An Efficient SDOF Sweep Wing Morphing Technology for eVTOL-UAV and Experimental Realization

Abstract

The presented study demonstrates that UAVs can be flown with a morphing wing to develop essential aerodynamic efficiency without a tail structure, which decides the operational cost and flight safety. The mechanical control for morphing is discussed, where the system design, simulation, and experimental realization of ±15° SDOF sweep motion for a 7 kg eVTOL wing are detailed. The methodology, developed through a mathematical modeling of the mechanism’s kinematics and dynamics, is explained using Denavit–Hartenberg (D-H) convention, Lagrangian mechanics, and Euler–Lagrangian equations. The simulation and MBD analyses were performed in MATLAB R2021 and by Altair Motion Solve, respectively. The experiment was conducted on a dedicated test rig with two wing variants fitted with IMUs and an autopilot. The results from various methods were analyzed and experimentally compared to provide an accurate insight into the system’s design, modeling, and performance of the sweep morphing wing. The theoretical calculations by the mathematical model were compared with the test results. The sweep requirement is essential for eVTOL to have long endurance and multi-mission capabilities. Therefore, the developed sweep morphing mechanism is very useful, meeting such a demand. However, the results for three-dimensional morphing, operating sweep, pitch, and roll together are also presented, for the sake of completeness.

1. Introduction

UAVs are often employed in fields that require multi-mission, multi-role operations. The reliability of the UAV during operations in various ambient conditions depends on its aerodynamic efficiency and flexibility. The most common “change” in flight conditions is the flight speed. Through the understanding of critical and drag divergence Mach numbers, it is observed that, during take-off, straight wings provide the highest lift and stability, whereas for higher speed operations, the swept back wing increases the critical Mach number, making it more efficient than straight wings at increased flight speeds [1].

The wing morphing system described in this paper is applicable to both swept-forward and swept-back configurations. However, the swept-back is more favored due to its stability characteristics. The existing variable sweep or swing-wing technology has a maximum of three sweep angle settings, corresponding to the three regimes of the mission profile. However, the morphing mechanism allows the sweep angle to be set anywhere between −15° to +15°. The wing sweep can be dynamically controlled; therefore, it is not restricted to any one angle [2,3]. Bird flight inspires researchers to adopt mid-flight shape-changing technology to improve the aerodynamic characteristics of the wing. The current status of wing morphing concepts, designs, and technology, including flight testing issues and performance insights, is reported by Gomez and Ephrahim [4]. The history of wing morphing and important government-sponsored research projects are discussed along with recent technological advancements in the field [5]. Birds seamlessly change their wing shapes by slight to large margins, depending upon ambient conditions [6]. The morphing system in the present study was developed for utilization on the CSIR-NAL 7 kg eVTOL UAV. These UAVs face a drastic drop in lift generation during transition mode. To overcome this discrepancy, the wing can be morphed accordingly without altering the attitude of the aircraft. The review of morphing-related literature aids the understanding of the current technological status, problems, trade-offs, and methodologies used to attain wing morphing. Binbin Yan et al. [7] clarify the performance analysis of a wing geometry and dynamic model of the morphing aircraft. The aerodynamic parameters are analyzed to understand their variations during morphing operation. The aircraft behavior is predicted through simulation for various wing sweep configurations.

The authors [8,9,10,11] have solved the problems of adaptable wing shape to enhance aircraft performance by creating a revolutionary under-actuated parallel mechanism. With just four actuators and active and passive linearly adjustable components, the developed mechanism provides a consistent sweep motion in all six spatial degrees of freedom. The design efficacy is illustrated through modeling and a multimode prototype for a wing tip morphing application. Amendola et al. [12] developed a ground-breaking under-actuated parallel mechanism to address the issue of adjustable wing shape to improve aircraft performance. Consistent sweep motion in all six spatial degrees of freedom is achieved by the mechanism. An optimal motion control technique for least energy actuation is proposed. Keisuke Otsuka et al. [13] have created a framework for MBD analysis based on a strain-based beam formulation. With the strain-based method, a velocity–transformation relationship is established by the constraint equation, which is stated using vector variables. Both open-loop and closed-loop rigid joints are evaluated using a 3D L-shaped beam simulation.

David Colemen et al. [14] utilize 4 bar and 5 bar systems to overcome the difficulties of wing design optimization and kinematics in order to achieve high lift generation. Xiaoming et al. [15] created an octahedron tension structure with intricate shape variations by utilizing five struts and eight wires. Segmented morphing of variable DOF allows the wing to bend spanwise and twist in multiple directions. The authors of [16] designed a bat-style morphing wing. The wing features a tilted mounting angle around the radius at the wrist joint to mimic the wrist supination and pronation effect of the bat.

A multi-fidelity, multibody modeling approach to develop a slender deployable wing, composed of multiple constituent bodies, is proposed [17]. The wing can be folded or extended spanwise while in flight.

The rigid body morphing, involving robotic couplings, is promising and requires the construction of a very high-performance UAV. However, this interesting area, so far, has received minimal attention. Hence, the focus of this research is to develop a single-axis robotic coupling for realizing a multi-axis morphing mechanism and study its progression with LMA’s. The CSIR-NAL’s Vertical Takeoff and Landing (eVTOL) 7 kg tilt-rotor-type UAV was considered the platform for implementation. The wing geometrical motions commencing from optimized leverage were observed through MATLAB simulation. A three-dimensional kinematic model was further developed using CATIA, and motion analysis was subsequently performed in Altair Motion Solve. A mathematical model was built using the Euler–Lagrangian equation, and the simulation results of the mechanism have been accordingly validated with the experimental results.

The design of a multi-axis morphing mechanism is used to overcome the deficiency of the eVTOL vehicle’s significant drop in lift generation after a vertical takeoff at cruise altitude when the front propellers are deployed in aircraft transition mode. In order to counteract this decreased lift and altitude drop during the transition flight with the established morphing system, it is suggested that the aerodynamic properties of the wing be improved to reinforce the lift generation during transition and unpredictable weather. To demonstrate the capabilities of wing morphing, the present study delves into the development of a sweep morphing system. The flow chart shown in Figure 1 indicates the work modules.

2. SDOF Morphing Wing Design and Integration

The 7 kg eVTOL UAV features wings with a total wingspan of 2000 mm and an effective wingspan of 1876 mm. The airfoil used for wing construction is the Selig 1210, a low-speed, low Reynolds number airfoil. The wing was constructed with nine ribs spaced equidistantly. The wing root chord is 362 mm, which tapers into a wing tip chord of 266 mm. The space segmented between the fuselage and wing is maintained at 100 mm to avoid interference during wing morphing. A novelty was introduced through the coupling of the front spar and the half-span rear spar through a single titanium rod, which acts as a centralized spar. The centralized spar bears the structural and aerodynamic loads without affecting the CG. The fuselage is 150 mm × 150 mm in dimension, providing adequate room for the robotic mechanism. The 3D CAD model of the wing is created using CATIA V5. The wing assembly featuring a centralized spar, having composite skin and skeleton variations, is depicted in Figure 2a,b.

The LH and RH wings are integrated with the fuselage through the centralized robotic coupling system. The coupling system is equipped at the fuselage CG position to achieve stable SDOF and multi-axis morphing. The sweep LMA is accommodated along with the roll and pitch LMAs at an inverted L position. It is placed at 90° from the roll LMA (roof-fixed), and parallel to the pitch LMA inside the fuselage. However, in a simulated ground test rig and mathematical modeling, the roll LMA is placed at the base of the fuselage. The conventional wing fixed assembly with tail and a tailless morphed wing assembly are differentiated in Figure 3a,b.

Aerodynamic Analysis of Wing

The wing aerodynamic characteristics have been computed for multi-mission morphed conditions. The combination of angle of attack (α), sweep angle (β), and dihedral angle (Ƴ) is considered to evaluate the lift coefficient (C_L), drag coefficient (C_D), and aerodynamic efficiency (C_L/C_D) in XFLR5 3D^® using Vortex Lattice methods. The simulation was carried out for a 20 m/s velocity at 5.2 × 105 Reynolds numbers. It was observed, the C_L/C_D increases steadily up to 2.5 units of aerodynamic efficiency from 16.1 of conventional fixed wing to 18.6 for different morphed wing conditions within the range of ±15 degrees, which provided a gain of 15% improvement in aerodynamic efficiency. This optimization was achieved due to the elimination of horizontal and vertical tails, leading to reduced parasite and induced drag associated with the tails. The higher C_L/C_D and controllability are achieved through the combination of sweep–pitch–dihedral actuations. The results are tabulated in

Table 1. Wing aerodynamic characteristics (conventional and morphed conditions).

Conditions		Wing Position	Max CL/CD
α        β        γ			16.137
0        0        0
Un-morphed (Conventional Fixed Wing)
CL                CD
0.69034        0.04278
α        β        γ			18.377
5        5        5
Morphed
CL                CD
0.64392        0.03504
α        β        γ			18.045
10        10        10
Morphed
CL                CD
0.64089        0.03552
α        β        γ			17.873
12        12        12
Morphed
CL                CD
0.63823	0.03571
α        β        γ			17.544
15        15        15
Morphed
CL                CD
0.63305        0.03571
α        β        γ			18.589
−5        −5        −5
Morphed
CL	CD
0.66998        0.03604
α        β        γ			18.477
−10        −10        −10
Morphed
CL                CD
0.65758        0.03559
α        β        γ			17.545
−12        −12        −12
Morphed
CL                CD
0.6881        0.03742
α        β        γ			17.051
−15        −15        −15
Morphed
CL                CD
0.67687        0.03711

and Figure 4. The optimized C_L/C_D was observed across the range of cruise-relevant angle of attack, thereby validating the benefits that morphing wing assembly can offer to trim the drag and enhance overall aerodynamic efficiency as compared to a conventional fixed wing.

3. Methodology for SDOF Sweep Mathematical Model

3.1. System Forward Kinematics Using D-H Convention

The kinematic analyses of a system are classified into forward and inverse kinematics [18]. The forward kinematic analysis of the sweep morphing mechanism provides the theoretical prediction of the system’s motion without considering external forces. The end position of the wing is determined using D-H parameters, while assuming that the system’s calibration is “one mm of LMA stroke achieves 1° wing travel in space”.

The wing displacement occurs in Cartesian space, and it is controlled in the joint space. The kinematic model provides the necessary correlation between the two spaces to achieve the desired end effector (wing) position accurately [6,19,20,21]. The current section covers the theoretical kinematic model of the wing and actuation assembly for SDOF sweep morphing. The sweep system is considered a serial manipulator in the PRR (Prismatic–Revolute–Revolute) arrangement. The joint frames that adhere to the D-H convention are used in robotic kinematics [22]. The local joint frames vary from the actual aircraft coordinate system. The joint frames for SDOF sweep morphing, assigned based on the D-H method, are shown in Figure 5.

Forward kinematics involves determining the position of the end-effector in space based on the joint variables, represented as F (θ1, θ2… θn) = [x, y, z, R]. Conversely, inverse kinematics focuses on finding the joint variables that correspond to a specific end-effector position and orientation, expressed as F [x, y, z, R] = F (θ1, θ2…θn). y, z, R]. The Denavit–Hartenberg (D-H) convention simplifies this process by breaking down the transformations into a sequence of four fundamental operations:

Hi (Homogeneous Transformation Matrices) = Rot (Z,θ) Trans(Z,b) Trans(X,a) Rot(X,γ), and can be expressed as ${\mathrm{H}}_0^3$ = ${\mathrm{H}}_0^1$× ${\mathrm{H}}_0^2$× ${\mathrm{H}}_2^3$, assuming the initial rotation about (Z, θ) is zero [23,24,25]. By employing this method, the sweep morphing is achieved, as illustrated in Figure 6, with a range of ±15°.

The D-H method used four parameters to describe the system; namely, as joint offset, joint angle, link length sequence, and twist angle. These defining parameters are derived from the Cartesian coordinates of the system through simple measurement and trigonometry. The derived parameters are fed into the “D-H table” and subsequently substituted into the general form of the Homogeneous Transformation Matrix (HTM) and multiplied in the sequence of joints. The final HTM provides the Cartesian coordinates of the end effector frame in the base frame. The coordinates within the HTM are dependent on all the joint operations and motions performed within the morphing system. The SDOF sweep morphing system is calibrated to achieve forward and backward sweep angles of ±15°. The HTMs obtained through direct measurement and the D-H method were identical. However, the D-H method proved to be simpler and less time-consuming.

The initial local joint frame (CS₀), located at the hinge of the sweep LMA, is the global frame of reference for the system. The base hinge of the sweep LMA is assumed to be fixed, as the system is only operated in SDOF mode. Therefore, α_0DH = 0. CS₁ is fixed at the central point where the spar and the spherical bearing of the rod end coupler are connected. Negative stroke of the LMA provides clockwise rotation (−α_1DH) about axis X₁. Therefore, negative LMA actuation provides sweep forward motion. The angle subtended by the wing is equal to the measure of LMA stroke in mm. It can be safely assumed that α_1DH = β (sweep angle), as they are opposite angles. Similarly, positive actuation of the LMA results in anti-clockwise ‘α_1DH’, resulting in a sweep back motion of the wing. The second joint frame is assigned to the fulcrum, CS₂. This frame is 60 mm from the origin of the global frame. The system is calibrated such that CS₁ and CS₂ move together without any change in their orientation or position with respect to each other. The joint frame CS₃ is placed at the wing tip (end effector), which is 876 mm from the fulcrum. CS₃ undergoes an independent motion, but moves as a result of actuation.

The D-H parameters are substituted in the general HTM for each link to obtain three HTMs.

$$T_{i-1}^i=\left[\begin{array}{cccc}cos{{\theta }_{DH}}_i& -sin{{\theta }_{DH}}_i·cos{{\alpha }_{DH}}_i& sin{{\theta }_{DH}}_i·sin{{\alpha }_{DH}}_i& a_{DH}·cos{{\theta }_{DH}}_i\\ sin{{\theta }_{DH}}_i& cos{{\theta }_{DH}}_i·cos{{\alpha }_{DH}}_i& -cos{{\theta }_{DH}}_i·sin{\alpha }_{DH}& a_{DH}·sin{{\theta }_{DH}}_i\\ 0& sin{{\alpha }_{DH}}_i& cos{{\alpha }_{DH}}_i& b_{DH}\\ 0& 0& 0& 1\end{array}\right]$$

(1)

where ‘9i’ is the link number.

The end effector position is ${\mathrm{T}}_0^1\times {\mathrm{T}}_1^2\times {\mathrm{T}}_2^3={\mathrm{T}}_0^3$.

Table 2. D-H table for sweep neutral position.

Link	Joint Offset (${\mathit{b}}_{\mathit{D}\mathit{H}}$)	Joint Angle (${\mathit{\theta }}_{\mathit{D}\mathit{H}}$)	Link Length (${\mathit{a}}_{\mathit{D}\mathit{H}}$)	Twist Angle (${\mathit{\alpha }}_{\mathit{D}\mathit{H}}$)
1	168	0	0	−90
2	0	0	60	0
3	0	0	876	0

presents the wing travel in space from the neutral position to end effector values.

It has been noted that when LMA is unactuated, it is absent, angles α1 = α2 = 0, indicating no rotation occurs. In the translational portion of the HTM, the values 168, 60, and 876 represent the actuator assembly height at neutral condition, distance to the fulcrum, and wingspan, measured in millimeters, respectively. The validation of HTM for 15° maximum swept-forward and swept-back angles is represented in

Table 3. Combined D-H data for ±15ᵒ sweep deflection.

Link	Joint Offset (${\mathit{b}}_{\mathit{D}\mathit{H}}$)		Joint Angle (${\mathit{\theta }}_{\mathit{D}\mathit{H}}$)		Link Length (${\mathit{a}}_{\mathit{D}\mathit{H}}$)		Twist Angle (${\mathit{\alpha }}_{\mathit{D}\mathit{H}}$)
	+15°	−15°	+15°	−15°	+15°	−15°	+15°	−15°
1	183	153	0	0	0	0	−90	90
2	0	0	15	−15	60*cos (15)	60*cos (−15)	0	0
3	0	0	0	0	876	876	0	0

. The DH tables for maximum wing deflection are shown below.

To achieve a general expression for the X, Y, and Z coordinates of the wing tip, an arbitrary sweep angle ‘β’ and stroke length ‘L₁’ are substituted in the HTMs [19]. The DH parameters are listed in

Table 4. D-H table for arbitrary stroke length and sweep angle β.

Link	Joint Offset (${\mathit{b}}_{\mathit{D}\mathit{H}}$)	Joint Angle (${\mathit{\theta }}_{\mathit{D}\mathit{H}}$)	Link Length (${\mathit{a}}_{\mathit{D}\mathit{H}}$)	Twist Angle (${\mathit{\alpha }}_{\mathit{D}\mathit{H}}$)
1	$L_1^{\mathrm{*}}$	0	0	−90
2	0	β*	60 cos(β*)	0
3	0	0	876	0

.

After multiplication of the joint-wise HTMs, we obtain the wing tip position,

$${\mathrm{T}}_0^3=\left[\begin{array}{cccc}\cos \mathsf{\beta }& -sin\mathsf{\beta }& 0& 876.\mathit{cos}\mathsf{\beta }+30·\mathit{cos}\left(2\mathsf{\beta }\right)-30\\ 0& 0& 1& 0\\ 0.26& -0.97& 0& {L_1}^{\mathrm{*}}+30·\mathit{cos}\left(2\beta \right)-876·sin\beta -30\\ 0& 0& 0& 1\end{array}\right]$$

where the spatial coordinates, X = $876·\mathit{cos}\mathsf{\beta }+30·\mathit{cos}\left(2\mathsf{\beta }\right)-30$,

Y = 0,

$$\mathrm{Z}={L_1}^{\mathrm{*}}+30·\mathit{cos}\left(2\beta \right)-876·sin\beta -30$$

(2)

The robotic coupling system maintains the ‘L’ shape in all actuator combinations for two and three LMA actuations. The system is adept at providing angular displacements, ‘α’, ‘β’, and ‘γ’, of the wing in all three axes.

3.2. Rigid Body Equations of Motion for Single-Axis Sweep Morphing

The dynamic mathematical model of the system is developed using Lagrangian mechanics. This method is compatible with any type of coordinate system, including user-defined systems. The equations of motion (EOM) of the system are derived through the simplification of Euler–Lagrange (E-L) equations specific to the sweep morphing system [24,25,26]. The E-L equation describes the forces due to LMA actuation, gravity, and damping effects. The external forces present in this study are actuation and gravity only.

The Lagrangian function is the difference between the kinetic energy (K.E) and potential energy (P.E) of the system.

Lagrangian L = K.E − P.E

(3)

The function L is differentiated with respect to the state variables of the system and substituted in the Euler–Lagrange equation to obtain the EOMs,

$${\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{\partial L}{\partial \dot{\mathrm{q}}}}\right]-{\displaystyle \frac{\partial L}{\partial \mathrm{q}}}=\mathrm{F},$$

(4)

where ‘q’ is a variable in the system, $\dot{\mathrm{q}}$ is the first time derivative of ‘q’ and F denotes the external force or required torque applied [27,28].

The coordinate system used to describe the sweep morphing system is the two-dimensional Cartesian coordinate system. The Cartesian frame encapsulates the top view of the SDOF sweep morphing system. To maintain the base of the sweep LMA at the origin, the fourth quadrant of the Cartesian frame is utilized. The morphing system is described using four major points to depict the base of the LMA, the fulcrum point, the point of the first rib, and the wing tip, respectively. The depiction of the sweep system in neutral configuration is shown in Figure 7a. In the neutral configuration, the coordinates of each point are obtained through direct measurement using known values of link lengths. The link L1, measures the distance between the LMA base hinge and the center of the rod end coupler. L1 is 168 mm at neutral or un-morphed condition. The second link L2 is 60 mm, portraying the distance between the fulcrum and the first wing rib. L2 remains constant through all morphed and un-morphed conditions. The final link L3, extends from the first wing rib to the wing tip (last wing rib), measuring 876 mm. L3 remains constant throughout the morphing operation. Within the sweep system, two parameters, L1 and β, are time-dependent. L1 is affected by the sweep LMA actuation, while ‘β’ denotes the sweep angle (Figure 7b). In contrast, L2 and L3 stay constant throughout the motion period. The point coordinates vary during the morphing operations. However, they can be obtained through straightforward trigonometry with careful consideration that the system lies in the fourth quadrant. The potential energy of the system is calculated based on its mounting height from the base of the fuselage (188 mm). The un-morphed, wing neutral position yields the following coordinates.

The coordinates of the sweep system for an arbitrary angle “β” are represented

• where x₀, y₀ = (0,0),

x₁, y₁ = (0, −L1)

$$x_2, {\mathrm{y}}_2=({\mathrm{L}}_2\mathit{cos} \mathsf{\beta },-\mathrm{L}1+({\mathrm{L}}_2\mathit{sin}\mathsf{\beta }\left)\right)$$

$${\mathrm{x}}_3, {\mathrm{y}}_3=\left(\right({\mathrm{L}}_2+{\mathrm{L}}_3) \mathit{cos} \mathsf{\beta },-{\mathrm{L}}_1+({\mathrm{L}}_2+{\mathrm{L}}_3) \mathit{sin}\mathsf{\beta })$$

K.E equation for the sweep equation is

$$\mathrm{K}.\mathrm{E}={\displaystyle \frac{1}{2}}{m}_1\left({\dot{x}}_1^2+{\dot{y}}_1^2\right)+{\displaystyle \frac{1}{2}}{m}_2\left({\dot{x}}_2^2+{\dot{y}}_2^2\right)+{\displaystyle \frac{1}{2}}{m}_3\left({\dot{x}}_3^2+{\dot{y}}_3^2\right)$$

(5)

where ${\dot{x}}_1$ = 0

$${\dot{x}}_2=-L_{2}sin\beta ·\dot{\beta }$$

$${\dot{x}}_3=-\left(L_2+L_3\right)sin\beta ·\dot{\beta }$$

$${\dot{y}}_1=-\dot{L_1}$$

$${\dot{y}}_2=-\dot{L_1}+L_{2}cos\beta ·\dot{\beta }$$

$${\dot{y}}_3=-\dot{L_1}+\left(L_2+L_3\right)cos\beta ·\dot{\beta }$$

The total kinetic energy can be written as follows:

K.E_T = K.E₁ + K.E₂ + K.E₃

(6)

where

$$\mathrm{K}.{\mathrm{E}}_1={\displaystyle \frac{1}{2}}{m}_1\left({\dot{x}}_1^2+{\dot{y}}_1^2\right)={\displaystyle \frac{1}{2}}{m}_1{\dot{L}}_1^2$$

$$\begin{array}{l}\mathrm{K}.{\mathrm{E}}_2={\displaystyle \frac{1}{2}}{m}_2\left({\dot{x}}_2^2+{\dot{y}}_2^2\right)=(-L_{2}sin\beta ·\dot{\beta }{)}^2+(-\dot{L_1}+L_{2}cos\beta ·\dot{\beta }{)}^2\\ =L_2^2{sin}^2\beta .{\dot{\beta }}^2+{\dot{L}}_1^2-2 (\dot{L_1}{L}_{2}cos\beta ·\dot{\beta })+L_2^2{cos}^2\beta ·{\dot{\beta }}^2\\ ={\displaystyle \frac{1}{2}}{m}_2{(L}_2\dot{\beta }(L_2\dot{\beta }-2\dot{L_1}cos\beta )+{\dot{L}}_1^2)\end{array}$$

$$\begin{array}{l}\mathrm{K}.{\mathrm{E}}_3={\displaystyle \frac{1}{2}}{m}_3\left({\dot{x}}_3^2+{\dot{y}}_3^2\right)=(-\left(L_2+L_3\right)sin\beta ·\dot{\beta }{)}^2+(-\dot{L_1}+\left(L_2+L_3\right)cos\beta ·\dot{\beta }{)}^2\\ ={\displaystyle \frac{1}{2}}{m}_3({\dot{L}}_1^2-2\dot{L_1}\left(L_2+L_3\right)cos\beta .\dot{\beta }+{\dot{\beta }}^2(L_2+L_3{)}^2)\end{array}$$

$$\mathrm{P}.\mathrm{E} \mathrm{equation}=\sum m_{i}g{h}_i$$

(7)

All the components of the system are placed at the same height above the real ground, $h_1=h_2=h_3=h=188 \mathrm{m}\mathrm{m}$, where

$$\mathrm{P}.\mathrm{E}=gh\left(m_1+m_2+m_3\right)$$

(8)

The Lagrangian,

$$\begin{array}{c}\mathrm{L}={\displaystyle \frac{1}{2}}{m}_1{\dot{L}}_1^2+{\displaystyle \frac{1}{2}}{m}_2{(L}_2\dot{\beta }(L_2\dot{\beta }-2\dot{L_1}cos\beta ))+{\displaystyle \frac{1}{2}}{m}_2{\dot{L}}_1^2+{\displaystyle \frac{1}{2}}{m}_3({\dot{L}}_1^2-\\ 2\dot{L_1}\left(L_2+L_3\right)cos\beta .\dot{\beta }+{\dot{\beta }}^2(L_2+L_3{)}^2)-gh(m_1+m_2+m_3)\end{array}$$

(9)

In sweep morphing the two variables are L₁ and $\mathsf{\beta },$

Considering the first variable L1, the E-L equation becomes the following:

$${\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{\partial L}{\partial \dot{L_1}}}\right]-{\displaystyle \frac{\partial L}{\partial L_1}}=F_{Sweep}$$

(10)

$${\displaystyle \frac{\partial L}{\partial \dot{L_1}}}=m_1\dot{L_1}-m_2{L}_{2}cos\beta ·\dot{\beta }+m_2\dot{L_1}+m_3\dot{L_1}-m_3(L_2+L_3)cos\beta ·\dot{\beta }$$

$${\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{\partial L}{\partial \dot{L_1}}}\right]=m_1\ddot{L_1}-m_2{L}_2(\ddot{\beta }·cos\beta -sin\beta ·{\dot{\beta }}^2)+m_2\ddot{L_1}+m_3\ddot{L_1}+m_3(L_2+L_3)(\ddot{\beta }·cos\beta -sin\beta ·{\dot{\beta }}^2)$$

(11)

$${\displaystyle \frac{\partial L}{\partial L_1}}=0 (becausepotentialenergydoesnotdependonthelengthL1)$$

The first EOM:

$$\ddot{L_1}\left[m_1+m_2+m_3\right]-(\ddot{\beta }·cos\beta -sin\beta ·{\dot{\beta }}^2)\left(m_3\right(L_2+L_3)-m_2{L}_2)=F_{Sweep}$$

(12)

Considering the second variable $\mathsf{\beta }$, the E-L equation becomes the following:

$${\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{\partial L}{\partial \dot{\beta }}}\right]-{\displaystyle \frac{\partial L}{\partial \beta }}={\tau }_{Sweep}$$

(13)

$$\left[{\displaystyle \frac{\partial L}{\partial \dot{\beta }}}\right]=m_2{L}_2^2\dot{\beta }-m_3\dot{L_1}\left(L_2+L_3\right)cos\beta +2\dot{\beta }(L_2+L_3{)}^2$$

$${\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{\partial L}{\partial \dot{\beta }}}\right]=m_2{L}_2^2\ddot{\beta }-m_3\left(L_2+L_3\right)(\ddot{L_1}cos\beta -sin\beta ·\dot{\beta }\dot{L_1})+2\ddot{\beta }(L_2+L_3{)}^2$$

(14)

and

$${\displaystyle \frac{\partial L}{\partial \beta }}=m_2{L}_2\dot{\beta }\dot{L_1}sin\beta +m_3\dot{L_1}\left(L_2+L_3\right)sin\beta ·\dot{\beta }$$

(15)

The second EOM is as follows:

$$[m_2{L}_2^2+2(L_2+L_3{)}^2]+\dot{L_1}sin\beta .\dot{\beta }(m_2{L}_2+m_3)+m_3(L_2+L_3)(\ddot{L_1}cos\beta -sin\beta ·\dot{\beta }\dot{L_1})={\tau }_{Sweep}$$

(16)

The representation of the EOMs is in vector form, and the equation of motions can also be represented in the general matrix form as follows:

$$\mathrm{H} (\ddot{q})+\mathrm{C} (\dot{\mathrm{q}}, \mathrm{q}) + \mathrm{f}(\mathrm{q}) = \mathrm{M}$$

(17)

where

• H ($\ddot{q}$) represents the matrix containing linear and angular acceleration terms;
• C ($\dot{q}$, q) is the matrix containing centripetal and Coriolis terms;
• f(q) is the matrix containing gravity terms;
• M is a matrix containing external forces and torques.

$$\begin{gathered} \left[\begin{array}{cc}(m_1+m_2+m_3)& -cos\beta (m_2{L}_2+m_3\left(L_2+{\mathrm{L}}_3\right))\\ -cos\beta (m_2{L}_2+m_3\left(L_2+{\mathrm{L}}_3\right))& (m_2{L}_2^2+m_3{L}_2^2+2m_3{L}_2{L}_3+m_3{L}_3^2)\end{array}\right]·\left[\begin{array}{c}\ddot{L_1}\\ \ddot{\mathsf{\beta }}\end{array}\right]+ \\ \left[\begin{array}{cc}0& sin\beta ·\dot{\beta }(m_2{L}_2+m_3\left(L_2+{\mathrm{L}}_3\right))\\ 0& 2\dot{L_1}sin\beta ·\dot{\beta }(m_2{L}_2+m_3(L_2+{\mathrm{L}}_3)\end{array}\right]\left[\begin{array}{c}\dot{L_1}\\ \dot{\beta }\end{array}\right]+\left[\begin{array}{c}0\\ 0\end{array}\right]=\left[\begin{array}{c}{F}_{Sweep}\\ {\tau }_{Sweep}\end{array}\right] \end{gathered}$$

(18)

4. Static Torque Calculation for Sweep Actuator Selection

The working capacity of the SDOF sweep morphing system relies on the power generation system. The sweep LMA is expected to overcome the forces of self-weight of the wing and aerodynamic forces to provide the desired morphing motion. To maintain the wing at neutral position, the static torque requirement must be calculated. Subsequently, the rating of the actuator can be referred to check if it is capable of providing the necessary torque at the fulcrum (optimized leverage position). The actuator specifications are presented in

Table 5. Specification of linear miniature actuator.

Type and Stroke Length	Operational Voltage	Maximum Load	Stroke Speed	Peak Current	Backlash	Feedback
Electric &30 mm	12VDC	45 N	3.5 mm/s at peak load	1 A at peak load	0.1 mm	Open Loop

.

Apart from the static torque at the fulcrum, a simple calculation of actuator force is explained in Figure 8. To realize the static torque at the fulcrum required to hold the neutral sweep position of the wing, two major parameters of the system play a determining role. The weight of the wing assembly and the distance between the fulcrum and the wing tip (where the self-weight of the wing acts). Firstly, the mass of the wing assembly, along with the coupling components, is approximated at 1 kg. The weight of the wing is calculated for two cases: 1 g conditions for rest and level operation, and 3 g conditions for maximum possible load during flight. The second parameter, ‘lever arm’, is calculated from the fulcrum to the wing tip, which is measured to be 876 mm. The static torque is calculated using ${\tau }_{static}$ = m × g × l.

$$\begin{array}{l}{\tau }_{static}=m·g·l \mathrm{at} 1 \mathrm{g} \mathrm{condition}\\ =1\times 9.81\times 0.876\\ =8.6 \mathrm{N}·\mathrm{m}\end{array}$$

(19)

$$\begin{array}{l}{\tau }_{static}=m·3g·l \mathrm{at} 3 \mathrm{g} \mathrm{condition}\\ =1\times 3\times 9.81\times 0.876\\ =25.8 \mathrm{N}·\mathrm{m}\end{array}$$

(20)

Therefore, to perform wing sweep morphing, the selected actuator must be capable of providing torque higher than the calculated values at both ‘g’ conditions.

5. LMA Stroke Visualization on MATLAB

The rotation and linear actuations from SDOF to MDOF have been represented by the authors [29]. The LMA provides linear motion, which is converted to rotary motion at the fulcrum, which works as a gimbal. The rotary motion is transferred to an efficient and smooth wing sweep motion through the centralized spar. The system is modeled and simulated on MATLAB to understand the validate the relationship between stroke length and sweep angle, and to simulate the backlash of the LMA. As calculated in previous chapters, the fulcrum is placed at an optimized leverage point where the self-weight of the wing is balanced

Through the simulation, it is noted that the wing sweep angle undergoes a change of 1° about the aircraft ‘z’ axis, for every 1 mm of LMA displacement, with a backlash of 0.1 mm. The maximum sweep angle of ±14.78° is achieved out of the predicted ±15° deflection, having a 1.5% error. The plot between LMA stroke (mm) and sweep angle is depicted in Figure 9a. The wing motion is stabilized about the fulcrum. The LMA experiences a load of 8.6 N at rest and a maximum of 25.8 N during 3G operation, without considering forces due to aerodynamic load. The wing sweep travel with emphasis on the fulcrum point is represented in Figure 9b. The wing sweep travel of forward and backward in the space is depicted in Figure 10a,b.

6. Single Axis to Multi-Axis Motion Simulation

The MultiBody Dynamic model is made with the assumption of the wing being a rigid body. The SDOF sweep motion animation of ±15° is performed on Altair Motion Solve. The set-up with required constraints has been extracted from the multi-axis morphing assembly. The resultant time vs. angular displacement graph is shown in Figure 11. The wing is made with multiple rigid components and joints, both prismatic and revolute joints. For rigid body transformation, the authors have factored in scleronomic constraints, which are the results of position and direction vectors of the ball, hinge, universal, and rigid joints.

The sweep morphing actuator is placed horizontally and coupled with the rod end coupler of the roll LMA using a specialized inverted ‘U’ bracket, forming an ‘L’ shape. The bracket is connected to the fulcrum (rod end coupler) using a 60 mm rod, which is attached to the rod end coupler of the Roll LMA. As the sweep and roll LMAs are coupled, the rod transfers the actuation of both the LMAs to the wing. The rod extends into the fulcrum point and connects to the wing’s central spar. The end of the LMA piston is connected to a rod end coupler/bearing of standard MW3 and arrested to provide a swivel of up to 8°. The authors analyzed the design of the morphing wing mechanism for its 6 DOF [2,29,30]. The system is calibrated to deliver 1° wing deflection for a 1 mm actuator stroke change. The positive stroke of the LMA provides swept-back angles, and the negative stroke of the LMA causes the wing to sweep forward. The robotic mechanism, along with the wing, is animated for SDOF sweep variation in ±15°.

All the wing components, including actuators, joints, and couplers, have been accounted for under rigid body considerations for the wing MBD simulation. The optimized leverage point known as the fulcrum is a rod end coupler that provides spherical motion with a constraint of 8° swivel in all directions. The spherical joint provides smooth travel of the wing to all desired morphing positions. The model set-up and simulation are shown in Figure 12.

The third LMA under the “L” coupling system is added in-plane with the fulcrum to perform pitch morphing of the wing. The MBD model system is suited to simulate SDOF, two-axis, and three-axis morphing. The x, y, and z components and angular displacements α, β, and γ from the initial position are utilized to determine the end position and orientation of the wing. In this study, the wing travel in all directions is restricted to ±15°. It is observed that the actuator and wing travel align with the theoretical calculations and the experimental test. The results of the motion simulation are plotted for time vs. angular displacement. For the purpose of technology demonstration, a case of three-axis morphing (yaw 13°, sweep 4°, and pitch 3°) is simulated and displayed in Figure 13.

7. Morphing Experiment on Ground Test Rig

The experimental test for demonstrating the SDOF sweep morphing is conducted on a dedicated multi-axis morphing ground test rig, as shown in Figure 14. The wing is equipped with a closed-loop PID control system, which provides stable motion of the wing during actuation, including a control strategy.

The extensive control procedures and their implementation are explained in reference [28].

Figure 15 shows the control block diagram, explaining the control strategy adopted in the morphing experiment. Here, three LMAs are employed with linear PID controls to realize the 3D rigid body motions on the UAV wing. The power management system provides the LMA with 12 VDC at maximum current consumption of 1A for optimal operation. The LMA converts the applied voltage into linear motion, resulting in required wing travel in 3D space. The wing is fitted with an inertial measurement unit (IMU) and autopilot to simulate desired mission conditions and obtain accurate readings of angular displacement from the system. The LMA is capable of actuation at 3.5 mm per second, and the stroke was limited using PID controller to establish 1 mm travel at the rate of 0.0285 s over an operational range of 0 to 12VDC. The experimental tests may be monitored through GCS (Ground Control Station).

8. Results and Discussions Experimental Test on Ground Test Rig

The mathematical modeling and simulation through MATLAB have been carried out initially. The kinematic motion simulation has been developed to understand the adoptability of SDOF wing sweep morphing to align with the multi-axis morphing requirements of the eVTOL UAV. The aim of validation is to implement 1° of wing space travel for an input of one mm LMA travel.

The linear actuation system is focused on achieving 6 DOF, is realized with 1 mm travel of the actuator, providing 1° of wing travel in space with respect to time. The other important observation is that pitch, yaw, and sweep actuators must be placed in an inverted position in the actual VTOL-UAV in order to be accommodated and integrated with the robotic coupling system.

The experimental realization of forward sweep travel of 13° and backward sweep of 14.5° is observed. The pitch value is attained with positive and negative angles of attack of 12°, and the sweep attitude of −10° is achieved with a dihedral angle of 21° and an anhedral angle of −14°. Considering the initial offset value of actuator travel ±15 mm, the actual values fall at ±15° angular displacement. The backward and forward sweeps have been achieved with an acceptable range of percentage error between 0.8% to 1.5%, which validates the linear actuator motion of ±15° in terms of wing travel in the space. The ±15° travel is depicted in a scaled-up version of the graph in Figure 16. Thus, the robotic L coupling system achieved the multi-axis morphing wing travel to meet the mission requirements. The technical advantage of using the proposed mechanism is envisaged as follows:

• SDOF can be independently employed with the proposed L coupling for sweep, pitch, and roll, respectively.
• Also in combination, namely sweep-Pitch, Pitch-Roll and Sweep, pitch-Roll may be achieved by L coupling.
• Linear motion is ensured, using LMA’s.
• Since the real-time change in flight parameters is possible to achieve, optimized aerodynamics configurations can be designed in real time for the required aerodynamic performances during flight.
• The designed morphing mechanism is stress and deflection cleared with respect to maneuvering and inertia loads, so that it can be directly scaled up and implemented.

9. Conclusions

Using an SDOF sweep morphing system and linear actuators, the methodology for a multi-axis morphing wing approach is developed. The theoretical calculations are validated using simulation and experimental tests, that has provided confidence. The theoretical findings are obtained by considering the system as a PRR serial robotic manipulator and assigning local axes to simplify the calculations. The method was confirmed to be straightforward and acceptably precise to explain the system. The MATLAB and Altair Motion Solve simulations also ensured that the system performs as predicted. The final validation is performed on a dedicated test rig, equipped with a PID controller and IMU to map the wing motion accurately within the error limitations. The aerodynamic efficiency of the wing is studied for the required range of ±15° at its optimized condition. It forms an important part of a multi-axis morphing scheme to provide precise wing configurations in all three axes, namely pitch, yaw, and roll, to equip the UAV for ‘all-terrain’ missions.

The system developed paves the way for future multi-axis morphing wing validation in flight. The developed technique can offer morphing in each axis and also in the combination of two and three axes. However, this paper focuses on the methodology used for SDOF sweep morphing about the aircraft ‘z’ axis alone. Further steps are presented, which include testing the system for two-axis and three-axis morphing, and implementation of an efficient control scheme to provide seamless wing morphing in any direction. The CFD analysis, dedicated wind tunnel tests for the morphing wing at CSIR-NAL MART facilities, and flight test demonstration will be performed in the future.