The UAV design process investigated in this paper primarily focuses on the conceptual design stage. The key tasks involve: Initially determining the primary parameters of the UAV, preliminarily establishing the geometric parameters of each component, and formulating the initial conceptual scheme of the UAV. Firstly, based on the performance index of the UAV, estimates are made for its maximum take-off weight, fuel weight, and other relevant data. By plotting a boundary diagram for the UAV within feasible limits, both thrust-to-weight ratio and wing load are determined; Subsequently obtaining engine thrust and wing area requirements according to their respective definitions. The fuselage length is estimated using an empirical formula from literature [6], while selecting an appropriate length-diameter ratio determines fuselage width. Thus concludes preliminary estimations for basic geometric parameters of the UAV. A conceptual model for the UAV scheme is obtained which will be utilized for subsequent multidisciplinary analysis and optimization.

This paper investigates the economic-based design optimization of low-cost UAVs, focusing on determining a reasonable design scheme and overall parameters for given UAV performance indicators. In the overall UAV design process, two key components are the aerodynamic performance analysis and weight analysis. Therefore, in this study, the lift-to-drag ratio is used to evaluate the UAV’s aerodynamic performance, while the basic empty weight serves as the output for the weight analysis. The problem of determining the overall parameters can be formulated as an optimization problem. The three main elements of the optimization model are design variables, constraints, and the optimization objective.

Optimization objective: min C d e v e l o p m e n t ( X )

Constraints: s . t . ( L / D ) ≥ 8

W e m p t y < 3

Design variables: X = { L f , A R , φ , S w i n g , λ }

Optimization objective: The objective is to minimize the UAV’s development cost.

Constraints: The constraints are the UAV’s lift-to-drag ratio and the basic empty weight.

Design variables: The design variables include the geometric parameters of the UAV, such as fuselage length(L_f), wing aspect ratio(AR), wing taper ratio(λ), wing sweep angle(φ), and wing area(S_wing).

Sequential Least Squares Quadratic Programming (SLSQP) is an iterative algorithm designed to solve constrained nonlinear optimization problems. As a special case of the Sequential Quadratic Programming (SQP) method, SLSQP combines least squares techniques with quadratic programming to address nonlinear issues, enabling it to manage both linear and nonlinear equality and inequality constraints.

SLSQP is primarily employed to solve constrained minimization problems, with the goal of identifying design variables that minimize the objective function while adhering to a set of equality and inequality constraints. The algorithm’s approach resembles that of SQP, wherein the problem is iteratively linearized or approximated as a quadratic programming problem. However, SLSQP distinguishes itself by utilizing least squares methods to handle constraints and the objective function.

The main steps of the algorithm are as follows:

Step 1: Initialization

SLSQP primarily solves nonlinear optimization problems with constraints, which can be formulated in the standard form as:

min C d ( X ) (1)

where C d ( X ) is the UAV’s development cost, subject to the following constraints:

Empty weight constraints:

W e m p t y ( X ) < 0 (2)

Lift-to-drag ratio constraints:

( L / D ) ( X ) ≥ 0 (3)

Select initial design variables X₀, such as the UAV’s geometric parameters (wing area, fuselage length, etc.). Define the objective function C d ( X ) , which represents the UAV’s development cost. Assume this function depends on the design variables X. Define inequality constraint functions. Initialize Lagrange multipliers μ₀ to handle the inequality constraints.

Step 2: Construct the Lagrangian Function

SLSQP utilizes the Lagrange multiplier method to handle constrained problems. The Lagrangian function L agrange ( X , μ ) is constructed, incorporating only the inequality constraints W_empty and (L/D).

L agrange ( X , μ ) = C d ( X ) + μ 1 W e m p t y ( X ) + μ 2 ( L / D ) ( X ) (4)

where: μ₁ corresponds to the Lagrange multiplier of the lift-drag ratio constraint (L/D). μ₂ corresponds to the Lagrange multiplier of the empty weight constraint W_empty.

Step 2: Linearize the Objective Function and Constraints

At the current iteration point X_k, perform a second-order approximation of the objective function C d ( X ) and a first-order linearization of the constraints:

Objective function approximation using Taylor series expansion:

C d ( X ) ≈ C d ( X k ) + ∇ C d ( X k ) T ( X − X k ) + 1 2 ( X − X k ) T H ( X k ) ( X − X k ) (5)

where ∇ C d ( X k ) is the gradient of the objective function at x_k, and H(x_k) is the Hessian matrix (the second-order derivative matrix).

Linearization of the constraints:

W e m p t y ( X ) ≈ W e m p t y ( X k ) + ∇ W e m p t y ( X k ) T ( X − X k ) ( L / D ) ( X ) ≈ ( L / D ) ( X k ) + ∇ ( L / D ) ( X k ) T ( X − X k ) (6)

where:

∇ W e m p t y ( X k ) and ∇ ( L / D ) ( X k ) are the gradients of the lift-to-drag ratio and empty weight constraints at the current design point, respectively.

Step 3: Construct and Solve the Quadratic Programming Subproblem

In each iteration, construct a local quadratic programming (QP) problem to update the design variables X:

Objective:

min ∇ C d ( X k ) T d + 1 2 d T H ( X k ) d (7)

subject to the linearized constraints:

∇ W e m p t y ( X k ) ( X k ) T d + W e m p t y ( X k ) ( X k ) < 0 ∇ ( L / D ) ( X k ) T d + ( L / D ) ( X k ) ≥ 0 (8)

By solving this quadratic programming subproblem, obtain the update d_k for the design variables.

Step 4: Update the Design Variables

Update the design variables X using the obtained update d_k:

X k + 1 = X k + d k (9)

Here, X_k+1 is the new design scheme for the next iteration.

Step 5: Update the Lagrange Multipliers

Update the Lagrange multipliers μ₁ and μ₂ according to the Karush-Kuhn-Tucker (KKT) conditions to ensure the feasibility of the inequality constraints:

If g(X_k) is inactive (i.e., the lift-to-drag ratio constraint is not tight), set μ₁ = 0.

If h(X_k) is inactive (i.e., the empty weight constraint is not tight), set μ₂ = 0.

Step 6: Check Convergence

Verify whether the convergence criteria are satisfied:

Change in development cost: | C d ( X k + 1 ) − C d ( X k ) | is sufficiently small.

Change in design variables: ‖ X k + 1 − X k ‖ is sufficiently small.

Constraint satisfaction: All constraints are met, i.e., ( L / D ) ( X k + 1 ) ≥ 0 and W_empty(X_k+1) < 0.

If the convergence criteria are satisfied, terminate the algorithm; otherwise, return to Step 2 and continue iterating.

Step 7: Output the Optimization Results

Upon convergence of the iterative process, output the optimized design variables X^∗. At this point, X^∗represents the optimal UAV design that minimizes the development cost while satisfying the lift-to-drag ratio and empty weight constraints.

In optimization algorithms, SLSQP is a robust and efficient choice. It performs well when addressing multi-constraint optimization problems, particularly those involving high-dimensional, nonlinear objective functions and complex constraints. Additionally, in practical applications, SLSQP demonstrates relatively fast convergence and effectively handles the feasibility of constraints. SLSQP is widely applied in multidisciplinary optimization (MDO), particularly in the aerospace field [7]-[9].

OpenVSP is developed from the Rapid Airplane Modeler (RAM) software for rapid aircraft modeling. Its early versions utilized a Command User Interface (CUI) for interaction; however, as the user base expanded, it gradually evolved into a Graphical User Interface (GUI). Additionally, it supports script-driven operation, which serves as the technical foundation for OpenVSP’s extensibility. This paper employs Python to invoke the API for conducting parametric modeling and aerodynamic analysis.

Utilizing the open-source parametric geometric modeling software OpenVSP [10], a three-dimensional model for the conceptual design is established. The geometric module’s inputs consist of geometric parameters, which are encoded into Python scripts. By invoking the API, the three-dimensional shape of the conceptual design is generated, along with the output of relevant data such as reference area and wetted area.

The weight composition of the UAV is shown in Figure 2. The primary distinction from conventional military aircraft is the reduction of crew and operational items, so the operating empty weight defined in this context is typically equal to the basic empty weight. The basic empty weight of the UAV comprises the weight of the airframe structure, propulsion system, and system equipment.

1) Fuselage Weight

The volume of the pressurized section of military aircraft is relatively small compared to the non-pressurized section, allowing it to be simplified as a non-pressurized fuselage. The design criteria for such fuselages primarily consider the bending moment of the rear fuselage associated with the design cruise speed. The estimation formula for the structural weight of a non-pressurized fuselage is [11]:

W f = 0.044 k 7 V D 0.74 ( L T 2 D ) [ S f 0.07 + 0.22 S f 0.45 N 0.32 V D 0.35 ] S f (10)

In the equation: V_D is the maximum design speed at sea level; L_T is the tail moment arm length, defined as the distance between the quarter-chord points of the wing and tail; for non-tail configurations, it is defined as the distance between the quarter-chord points of the wing and control surface; D is the fuselage width; S_f is the fuselage surface area; N is the design limit maneuvering factor; and k₇ is a complexity factor based on different types of aircraft and design characteristics, with the calculation formula as follows:

k 7 = k 8 + 0.2 ε + 0.4 α + k 9 δ + 2.7 k 10 + 0.1 η + 0.3 ω (11)

where:

k₈―For medium to large military aircraft, it is taken as 1.8; for twin-engine general aircraft or engine-mounted configurations, it is taken as 2.0; for single-engine light aircraft with a wetted area less than 20 m², it is given by 2.0 + 1.5 ( 20 − S f ) / S f

k₉―For main landing gear connected to the wing and retracted into the fuselage, it is taken as 0.35; for main landing gear connected to the fuselage and retracted into the fuselage, it is taken as 0.7.

k₁₀―The ratio of weapon bay length to fuselage length.

𝜀―Taken as 1 when the engine is embedded in the fuselage, and 0 in other cases.

α―Taken as 1 when the intake duct is located on the fuselage, and 0 in other cases.

δ―Taken as 1 when the main landing gear intrudes into the fuselage structure, and 0 in other cases.

η―Taken as 1 for carrier-based aircraft, and 0 in other cases.

ω―Taken as 1 for truss-supported wings, and 0 in other cases.

2) Wing Weight

The main structure of the wing refers to the wing sections that bear the loads. The primary components of the box section include beams, spars, skins, and ribs. The estimation formula for the weight of the wing’s main structure is:

W I P S = ( m c + m r ) W T O (12)

m C = 1920 A 1.5 S 0.5 N ¯ r ( 1 + λ ) sec ϕ sec φ / τ f a (13)

m r = 3 S 1.25 τ 0.5 M T O A 0.25 [ ( 1 − 0.34 λ + 0.44 λ 2 ) + 2.2 τ ( S A ) 0.5 ( 1 − λ + 0.72 λ 2 ) ] (14)

In the equation: W_TO is the maximum takeoff weight; m_c is the weight coefficient for the box section skin and beam web; m_r is the weight of the main wing ribs; A is the wing aspect ratio; S is the wing reference area; λ is the taper ratio; ϕ is the effective sweep angle (at the quarter-chord line); φ is the structural sweep angle (generally approximated as equal to ϕ; τ is the relative thickness at the wing root; f_a is the allowable working stress of the wing material.

The above weight calculation formula does not account for weight increases due to factors such as the connection structures between the wing and other components, structural openings, variable sweep, and wing folding, nor does it include the weight of secondary structures such as flaps and ailerons. Therefore, it requires correction. The values of the correction factors C₁, C₂, and C₃ are obtained from the literature [6]. The total weight of the wing structure is:

W w = ( C 1 + C 2 + C 3 ( m C + m r ) ) W T O (15)

3) Tail Weight

The tail weight typically accounts for less than 3% of the total aircraft weight, but it significantly impacts the longitudinal position of the center of gravity. The primary factors influencing tail weight are the tail area and the maximum design speed.

The estimation formula for the weight of a single vertical tail is:

W v = 0.065 k 12 V D S h 1.15 (16)

In the equation, S_h is the vertical tail area; V_D is the maximum design speed at sea level;

The value of k₁₂ depends on the tail configuration type: for vertical tails not mounted on a horizontal tail, it is taken as 1; for T-tail configurations, it is taken as 1.5.

4) Landing Gear Weight

According to statistical data, the weight of the landing gear assembly generally accounts for 3% to 5% of the maximum takeoff weight and 8% to 15% of the total aircraft structural weight. The weight of the nose landing gear constitutes 15% of the total landing gear assembly weight, while the main landing gear accounts for 85%.

5) Propulsion System Weight

The propulsion system weight refers to the total weight of the complete propulsion system, including the short nacelle and engine accessories, calculated as shown in the following equation:

W P P = n C e n g W e n g (17)

In the equation: n is the number of engines; C_eng is the propulsion system installation coefficient; W_eng is the bare engine weight, which can be provided by the supplier. The installation coefficient depends on the type of UAV and is taken as 1.4.

6) System Equipment Weight

The onboard systems mainly include the fuel system, flight control system, hydraulic system, avionics system, power system, anti-icing system, and auxiliary power system. Fixed equipment primarily consists of kitchens, restrooms, and other supply equipment, which can be disregarded for UAVs since there are no personnel onboard. The fuel system has already been accounted for in the propulsion system weight module. The total weight of the onboard systems, excluding the fuel system, W_sys can be expressed as:

W s y s = C s y s W T O (18)

In the equation, C_sys is the sum of the coefficients of the onboard systems relative to the maximum takeoff weight. It can be calculated by summing the weight coefficients for each onboard system of the UAV using empirical formulas, as detailed in the literature [11].

Currently, there are numerous common aerodynamic analysis methods. Based on computational accuracy, these can be categorized into low-accuracy engineering estimation methods, medium-accuracy rapid numerical methods, and high-accuracy numerical solutions of the Euler equations and Navier-Stokes equations. As computational precision increases, the time cost also rises correspondingly. During the preliminary optimization phase of an aircraft, considering both computational speed and accuracy, rapid numerical methods are primarily selected for aerodynamic analysis. The vortex lattice method is a classic approach within medium-accuracy rapid numerical methods for calculating aircraft aerodynamic characteristics. It is based on the lifting surface theory and is a practical numerical computation method. This approach has advantages such as clarity of physical quantities, rapid computation, high accuracy, and ease of implementation through programming.

Mainstream open-source software based on the vortex lattice method includes AVL, VLM, and Tornado. The parametric modeling software OpenVSP (Open Vehicle Sketch Pad), used in this study, integrates an aerodynamic analysis program based on the vortex lattice method within its VSPAERO analysis module and provides engineering algorithms for viscous corrections. Using the API interface of VSPAERO, aerodynamic characteristic analysis of the UAV is performed in Python, resulting in the lift and drag characteristics of the UAV.

The cost of UAV development refers to the expenses necessary for completing the entire project, encompassing activities such as design, prototype sampling, trial production, and test verification conducted from project initiation to completion.

Based on domestic aviation equipment development practices, management of development funding, and statistical analysis of development costs for hundreds of types of aviation equipment in the United States, the primary factors influencing aviation equipment development costs include the level of new technology content, technology maturity, operational capability, development workload, number of prototypes, quality of prototypes, and development cycle.

In the early stages of development, the equipment development plan is not specific enough to utilize engineering estimation methods for calculating development costs. For estimating the total development costs, this paper employs a parametric method, taking into account factors such as equipment weight and development cycle. For the overall estimation of development costs, the partial least squares regression method is selected based on the aforementioned introduction and literature review.

Using the original data from literature [12] for partial least squares regression fitting, the regression equation obtained is:

y = − 8943713.731 + 140083.2912 x 1 + 134906.3058 x 2     − 128289.2566 x 3 + 9345.2726 x 4 + 87169.3217 x 5     − 140706.8749 x 6 − 170687.4218 x 7 (19)

The fitted regression equation was tested, and the results are shown in Table 1. The results demonstrate that the estimation model achieves a prediction accuracy of 98.9%, confirming its effectiveness in accurately estimating the development cost.

Assuming the performance requirements of a certain low bypass ratio, UAV are as shown in Table 2.

Before performing overall parameter estimation, some empirical parameters need to be preliminarily estimated. The aerodynamic parameters involved in constraint analysis are the zero-lift drag coefficient (C_D0), the induced drag factor (K₁), and the maximum lift coefficient. For subsonic aircraft, the C_D0 value for the UAV in this study is taken as 0.018. The maximum lift coefficients for takeoff and landing are set to 1.6 and 1.8, respectively, and the UAV’s aspect ratio is 5, from which the induced drag factor (K1) can be calculated. Based on the UAV performance requirements, a specific model of low bypass ratio turbofan engine is adopted, with a sea-level thrust of approximately 5 tons and a bypass ratio of 0.56. Based on the above analysis, the input parameter values for the parameter estimation module are shown in Table 3.

The above parameters are input into the parameter estimation module, and the program calculates the boundary charts, initial weight estimation data, and geometric parameters. The interface is shown in Figure 3.

The results are presented as shown in Table 4.

The estimated maximum takeoff weight of the UAV is 3293.7 kg, with an empty weight of 1324.86 kg and a fuel weight of 1418.83 kg. The empty weight fraction is 0.4, and the fuel fraction is 0.43. The wing loading is 226.13 kg/m², and the thrust-to-weight ratio is 0.32. Based on the definitions of wing loading and thrust-to-weight ratio, the reference wing area is 14.57 m², the fuselage length is 9.16 m, and the fuselage width is 1.14 m.

The geometric parameters are input into the geometry module to obtain the UAV’s geometric model. Figure 4 shows the interface of the geometry module. The input parameters are shown in Table 5. The outputs of the geometry module are the UAV geometric model files and the fuselage surface area. The calculated fuselage surface area is 20.42 m².

Import the model generated by the geometry module, set the Mach number to 0.8 and the angle of attack to 2 degrees. By running the VSPAERO vortex lattice method, the lift-to-drag ratio of the UAV is calculated to be 14. Figure 5 illustrates the interface of the aerodynamic module.

Using the weight analysis program for iterative calculations, Figure 6 shows the interface of the weight module. The input parameters are listed in Table 6. The UAV’s maximum takeoff weight is found to be 3812.46 kg, with a fuel weight of 1212.91 kg and a zero-fuel weight of 2599.56 kg. The basic empty weight is 1599.56 kg, structural weight is 640.82 kg, landing gear weight is 171.56 kg, propulsion system weight is 350 kg, system equipment weight is 362.19 kg, and miscellaneous weight is 38.12 kg.

By using the development cost estimation module, as shown in Figure 7, with input parameters listed in Table 7, the calculated development cost is 358.51 million CNY.

Figure 8 shows the interface of the optimization module. The design variable ranges are set as shown in Table 8.

In the optimization module, the constraints were set as a lift-to-drag ratio greater than 8 and an empty weight less than 3 tons.

The design variables, constraint values, and objective function values before and after optimization are shown in Table 9.

The iterative changes of UAV parameters during the optimization process are shown in Figure 9-11. we observe that in the initial phase of the optimization module, various design points are explored, resulting in significant fluctuations in the UAV parameters. As the number of iterations increases, the search range of the optimization module gradually narrows. After reaching the optimal value, several subsequent attempts stabilize at the same value, indicating that the

iteration has converged and the optimization process has concluded. The optimal value of the objective function has been obtained. The variation trend of UAV air weight and development cost exhibits a high degree of consistency, in accordance with the objective law. However, during the sixth iteration, the lift-drag ratio approaches its constraint lower limit while the development cost is not yet optimal. This indicates that there exists a complex relationship between the lift-drag ratio and development cost; achieving optimal cost does not necessarily entail reducing the lift-drag ratio to its lowest level.

The geometric shapes before and after optimization are shown in Figure 12.

The red wireframe model represents the UAV’s geometric shape before optimization, while the blue solid model represents the geometric shape after optimization. By overlapping them within the same coordinate system in the figure, the changes in the UAV’s geometry before and after optimization become more apparent. From the figure, it can be observed that compared to the initial model, the optimized UAV has an increased fuselage length and a reduced wing area.

The optimization results indicate that the development cost of the optimized design is reduced by 31.55% compared to the initial design, while the empty weight is reduced by 0.22 tons, and the lift-to-drag ratio is only reduced by 1.69. This demonstrates effective control over the UAV development cost with minimal aerodynamic performance loss. The reduction in wing area after optimization is the main factor contributing to the decrease in the lift-to-drag ratio. However, the smaller wing area also resulted in a reduction in the UAV’s empty weight, which in turn impacted the development cost. On balance, the benefits outweigh the drawbacks. From the optimization iteration chart, it is evident that at the initial stage, the optimizer explored various design points, leading to significant fluctuations in the UAV’s development cost. As the number of iterations increased, the objective function gradually stabilized, and the optimization process converged to a value near equilibrium. The objective function reached its minimum value at the 8^th iteration, with optimization converging at the 10^th iteration.