Design Optimization of a Pseudo-Rigid-Compliant Mechanism for Large, Continuous, and Smooth Morphing of Airfoil Camber

Abstract

This work introduces a novel variable camber mechanism that combines the high-load capacity, structural stability, and mechanical efficiency of rigid-body mechanisms with the adaptability, lightweight design, and continuous and smooth motion of compliant mechanisms. The proposed mechanism, featuring an articulated airfoil structure with revolute joints and a cantilever beam that models and controls airfoil camber morphing, employs both standard and higher kinematic pairs to constrain mobility and facilitate camber adjustments through beam deflection and coordinated kinematic interactions. Through multidisciplinary optimization, this study determined the optimal mechanism configuration and airfoil shapes for a small fixed-wing UAV (Unmanned Aerial Vehicle), meeting its morphing and mission requirements, showing the potential for drag reduction by up to 13% across various cruise conditions, thus lowering overall mission drag and energy usage. 2D (airfoil) and 3D (wing) prototypes were built to demonstrate the working principle of the proposed mechanism and to highlight its morphing capabilities. It can morph into multiple airfoil configurations, producing continuous, smooth and efficient airfoil shapes. Moreover, the mechanism is robust, simple, and easy to manufacture, effectively harnessing the strengths of both rigid-body and compliant mechanisms.

1. Introduction

Morphing wing technology represents a significant advancement in aviation, offering the potential to enhance aerodynamic performance and efficiency in both manned and unmanned aircraft by tailoring wing shape to varying flight conditions [1,2]. This adaptability can be achieved by incorporating variable camber mechanisms into morphing wing systems [3], which play a pivotal role by enabling dynamic changes in the curvature of airfoil sections along the wingspan.

By adjusting airfoil curvature, variable camber mechanisms optimize the lift-to-drag ratio [4], effectively minimizing drag penalties associated with suboptimal flight conditions and enhancing flight efficiency [5]. Furthermore, variable camber systems promote natural laminar flow over wing surfaces, contributing to reduced skin friction drag [6]. The integration of these mechanisms introduces active flow control capabilities [7], allowing real-time adjustments to wing shape to mitigate turbulence and optimize airflow, thereby improving aerodynamic performance. The flexibility inherent in variable camber morphing mechanisms also supports improved maneuverability, particularly beneficial in applications requiring heightened agility, such as small flying robots [8,9]. Additionally, these mechanisms enhance payload capacity and adaptability to varying operational conditions, underscoring their significance in optimizing flight efficiency across diverse mission profiles [10].

Within the extensive literature on variable camber morphing wing technology, numerous mechanisms have been proposed and studied, ranging from conventional hinged systems [11] to flexible structures [12] and bio-inspired designs [13]. These mechanisms differ in design, materials, and actuation methods, each with distinct advantages and disadvantages. Based on their physical principles, mechanisms available in the literature can be classified into two main categories: rigid-body mechanisms [14,15,16,17,18] and compliant mechanisms [12,19,20,21,22].

Rigid-body variable camber mechanisms utilize discrete actuators and hinges to achieve wing shape adjustments [14,15,16,23]. Their primary advantages include robust load-carrying capacity, making them suitable for applications requiring structural integrity [14], and mechanical efficiency, as discrete actuators enable precise control over camber changes [17]. However, rigid-body mechanisms often exhibit limited adaptability due to fixed geometries, potentially resulting in suboptimal performance under certain flight conditions. Additionally, discrete hinges and actuators may increase aircraft weight and disrupt smooth motion during transitions [24,25]. Furthermore, an increase in the number of rigid links necessitates additional actuators to achieve more pronounced cambered shapes, intensifying control complexity.

In contrast, compliant variable camber mechanisms use flexible materials and continuous actuators to adjust airfoil curvature [12,19,26]. These mechanisms are valued for their lightweight design, enhancing flight efficiency and contributing to aircraft weight reduction [20,21]. Continuous actuation enables smoother, more gradual transitions in wing shape, promoting greater drag reduction and mitigating undesired flow phenomena such as separation and stall [12,22]. Nevertheless, compliant mechanisms present certain limitations, including: (i) concerns regarding load capacity and structural stability under high-stress conditions, (ii) the need for specialized materials and intricate manufacturing processes to achieve the requisite flexibility, limiting applicability in larger structures such as entire wing systems, and (iii) studies indicating that compliant airfoils require additional energy for deformation compared to articulated airfoils under identical flight conditions [24,25].

It is important to acknowledge that the selection between rigid-body and compliant variable camber mechanisms depends upon specific application requirements and design objectives. Researchers continue to explore methods to combine the benefits of both types, aiming to develop hybrid mechanisms offering structural robustness, adaptability, and efficient aerodynamic performance [27,28]. Designing variable camber morphing wing mechanisms thus represents a multifaceted challenge, requiring careful balance among conflicting characteristics. In pursuing high morphing capability, designers must address the necessity of minimizing actuation power and complexity [24]. For instance, References [5,29] underscore efforts to achieve optimal designs that provide extensive morphing with minimal energy consumption and mechanical complexity. Additionally, variable camber mechanisms must account for the complex interplay between wing structures and surrounding airflow. This fluid-structure interaction is a crucial aspect of morphing wing design, significantly influencing both aerodynamic performance and structural behavior [30].

In this study, a novel variable-camber mechanism is introduced, combining the robust characteristics of rigid-body mechanisms—such as high load capacity, structural stability, and mechanical efficiency—with the adaptive features, lightweight construction, and seamless motion inherent to compliant mechanisms. The proposed mechanism comprises an articulated airfoil structure as the rigid-body component, interconnected via revolute joints (lower kinematic pairs) and higher-degree pairs (contact line), combined with a cantilever beam representing the compliant component. This beam emulates airfoil camber and facilitates morphing in the envisioned pseudo rigid-compliant design. Addressing the complex challenges of designing variable camber morphing wing mechanisms, a multidisciplinary and multipoint optimization methodology augmented by fluid-structure-interaction analysis is employed to develop an optimized design tailored to a specific case-study application. The following sections detail the outcomes of the conceptual mechanism design, present findings from multidisciplinary and multipoint optimization applied to a small fixed-wing UAV case study, and describe the experimental prototype demonstration. Subsequently, the results are discussed, conclusions and avenues for future research are presented, and finally, the computational framework developed for designing and optimizing the mechanism is described (Supplementary Materials).

2. Methodology

2.1. Conceptual Design of the Pseudo Rigid-Compliant Mechanism

The proposed concept of a pseudo rigid-compliant mechanism for Variable Camber Morphing Wings seamlessly integrates the strengths of articulated rigid mechanisms and compliant mechanisms. This integration strategically addresses challenges such as achieving smooth deformations, minimizing actuator count, reducing energy consumption, avoiding fatigue failure, and ensuring ease of manufacturing. The proposed design envisions an articulated airfoil structure (rigid body) intricately connected through ideal frictionless rotational joints. Additionally, the design incorporates higher-degree pairs (line-contact joints), coordinated by a single elastic element (compliant component: cantilever beam), emulating a flexible camber line. Figure 1 introduces the conceptual layout of the proposed mechanism, illustrating separately the rigid-body component (Figure 1(ii)), compliant component (Figure 1(iii)), their assembly, and the simulated camber deformation (Figure 1(iv)).

Within the proposed rigid-compliant mechanism, the following key features stand out:

• Rigid-body links employing low-friction rotational connections, enabling smooth and continuous deformation of the wing surface, critical for optimizing aerodynamic performance.
• The use of only a single actuator (e.g., servomotor) simplifies complexity, reduces overall weight, and enhances energy efficiency in the variable camber system.
• A single elastic element in the form of a cantilever beam minimizes elastic joints, thereby reducing fatigue risk and enhancing long-term durability and reliability.
• The proposed design facilitates easy manufacturing using conventional methods and materials, essential for practical integration into existing aircraft structures.
• The rigid-body component ensures structural integrity suitable for aircraft wing ribs, providing both aerodynamic flexibility through articulation and effective load transfer through rigidity.

2.2. Rigid-Body Component: Articulated Airfoil

The rigid-body component of the pseudo rigid-compliant airfoil is detailed in Figure 2. This component comprises an articulated structure composed of rigid links interconnected by frictionless rotational joints. Rigid-body articulated mechanisms tend to generate non-smooth and discontinuous deformations, as depicted in Figure 2(ii), due to the high degree of mobility inherent in articulated structures. Employing Gruebler’s equation, it is determined that the rigid-body component possesses mobility equivalent to $F=n-1$, requiring $n-1$ actuators to control an articulated structure of n rigid links. This analysis highlights two primary concerns: (i) non-smooth and discontinuous deformations, and (ii) a substantial number of actuators required for morphing.

To address non-smooth morphing, the baseline airfoil can be subdivided into additional rigid links. Increasing the number of rigid links reduces morphing errors by better aligning actual deformation with the ideal aerodynamic shape.

To address excessive actuator requirements, this study incorporates higher-degree kinematic pairs (line-contact pairs), strategically positioned at the center of mass of each rigid link (Figure 2(iii)). This integration reduces the mobility of the articulated structure to $F=0$, indicating constrained relative movements between adjacent rigid links, thus ensuring precise control and stability in morphing wing applications. This does not imply immobility; rather, a unique deformation is determined by external forces and constraints. A single elastic joint (cantilever beam) actuated by a servomotor induces deformation, ensuring a statically determined solution. Subsequent sections elaborate further on the compliant component.

2.3. Compliant Component: Cantilever Beam

The compliant component of the pseudo rigid-compliant design is illustrated in Figure 3. Modeled as a cantilever beam, this component mirrors the flexibility of the airfoil camber line. Consequently, deflecting the cantilever beam morphs the airfoil camber line and the airfoil’s outer contour. Deformation of both leading and trailing edges occurs autonomously, requiring distinct cantilever beam models and articulated rigid mechanisms for each edge.

As previously stated, incorporating higher-degree pairs ensures zero mobility ($F=0$). Consequently, the cantilever beam becomes the sole elastic joint responsible for inducing deformation in the mechanism, influenced by aerodynamic forces and external actuator torque. Beam deflection, symbolizing airfoil camber, is transmitted to the articulated structure via line-contact pairs, with rotational joints facilitating relative movement between rigid links. Through this coordinated workflow, the mechanism achieves negative and positive camber morphing.

Figure 3 depicts camber morphing of the trailing edge. The fixed position of the cantilever beam model aligns with the camber line at maximum airfoil thickness, serving as the reference point for camber deformations. A similar approach is required for morphing the leading edge. Subsequent sections address loading conditions and numerical modeling of the cantilever beam.

2.4. Computational Framework

A computational framework has been developed for the design and optimization of pseudo-rigid-compliant mechanisms. This framework includes an aerodynamic solver (panel method), a finite element solver for elasticity, a geometric parameterization module, and an optimization solver. The following sections examine each component thoroughly.

2.4.1. CMPLXFOIL: Aerodynamic Solver

The aerodynamic solver CMPLXFOIL, developed by MDO-Lab at the University of Michigan [31], is employed in this study. Derived from the well-established XFOIL, CMPLXFOIL integrates a viscous panel method with a boundary layer solver, enabling efficient aerodynamic predictions in subsonic flow regimes at low Reynolds numbers. Notably, it extends the original capabilities of XFOIL by incorporating the Complex-Step Method for gradient-based optimization and supporting flexible geometry parameterizations such as Free-Form Deformation (FFD).

The selection of CMPLXFOIL is motivated by the early-stage, conceptual nature of the present design exploration, which focuses on introducing and assessing a novel morphing airfoil mechanism through multipoint and multidisciplinary design optimization (MDO). While drag reduction is considered as an indicator of potential aerodynamic benefit, the primary goal of this work is to demonstrate the morphing capabilities and integration potential of the proposed mechanism. In this context, CMPLXFOIL offers an effective trade-off between computational efficiency and fidelity, aligning well with typical practices in preliminary aircraft and airfoil design.

Although more accurate aerodynamic predictions can be achieved with high-fidelity CFD tools in later design stages, several studies confirm that XFOIL and its derivatives (including CMPLXFOIL) provide sufficiently accurate lift and drag estimates, especially for low Reynolds numbers and angles of attack below stall. These tools have been widely validated against FLUENT, OpenFOAM, and experimental data in similar contexts [32,33,34], supporting their continued use in morphing airfoil research where the emphasis is on mechanism innovation and design feasibility [30,35].

Nonetheless, the known limitations of XFOIL-based solvers are fully acknowledged in this work. Their applicability is restricted to low-speed, attached-flow conditions and may be less reliable near stall or under separated flow, conditions that are not assessed in this work. These constraints are taken into account in the interpretation of the results, which are presented as indicative trends rather than precise performance metrics.

A grid refinement study (Figure 4(i)) confirmed that accurate drag and pressure coefficient predictions require at least 150 airfoil surface points, distributed using a half-cosine spacing to resolve boundary layer effects effectively.

2.4.2. Finite Element Solver for Elasticity

A straightforward finite element code for a one-dimensional cantilever beam was developed. First-order 1D line elements were used to reduce computational costs during the optimization process. A simple routine was implemented to generate the beam mesh using these elements. Given the sensitivity of the cantilever beam model to mesh resolution, a grid refinement study was conducted to determine the appropriate number of nodes for accurately estimating maximum beam deflection and stress.

Figure 4(ii) illustrates the results of this refinement study for a cantilever beam subjected to a point load at its free end. The figure shows the relative errors in maximum deflection and maximum stress compared to analytical solutions as a function of the number of degrees of freedom ($2,n_x$). The baseline mesh used $n_x=10$ nodes in the X-direction. This mesh was progressively refined by doubling the node count. A mesh with $n_x=41$ (82 DOF) yielded a relative error of just 0.6%, offering a balanced trade-off between accuracy and computation time—especially beneficial for optimization studies.

Figure 4. Grid refinement study for the (i) aerodynamic, and (ii) elasticity solvers.

Figure 4. Grid refinement study for the (i) aerodynamic, and (ii) elasticity solvers.

2.5. FSI: Fluid-Structure Interaction

A partitioned approach was used for the FSI analysis, where the aerodynamic forces and structural displacements were computed separately using two independent solvers. In this study, CMPLXFOIL and the 1D-FEM solver were employed to build the FSI routine, both chosen for their computational efficiency and reliability. Figure 5a presents the XDSM (eXtended Design Structure Matrix), which illustrates the interaction between solvers and the interface matcher.

Coupling iterations are essential in strongly coupled FSI problems, particularly in morphing-wing structures. The Aitken relaxation method was used to enhance convergence and ensure stability by applying numerical damping based on past iterations. This approach has been successfully applied to morphing-wing analyses in previous works [30,36].

Validation of the FSI Routine

The implemented FSI routine is based on the method developed and validated by Woods et al. for the Fish-BAC mechanism [30,37]. Although their work validates the general approach, additional validation was performed for the specific configuration and loading conditions used in this study.

An experimental test bench was constructed using a bridge-like setup. Standardized weights were suspended from a digital scale via a small metallic interface to approximate point loads, validating the assumption that distributed aerodynamic loads can be represented as equivalent point loads. The mechanism was mounted on a support, and deflections were measured at three locations along the beam: the free end, 75%, and 50% of the beam length, using rulers. Figure 5b illustrates the experimental setup.

Two types of tests were conducted. The first involved incrementally increasing point loads applied at the three measurement locations (Figure 5c (i)–(iii)), while the second combined a point load and an opposing torque applied at the free end of the compliant beam using a servomotor actuator. Both test scenarios were simulated using the FSI routine for validation.

Figure 5c presents the numerical results alongside the experimental measurements, shown as error bars. The simulated beam deflections closely match the experimental data, exhibiting very small relative errors. This confirms the accuracy of the FSI routine in capturing the structural response, both with and without the application of actuator torque.

Table 1. Standard deviation of experimental errors for each test case (in mm).

Test Case	Standard Weight (g)	Std. Dev. Range	Average Std. Dev.
Case 1 (100% span)	50, 100, 200	0.016–0.106	0.0496
Case 2 (75% span)	50, 100, 200	0.010–0.068	0.0317
Case 3 (50% span)	50, 100, 200	0.007–0.045	0.0202
Point Load + Torque	50, 100, 200 †	0.005–0.042	0.0272

† Servo torque: 5 kg·cm.

reports the standard deviation (Std. Dev.) of the experimental errors for each test case. As observed, the errors are minimal and can be practically neglected for the purposes of this preliminary design study.

Subsequent sections describe the loading conditions, load transfer, and solver coupling mechanisms.

Figure 5. FSI routine: (a) XDSM diagram; (b) Experimental setup for the validation; (c) Validation results.

Figure 5. FSI routine: (a) XDSM diagram; (b) Experimental setup for the validation; (c) Validation results.

2.6. Loading Conditions, Load Transfer, and Solver Matching

Figure 6(i) shows the loads applied to the cantilever beam, primarily consisting of aerodynamic pressure on the airfoil surface. Modeling the airfoil as a rigid-body linkage allows pressure loads to be transferred to the cantilever beam (Figure 6(ii)(c,d)). The normalized pressure coefficient $C_p$ (Figure 6(ii)(b)) was fitted using a piecewise spline model to align CFD and FEM grids efficiently.

The pressure distribution was discretized into point loads applied perpendicular to the beam surface. Their application points were updated at each FSI iteration using local normal and curvature vectors (Figure 6(ii)(d)).

Additional loads on the beam include the weight of the rigid-body links, represented as point loads located at their respective centers of mass (Figure 6(i)). An external torque, produced by a servomotor, was applied at 95% of the beam length on both the leading and trailing edge mechanisms.

2.7. Optimization Methodology

The objective of optimizing the pseudo rigid-compliant mechanism is to demonstrate aerodynamic advantages achievable through morphing. A multidisciplinary, multipoint optimization strategy incorporating FSI analysis was employed to address the complexity of the coupled problem. This approach ensures practical and effective solutions across a variety of mission profiles. Figure 7 shows the XDSM diagram of the optimization framework, composed of the optimizer, parameterization module, and FSI solver. The gradient-based SLSQP algorithm from the SCIPY package was selected for its efficiency in constrained optimization.

The following sections present details on the airfoil and mechanism parameterizations, optimization formulation, objective function, design variables, and constraints.

2.7.1. Geometry Parameterization

Two methods were implemented for parameterizing geometry: Free-Form Deformation (FFD) for airfoil shapes, and a direct approach for the mechanism’s geometry.

2.7.2. Free-Form Deformation (FFD): Airfoil Parameterization

FFD is a widely used technique for deforming 2D and 3D geometries using a control mesh. A box-type mesh conforms to the shape, and its control points influence the geometry. Adjustments to these points result in smooth shape changes. FFD is especially suitable for aerodynamic shape optimization due to its flexibility and compatibility with gradient-based methods.

Figure 8a displays the baseline (left) and optimized (right) airfoil within the FFD bounding box. While only five control points are shown, nine active control points were used in the optimization (ten control points with two of them collapsed into a single one) as suggested in Ref. [38]. Because the baseline airfoil is symmetrical, only nine design variables are needed due to enforced upper–lower symmetry. FFD points define the optimal thickness distribution but do not control camber directly, which is instead governed by aerodynamic and structural loading.

2.7.3. Mechanism Parameterization

A direct parameterization approach was used for the mechanism. Only the lengths of the rigid links were treated as design variables. Properties of the cantilever beam (e.g., cross-section and stiffness) were held constant to focus on demonstrating the mechanism’s potential.

Figure 8b shows the parameterized trailing-edge mechanism with six links; the leading edge features four. Link lengths are defined from the beam’s fixed end, aligned with the location of maximum airfoil thickness. The diameter of circular contact surfaces is defined by tangency conditions, ensuring a fit with the upper and lower airfoil surfaces.

2.7.4. Optimization Problem

The optimization aims to resolve numerical challenges and highlight drag reduction and morphing potential using the proposed mechanism. Unlike prior works focused solely on kinematic feasibility [39,40,41,42], this study formulates and solves a constrained multipoint, multidisciplinary problem accounting for realistic morphing behavior, loads, and requirements.

The mathematical formulation is given in

Table 2. Formulation of the multipoint and multidisciplinary optimization problem.

Objective	${\displaystyle \underset{\mathcal{X},\mathcal{Y}}{min}\mathcal{F}(\mathcal{X},\mathcal{Y})=\prod _{i=1}^3\frac{C_D^{\mathrm{morphing}}@{\mathrm{OP}}_i}{C_D^{\mathrm{fixed}}@{\mathrm{OP}}_i}}$
Design variables	$\mathcal{X}:$ FFD control points (airfoil parameterization)
	$\mathcal{Y}:$ Rigid link lengths (mechanism parameterization)
Constraints	$g_1(\mathcal{X},\mathcal{Y})=C_L@{\mathrm{OP}}_1$
	$g_2(\mathcal{X},\mathcal{Y})=C_L@{\mathrm{OP}}_2$
	$g_3(\mathcal{X},\mathcal{Y})=C_L@{\mathrm{OP}}_3$
	$x_{\mathrm{lower}}<x_i<x_{\mathrm{upper}}\forall x_i\in \mathcal{X}$
	$y_{\mathrm{lower}}<y_i<y_{\mathrm{upper}}\forall y_i\in \mathcal{Y}$
	$t@95\%\mathrm{chord}>10\%t_{\max }$
	$t@75\%\mathrm{chord}>25\%t_{\max }$

, with further discussion of the objective function, constraints, and design variables following.

2.7.5. Objective Function

The motivation behind developing variable-camber morphing wings lies in the potential to optimize airfoil shapes for specific operating conditions, enhancing aircraft efficiency through reduced drag compared to fixed-wing configurations. Besides weight and complexity, the primary operational challenge of morphing wings is designing a mechanism capable of transforming a baseline airfoil into multiple optimal shapes from a single configuration. Thus, the mechanism must morph into various shapes solely in response to aerodynamic loads and external actuator forces, such as servomotor torque.

To address this challenge, a multipoint and multidisciplinary optimization approach is adopted. Previous research has demonstrated the effectiveness of multipoint optimization when designs must excel across diverse operating conditions, often involving distinct lift and altitude requirements [43,44].

In this study, the multipoint objective function is formulated as the weighted product of the drag coefficient of the morphed airfoil divided by the drag coefficient of the fixed airfoil at each operating condition. Three typical operating points for small UAVs used in remote monitoring are considered.

Table 3. Operating points for the multipoint and multidisciplinary optimization.

Operating Point	${\mathit{C}}_{\mathit{L}}$	Vcruise [m/s]	$\mathit{WTO}$ [kg]	Wpayload [kg]	$\mathit{\rho }$ [kg/m3]	Flight Condition
Cruise 1	0.5428	16	5	1	1.015	Nominal conditions
Cruise 2	0.6022	16	6	2	1.015	More payload Same speed Same altitude
Cruise 3	0.6514	16	5	1	0.978	Same payload Same speed Higher altitude

details the required lift coefficient, take-off weight ($WTO$), payload weight, cruise speed, and air density at each cruise condition. Equal weighting is applied to the three operating points; the effects of non-uniform weighting will be explored in future research. Equation (1) expresses the objective function mathematically, aligning with the optimization formulation.

$$\mathcal{F}(\mathcal{X},\mathcal{Y})=\prod _i^3{\displaystyle \frac{{C_D}^{\mathit{morphing}};@;{\mathit{OP}}_i}{{C_D}^{\mathit{fixed}};@;{\mathit{OP}}_i}}$$

(1)

2.7.6. Design Variables

The design variables considered in this study are the FFD control points for airfoil parameterization and the lengths of rigid links in the mechanism. Optimization of airfoil thickness distribution, serving as the baseline for camber morphing, relied on the FFD control points. Equation (1) denotes these control points as $\mathcal{X}$, each having two degrees of freedom in the Cartesian plane. As previously stated, symmetry constraints between upper and lower airfoil surfaces were enforced, reducing the number of actual design variables. Thus, optimal airfoil shapes for different operating conditions primarily arise from camber deformation driven by aerodynamic loads and actuator torque, rather than direct changes in FFD points.

The second group of design variables comprises rigid link lengths in the leading and trailing edges of the mechanism, denoted as $\mathcal{Y}$ in Equation (1). These variables determine the optimal mechanism configuration, aiming to minimize drag at multiple operating conditions by permitting significant camber morphing. Initially, the mechanism used uniform-length rigid links, but subsequent analysis revealed optimal configurations require variable lengths. The multidisciplinary optimization approach produces four distinct airfoil shapes: one symmetrical baseline shape and three additional shapes obtained through camber morphing facilitated by FSI analysis. Design variables were constrained with upper and lower bounds based on the relevant geometric scales of the problem.

2.7.7. Constraints and Requirements

The multidisciplinary optimization problem involves constraints related to the three lift conditions ($g(\mathcal{X},\mathcal{Y})=C_L$ at each operating point). Compliance with lift constraints was evaluated through FSI analysis during each optimization iteration. This ensured that intermediate geometry changes due to camber morphing consistently satisfied lift requirements.

Alongside morphing and mission requirements, two thickness constraints were included in the optimization problem (Table 2). These constraints prevent excessively thin trailing-edge geometries in the optimized airfoils. Specifically, at 75% of the chord length, the minimum thickness must be 25% of $t_{\max }$, and at 95% chord length, the minimum thickness must be 10% of $t_{\max }$.

3. Results and Discussion

This section presents the outcomes of the multidisciplinary and multipoint optimization study for a small fixed-wing UAV used as a case study, aiming to identify optimal airfoil shapes and the most effective mechanism configuration. Finally, the results from experimental tests conducted with 2D (airfoil) and 3D (wing) prototypes are showcased, emphasizing the morphing capabilities and the proof-of-concept of the proposed mechanism.

3.1. Case Study (Small UAV)

An in-house fixed-wing UAV has been used as the case study. Figure 9 shows two photographs of this UAV. The objective is to use this UAV as a baseline platform to demonstrate the morphing capabilities and the potential benefits that a hypothetical implementation of the proposed mechanism could provide. The aforementioned UAV has a wingspan of 2.6 m and a nominal takeoff weight of $WTO=5$ kg, carrying a payload of $W_{\mathrm{payload}}=1$ kg, which includes an RGB camera and an additional battery for extended flight time. Table 3 presents in more detail the current operating conditions of this UAV (Cruise 1 in Table 3).

Based on these nominal conditions, two additional cruise scenarios have been defined to evaluate situations where the proposed mechanism could play a significant role. The first scenario considers the need to carry a heavier payload while flying at the same speed and altitude (Cruise 2 in Table 3). The second scenario involves flying at the same speed but at a higher altitude, while maintaining the baseline payload (Cruise 3 in Table 3). These scenarios are representative of typical mission profiles for UAVs used in remote sensing operations.

The current airfoil used in the baseline UAV is shown in Figure 11d–f, where the optimized cambered morphing airfoils are compared against the baseline fixed airfoil.

3.2. Multidisciplinary and Multipoint Optimization

Figure 10 presents the convergence history of the multidisciplinary and multipoint optimization problem. The figure shows on the X-axis the number of iterations of the FSI routine executed during the MDO formulation, and on the Y-axis, the drag coefficient. The objective function value, as defined in Equation (1), is represented in black, while the drag coefficients for each cruise condition outlined in Table 3 are also presented in the same plot. The oscillating behavior of the objective function is attributed to the convergence characteristics of the FSI routine, which runs throughout the multidisciplinary and multipoint optimization process. On average, each FSI step involved between twenty and thirty iterations, with the optimization taking 64 iterations to converge, resulting in a total of approximately 1700 FSI evaluations.

Figure 11 depicts the outcomes of the multidisciplinary and multipoint optimization, presenting the resulting airfoil shapes and mechanism configuration. The topmost figure illustrates the optimized symmetric airfoil shape (solid line), which serves as the initial configuration for morphing into the optimal cambered airfoils. Notably, the optimized symmetric airfoil has a thinner trailing edge compared to the baseline NACA-0012 airfoil (segmented line). Additionally, the maximum thickness of the optimized symmetric airfoil has shifted slightly downstream, with minor adjustments in the leading edge shape.

The middle diagram in Figure 11 illustrates the optimal configuration of the mechanism for both leading and trailing edge morphing, superimposed on the optimal symmetric airfoil profile. It is evident that the optimal mechanism comprises rigid links of varying lengths, contrasting with the baseline configuration characterized by uniform link lengths (i.e., the distance between the centers of adjacent circles). This suggests that the rigid link lengths adjust proportionally to the curvature of the airfoil camber during morphing. Furthermore, the results indicate that the maximum camber curvature occurs approximately halfway along the cantilever beam for both leading and trailing edges.

The lower graph in Figure 11 illustrates the optimal cambered airfoils for the three operational conditions considered in this study. Additionally, it shows the baseline cambered fixed airfoil, represented by a black segmented line, which was utilized for quantifying drag reduction. Initially designed for the first cruise condition detailed in Table 3, the baseline airfoil for the fixed-wing UAV closely resembles the optimized cambered airfoil for this specific cruise condition. Both airfoils exhibit similar camber in both leading and trailing edges, with the primary difference lying in the optimized airfoil’s distinct thickness distribution, particularly noticeable at the leading edge.

Conversely, the optimized cambered airfoils for the second and third cruise conditions detailed in Table 3 exhibit higher camber, requiring greater camber deflection to achieve the desired lift coefficient. Compared with the baseline fixed airfoil under identical conditions, the optimized cambered airfoils enable cruise drag reductions of $10.3\%$, $12.9\%$, and $11.7\%$ for the first, second, and third cruise conditions, respectively.

Table 4. Drag Reduction and Optimized Drag Coefficients.

Cruise Condition	Drag Reduction [%]	Baseline ${\mathit{C}}_{\mathit{D}}$	Optimized ${\mathit{C}}_{\mathit{D}}$
Cruise 1	$-10.3$	$0.0485$	$0.0435$
Cruise 2	$-12.9$	$0.0492$	$0.0428$
Cruise 3	$-11.7$	$0.0489$	$0.0432$

presents the drag coefficients and the corresponding reduction percentages. The drag reduction is computed by comparing the optimized drag obtained with the morphing airfoil for each cruise condition against the drag coefficients of the cambered airfoil used in the UAV case study, as determined for each respective condition. The baseline airfoil used for comparison is depicted as a black dashed line ($---$) in the lower plots of Figure 11.

The bottom plots in Figure 11 depict the transformation from the baseline symmetrical airfoil into the optimal cambered airfoils through the proposed pseudo rigid-compliant mechanism. Notably, distinct mechanisms facilitate the morphing of leading and trailing edges, with their fixed supports aligned with the maximum airfoil thickness position (indicated by the green filled circle in the middle plot). This strategic placement ensures precise induction and control of the morphing process for both leading and trailing edges.

3.3. Experimental Prototype Demonstration

A prototype of a 2D airfoil was constructed to exemplify the operational principles of the proposed variable camber mechanism. The prototype was fabricated using lightweight plywood, 3D printed components, nylon revolute joints, and carbon fiber plates. Plywood was selected for the rigid components due to its favorable strength and stiffness characteristics provided by its cross-laminated structure, closely approximating the mechanical properties of more rigid materials. Compared to other rigid materials, plywood’s lightweight nature facilitated easier handling, reduced prototype weight, and simplified assembly. Additionally, plywood’s ease of machining made it suitable for manufacturing the articulated airfoil structure.

Carbon fiber was selected for the compliant cantilever beam due to its exceptional strength-to-weight ratio and high stiffness, enabling it to withstand aerodynamic loads while remaining lightweight—critical for aerospace and robotic applications. Its high stiffness ensures precise deflection control, essential for achieving the desired motion in compliant mechanisms. Furthermore, carbon fiber’s excellent fatigue resistance enables repeated camber deformations without significant performance degradation, making it suitable for dynamic applications like morphing airfoils. Additionally, extensive testing demonstrated that the mechanism did not exceed the maximum deflection capacity of the carbon fiber beam, eliminating the risk of flexural or fatigue failure.

Figure 12 visually illustrates the actual implementation of the proposed mechanism for leading and trailing edge camber morphing. Specifically, Figure 12a depicts the rigid articulated structure assembled solely with revolute joints, highlighting the relative and arbitrary linkage movement before constraining its degrees of freedom with higher-degree joints (line contacts).

Figure 12b illustrates the compliant cantilever beam and the integration of higher-degree joints into the rigid articulated structure. This figure comprehensively represents the proposed pseudo rigid-compliant mechanism. The inclusion of higher-degree joints and the cantilever beam reduces the degrees of freedom of the rigid articulated structure, effectively eliminating relative linkage movement. However, this constraint does not prevent mechanism deformation. Instead, deformation occurs through higher-degree pairs, strategically positioned to follow a predefined trajectory, such as the airfoil camber curvature.

Figure 12c depicts the mechanism enabling trailing edge camber morphing. During morphing, higher-degree pairs establish line contacts with the cantilever beam surface, facilitating proportional deformation transfer (camber deformation) to the articulated structure through rotational joints connecting the rigid links. This mechanism generates smooth, continuous camber morphing using a single actuator—specifically, an electric servomotor. Thus, Figure 12 experimentally validates the conceptual and numerical modeling shown in Figure 1, Figure 2 and Figure 3.

Additionally, a 3D wing prototype was constructed to demonstrate the integration of the proposed mechanism into conventional wing structures found in manned and unmanned aircraft. Figure 13a illustrates the internal configuration of a finite wing incorporating the proposed camber morphing mechanism. The morphing of the leading and trailing edges is independently controlled by electric servomotors installed on a fixed element—the wing spar—providing reference points for the deflection of both cantilever beams. During tests, beam deflection was induced solely by servomotor torque transmitted through a rigid bar. In flight, deflection will result from servomotor torque combined with aerodynamic pressure loads. Figure 13a visually demonstrates the morphing capability of the 3D wing prototype, confirming the practicality of the proposed mechanism for developing fully adaptable aerial robots and potentially manned aircraft.

Similarly, Figure 13b elaborates on the design’s morphing capabilities, enabling adjustments to both negative and positive camber. These capabilities highlight the versatility of the proposed mechanism in adapting the wing shape to diverse flight conditions through coordinated and independent manipulation of leading and trailing edges, allowing for moderate to extreme deformations.

3.4. Energy Consumption and Impact on Aircraft Performance

The power and energy consumed to morph from one airfoil shape to another, and the energy required to maintain a morphed shape for a given interval, were experimentally determined using the same test bench and procedure developed for FSI routine validation. Specifically, the case of Point-load plus torque of the servomotor was used for the power and energy test. the Figure 14 illustrates the power consumption and cumulative current draw of a morphing airfoil mechanism during a simulated 20-min mission. Three distinct morphing events occur at approximately 5, 10, and 15 min, each producing a brief power spike of approximately 4–4.5 W. These are followed by stabilized power levels associated with progressively higher camber configurations (Cruise 1 to Cruise 3). During the idle phase, corresponding to the symmetric airfoil configuration, the mechanism consumes minimal power (approximately 0.5 W). Even under active morphing and holding conditions, the cumulative current draw remains low. As shown on the secondary vertical axis, the total energy consumption represents only a small fraction of the available 8000 mAh capacity of the battery used in the test (indicated by the dashed line), demonstrating that the morphing mechanism consumes minimal energy and has negligible impact on the UAV’s overall energy budget while the benefits could be significant. In future stages, these estimations will need to be refined, by taking into account the integration of the mechanism with the aerodynamic and electronics of an actual UAV to quantify the final benefits the mechanism could produce for a full-aircraft configuration.

4. Conclusions

This paper presents a novel variable-camber mechanism that effectively amalgamates the advantages of rigid-body and compliant mechanisms. By combining the high-load capacity, structural stability, and mechanical efficiency characteristic of rigid-body mechanisms with the adaptability, lightweight design, and continuous and smooth morphing inherent in compliant mechanisms, the proposed pseudo rigid-compliant design offers a versatile solution for morphing aircraft. This innovative mechanism comprises an articulated airfoil structure, serving as the rigid-body component, assembled using revolute joints, and coupled with a cantilever beam, the compliant component, which models the airfoil camber and drives the morphing of the pseudo rigid-compliant design. Furthermore, the mobility of the rigid-body component is augmented through the strategic incorporation of higher kinematic pairs within the non-fixed rigid-body links. This multifaceted approach facilitates precise control over the mechanism’s motion and ensures its compliance with stringent performance requirements.

Utilizing a multidisciplinary and multipoint optimization strategy, the proposed mechanism demonstrates the potential to reduce aircraft drag by up to $13\%$ across various cruise conditions when compared to the baseline airfoil of the fixed-wing UAV. This reduction in drag holds promise for diminishing total mission drag and consequent energy consumption. A notable attribute of the proposed design is its ability to seamlessly morph into multiple airfoil shapes, accentuating its versatility and adaptability. Additionally, the suggested design seamlessly integrates with traditional wingbox structures utilized in both manned and unmanned aircraft. The incorporation of the proposed variable camber system into the wing ribs necessitates minimal adjustments to the structural design and overall wing mass. This adaptability positions the proposed design as a promising solution for morphing aircraft development. Its efficacy is further underscored by its capacity to leverage aerodynamic and flight benefits, its cost-efficiency, straightforward manufacturing process, low mechanical complexity, and the essential robustness required for such mechanisms.

The proposed mechanism facilitates both local adjustments (involving either the leading or trailing edge exclusively) and comprehensive airfoil morphing (affecting both the leading and trailing edges). Such versatility broadens its potential applications across manned and unmanned aircraft, enhancing their flight dynamics and maneuverability while optimizing performance across diverse operating scenarios. However, the most substantial advantages are foreseen within the domain of unmanned aircraft, colloquially known as drones or flying robots, encompassing a spectrum from miniature to large-scale UAVs. This is primarily attributable to the unmanned aviation industry’s agility in swiftly integrating innovative technologies such as variable camber morphing wings.

Even though the integration of the proposed mechanism with conventional wing designs and structures presents several challenges, there is plenty of room for improvement, such as using materials with a high strength-to-weight ratio, applying topology optimization to reduce the weight penalty, and employing multidisciplinary optimization to meet design requirements and ensure seamless integration with existing wing designs and structures. Structurally, the added weight, ensuring structural integrity, and managing fatigue and durability are key concerns. Aerodynamically, managing power consumption, maintaining control and stability, and optimizing performance are critical to fully realizing the potential benefits of a morphing mechanism in terms of flight efficiency. Operationally, ensuring maintenance and reliability, as well as meeting certification and safety standards (especially in the case of transport aircraft), are essential. Overcoming these challenges requires a multidisciplinary approach to balance the potential performance benefits against the added complexity and risks.