into usable propulsive force by incorporating the propeller efficiency, $e$ , which accounts for aerodynamic losses in the power-to-thrust conversion process. Finally, the net thrust available for aircraft acceleration and forward motion is evaluated as ${F_{{x_T}}} = e\,550\,HP/{V_{TAS}}$ , where the factor 550 represents the unit conversion between horsepower and foot-pounds per second, and ${V_{TAS}}$ denotes the true airspeed. This formulation provides a physically consistent representation of the coupling between engine performance, propeller aerodynamics and flight speed in the propulsion model.

2.2 Geometric modeling of the X-wing configuration

The investigated X-wing configuration is presented in Fig. 3, highlighting both its fixed and variable geometric parameters. The yellow square outlines the boundary as viewed from the X-tail, while the green square delineates the boundary from the X-wing perspective. Within this framework, ${X_{ac}}$ denotes the aerodynamic centre, which is defined as the point along the mean chord where the aerodynamic pitching moment remains essentially invariant with changes in the angle-of-attack. In contrast, ${X_{cg}}$ identifies the position of the centre of gravity of the aircraft. The longitudinal separation between these two reference points is critically important, as both its magnitude and sign govern the inherent tendency of the aircraft toward static longitudinal stability or instability. To facilitate comparisons across different configurations, this distance is normalised by the mean aerodynamic chord ( $\bar c$ ), producing a nondimensional parameter that is expressed in a percentage form as follows: $[{({{X_{ac}} - {X_{cg}}})/\bar c}] \times 100\% $ . A positive ratio value, as employed in the present study, indicates that the centre of gravity lies ahead of the aerodynamic centre, thereby ensuring that the aircraft maintains longitudinal static stability. The geometric and physical parameters, together with the reference geometry and inertial properties adopted in this study for the front-wing and rear-tail configurations with combined anhedral–dihedral angles of 50°, 60°, 70°, 80° and 90°, are presented in Tables 1 and 2, respectively. It is important to note that the reference area ${S_{ref}}$ , defined as the projected surface area of the X-wing configuration onto the $x$ – $y$ plane, is maintained at a constant value for all combined anhedral–dihedral angles of 50°, 60°, 70°, 80° and 90° listed in Table 1. This consistent definition ensures a fair and meaningful comparison of aerodynamic performance across different geometric configurations. Additionally, in Table 1, as the combined dihedral and anhedral angles increase, the centre-of-gravity location $({{X_{cg}}})$ is adjusted to maintain a consistent static stability margin. Specifically, both the centre of gravity and the aerodynamic centre $({{X_{ac}}})$ shift rearward, ensuring that the aircraft preserves a similar level of static stability across different wing configurations. This coordinated adjustment allows comparable longitudinal stability characteristics to be maintained despite variations in geometric arrangement.

Figure 3.

Illustration of the studied X-wing configuration with both fixed and variable geometric parameters. The yellow square indicates the boundary from the X-tail perspective, while the green square denotes the boundary from the X-wing perspective.

Table 1.

Geometric and physical parameters adopted in this study for front wing and rear tail configurations with combined anhedral–dihedral angles of 50°, 60°, 70°, 80° and 90°

Table 1 Long description

A table with five rows and seven columns comparing geometric and physical parameters for front wing and rear tail configurations at angles of 50, 60, 70, 80, and 90 degrees. The columns are labeled as follows: 50, 60, 70, 80, 90, and Unit. The rows are labeled as follows: c̄, Sref, b, Xac, and Xcg. Row 1: c̄, 0.87, 0.87, 0.87, 0.87, 0.87, ft. Row 2: Sref, 4.707, 4.707, 4.707, 4.707, 4.707, ft squared. Row 3: b, 5.41, 5.41, 5.41, 5.41, 5.41, ft. Row 4: Xac, 76.85, 81, 85.1, 88.37, 88.9, percent. Row 5: Xcg, 60, 65, 70, 73, 73, percent.

Table 2.

Reference geometry and inertial properties adopted in this study for front wing and rear tail configurations with combined anhedral–dihedral angles of 50°, 60°, 70°, 80° and 90°

Table 2 Long description

The table presents geometric and inertial properties for front wing and rear tail configurations with combined anhedral/dihedral angles of 50, 60, 70, 80, and 90 degrees. It has 8 rows and 6 columns. The columns are labeled with the angles and the units. The rows are labeled with different properties: m, g, E, HP_max, I_xx, I_xy, I_xz, I_yy, I_yx, I_yz, and I_z. Each row provides values for each angle. For example, the first row under the column 50 degrees shows m as 0.8216 slug, g as 32.2 ft/s^2, E as 0.73, HP_max as 4 hp, I_xx as 1.46 slug*ft^2, I_xy as 0 slug*ft^2, I_xz as 0.18 slug*ft^2, I_yy as 5.46 slug*ft^2, I_yx as 0 slug*ft^2, I_yz as 0 slug*ft^2, and I_z as 6.38 slug*ft^2.

The mechanisms governing STT and BTT can be understood through their distinct aerodynamic pathways for generating centripetal acceleration, as shown in Fig. 1. In the STT mode, increasing the projected lateral area of the vehicle enhances the derivative of the lateral force with respect to the sideslip angle ( ${C_{{Y_\beta }}}$ ), thereby amplifying the lateral force coefficient ( ${C_Y}$ ). This stronger lateral force produces greater centripetal acceleration ( ${a_c}$ ), which in turn reduces the turning radius ( $R$ ). In contrast, in the BTT mode, the vehicle achieves turning through an increased roll angle (ψ), which effectively redirects the lift vector to generate greater centripetal acceleration. As with the STT mode, the increase in ${a_c}$ leads to a corresponding decrease in the turning radius. Although both mechanisms ultimately achieve tighter manoeuvring by enhancing the centripetal force, STT relies on an augmented side force generation procedure, whereas BTT exploits lift vector tilting through controlled banking.

Interestingly, the present X-wing configuration introduces a remarkable degree of agility by enabling flexible manoeuvring through the coordinated mixing of the control surface deflections observed across its three tail surfaces – elevator, aileron and rudder. Unlike conventional UAVs, which are constrained to relatively simple control strategies – such as BTT, which is achieved by rotating one aileron upward while deflecting the opposite aileron downward, or STT, which is executed by synchronously deflecting the rudder and/or elevator – the X-wing design substantially expands the manoeuvring envelope. The deflection of an individual control surface $({di})$ is obtained by algebraically combining the three commanded inputs – elevator, aileron and rudder – with appropriate sign conventions: $di = \pm da \pm de \pm dr$ , where the $ \pm $ signs indicate that each command may be added or subtracted depending on the specific surface location and the adopted control-mixing scheme, as shown in Fig. 2. In addition, each control input is limited to a maximum magnitude of ${20^ \circ }$ to ensure that the resulting surface deflection does not exceed the mechanical constraints of the actuator. By blending the control inputs across multiple surfaces, the vehicle can distribute aerodynamic loads more effectively, reduce the degree of control saturation and achieve smoother trajectory shaping. This capability not only enhances the controllability of the path toward a target destination when precision manoeuvres are executed but also introduces redundancy and robustness during control allocation, ensuring stable operations even under adverse aerodynamic conditions. Surprisingly, this mixing strategy allows the X-wing to exploit cross-axis coupling in beneficial ways, offering manoeuvre combinations that are simply unattainable in the conventional fixed-wing UAV architectures. Such versatility underscores the potential of the X-wing configuration as a next-generation platform that is capable of highly adaptive flight behaviour.

2.3 Aerodynamic coefficients

Flight mechanics modeling is built on nondimensional aerodynamic coefficients (Schmidt [Reference Schmidt44]; Brandt et al. [Reference Brandt, Stiles, Bertin and Whitford45]) that are referenced to the dynamic pressure ${q_{dp}} = 1/2\rho V_{TAS}^2$ and geometry $({{S_{ref}},\bar c,b})$ . The forces include ${C_L} = {F_{li}}/{q_{dp}}{S_{ref}}$ , ${C_D} = {F_{dg}}/{q_{dp}}{S_{ref}}$ , and ${C_Y} = {F_{sf}}/{q_{dp}}{S_{ref}}$ , whereas the moments are ${C_m} = {M_{{C_m}}}/{q_{dp}}{S_{ref}}\bar c$ (pitch), ${C_l} = {M_{{C_l}}}/{q_{dp}}{S_{ref}}b$ (roll), and ${C_n} = {M_{{C_n}}}/{q_{dp}}{S_{ref}}b$ (yaw). In the linearised small-disturbance form, each coefficient is expanded about a trim state in terms of state and control derivatives – for example,

(3) \begin{align}{C_L} \cong {C_{L0}} + {C_{{L_\alpha }}}\alpha + {C_{{L_q}}}\frac{{{q_{dp}}\bar c}}{{2{V_{TAS}}}} + {C_{{L_{de}}}}de,\end{align}

(4) \begin{align}{C_Y} \cong {C_{{Y_\beta }}}\beta + {C_{{Y_p}}}\frac{{pb}}{{2{V_{TAS}}}} + {C_{{Y_r}}}\frac{{rb}}{{2{V_{TAS}}}} + {C_{{Y_{dr}}}}dr,\end{align}

with analogous expressions for ${C_D}$ , ${C_m}$ , ${C_l}$ and ${C_n}$ . These derivatives quantify how the angle-of-attack $\alpha = ta{n^{ - 1}}({w/u})$ ; sideslip $\beta = ta{n^{ - 1}}({v/\sqrt {{u^2} + {w^2}}})$ ; and body rates $p$ , $q$ and $r$ reshape the aerodynamic forces and moments, whereas the control effectiveness terms (the elevator $de$ , aileron $da$ , rudder $dr$ ) map the pilot/autopilot inputs to dynamic responses, as shown in Fig. 4. Coefficients may be defined in the stability axis (convenient for $\alpha ,\;\beta $ ) and transformed to the body axis for use in the 6-DOF equations, ensuring consistent coupling with the mass and inertia. Together, this coefficient set underpins performance metrics (e.g. ${F_{li}}/{F_{dg}}$ ), stability characteristics (e.g. the static margin via ${C_{{m_\alpha }}}$ ), and manoeuvring modes (e.g. STT via ${C_{{Y_\beta }}}$ , ${C_{{n_\beta }}}$ and BTT via lift-tilt and ${C_{{l_{da}}}}$ ), providing a compact yet comprehensive interface between aerodynamics, controls and flight dynamics. The aerodynamic parameters employed in this study are derived using the USAF Digital DATCOM (https://www.pdas.com/datcom.html) and are listed in the nomenclature section.

Figure 4.

Aerodynamic coefficients (Schmidt [Reference Schmidt44]; Brandt et al. [Reference Brandt, Stiles, Bertin and Whitford45]) employed in this work. The terms highlighted in purple are state-/rate-dependent (dynamic) quantities. The entries in green depend on control surface deflections – elevator (Δe or de), aileron (Δa or da) and rudder (Δr or dr). The coefficients shown in red are predominantly set by the airframe geometry, including the wing, fuselage and vertical/horizontal tail shapes. A comprehensive description of the aerodynamic coefficients is included in the nomenclature section. Any variable indicated by an overdot denotes the first-order derivative of that quantity with respect to time.

Although the applicability of the USAF Digital DATCOM semi-empirical tool under the present operating conditions may be subject to certain limitations, its use in this study provides an efficient and practical approach for obtaining preliminary aerodynamic predictions and design guidance. In particular, DATCOM facilitates rapid assessment of the effects of dihedral and anhedral angles on the lateral stability derivative ${C_{{Y_\beta }}}$ , which plays a key role in generating lateral aerodynamic forces and reducing turning radius. This capability supports timely and well-informed decision-making during the early stages of X-wing UAV development. Moreover, the application of the USAF Digital DATCOM tool substantially reduces the time and financial resources required to establish reliable design guidelines when compared with the development and operation of dedicated experimental facilities. As a result, the proposed methodology represents a cost-effective and technically robust framework for preliminary aerodynamic evaluation and configuration optimisation of X-wing UAVs.

The classification of the aerodynamic force and moment coefficients for a UAV in 6-DOF motion is illustrated in Fig. 5. The blue components represent the forces acting on the airframe, including the lift coefficient ${C_L}$ , drag coefficient ${C_D}$ and side-force coefficient ${C_Y}$ , together with their derivatives with respect to the angle-of-attack, sideslip angle, angular velocities and control surface deflections. The green components represent the moments acting on the airframe, including the rolling moment coefficient ${C_l}$ , pitching moment coefficient ${C_m}$ and yawing moment coefficient ${C_n}$ , along with the corresponding stability and control derivatives. The orange symbols $p$ , $q$ and $r$ denote the angular velocity components of the airframe, corresponding to roll, pitch and yaw motions, respectively.

Figure 5.

Aerodynamic coefficients referenced to their respective body/stability axes.

To generate the aerodynamic coefficients of the X-wing aircraft, this study employs the USAF Digital DATCOM, using a detailed set of flight condition inputs, geometric definitions and control surface configurations. The $FLTCON block specifies the operating environment for the aerodynamic sweep, including a single Mach number (M = 0.1), an altitude (10,000 ft) and a 20-point angle-of-attack schedule ranging from −7° to 11°, enabling the DATCOM to compute the lift, drag and pitching moment characteristics over the relevant low-speed flight regime. The $SYNTHS section defines the aircraft-level reference geometry and mass properties, including the longitudinal and vertical positions of the centre of gravity and the leading-edge reference locations of the wing, horizontal tail and vertical tail. The fuselage shape is described through the $BODY block, where six longitudinal stations with corresponding radii and upper/lower offsets reconstruct a streamlined, symmetric body that is suitable for stability derivative estimation. Similarly, the wing and tail surfaces are defined using $WGPLNF and $HTPLNF, respectively, which prescribe platform parameters such as span, chord, dihedral and aerofoil characteristics. The effectiveness of the control surface is modeled using the $SYMFLP and $ASYFLP blocks, which specify symmetric (elevator-like) and differential (aileron-like) flap schedules from −20° to +20°, allowing the DATCOM to compute the associated aerodynamic increments and stability/control derivatives. Collectively, these parameter blocks enable the DATCOM to output a complete set of aerodynamic coefficients – including static stability derivatives, control surface effectiveness coefficients and force/moment characteristics – that form the foundation for the subsequent flight dynamic modeling process and performance analysis of the X-wing UAV. The resulting aerodynamic coefficients, along with the stability, control and dynamic derivatives for the present X-wing configuration, are summarised in Tables 3 and 4.

Table 3.

Parameters of aerodynamic coefficients and stability derivatives obtained from DATCOM for the present X-wing configuration, evaluated for combined anhedral–dihedral angle settings of $50$ °, $60$ °, $70$ °, $80$ ° and $90$ °, and applied to both the front wings and rear tails

Table 3 Long description

A table comparing aerodynamic coefficients and stability derivatives for an X-wing aircraft at different anhedral-dihedral angle settings. The table is divided into two sections: Aerodynamic coefficients and Stability derivatives. The Aerodynamic coefficients section has 7 rows and 6 columns, including headers. The columns are labeled with angle settings: 50, 60, 70, 80, and 90 degrees, and Unit. The rows are labeled with different coefficients: C_L alpha, C_D zero, K, alpha_max, and C_m zero. The Stability derivatives section has 4 rows and 6 columns, including headers. The columns are labeled with angle settings: 50, 60, 70, 80, and 90 degrees, and Unit. The rows are labeled with different stability derivatives: C_Y beta, C_l p, C_l r, and C_n r. Each cell contains specific values corresponding to the angle settings and units.

Table 4.

Parameters of control and dynamic derivatives obtained from DATCOM for the present X-wing configuration, evaluated for combined anhedral–dihedral angle settings of $50$ °, $60$ °, $70$ °, $80$ ° and $90$ °, and applied to both the front wings and rear tails

Table 4 Long description

The table presents control and dynamic derivatives for an X-wing configuration evaluated at angles of 50, 60, 70, 80, and 90 degrees. The table is divided into two sections: Control derivatives and Dynamic derivatives. Each section contains multiple rows and columns with specific values for each angle. Control derivatives include parameters like C_T_delta, C_D_delta, C_Y_delta, and others, measured in degrees inverse. Dynamic derivatives include parameters like C_L_q, C_m_q, C_Y_q, and others, measured in radians inverse. Each row represents a different derivative, and each column represents a different angle. The values in the table indicate the specific derivative values at each angle.

In Table 3, as the combined dihedral and anhedral angles increase, the zero–angle-of-attack lift coefficient ${C_{{L_0}}}$ exhibits a steady rise. This trend is primarily attributed to the larger inter-wing inclination, which modifies the aerodynamic interaction between the fore and aft wings. Specifically, the effective aspect ratio increases, leading to a reduction in mutual downwash interference and an improvement in overall lift efficiency. Simultaneously, the lift-curve slope ${C_{{L_\alpha }}}$ also increases, indicating that wing configurations with larger inclination angles enhance the sensitivity of lift generation to variations in angle-of-attack. Moreover, increasing the combined dihedral and anhedral angles enlarges the lateral projected area of the wings. Under sideslip conditions, the airflow acting on these more ‘verticalized’ wing surfaces generates stronger lateral aerodynamic forces. Consequently, the absolute value of the sideslip derivative ${C_{{Y_\beta }}}$ increases, reflecting an enhanced lateral-force response to sideslip disturbances.

In Table 4, as the combined dihedral and anhedral angles increase, the effectiveness of longitudinal control (elevator) and roll control remains relatively stable. In contrast, the effectiveness of directional control (rudder) and the associated aerodynamic drag increase significantly. This trend indicates that larger inter-wing inclination angles enhance the sensitivity of yaw control, making the rudder response more pronounced. However, this improvement in control authority is accompanied by increased yaw-related drag, which may adversely affect aerodynamic efficiency.

In this study, USAF Digital DATCOM is adopted as the primary tool for aerodynamic modeling, motivated by several key technical advantages. The aerodynamic characteristics of the X-wing configuration are approximated by superimposing the results obtained from separate analyses of upward- and downward-dihedral wing sets, thereby constructing a representative full-vehicle aerodynamic model. The built-in body–wing interference correction modules and cruciform-wing superposition logic within DATCOM enable efficient and reasonably accurate capture of induced downwash effects and aerodynamic trends associated with varying dihedral and anhedral angles of both the wings and tail surfaces. In addition, DATCOM provides rapid access to both linear and nonlinear aerodynamic coefficients during the preliminary design stage, offering significant engineering value for the early assessment of the distinctive roll–yaw coupling behaviour inherent to X-wing configurations. The resulting semi-empirical aerodynamic database facilitates the initial screening and reduction of candidate configurations, thereby substantially reducing the computational cost and development time required for subsequent high-fidelity CFD analyses and control law design, including the implementation of BTT and STT manoeuvring strategies. While alternative tools could be employed for such a complex configuration, DATCOM has demonstrated its ability to provide a consistent and reasonably representative aerodynamic dataset for the present investigation. Moreover, the existing literature on X-wing configurations remains extremely limited, with only a single study adopting CFD for aerodynamic characterisation (Bunescu et al. [Reference Bunescu, Hothazie, Pricop and Negoiţă30]). In light of these considerations, the use of DATCOM in this work is deemed appropriate for capturing the essential qualitative aerodynamic features of the X-wing configuration, thereby effectively supporting the primary objective of this study – namely, to establish a physics-based understanding of its flight-mechanics behaviour in a scientifically rigorous and meaningful manner.

Notably, unlike several previous studies that explicitly solve the transient Navier–Stokes equations coupled with rigid-body dynamics to resolve full 6-DOF motion – thereby directly determining the position and orientation of flying bodies from flow-induced forces (e.g. Cheng, Wang and Hashino [Reference Cheng, Wang and Hashino46]; Hashino et al. [Reference Hashino, Cheng, Chueh and Wang47]; Chueh, Wang and Hashino [Reference Chueh, Wang and Hashino48]) – the present study adopts a flight-mechanics-based approach. Specifically, the 6-DOF motion of the X-wing aircraft is computed using aerodynamic force and moment coefficients, which enables an efficient and physically consistent evaluation of manoeuvring dynamics while avoiding the substantial computational cost associated with fully coupled flow–structure simulations.

Table 5.

Comparison of lateral–directional aerodynamic derivatives ( ${C_{{Y_\beta }}}$ , ${C_{{n_\beta }}}$ , ${C_{{l_\beta }}}$ ) predicted by USAF Digital DATCOM and the present CFD simulations

Table 5 Long description

A table comparing aerodynamic derivatives predicted by DATCOM and CFD simulations. The table has three rows and four columns. The columns are labeled DATCOM, CFD, and Unit. The rows are labeled with the aerodynamic derivatives ${C_{{Y_beta }}}$, ${C_{{n_beta }}}$, and ${C_{{l_beta }}}$. Row 1: ${C_{{Y_beta }}}$, DATCOM: 0.03248, CFD: 0.16014, Unit: deg1. Row 2: ${C_{{n_beta }}}$, DATCOM: 0.0019, CFD: 0.001476, Unit: deg1. Row 3: ${C_{{l_beta }}}$, DATCOM: 0.00130.00096( ${rm{alpha }}$ = 1 $^circ$ ), CFD: 0.000792, Unit: deg1.

Interestingly, after establishing the geometric configurations proposed by Bunescu et al. [Reference Bunescu, Hothazie, Pricop and Negoiţă30] and deriving the corresponding aerodynamic coefficients using DATCOM, it is observed that the predicted values of ${C_{{Y_\beta }}}$ , ${C_{{l_\beta }}}$ and ${C_{{n_\beta }}}$ are in close agreement with those obtained from the CFD simulations reported by Bunescu et al. [Reference Bunescu, Hothazie, Pricop and Negoiţă30]. This consistency indicates that the present aerodynamic coefficients are comparable to high-fidelity numerical results, thereby demonstrating that the DATCOM-based approach provides a reliable and physically credible representation of the lateral–directional aerodynamic characteristics. Consequently, these findings support the validity and suitability of using DATCOM for the aerodynamic modeling adopted in this study.

To further substantiate the reliability of the aerodynamic database employed in this study, a cross-validation is performed by comparing the coefficients predicted by USAF Digital DATCOM with those obtained from high-fidelity CFD simulations. These coefficients serve as critical inputs for accurately capturing the manoeuvring characteristics of the X-wing configuration within the flight-dynamics framework. The CFD simulations are conducted using a rigorously defined numerical setup, with detailed descriptions of modeling parameters, boundary conditions and solver configurations provided in the Supplementary Material to ensure reproducibility and transparency. Overall, the comparative analysis demonstrates that the aerodynamic coefficients derived from CFD are generally consistent in both magnitude and trend with those predicted by DATCOM, particularly for key parameters governing lift, drag and moment characteristics across the range of operating conditions considered.

As shown in Table 5, a more detailed examination of the lateral–directional stability derivatives reveals several important distinctions between the two prediction approaches. A particularly substantial discrepancy is observed in the side-force derivative ${C_{{Y_\beta }}}$ , for which DATCOM predicts a value of (−0.03248), whereas the CFD analysis yields a significantly larger magnitude of (−0.16014). This difference indicates that DATCOM provides a considerably more conservative estimate of the lateral force response. The discrepancy is likely attributable to the inherent simplifications employed in DATCOM, where the aerodynamic effects of dihedral and anhedral surfaces are largely treated through superposition methods, potentially overlooking the complex flow interactions, vortex structures and interference effects associated with unconventional cruciform configurations. Consequently, the CFD results are considered to provide a more faithful representation of the lateral force generation mechanisms governing STT manoeuvring. More importantly, this finding highlights that semi-empirical design tools may substantially underpredict the lateral force authority of cruciform aircraft geometries during high-sideslip manoeuvres, thereby offering an important cautionary insight for future aircraft designers relying on preliminary aerodynamic estimation methods for unconventional configurations.

In contrast, the yawing-moment derivative ${C_{{n_\beta }}}$ predicted by DATCOM (0.0019) exhibits close agreement with the CFD result (0.001476), indicating a consistent assessment of directional static stability. Regarding the rolling-moment derivative ${C_{{l_\beta }}}$ , both approaches demonstrate good agreement at low angles of attack (e.g. α = −1°), yielding values of (−0.00096) (DATCOM) and (−0.000792) (CFD). However, under varying operating conditions, DATCOM tends to overpredict the magnitude of the derivative (e.g. (−0.0013) versus (−0.000792)), which may increase the aileron control effort required to counteract lateral moments and could introduce a slight right-rolling tendency during STT simulations. Nevertheless, such effects can be readily compensated through adequate aileron authority and do not materially influence the overall manoeuvring characteristics or turning performance in either STT or BTT flight modes.

While minor discrepancies arise due to the fundamental differences between semi-empirical methods and high-fidelity numerical simulations, the overall level of agreement confirms that DATCOM provides a sufficiently accurate and physically representative approximation of the aerodynamic behaviour for the purposes of this study. This validation lends confidence to the use of DATCOM-generated data in subsequent flight-mechanics analyses and supports the robustness of the conclusions drawn regarding the manoeuvring performance of the X-wing UAV.

In summary, the mathematical formulations introduced above are systematically correlated and synthesised in the following analysis.

(5) \begin{align}\left[ {\begin{array}{c}{{F_{dg}}}\\[5pt] {{F_{sf}}}\\[5pt] {{F_{li}}}\end{array}} \right] = \frac{1}{2}\rho {V_{TAS}}^2 \times {S_{ref}} \times \left[ {\begin{array}{c}{{C_D}}\\[5pt] {{C_Y}}\\[5pt] {{C_L}}\end{array}} \right]\!,\end{align}

where ${F_{dg}}$ represents the aerodynamic drag force, ${F_{sf}}$ denotes the side force and ${F_{li}}$ corresponds to the lift force acting on the aircraft. The relationships among ${F_{{x_A}}}$ , ${F_{{y_A}}}$ , ${F_{{z_A}}}$ , ${F_{dg}}$ , ${F_{sf}},$ and ${F_{li}}$ are presented as follows:

(6) \begin{align}\left[ {\begin{array}{c}{{F_{{x_A}}}}\\[5pt] {_{{y_A}}}\\[5pt] {{F_{{z_A}}}}\end{array}} \right] = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{cos\alpha } {}& 0 & {}{ - sin\alpha }\\[5pt] 0& {}1& {}0\\[5pt] {sin\alpha } & {}0 & {}{cos\alpha }\end{array}} \right]\left[ {\begin{array}{c}{ - {F_{dg}}}\\[5pt] {{F_{sf}}}\\[5pt] { - {F_{li}}}\end{array}} \right]\!.\end{align}

The definitions of ${\boldsymbol M_{{C_l}}}$ , ${\boldsymbol M_{{C_m}}}$ and ${\boldsymbol M_{{C_n}}}$ are given as follows:

(7) \begin{align}\left[ {\begin{array}{c}{{\boldsymbol M_{{C_l}}}}\\[5pt] {{\boldsymbol M_{{C_m}}}}\\[5pt] {{\boldsymbol M_{{C_n}}}}\end{array}} \right] = \frac{1}{2}\rho {V_{TAS}}^2 \times {S_{ref}} \times \left[ {\begin{array}{c}{b \times {C_l}}\\[5pt] {\bar c \times {C_m}}\\[5pt] {b \times {C_n}}\end{array}} \right]\!.\end{align}

The relationships among ${\boldsymbol M_{{x_A}}}$ , ${\boldsymbol M_{{y_A}}}$ , ${\boldsymbol M_{{z_A}}}$ , ${\boldsymbol M_{{C_{}}}}$ , ${\boldsymbol M_{{C_m}}}$ and ${\boldsymbol M_{{C_n}}}$ are presented as follows:

(8) \begin{align}\left[ {\begin{array}{c}{{\boldsymbol M_{{x_A}}}}\\[5pt] {{\boldsymbol M_{{y_A}}}}\\[5pt] {{\boldsymbol M_{{z_A}}}}\end{array}} \right] = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{cos\alpha } {}& 0 & {}{ - sin\alpha }\\[5pt] 0& {}1 {}&0\\[5pt] {sin\alpha } {}&0& {}{cos\alpha }\end{array}} \right]\left[ {\begin{array}{c}{{M_{{C_l}}}}\\[5pt] {{M_{{C_m}}}}\\[5pt] {{M_{{C_n}}}}\end{array}} \right]\!.\end{align}

The rotation matrix ${\boldsymbol M_{EB}}$ , which transforms vectors from the Earth-fixed reference frame to the body-fixed coordinate system, is defined as follows:

(9) \[\begin{array}{*{20}{c}}{{\boldsymbol M_{EB}} = \left[ {\begin{array}{*{20}{c}}{cos\psi cos\theta } & {sin\psi cos\theta } & { - sin\theta } \\{cos\psi sin\theta sin\phi - sin\psi cos\phi } & {sin\psi sin\theta sin\phi + cos\psi cos\phi } & {cos\theta sin\phi } \\{cos\psi sin\theta cos\phi + sin\psi sin\phi } & {sin\psi sin\theta cos\phi - cos\psi sin\phi } & {cos\theta cos\phi } \\\end{array}} \right].} \hfill \\\end{array}\]

The three-dimensional translational velocity components expressed in the Earth-fixed and body-fixed reference frames are related as follows:

(10) \begin{align}\left[ {\begin{array}{c}{{U_E}}\\[5pt] {{V_E}}\\[5pt] {{W_E}}\end{array}} \right] ={\boldsymbol M}_{EB}^T\left[ {\begin{array}{c}u\\[5pt] v\\[5pt] w\end{array}} \right]\!,\end{align}

where the superscript $T$ indicates the transpose of the transformation matrix.

The Euler-rate transformation matrix is defined as follows:

(11) \begin{align}\left[ {\begin{array}{c}{\dot \phi }\\[5pt] {\dot \theta }\\[5pt] {\dot \psi }\end{array}} \right] = \left[ {\begin{array}{c@{\quad}c@{\quad}c}1 & {}{sin\phi tan\theta } & {}{cos\phi tan\theta }\\[5pt] 0 & {}{cos\phi } & {}{ - sin\phi }\\[5pt] 0 & {}{sin\phi sec\theta } & {}{cos\phi sec\theta }\end{array}} \right]\left[ {\begin{array}{c}p\\[5pt] q\\[5pt] r\end{array}} \right]\!.\end{align}

2.4 Numerical methods

In the present study, the classic fourth-order Runge–Kutta (RK-4) method is adopted to numerically integrate the system of ordinary differential equations that govern the translational velocities, angular rates and position states of the aircraft. RK-4 is selected because of its well-established reliability and effectiveness in flight dynamics simulations (Schmidt [Reference Schmidt44] and Butcher [Reference Butcher49]). The method provides fourth-order accuracy, enabling high numerical precision to be achieved while allowing relatively large time-step sizes, which is advantageous for simulations that require both accuracy and computational efficiency. Within each time step, RK-4 evaluates the system derivatives at four intermediate points, yielding a weighted average that more accurately captures the local curvature of the state trajectories. This multislope evaluation significantly improves the solution accuracy over that of lower-order schemes that rely on a single derivative estimate. Moreover, RK-4 is a single-step method in which the state at the next time stamp depends only on the current state, simplifying the implementation and reducing the imposed memory requirements while maintaining robust numerical stability.

To implement the classical fourth-order RK4 time-integration scheme adopted in this study, the state vector is first defined as

(12) \begin{align}{\boldsymbol Q} = {[{u\,\, {}v\,\, {}w\,\,p\,\, {}q\,\, {}r}]^T},\end{align}

which contains the translational and rotational velocity components in the body-fixed reference frame. Based on Equations (1) and (2), all time-derivative terms are collected on the left side, while the remaining terms are arranged on the right side, yielding the following first-order ordinary differential equation:

(13) \begin{align}{{d{\boldsymbol Q}} \over {dt}} = f({\boldsymbol Q}),\end{align}

where $f({\boldsymbol Q})$ represents the nonlinear force and moment contributions.

The state vector at the next time step, ${{\boldsymbol Q}_{j + 1}}$ , is then evaluated using the RK4 formulation as

(14) \begin{align}{{\boldsymbol Q}_{j + 1}} = {\boldsymbol Q}({{t_j} + \Delta t}) = {{\boldsymbol Q}_j} + \Delta t\left( {{{{{\boldsymbol k}_1}} \over 6} + {{{{\boldsymbol k}_2}} \over 3} + {{{{\boldsymbol k}_3}} \over 3} + {{{{\boldsymbol k}_4}} \over 6}} \right)\end{align}

where $\Delta t$ denotes the time-step size, and the intermediate slope vectors ${{\boldsymbol k}_1}$ , ${{\boldsymbol k}_2},$ ${{\boldsymbol k}_3}$ and ${{\boldsymbol k}_4}$ are computed as follows:

(15) \begin{align}{{\boldsymbol k}_1} = f\left( {{{\boldsymbol Q}_j}} \right) \to {{\boldsymbol Q}_1} = {{\boldsymbol Q}_j} + {{\Delta t} \over 2}{{\boldsymbol k}_1},\end{align}

(16) \begin{align}{{\boldsymbol k}_2} = f({{{\boldsymbol Q}_1}}) \to {{\boldsymbol Q}_2} = {{\boldsymbol Q}_j} + {{\Delta t} \over 2}{{\boldsymbol k}_2},\end{align}

(17) \begin{align}{{\boldsymbol k}_3} = f({{{\boldsymbol Q}_2}}) \to {{\boldsymbol Q}_3} = {{\boldsymbol Q}_j} + {{\Delta t} \over 2}{{\boldsymbol k}_3},\end{align}

(18) \begin{align}{{\boldsymbol k}_4} = f({{{\boldsymbol Q}_3}}).\end{align}

This procedure enables accurate and stable numerical integration of the 6-DOF equations of motion over each time step.

To validate the 6-DOF flight dynamics model developed in this study, a reference flight mechanics model of the de Havilland Canada DHC-2 Beaver was first constructed using MATLAB [50], and its simulation results were systematically compared with benchmark solutions. The model is formulated based on the 6-DOF governing equations for fixed-wing aircraft, incorporating control inputs such as aileron, elevator, rudder and flap deflections, as well as propeller rotational speed and propulsion-related parameters. In addition, atmospheric variables – including temperature, speed of sound, altitude, pressure and air density – are integrated into the dynamics solver to compute the state derivatives. These are subsequently integrated to obtain the full state vector $[{p,q,r,\phi ,\theta ,\psi ,u,v,w,x,y,z}]$ , from which derived flight quantities such as attitude angles, three-dimensional position, angle-of-attack, sideslip angle and true airspeed are evaluated. This modeling framework comprehensively captures translational and rotational motion, attitude kinematics and flight parameter estimation, thereby providing a robust foundation for subsequent dynamic analysis. Under steady straight-flight conditions, trim solutions satisfying force and moment equilibrium are first obtained, followed by the computation of associated flight parameters for comparison. The results of Figs. 6 and 7 demonstrate excellent agreement between the MATLAB model and the reference 6-DOF solutions in both trend and magnitude, confirming the accuracy of trim prediction and post-trim parameter evaluation. Furthermore, trajectory comparisons reveal that the simulated flight paths from both models nearly overlap, forming smooth and stable closed-loop elliptical patterns under constant control inputs. This high level of consistency indicates that the present model accurately reproduces the coupled dynamics of attitude evolution, velocity interactions, coordinate transformations and position integration, thereby validating its capability for reliable flight simulation and trajectory analysis.

Figure 6.

Numerical validation through comparison of straight-flight trajectories obtained from the present implementation and the MATLAB Aerospace Toolbox [50].

Figure 7.

Numerical validation through comparison of circular flight trajectories obtained from the present implementation and the MATLAB Aerospace Toolbox [50].

A comprehensive 6-DOF aircraft simulation framework is developed in the present study to investigate flight dynamics and control behaviours under a wide range of manoeuvring conditions. The model integrates a proportional–derivative (PD) control architecture (Etkin and Reid [Reference Etkin and Reid51] and Stevens et al. [Reference Stevens, Lewis and Johnson52]), shown in Fig. 8, with a classic RK-4 time integration scheme to update the translational velocities, angular rates and position states of the aircraft at each simulation step. The PD control system adopted in this study is organised in a hierarchical three-loop architecture, as illustrated in Fig. 8, which integrates guidance, attitude regulation and actuator control within a unified framework. In the outer guidance loop, navigation commands are generated based on the positional errors between the UAV and the target waypoint, allowing the desired heading angle to be continuously updated. This heading command is subsequently processed in the intermediate loop, where it is converted into either a commanded roll angle for BTT operation or a commanded sideslip angle for STT manoeuvring. Through this cascaded structure, the controller effectively decouples trajectory planning from low-level attitude regulation, thereby enhancing the stability and robustness of autonomous navigation under varying flight conditions.

Figure 8.

Proportional–derivative (PD) control architecture (Etkin and Reid [Reference Etkin and Reid51] and Stevens et al. [Reference Stevens, Lewis and Johnson52]) used in the present study.

In the inner control loop, the PD controller (Etkin and Reid [Reference Etkin and Reid51] and Stevens et al. [Reference Stevens, Lewis and Johnson52]) regulates the UAV’s attitude, altitude and velocity by minimisng the tracking errors between commanded and measured states. Thrust compensation is applied using altitude error feedback to ensure smooth convergence to the target height without inducing excessive oscillations. Simultaneously, elevator deflection is adjusted based on airspeed and acceleration errors, while pitch-rate feedback provides additional damping to improve longitudinal stability. For lateral–directional control, aileron inputs are generated from roll-angle and roll-rate errors in BTT mode, whereas in STT mode, rudder deflections are determined from sideslip-angle tracking errors. These control signals are directly coupled with the 6-DOF flight-dynamics model, forming a closed-loop feedback system that continuously updates the UAV states. This multi-loop PD-based architecture ensures accurate trajectory tracking, reliable attitude stabilisation and effective coordination between BTT and STT manoeuvring modes, thereby providing a practical and robust control solution for high-agility X-wing UAV operations.

Thrust compensation (inner loop), shown in Fig. 8, is implemented by evaluating the error between the commanded altitude and the measured altitude. This altitude tracking error is processed through the proportional gain ${G_{P,H}}$ and the derivative gain ${G_{D,H}}$ to generate appropriate corrective signals. These compensation terms are designed to regulate the thrust output and ensure smooth convergence toward the target altitude while suppressing excessive oscillations and transient instabilities. As a result, the UAV is able to maintain stable vertical motion and reliable altitude tracking during manoeuvring:

(19) \begin{align}\Delta {\delta _{th}} &= ({ - H + {H_{CMD}}}) \times {G_{P,H}} + \dot H \times {G_{D,H}}\nonumber \\ &= ({-{Z_E} + {H_{CMD}}}) \times {G_{P,H}} - {\dot Z_E} \times {G_{D,H}}\nonumber \\ &= ({-{Z_E} + {H_{CMD}}}) \times {G_{P,H}} - {W_E} \times {G_{D,H}}.\end{align}

Elevator deflection compensation (inner loop), shown in Fig. 8, is implemented by adjusting the elevator angle based on the indicated airspeed error and longitudinal acceleration to maintain the desired flight speed. In addition, a damping control term based on the pitch-rate feedback $q$ is incorporated through the stability augmentation gain ${G_{SAS}}$ to enhance longitudinal stability and suppress undesirable oscillations. This combined control strategy enables effective velocity regulation and improved dynamic response of the UAV:

(20) \begin{align}de = ({{V_{IA}} - {V_{IA{S_{CMD}}}}}){ \times G_{P,V}} + \dot u \times {G_{D,V}} + q \times {G_{SAS}}.\end{align}

In the BTT mode, the inner loop is responsible for the automatic regulation of altitude, airspeed and roll angle. Based on the commanded altitude, airspeed and roll angle, the proportional–derivative controller (PD_CTL) generates the corresponding actuator commands, including throttle and elevator deflections. The intermediate loop performs heading-angle regulation by converting the heading error into a commanded roll angle, which enables the aircraft to turn through banking motion. The outer loop provides waypoint-based navigation by computing the desired heading angle from the positional error between the UAV and the target waypoint. This outer loop activates the intermediate loop, which in turn calls the inner loop, forming a cascaded control structure.

Aileron control command (inner loop), shown in Fig. 8, is implemented for roll-angle regulation by evaluating the tracking error between the measured roll angle $\phi $ and the commanded roll angle ${\phi _{{\rm{CMD}}}}$ . This roll-angle error is combined with roll-rate feedback $p$ to generate the appropriate aileron deflection. The inclusion of roll-rate feedback provides additional damping and improves dynamic stability, thereby ensuring accurate and responsive roll control. This control strategy enables precise attitude regulation during manoeuvring:

(21) \begin{align}da = ( {\phi - {\phi _{CMD}}}) \times {G_{P,\phi }} + p \times {G_{D,\phi }}.\end{align}

Roll-angle command generation (intermediate loop), shown in Fig. 8 in the BTT mode, is implemented by converting the heading error $({\psi-{\psi _{{\rm{CMD}}}}})$ into the required roll-angle command. This command enables the aircraft to execute turning manoeuvres through coordinated banking motion, thereby redirecting the lift vector to produce the necessary centripetal force. Through this mechanism, the UAV achieves smooth and efficient directional changes:

(22) \begin{align}{\phi _{CMD}} = ({\psi - {\psi _{CMD}}})\; \times {G_{P,\psi }} + r \times {G_{D,\psi }}.\end{align}

The STT mode adopts a similar three-loop architecture; however, key differences exist in the intermediate and inner loops. In this mode, the inner loop tracks the commanded altitude, airspeed and a zero-roll-angle command, and determines the throttle input through the PD controller. The intermediate loop converts the heading error into a commanded sideslip angle rather than a roll-angle command. This sideslip command is then processed by the PD controller to generate appropriate rudder deflections. Through this structure, turning manoeuvres are achieved primarily via lateral aerodynamic forces rather than banking, allowing the UAV to perform rapid heading changes while maintaining a nearly level attitude.

Sideslip-angle command generation (intermediate loop), shown in Fig. 8, in the STT mode differs from that in the BTT mode in that the roll-angle command is replaced by a sideslip-angle command ${\beta _{{\rm{CMD}}}}$ . This command is generated from the heading error and serves as the primary reference for lateral manoeuvring through lateral aerodynamic force production rather than banking motion. By regulating the sideslip angle, the UAV is able to achieve rapid directional changes while maintaining a nearly level attitude:

(23) \begin{align}{\beta _{CMD}} = ({\psi - {\psi _{CMD}}})\; \times \overline {{G_{P,\psi }}} + r \times \overline {{G_{D,}}} \end{align}

Rudder control command (inner loop), shown in Fig. 8, is implemented by converting the commanded sideslip angle into the corresponding rudder deflection required to achieve the desired lateral motion. This transformation is formulated based on the aerodynamic characteristics of the UAV. Specifically, the commanded sideslip angle $({{\beta _{CMD}}})$ is directly mapped to the required rudder deflection $({{\delta _r}})$ by utilising the ratio between the yawing-moment derivative with respect to sideslip $({{C_{{n_\beta }}}})$ and the rudder control effectiveness coefficient $({{C_{{n_{{\delta _r}}}}}})$ . Through this relationship, the aerodynamic sensitivity of the vehicle to sideslip and rudder inputs is explicitly incorporated into the control formulation. As a result, the commanded sideslip angle can be systematically converted into the corresponding rudder deflection required to generate the desired yawing moment, thereby enabling the UAV to achieve the intended lateral aerodynamic response while maintaining consistency with its inherent stability and control characteristics. Through this mechanism, precise regulation of lateral aerodynamic forces is achieved, enabling effective STT manoeuvring:

(24) \begin{align}dr = {\beta _{CMD}} \times \frac{{{C_{{n_\beta }}}}}{{{C_{{n_{dr}}}}}}.\end{align}

The incorporation of the ratio ${C_{{n_\beta }}}/{C_{{n_{{\delta _r}}}}}$ in the control law of Equation (24) is designed to ensure that different configurations produce consistent physical responses under an identical yaw command ( ${\psi _{{\rm{CMD}}}}$ ). Due to inherent differences in stability derivatives among configurations, applying a uniform rudder deflection would result in uneven responses, with some configurations exhibiting overly strong yaw motion while others respond too weakly. Such disparities would hinder the ability to maintain steady flight at constant altitude and airspeed, and could potentially lead to instability. By introducing this ratio, the control input is effectively scaled according to the aerodynamic characteristics of each configuration, enabling a more uniform, balanced and physically meaningful comparison across different geometries.

Guidance command generation (outer loop), shown in Fig. 8, is implemented through a navigation logic that computes the desired heading angle based on the positional difference between the target waypoint and the current UAV location. By evaluating the relative coordinates in the horizontal plane, the guidance module determines the optimal direction of travel toward the target point. This commanded heading angle serves as the primary reference for the intermediate and inner control loops, enabling coordinated and accurate trajectory tracking:

(25) \begin{align}{\psi _{CMD}} = {\tan ^{ - 1}}\left( {\frac{{{X_{{E_{CMD}}}} - {X_E}}}{{{Y_{{E_{CMD}}}} - {Y_E}}}} \right)\!.\end{align}

The proposed numerical framework is rigorously verified and validated through its integration with a PD-based closed-loop control system that regulates throttle settings as well as elevator, aileron and rudder deflections, enabling accurate reproduction of realistic flight responses and dynamic behaviours over a wide range of operating conditions. Representative pitching, rolling and yawing motions are used to assess model fidelity, and close agreement between the predicted responses and expected control performance confirms the consistency, robustness and high reliability of the numerical implementation. To accommodate diverse mission objectives, the simulation incorporates six command modes: Mode-0 enables direct control of throttle, aileron and elevator deflections; Mode-1 regulates altitude and airspeed using roll-based control; Mode-2 achieves altitude and airspeed regulation through yaw-based control; Mode-3 provides autonomous waypoint navigation via an inertial navigation system while maintaining prescribed altitude and speed; Mode-4 executes STT manoeuvres using yaw-based control; and Mode-5 enables autonomous STT waypoint navigation based on inertial guidance. The unified integration of these control modes allows precise and coordinated manipulation of propulsion and control surfaces at specified intervals, ensuring stable attitude regulation, reliable trajectory tracking and smooth manoeuvre execution. Through extensive simulated flight tests, the framework provides comprehensive insight into aircraft response characteristics during attitude transitions, offering a robust and high-fidelity platform for evaluating advanced flight-control strategies and mission-level performance. The optimised gain parameters of the adopted PD controller are summarised in Table 6.

Table 6.

Parameters of gain parameters of the PD control system for the present X-wing configuration, evaluated for combined anhedral–dihedral angle settings of $50$ °, $60$ °, $70$ °, $80$ ° and $90$ °, and applied to both the front wings and rear tails

Table 6 Long description

The table presents the gain parameters of a PD control system for an X-wing configuration, evaluated for combined anhedral-dihedral angle settings of 50, 60, 70, 80, and 90 degrees, applied to both the front wings and rear tails. The table has 11 rows and 6 columns. The columns are labeled as 50, 60, 70, 80, 90, Unit, and Loop. The rows are labeled as G_DH, G_P_H, G_D_V, G_P_V, G_SAS, G_D_phi, G_P_phi, G_D_psi, G_P_psi, G_D_phi, and G_P_phi. Each cell contains a numerical value corresponding to the gain parameter for the specified angle setting. The Loop column categorizes the parameters into Inner, Mid, or blank. Row 1: G_DH, 0.004, 0.004, 0.004, 0.004, 0.004, s/ft, Inner. Row 2: G_P_H, 0.0002, 0.0002, 0.0002, 0.0002, 0.0002, 1/ft, Inner. Row 3: G_D_V, 0.005, 0.005, 0.005, 0.005, 0.005, deg/(knot/s), Inner. Row 4: G_P_V, 0.001, 0.001, 0.001, 0.001, 0.001, deg/knot, Inner. Row 5: G_SAS, 0.6, 0.6, 0.6, 0.6, 0.6, s, Inner. Row 6: G_D_phi, 0, 0, 0, 0, 0, s, Inner. Row 7: G_P_phi, 10, 10, 10, 10, 10, -, Inner. Row 8: G_D_psi, 0, 0, 0, 0, 0, s, Mid. Row 9: G_P_psi, 0.15, 0.15, 0.15, 0.15, 0.15, -, Mid. Row 10: G_D_phi, 0, 0, 0, 0, 0, s, Mid. Row 11: G_P_phi, 0.13, 0.13, 0.13, 0.13, 0.13, -, Mid.

It is important to note that the primary aim of this study is not to optimise control performance for each individual configuration – such as reducing response time or minimising tracking error – but to establish a consistent control baseline for identifying steady-state equilibrium solutions under specified flight conditions, including altitude, airspeed and roll angle. Accordingly, the use of fixed PD gains is a deliberate choice, ensuring a uniform control framework that allows the inherent influence of geometric variations (50°–90°) on the flight-mechanics behaviour to be examined in a direct and unbiased manner. In contrast, tuning the controller separately for each configuration could introduce additional artificial variability, potentially compensating for or obscuring the underlying aerodynamic differences and thereby limiting the clarity of the physical interpretation.