Design and Rapid Prototyping of Deformable Rotors for Amphibious Navigation in Water and Air

Abstract

This paper aims to report the design of a mechanism to drive a propeller to deform between an aerial and one aquatic shape. This mechanism can realize the deformation of blade angle, radius, blade twist angle distribution and blade section thickness. Inspired by the Kresling origami structure and utilizing its rotation-folding motion characteristics, a propeller hub structure with variable blade angle is designed. A blade deformation unit (S-unit) with extensional-torsional kinematic characteristics is designed through the motion analysis of a spherical four-bar mechanism. A rib support structure fixed to the linkages of the s-unit is designed to achieve the change in blade section thickness. Based on motion analysis, the coordinate transformation method has been used to establish the relationship between propeller shape and deformation mechanism. The deformation of blade extension, blade twist distribution, and blade section thickness are analyzed. The deformation ability of the proposed structure can be verified then by kinematic simulation and rapid prototyping based on 3-D printing. It is proved that the proposed mechanism is applicable to deformable propeller design. The rapid prototype testing validates the stable motion of the mechanism. However, due to the relatively large self-weight of the structure, the blade has a slight deformation. In the subsequent work, the structural strength issue needs to be emphasized.

1. Introduction

The hybrid aerial and aquatic vehicle (HAAV) is a new type of vehicle capable of navigating in both air and water and has broad application prospects in many fields [1,2]. In marine scientific research, HAAVs are designed to collect water and air samples at different heights and depths to monitor changes in the marine environment [3,4]. This kind of vehicle is able to quickly reach the target sea area and then dive into the water for data collection [5,6], providing comprehensive data support for studying changes in the marine and atmospheric environments [7,8]. In the field of seabed resource exploration, it is used for ocean terrain mapping and resource exploration. With the ability to fly, it conducts preliminary surveys of large sea areas to determine potential resource areas [9]. In the field of marine rescue, it is expected to quickly reach the accident area by flying and explore underwater accidents by underwater navigation [10,11].

In the field of driving methods of vehicles, some new-concept propulsion and driving systems are highly inspiring. Among them, the flapping-wing propulsion method, inspired by marine and aerial creatures, has great research potential. Pawel Piskur et al. [12] have proposed an innovative propulsion system designed for the bionic unmanned underwater vehicle (BUUV). In this novel system, an additional joint has been incorporated. This joint enables the vehicle to make a seamless quarter-turn to a position where the drag is minimized. Moreover, this additional joint does not necessitate a separate servo motor. The dynamic performance of the BUUV system has been verified through particle image velocimetry (PIV) experiments [13]. Qian Li et al. [14] designed a feathered flapping-wing robotic. This robot is able to sense the speed and direction of air flow with natural feathers and a piezoelectric system equipped. In some other cases that place more emphasis on engineering applications, multi-rotor HAAVs have garnered extensive attention based on their excellent maneuverability and cross-media capabilities [15,16,17]. Drews et al. [18,19] applied the multi-rotor configuration to hybrid air and aquatic flight. They installed four aerial shape propellers and four aquatic shape propellers on the vehicle. Since the propellers are selected according to air and underwater working conditions, respectively, this kind of vehicle has good working efficiency in both media. However, the vehicle is relatively heavy. Alzu’bi et al. [20,21] use aerial shape propellers as the driving system for the vehicle both underwater and in the air. This kind of vehicle is lighter and has a simple structure. However, due to the density difference between the two media, the rotational speed of the propeller when working underwater is much lower than the design rotational speed of the propeller, so the working efficiency is not high. Tan et al. [22,23,24] compared and studied the power configuration of underwater vehicles and air vehicles. They proposed a deformable multi-rotor amphibious vehicle that realizes flying and underwater navigation by changing the axis direction of the propeller.

Although there have been a large number of studies, the “driving compatibility [1]” of multi-rotor HAAVs still remains a difficulty [25,26]. When the propeller works in water, the viscosity and density of the medium are much greater than those in air [27]. This increases the working efficiency of the propeller and also requires the propeller to overcome greater resistance. Therefore, the radius of the water propeller is usually designed to be small. As the working efficiency of the water propeller is also affected by cavity, the propeller is designed to have a shape with a larger blade–area ratio [28]. Since underwater vehicles are usually designed to have equal buoyancy and gravity, additional power is not required to remain stationary. Thus, water propellers are usually designed under working conditions with a higher advance coefficient, so water propellers are generally more twisted. Conversely, air propellers usually adopt a slender blade shape to reduce air resistance, to increase rotational speed and propulsion efficiency [29]. For multi-rotor aircraft, hovering is the most important working condition. This type of propeller is usually designed according to working conditions of a lower advance coefficient, and the blades are more flat. Due to the significant shape disparity between air propellers and underwater propellers, a vehicle relying solely on air propellers cannot navigate efficiently underwater. Conversely, a vehicle using only water propellers cannot supply the power necessary for flight. This constitutes the difficulty in attaining driving compatibility for multi-rotor hybrid air and aquatic vehicles (HAAVs).

In terms of achieving the “driving compatibility [1]” of HAAVs, deformable drive mechanisms have great application potential [30]. This paper presents a deformation driving mechanism for a propeller that can actively change the radius, blade angle, attack angle distribution, and blade thickness. Inspired by the Kresling origami structure and utilizing its rotation-folding motion characteristics, a propeller hub structure with variable blade angle is designed. Through the motion analysis of the spherical four-bar mechanism, a blade deformation unit (s-unit) with extensional-torsional deformation ability is designed. The deformation driving mechanism of the propeller blade is obtained by connecting s-units in series. Driven by the proposed mechanism, the propeller blade angle, radius, twist angle distribution, and blade thickness can be deformed actively. The relationship between the blade shape and the shape parameters of the deformation mechanism is established by the coordinate transformation method. In aerial mode, the blade angle and twist angle are reduced while the radius is increased, thereby the thrust of the propeller is enhanced. In aquatic mode, the blade angle and twist angle is increased, the radius is decreased. This reduces the working torque of the propeller and enables it to achieve good working efficiency at a higher advance coefficient. A rapid prototyping machine based on 3-D printing is made to verify the deformation characteristics of the proposed propeller.

The paper is organized as follows. Section 2 presents the geometric principles of the deformation mechanism, including the Kresling structure and spherical four-bar mechanism. Section 3 details the deformation driving mechanism components like the variable blade angle hub, extensional-torsional deformable blade, and variable thickness rib structure. Section 4 focuses on kinematic analysis, covering blade section and s-unit parameterization and deformation analysis. Section 5 conducts simulation and rapid prototyping experiments to validate the mechanism. Section 6 concludes with a summary of findings, limitations, and future research directions.

2. Geometric Principles of the Deformation Mechanism

This section introduces the geometric characteristics of the Kresling origami structure and the spherical four-bar mechanism, which is the basis for constructing the deformation mechanism.

2.1. Kresling Origami Structure

The Kresling structure, arranged alternately by a series of triangular peaks and valleys to connect the upper and lower planes, is spiral in shape. It has a large folding ratio and good deployability. It has great advantages in applications requiring a large folding ratio. During folding and unfolding, the angle between each side valley crease and the upper and lower planes changes. The angle increases when the mechanism is extended and decreases when it is retracted. The movability of the Kresling structure is based on the flexible deformation of paper. Its deformation presents a bistable form [31]. If the paper is assumed to be rigid and the crease is a revolute joint, then the entire structure is immovable. Considering that the rotor hub rotates at high speed during operation and the fluid’s force on the structure is complex and variable, the bistable structure relies on material flexibility to ensure the movability of the mechanism. When applied to a rotor, it is difficult to ensure stability. Therefore, the Kresling structure is improved. The revolute joint connection between the triangular piece and the upper and lower planes in the Kresling structure is changed to a cylindrical joint connection. A cylindrical joint connection is added between the upper and lower planes. The Kresling origami and the modified structure are shown in Figure 1a,b.

The improved Kresling structure is a single-degree-of-freedom structure. The angle between the side link and the base plane is defined as ${\beta }_0$, the angle between the axis of the side revolute joint and the upper and lower planes is defined as $\varphi$, and the angle between the axes of the two revolute joints fixedly connected with the side link is defined as $\tau$, as shown in Figure 2. The relationship among these angles can be described by Equation (1).

$$sin{\beta }_0=sin\varphi ·sin\tau$$

(1)

2.2. Spherical Four-Bar Mechanism

The spherical four-bar mechanism is a spatial mechanism in which the axes of the revolute joints intersect at a point, as is shown in Figure 3a. The trajectory of the constructed point is located on a concentric spherical surface. The geometric relationship is expressed in Figure 3b. Let ${\mathbf{L}}_A$, ${\mathbf{L}}_B$, ${\mathbf{L}}_C$, and ${\mathbf{L}}_D$ represent the axes of each revolute joint, which are intersected at the center of the sphere O. AB, BC, CD, and DA denote the four linkages that constitute the spherical mechanism. AOB, BOC, COD, and DOA denote the plane formed by ${\mathbf{L}}_A{\mathbf{L}}_B$, ${\mathbf{L}}_B{\mathbf{L}}_C$, ${\mathbf{L}}_C{\mathbf{L}}_D$, ${\mathbf{L}}_D{\mathbf{L}}_A$. Let ${\alpha }_i(\mathit{i}=1,2,3,4)$ represent the torsion angles between the revolute joints. Let the angle between ${\mathbf{L}}_A$ and ${\mathbf{L}}_C$ be $\psi$, and form a plane $\Pi$. ${\beta }_i(i=1,2,3,4)$ respectively represent the angles between $\Pi$ and the planes AOB, BOC, COD, DOA. Let ${\theta }_s={\beta }_1$ + ${\beta }_4$.

Rich spatial deformations can be produced by the spherical four-bar mechanism. Considering modular design and for the purpose of simplifying the design steps, in this paper, two specific forms of spherical four-bar mechanisms, namely the oblique symmetric spherical four-bar mechanism and the equilateral spherical four-bar mechanism, are employed to construct a spatial scissor mechanism.

The oblique symmetric spherical four-bar mechanism has equal opposite side links. Its geometric constraint can be described as Equation (2).

$$\left\{\begin{array}{c}{\alpha }_1={\alpha }_3\hfill \\ {\alpha }_2={\alpha }_4\hfill \\ {\beta }_1={\beta }_3\hfill \\ {\beta }_2={\beta }_4\hfill \end{array}\right.$$

(2)

$\psi$ can be obtained by Equation (3).

$$c\psi ={\displaystyle \frac{c{\alpha }_{3}c{\alpha }_4-c{\beta }_{1}c{\beta }_{2}s{\alpha }_{3}s{\alpha }_4}{1-s{\beta }_{1}s{\beta }_{2}s{\alpha }_{3}s{\alpha }_4}}$$

(3)

$$\left\{\begin{array}{c}tan{\beta }_1={\displaystyle \frac{s{\theta }_{s}s{\alpha }_4}{s{\alpha }_3+c{\theta }_{s}s{\alpha }_4}}\hfill \\ {\beta }_4={\theta }_s-{\beta }_1\hfill \end{array}\right.$$

(4)

where s represents sin and c represents cos. The geometric constraint of the equilateral spherical four-bar mechanism can be described as (5). The remaining shape parameters have a similar form to those of the oblique symmetric type.

$$\left\{\begin{array}{c}{\alpha }_1={\alpha }_3={\alpha }_2={\alpha }_4\hfill \\ {\beta }_1={\beta }_3={\beta }_2={\beta }_4\hfill \end{array}\right.$$

(5)

3. Deformation Driving Mechanism

The variable blade angle hub and spherical linkages are composed to form the deformation drive mechanism. The design of the hub is applied with the change in the angle ${\beta }_0$ between the axis of the revolute joint of the Kresling origami structure and the base plane, as shown in Figure 2. The blade deformation drive structure is formed by the oblique symmetric and equilateral spherical four-bar structures which act as the deformation units (s-units) of the blade and are connected in series.

3.1. Variable Blade Angle Hub

Six spatial linkages are composed to form the hub. The upper and lower linkages are connected by a cylindrical pair. A cylindrical pair connection exists between the side linkage and the base linkage. Additionally, there is a revolute joint connection between the side connecting rods. The axis of the revolute joint is defined as ${\mathbf{L}}_P$. The mechanism schematic of the propeller hub and its comparison to the Kresling structure is presented in Figure 4. In the Kresling structure, the side creases are manifested as revolute joints in the Hub structure, and the creases on the upper and lower surfaces are manifested as cylindrical joints.The drive shaft of the propeller is fixedly connected to the lower plane of the hub and is connected to the upper plane of the hub by a cylindrical joint.

When the hub deforms, the angle between ${\mathbf{L}}_P$ and the base plane changes. The chord line of the root section of the propeller blade is set parallel to this, enabling the hub to generate blade angle deformation. The deformation of the hub and blade angle is shown in Figure 5. It can be seen that when the top and bottom linkages of the hub structure undergo translational and rotational motions (not helical motions) around the drive shaft during deformation. The angle between the side rotation axis and the base plane changes during deformation, and this characteristic is applied to the design of the hub with a variable blade angle.

3.2. Extensional-Torsional Deformable Blade

The driving mechanism of the deformable blade is formed by alternately arranging and serially connecting equilateral s-units and oblique symmetric s-units, as is shown in Figure 6. In the series sequence, a four-bar mechanism is referred to as an s-unit. The shape parameters described in Figure 3 are added with a superscript $\mathit{i}$ to depict the shape of the i-th s-unit. The ${\mathbf{L}}_C^i$ of the i-th s-unit coincides with the ${\mathbf{L}}_A^{i+1}$ of the $(i+1)$-th s-unit. The linkage $C^{\mathit{i}}{D}^{\mathit{i}}$ of the i-th is fixed connected to the linkage $C^{\mathit{i}}{D}^{\mathit{i}}$ of the $(i+1)$-th unit. The spherical centers O of s-units are alternately arranged on both sides of the s-sequence, which makes the overall deformation of the mechanism into an extensional deformation. The s-sequence is constructed by using two types of s-units: equilateral and oblique symmetric. This makes each $\Pi$ plane undergo folding deformation during deformation, which causes the mechanism to produce torsional deformation.

When the propeller deforms from the aerial mode to the underwater mode, the distance between the chord lines of the propeller blade sections is reduced and the angle between the chord lines is increased. The serially connected s-units possess the coupled motion characteristics of extension and torsion, which are necessary for the deformation of the aerial-aquatic propeller, as is depicted in Figure 7. Figure 7a,b shows the aerial shape of an s-sequence. The mechanism is in the extensional state: the distance between the chord lines is relatively large. The torsional angle $\Delta$ of the two chord line, which is defined in Section 4.3.2, is relatively small. The mechanism is in a long and flat shape. When the mechanism is folded, the length of the s-sequence becomes smaller and the $\Delta$ angle increases. The mechanism presents a short and twisted shape, as is depicted in Figure 7c,d.

3.3. Variable Thickness Rib Structure

The blade rib support is segmented into upper and lower parts along the chord line and is fixed connected to linkage AB and linkage DA of the equilateral s-unit, respectively. The chord line of the blade rib coincides with ${\mathbf{L}}_C$, thus the blade thickness undergoes deformation as the s-sequence deforms, as is depicted in Figure 8. When the ${\theta }_s$ of the equilateral s-unit decreases, the rib support structure is in a folded state and the section thickness is relatively small, as is shown in Figure 8a. When ${\theta }_s$ increases, the rib support structure is in an extended state and the section thickness is relatively large, as is depicted in Figure 8b.

3.4. Propeller Deformation Driving Mechanism

The propeller deformation mechanism is assembled by the aforementioned deformable hub and blade. The side linkages of the deformable hub act as the linkage AB and linkage DA of the blade root unit. As is shown in Figure 9, the axis of the revolute joint of the hub coincides with ${\mathbf{L}}_A^1$. The blade rib support structure is installed on each equilateral s-unit. Linkage AB in the equilateral s-unit, linkage DA in the oblique symmetric s-unit, and the upper part of the rib support are fixedly connected. Correspondingly, linkage DA in the equilateral s-unit, linkage AB in the oblique symmetric s-unit, and the lower part of the rib support are fixedly connected. The shapes of the deformation mechanism in the air and underwater are presented in Figure 10. It can be observed that the designed mechanism fits the shapes of the propeller in the aerial mode and aquatic mode rather well. From Figure 10a,c, it can be seen that the propeller has a larger radius and a smaller disk ratio in aerial mode. When folded to an aquatic shape, the radius decreased and the disk ratio increased. From Figure 10b,d, it can be seen that the aerial shape blade is flat and thin. On the contrary, the aquatic blade is more twisted and thicker.

4. Kinematic Analysis

To describe the relationships among the s-unit, the shape of the blade rib, and the shape of the propeller, this section presents the kinematic analysis of the proposed machinery. Based on this, the deformation characteristics of the propeller are analyzed.

For the clarity of the kinematic description of the mechanism, a rectangular coordinate system is defined on each s-unit with the installed blade rib support structure. The origin of the coordinate system coincides with the leading edge of the blade section. The x-axis is set on the $\Pi$ plane of the s-unit and perpendicular to the chord line of the blade rib. The y-axis coincides with the chord line of the blade rib, and the z-axis is perpendicular to the plane. The angle between the ${\Pi }^{\mathit{i}}$ plane and the ${\Pi }^{\mathit{i}+\mathit{1}}$ plane is defined as ${\delta }_{\mathit{i}}$, as is shown in Figure 11. Since the upper and lower planes of the rib are coplanar with the planes AOB and DOA, respectively, the angle between the rib plane and the Pi plane is $\theta /2$. $c^{\mathit{i}}$ is defined as the chord length of the i-th airfoil. $h^{\mathit{i}}$ is defined as the distance from $O^{\mathit{i}}$ to the leading point of the i-th airfoil. The definition of $\Pi$ and $\Psi$ can be found in Figure 3b.

4.1. Shape Parameterization of Blade Section

The section shape of the propeller blade is determined by the shape and folding degree of the blade rib support structure. The shapes of the upper and lower half blade rib support structures $S_u^{\mathit{i}}$ and $S_l^{\mathit{i}}$ can be respectively described as two B-spline curves of degree 3 with 15 control points, as shown in Equation (6).

$$\left\{\begin{array}{c}{S}_u^{\mathit{i}}\left(\xi \right)=\sum _{\mathit{t}=0}^{14}{N}_{\mathit{t}}\left(\xi \right){\mathbf{P}}_u^{\mathit{t}}\hfill \\ S_l^{\mathit{i}}\left(\xi \right)=\sum _{\mathit{t}=0}^{14}{N}_{\mathit{t}}\left(\xi \right){\mathbf{P}}_l^{\mathit{t}}\hfill \end{array}\right.$$

(6)

where $N_{\mathit{t}}\left(\xi \right)$ is the B-spline basis function, ${\mathbf{P}}_u$ and ${\mathbf{P}}_l$ represent the coordinates of the control points. The folding degree of the blade rib can be described by ${\theta }_s$ of the equilateral s-unit. Therefore, in the coordinate system, the control points of the real section shape can be described as Equation (7).

$$\left\{\begin{array}{c}{\mathbf{P}}_u^{\mathit{i}}=\left[\begin{array}{ccccc}0,& 0,& 0,& \cdots ,& 0\\ Y_0,& Y_1,& Y_2,& \cdots ,& Y_{14}\\ Z_{0}sin{\theta }_s^i,& Z_{1}sin{\theta }_s^i,& Z_{2}sin{\theta }_s^i,& \cdots ,& Y_{14}sin{\theta }_s^i\\ 1,& 1,& 1,& \cdots ,& 1\end{array}\right]\hfill \\ {\mathbf{P}}_l^{\mathit{i}}=\left[\begin{array}{ccccc}0,& 0,& 0,& \cdots ,& 0\\ Y_0,& Y_1,& Y_2,& \cdots ,& Y_{14}\\ -Z_{0}sin{\theta }_s^i,& -Z_{1}sin{\theta }_s^i,& -Z_{2}sin{\theta }_s^i,& \cdots ,& -Y_{14}sin{\theta }_s^i\\ 1,& 1,& 1,& \cdots ,& 1\end{array}\right]\hfill \end{array}\right.$$

(7)

4.2. Shape Parameterization of s-Units

If a rib support is equipped on the i-th section, the next rib support be arranged on the $(i+2)$-th section. The relative positions of the chord lines at the two blade rib sections can be described by ${\Psi }^i$, ${\Psi }^{i+1}$, and ${\delta }^i$. As shown in Figure 11, the relative positions of the two s-units can be described by the coordinate transformation method as Equation (8).

$${\mathbf{T}}_{\mathit{i}}^{\mathit{i}+\mathit{2}}={\mathbf{T}}_y(-h^{\mathit{i}}){\mathbf{R}}_z(-{\Psi }^{\mathit{i}}){\mathbf{T}}_y\left(l^{\mathit{i}}\right){\mathbf{R}}_y\left({\delta }^{\mathit{i}}\right){\mathbf{R}}_z(-{\Psi }^{\mathit{i}+\mathit{1}}){\mathbf{R}}_y(-{\delta }^{\mathit{i}+\mathit{1}}){\mathbf{T}}_y(-c^{\mathit{i}})$$

(8)

where ${\mathbf{T}}_y$ represents the SE(3) translation transformation matrix in the y-axis direction, as is shown in Equation (9).

$${\mathbf{T}}_y\left(h\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& h\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(9)

${\mathbf{R}}_z$ represents the SE(3) rotation transformation matrix around the z-axis, as is shown in Equation (10).

$${\mathbf{R}}_z\left(\theta \right)=\left[\begin{array}{cccc}cos\theta & -sin\theta & 0& 0\\ sin\theta & cos\theta & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(10)

${\mathbf{R}}_y$ represents the SE(3) rotation transformation matrix around the y-axis, as is shown in Equation (11).

$${\mathbf{R}}_y\left(\theta \right)=\left[\begin{array}{cccc}cos\theta & 0& sin\theta & 0\\ 0& 1& 0& 0\\ -sin\theta & 0& cos\theta & 0\\ 0& 0& 0& 1\end{array}\right]$$

(11)

The control points of the i-th section shape can be described as Equation (12). Then, the blade shape can be described by a B-spline surface composed of the control points of the blade rib shape.

$${\mathbf{P}}_s^{\mathit{i}}={\mathbf{T}}_1^3{\mathbf{T}}_3^5\cdots {\mathbf{T}}_{\mathit{i}}^{\mathit{i}+\mathit{2}}{\mathbf{P}}^{\mathit{i}}$$

(12)

where $\mathbf{P}$ is composed of upper and lower half rib control points, as is shown in Equation (13).

$${\mathbf{P}}^{\mathit{i}}=\left[{\mathbf{P}}_u^{\mathit{i}},{\mathbf{P}}_l^{\mathit{i}}\right]$$

(13)

4.3. Deformation Analysis

Since the s-sequence is constructed using only oblique-symmetric and equilateral s-units, according to (2), (3), and (5), the $\Psi$ angle can be obtained by ${\theta }_s$ and spherical linkage angle ${\alpha }_1$, ${\alpha }_2$. Then, the $R_z(\Psi )$ term can be obtained. Thus, the whole propeller deformation characteristic can be obtained through parameters listed in

Table 1. Table of shape coefficients in example 1.

Deformation Characteristic	Parameters
Extension	${\theta }_s$, ${\alpha }^{eq}$, ${\alpha }_1^{ob}$, ${\alpha }_2^{ob}$, l
Torsion	${\theta }_s$, ${\alpha }_1^{ob}$, ${\alpha }_2^{ob}$
Thickness	${\theta }_s$

below. The relationship between parameters and deformation characteristics is analyzed in the following subsections.

4.3.1. Extension Deformation

Referring to Figure 11, two adjacent blade rib supports are connected by an equilateral and an oblique symmetric spherical four-bar linkage. During the deformation process, the change in $\Psi$ results in extensional deformation. As the shape coefficients difference in oblique symmetric unit and equilateral unit, two adjacent section are not parallel, thus, $O^{\mathit{i}}{O}^{\mathit{i}+2}$ is used to define the distance of two sections, as is shown in Equation (14).

$$O_{\mathit{i}}^{\mathit{i}+\mathit{2}}={\mathbf{T}}_y(-h^{\mathit{i}}){\mathbf{R}}_z(-{\Psi }^{\mathit{i}}){\mathbf{T}}_y\left(l^{\mathit{i}}\right){\mathbf{R}}_y(\pi -\Delta ){\mathbf{R}}_z({\Psi }^{\mathit{i}}+\mathit{1}){\mathbf{T}}_y(-c^{\mathit{i}}+\mathit{2}){[0,0,0,1]}^T$$

(14)

where $\Psi$ can be obtained by Equation (3). Obviously, $O^{\mathit{i}}{O}^{\mathit{i}+2}$ directly influences the size of the blade, and an extension ratio $ϵ$ is defined to describe the deformation characterization, as is shown in Equation (15).

$$ϵ={\displaystyle \frac{O^{\mathit{i}}{O}^{\mathit{i}+2}}{l^{\mathit{i}+1}}}$$

(15)

To investigate the influence of the shape parameters of the s-unit on the expansion rate, three calculation examples are set up. The parameter settings are shown in the following

Table 2. Table of shape coefficients in example 2.

	Example 1	Example 2	Example 3
${\alpha }^{eq}$	15	20	10
${\alpha }_1^{ob}$	18.5	23.5	13.5
${\alpha }_2^{ob}$	11.5	16.5	6.5

.

The relationship between $\epsilon$ and ${\theta }_s$ is shown in Figure 12. As can be seen from the figure, for a given ${\theta }_s$, when the linkage angle is larger, the expansion rate is larger. When ${\theta }_s$ is around 125 degrees, the derivative of the expansion rate with respect to ${\theta }_s$ can obtain a relatively large value. When the mechanism is fully folded or extended, the amplitude of extension deformation is relatively small.

4.3.2. Torsion Deformation

Planes ${\Pi }^{\mathit{i}}$ and ${\Pi }^{\mathit{i}}+\mathit{1}$ perform a folding motion around the axis of ${\mathbf{L}}_C^{\mathit{i}}$, resulting in the torsional deformation of the blade. As mentioned above, the relative motion between sections is not strictly translational motion; thus, there is a changing angle between two adjacent sections. The torsion angle ${\Delta }^{\mathit{i}}$ can be defined as angle between $y^{\mathit{i}}$ and ${\mathit{Pr}}_{{\mathit{y}}^{\mathit{i}}{\mathit{z}}^{\mathit{i}}}\left({\mathit{y}}^{i+2}\right)$. ${\mathit{Pr}}_{{\mathit{y}}^{\mathit{i}}{\mathit{z}}^{\mathit{i}}}\left({\mathit{y}}^{i+2}\right)$ represents the projection of ${\mathit{y}}^{\mathit{i}+2}$, as is shown in Figure 13.

In coordinate system $\mathit{i}$, ${\mathit{Pr}}_{{\mathit{y}}^{\mathit{i}}{\mathit{z}}^{\mathit{i}}}({\mathit{y}}^i+2)$ can be obtained by Equation (16).

$${\mathit{Pr}}_{{\mathit{y}}^{\mathit{i}}{\mathit{z}}^{\mathit{i}}}\left({\mathit{y}}^{i+2}\right)=[0,1,1,1]·{\mathit{y}}_{\mathit{i}}^{\mathit{i}+2}$$

(16)

where ${\mathit{y}}_{\mathit{i}}^{\mathit{i}+2}$ can be obtained by Equation (17).

$${\mathit{y}}_{\mathit{i}}^{\mathit{i}+2}={\mathbf{R}}_z(-{\Psi }^{\mathit{i}}){\mathbf{R}}_y(\pi -{\delta }^{\mathit{i}}){\mathbf{R}}_z(-{\Psi }^{\mathit{i}}){[0,1,0,1]}^T$$

(17)

The folding angle ${\delta }^{\mathit{i}}$ can be obtained by Equation (18).

$${\delta }^{\mathit{i}}=|{\alpha }_1^{\mathit{i}}-{\alpha }_2^{\mathit{i}}|$$

(18)

To investigate the influence of the shape parameters of the s-unit on the torsional angle, three calculation examples are set up. The parameter settings are shown in

Table 3. Table of shape coefficients in example 3.

	Example 1	Example 2	Example 3
${\alpha }^{eq}$	15	15	15
${\alpha }_1^{ob}$	18.5	20.5	16.5
${\alpha }_2^{ob}$	11.5	9.5	13.5

.

The relationship between $\Delta$ and s-unit coefficients is shown in Figure 14.

It can be seen that the greater the difference between ${\alpha }^1$ and ${\alpha }^3$ of the oblique symmetric unit, the more intense the torsional deformation. When the mechanism is fully extended, the derivative of the torsion angle is close to 0. The blade angle ${\beta }_0$ is driven by the modified Kresling hub, which can be obtained by Equation (1). The attack angle distribution can be obtained by Equation (19).

$${\beta }^{\mathit{i}}={\beta }_0-\sum _{\mathit{i}=1}^N{\Delta }^{\mathit{i}}$$

(19)

4.3.3. Thickness Deformation

As the upper and lower rib structure are connected to BC and CD linkages, respectively, their rotation around ${\mathbf{L}}_C$ results in thickness deformation, as is shown in Figure 8. If an angle is set between the blade rib plane and the connecting rod plane, more abundant thickness changes can be achieved. However, for the consideration of improving the strength of the mechanism, the rib plane is set to coincide with the linkage plane. Thus, points on the rib structure ${\mathbf{P}}_{rib}=[{\mathit{x}}_{rib},{\mathit{y}}_{rib},{\mathit{z}}_{rib}]$ can be expressed in the section coordinate system in the form of Equation (20). The schematic diagram of thickness variation is shown in Figure 15. As is shown in Figure 15, the real shape of the rib is described by the projection of the rib shape on the blade section. The folding deformation of the rib can be approximately regarded as a scaling deformation in the z-direction of the section coordinate system. The shape of the real airfoil can be obtained by Equation (7).

$$\mathbf{P}=[{\mathit{x}}_{rib}cos{\theta }_s,{\mathit{y}}_{rib},{\mathit{z}}_{rib}sin{\theta }_s]$$

(20)

5. Simulation and Rapid Prototyping Experiment

To verify the deformation characteristics of the deformation driving mechanism, kinematic simulation based on the digital mock-up module of CATIA V5R20 was carried out. The stress analysis of the s-unit-rib supporting structure was carried out based on the finite element method. Also, a rapid prototype based on 3-D printing was manufactured. The prototype is composed of seven identical equilateral s-units and seven identical skew-symmetrical s-units. A blade rib support structure is installed on each equilateral s-unit. The shape parameters are shown in

Table 4. Table of design specifications.

Shape Coefficient	Value	Meaning
$\tau$	$42.8^{°}$	Angle of Kresling revolute joint and cylindrical joint
${\alpha }_{\mathit{j}}^{\mathit{i}}(\mathit{i}=2,4,6,8,10,12\mathit{j}=1,2,3,4)$	${15}^{°}$	Angle of equilateral s-unit linkage
${\alpha }_{\mathit{j}}^{\mathit{i}}(\mathit{i}=1,3,5,7,9,11\mathit{j}=1,3)$	$11.2^{°}$	Angle of oblique symmetric s-unit linkage (AB and CD)
${\alpha }_{\mathit{j}}^{\mathit{i}}(\mathit{i}=1,3,5,7,9,11\mathit{j}=2,4)$	$18.{42}^{°}$	Angle oblique symmetric s-unit linkage (BC and DA)
${\theta }_s^{air}$	$57.{377}^{°}$	Angle between AB and DA linkage in aerial shape
${\theta }_s^{wtr}$	$130.{298}^{°}$	Angle between AB and DA linkage in aquatic shape
H	50 mm	Size of hub top and bottom linkage
t	5 mm	Length of hinge
$D_{air}$	263.872 mm	Propeller diameter in aerial mode
$D_{wtr}$	178.285 mm	Propeller diameter in aquatic mode
${\Delta }_{air}^{\mathit{i}}$	$3.{432}^{°}$	Twist angle between sections in aerial mode
${\Delta }_{wtr}^{\mathit{i}}$	$6.{254}^{°}$	Twist angle between sections in aquatic mode
${\beta }_{air}^0$	$24.{862}^{°}$	Blade angle in aerial mode
${\beta }_{wtr}^0$	$41.{476}^{°}$	Blade angle in aquatic mode

and Figure 16.

In kinematic simulation, an displacement driving point is set on the upper linkage of the propeller hub, as is shown in Figure 17. The deformation process is shown in Figure 18.

As can be seen from Figure 19, Figure 20 and Figure 21, the simulation results are in agreement with the calculation results of the proposed kinematic model. The simulation results show that the proposed mechanism is a single-degree-of-freedom mechanism and there is no interference during deformation. According to simulation results, the radius of the propeller in the aerial form is 263.872 mm, and the disk ratio is 0.204. In the underwater form, the radius is 178.285 mm, and the disk ratio is 0.346. The attack angle distribution of the rotor in the air and underwater are shown in Figure 19. It can be seen that under the drive of the designed mechanism, the propeller presents a more slender shape in the air, with a larger radius and a smaller disk ratio, which helps to increase the rotational speed during flight and increase lift. When working underwater, the radius of the propeller decreases and the disk ratio increases, which helps to reduce cavitation effects and improve work efficiency. Driven by the Kresling hub, the blade angle of the blade in the aerial shape is 24.${862}^{°}$, and it increases to 41.${476}^{°}$ underwater. This makes the propeller adapt to working conditions with a higher advance coefficient when working in water. Since the shapes of the blade deformation units are the same, the attack angle distribution of the blade is linearly distributed. The relation extension ratio $ϵ$ is shown in Figure 20. It can be seen that the results obtained by the coordinate transformation method are consistent with the simulation results. It can also be seen that, when ${\theta }_s$ is small, the derivative of the expansion rate is small. When theta is large, the folding motion has a greater impact on the expansion and contraction rate. Therefore, when the deformation of the propeller radius is more important, the mechanism should be designed to be in a more folded state (both underwater and in the air). The relationship between attack angle distribution and ${\theta }_s$ is shown in Figure 21. It can be seen that when ${\theta }_s$ is smaller, the torsional deformation between the sections is greater. Therefore, when the twist angle distribution of the propeller is more important, the mechanism should be designed to be in a more unfolded shape. It can be seen that the designed mechanism can effectively drive the propeller to deform between the aerial shape and the underwater shape.

The stress analysis of the section unit, including a oblique symmetric s-unit, a equilateral s-unit, and a pair of rib support structures in its aerial shape and aquatic shape, was conducted using the finite element method. The structure is fixed by adding revolute joints with fixed positions to the AB and AD connecting rods of the skew-symmetric s-unit. A distributed force of 10 N in the direction of gravity was applied to the lower half of the blade rib supporting structure to simulate the working condition. The mesh structures of the section unit in the aerial and aquatic shapes are shown in Figure 22.

The stress distribution of the section unit is shown in Figure 23.

To verify the feasibility of the designed mechanism, a trial production of a rapid prototype was carried out. The shape parameters of the rapid prototype are the same as those in the simulation. The diameter of the revolute joint shaft is 2 mm, and the wall thickness of each part is 5 mm. The rapid prototype mechanism is manufactured by 3-D printing technology. Some parts of the prototype are shown in Figure 24. The rapid prototype machine is made of ABS plastic and weighs 98.428 g. The part shapes are consistent with those of the simulated parts. The axis of the revolute pairs are marked with dotted lines. Figure 24a,b depicts linkage $B{C}^{eq}-A{D}^{ob}$ and linkage $B{C}^{eq}-A{D}^{ob}$. The revolute joint is designed to be an h shape. Figure 24c,d shows the $A{B}^{eq}-C{D}^{ob}$-upper rib and $D{A}^{eq}-B{C}^{ob}$-lower rib parts of the 6th section. The rib structure and the linkages of the s-unit are printed as a whole part to increase strength and reduce the weight of connectors. The shape parameters of the other blade ribs are the same as those of the third blade rib, and the sizes of the blade ribs are slightly different. The shapes of the prototype in the air and underwater are shown in Figure 25.

It can be seen from Figure 25a that the mechanism has a larger radius, a slender shape in the air. Both section twist angle and blade angle are small. When deformed into an underwater shape, as can be seen in Figure 25b, the blade radius is significantly reduced, the disk ratio is increased, the section twist angle is increased, and the blade angle is increased. The separated rib support structure stably forms the blade airfoil shape in both aerial and aquatic mode without interference during deformation. This indicates that the designed mechanism can effectively drive the propeller to deform between the two shapes in the air and underwater.

However, as can be seen from Figure 25a, due to manufacturing technology limitations, the overall size and weight of the blade are relatively large. Although the blade has better stability and self-supporting properties in the underwater shape, when the blade is deformed into the aerial shape, there is a slight deformation at the blade root, and the overall shape droops. In addition, in the folded state, the s-sequence has a large thickness, which forces the blade shape to be designed as a thicker shape. In this paper, NACA-0015 airfoil is used. In subsequent work, attention should be paid to structural strength issues and structure simplification.

The proposed deformation mechanism is able to drive the blade to produce radius deformation (from 263.872 mm to 178.285 mm), blade angle deformation (from $24.{862}^{°}$ to $41.{476}^{°}$), and section torsional deformation (from $3.{622}^{°}$ to $6.{235}^{°}$) under ideal conditions. The entire mechanism has been verified to be a single-degree-of-freedom mechanism, and only one actuator is required to drive the overall deformation of the blade. There is no interference or singularity during the movement of the mechanism.

Currently, practical aerial-aquatic power systems consist of an aerial-shaped propeller and an electric machine with a relatively low Kv value. When in the air, it operates at a relatively high speed, and when underwater, the revolute speed is greatly reduced to overcome high torque. This makes it impossible to take into account the power characteristics of both working conditions during the motor selection and the propeller design. The proposed deformable propeller has great application potential in improving the aerial-aquatic drive compatibility. In the flight state where hovering and low-speed forward flight are the main working conditions, the medium density is relatively small, the advance ratio is relatively low, and the propeller is mainly used to overcome gravity, so the power demand is relatively high. The propeller deforms to obtain a larger radius, a smaller setting angle, and a smaller blade-twist angle. This helps the motor-propeller system to obtain a higher speed and thus increase the lift. In the underwater state where forward navigation is the main working condition, the medium density is relatively large, the advance ratio is relatively high, and the power demand is relatively low. The propeller deforms to obtain a smaller radius, a larger setting angle, and a larger blade-wist angle to adapt to the higher advance ratio. The advantages and disadvantages of the deformable propeller and a fixed shape one are list in

Table 5. Table of comparison between deformable and fixed shape propeller.

	Deformable Propeller	Fixed Shape Propeller
Weight	Heavy	Light
Mechanical complexity	Complex	Simple
Drive requirement	Shaft/deformation drive	Shaft drive
Drive compatibility	Active design	Compromise design

.

Although the designed propeller benefits aerial-aquatic drive compatibility through deformation, its complex structure and relatively large weight may increase the manufacturing cost of the propeller. Since the mechanism is formed by connecting multiple units in series, its strength is lower than that of common one-piece-molded propellers, especially in the unfolded aerial shape. In addition, folding makes the blade have a relatively large thickness in the underwater shape, which forces us to select an airfoil with a relatively large thickness. In subsequent work, the simplification of the structure and the optimization of strength need to be emphasized.

6. Conclusions

In this paper, according to the shape characteristics of aerial propellers and underwater propellers, a driving mechanism with the deformation capabilities of blade angle, radius, twist angle distribution, and section thickness is designed. Through motion analysis, simulation, and rapid prototype testing, the following conclusions are obtained.

• The modified Kresling structure can be effectively applied to the design of hubs with variable blade angles. The spherical space scissor structure composed of equilateral s-units and oblique symmetric s-units alternately has the deformation ability of extensional-torsional coupling and is suitable as the deformation driving structure of the deformable propeller blades for amphibious applications in water and air.
• By comparing the motion simulation results and the calculation structure of the proposed motion model, it is proved that the mechanism kinematics model based on the coordinate transformation method can accurately describe the deformation characteristics of the propeller.
• According to the deformation analysis, it can be found that when the mechanism is in the extended state, the blade torsional deformation amplitude is relatively large, and the telescopic deformation amplitude is relatively small. When the mechanism is in the folded state, the blade torsional deformation amplitude is relatively small, and the telescopic deformation amplitude is relatively large.
• Through kinematic analysis and simulation, it is found that the designed structure can significantly change the radius and blade angle of the propeller. When equilateral s-units and oblique-symmetrical s-units with the same design variables are used to form the blade deformation mechanism, the blade attack angle is linearly distributed before and after deformation.
• Through rapid prototype testing, the motion characteristics of the deformation mechanism are verified. There is no interference or singularity in the deformation process, and the mechanism operates stably.
• The overall structure is relatively heavy, and there is a slight deformation at the blade root, resulting in a drooping phenomenon of the blade in the aerial state. Compared with common propellers, an additional drive is required to control the deformation of the propeller, which increases the mechanical complexity. The series-connected linkage mechanism causes a certain degree of decrease in blade strength. In addition, the manufacturing cost is higher, the design process is more complicated, and higher requirements are imposed on the machining accuracy.
• Compared with the common fixed propeller, the advantage of the deformable propeller is that it can incorporate the variables of blade angle, twist angle distribution, and diameter in both underwater and aerial working conditions into the design space, rather than making a compromise design. This is the potential of the proposed deformable propeller.