Design and flight performance of hybrid underwater glider with controllable wings

Abstract

Hybrid underwater glider combines motion modes of traditional autonomous underwater glider and those of autonomous underwater vehicles. Different motion modes need different flight performance, including flight efficiency, static stability, and maneuverability. Conventional hybrid underwater glider with fixed wings can’t achieve optimal flight performance in one flight mission demanding various motion modes. In this article, controllable wings for hybrid underwater glider Petrel II are designed. Angle of attack, sweep angle, and aspect ratio of controllable wings can be changed to adapt to different motion modes. Kinematics and dynamics models of Petrel II are established based on multibody theory. Motion simulations of Petrel II with different wing configurations are conducted in three motion modes, including glide motion, spiral motion, and horizontal turning motion. The simulation results show the impact of wing parameters on flight performance. Field trials demonstrate that the controllable wings can improve the flight performance.

Introduction

Autonomous underwater glider (AUG) is an economic and promising marine observation platform in terms of small size, long duration, and a harvest of oceanographic data. Volume-varying bladder provides buoyancy. Movable and rotary battery pack distributes mass to change pitch and roll angles. Wings convert vertical force into forward force. Therefore, AUG achieves saw-tooth trajectory. ¹ At present, widely recognized AUGs include Slocum, ² Seaglider, ³ and Spray. ⁴ Compared with AUG, hybrid underwater glider (HUG) with an auxiliary propeller gets the high speed and maneuverability at cost of duration but performs excellently in horizontal cruise, obstacle avoidance, and fast track. HUG Petrel II developed by Tianjin University has been put into use since 2014. It is 2.3 m in length without regard to antenna, 0.22 m in diameter, and 69 kg in weight. During the latest sea trial, Petrel II sailed 1108.4 km, dived 1514.2 m deep, and had the maximum glide speed of 0.82 m/s and propeller speed of 1.73 m/s, indicating good flight performance.

Flat, thin, and fixed wings are adopted by Petrel II in view of the glider’s low-speed and saw-tooth motion. However, fixed wings can’t achieve optimal flight performance to meet the long endurance demand of large-scale environmental monitoring or maneuverability demand of sub-mesoscale eddy tracking, adaptive sampling, or obstacle avoidance at the same time. ^{5 –7} Thus, controllable wings become a solution to satisfy HUG’s various flight performance demands. Some advanced materials, such as shape memory alloy and ionic conducting polymer film, are used to achieve the transformation of the wings. ^{8 –10} However, mechanical structures are still the most popular among researchers. M. Arima developed a glider prototype named ALEX. ¹¹ With independently controllable wings, which can change angle of attack (AOA), ALEX has high maneuverability. Isa K established a dynamic model of HUG named USM with controllable wings and designed a neural network predictive control method. Gliding and hovering motions are analyzed. ^12,13 Xie focused on glide efficiency of underwater gliders with passively rotatable wings. Simulation shows that the lift-drag ratio and AOA present a quadratic function relation. ¹⁴ Researches shown earlier focus on AOA only. In this article, parameters of controllable wings are extended. AOA, sweep angle, and aspect ratio all can be adjusted as shown in Figure 1.

Figure 1.

Three wing configurations.

This article is organized as follows. “System design” section introduces the basic structure of Petrel II and controllable wings system. “Multibody kinematics” section establishes the kinematic model of Petrel II with controllable wings. “Multibody dynamics” section analyzes the forces and moments acting on Petrel II and establishes the dynamic model. “Motion simulation” section simulates five motions with different wing configurations. “Field trial” section shows results of field trials. Finally, conclusions and future work are discussed in “Conclusions and future work” section.

System design

Petrel II with controllable wings consists of six subsystems: buoyancy engine system, attitude adjustment system, controllable wings system, control system, propeller engine system, and communication and location system as shown in Figure 2. An independent cabin is added to contain the controllable wings system. The wing on the right side is described in detail.

Figure 2.

System layout of Petrel II with controllable wings.

The controllable wings system consists of two parts: wings and drive mechanism. Each wing is made up of three separate parts: wing A, wing B, and wing C, as shown in Figure 3. Wing A and glider body form a revolute pair. Wing B and wing A form a revolute pair. Wing C and wing B form a sliding pair. Motions of wing A and wing B will be transmitted to wing C. Thus, wing C has three degrees of freedom (DoFs) relative to the glider body. ¹⁵

Figure 3.

Structure of controllable wings.

The drive mechanism includes mechanism I and mechanism II as shown in Figure 4. Wing A is fixed to the double rocker mechanism. Wing B is fixed to the crank slider mechanism. Wing C matches with the output slider C. Mechanism I is a combination of double parallelogram linkages ABCDHG and plane five-bar linkages DFEGH using linkage synthesis. ¹⁶ Single plane five-bar linkages can achieve two linear movements. However, when any output bar moves along one axis, revolution always exists. ¹⁷ Double parallelogram linkages’ output bar CD possesses two linear movements which are yet not certain. Mechanism I realizes certain linear motion of output bar CD by combining two linkages as shown in Figure 5.

Figure 4.

Drive mechanism.

Figure 5.

Motion course of mechanism I.

Mechanism II includes displacement multiplication mechanism, crank slider mechanism, and double rocker mechanism. At the initial stage, the output bar CD is separated from mechanism II. When CD moves in specific ways, specific output slider pairs will match. When output slider A and A′ match, the crank slider mechanism fixed with wing B will rotate to change sweep angle. When output slider B and B′ match, the double rocker mechanism fixed with wing A will rotate to change AOA. When output slider C moves linearly forward along wing B, the displacement multiplication mechanism using duplicate gear will make wing C move 3 times the distance of slider C to change wingspan. The variation ranges of AOA, sweep angle, and aspect ratio are 0–55°, 0–70°, and 6–8, respectively. ¹⁵

Multibody kinematics

A multibody system is a system consisting of a set of bodies connected to one another through internal joints and to the rest of the world through external joints or contacts. ¹⁸ Traditional methods take HUG as a single particle and miss some HUG’s dynamic characteristics. With the help of multibody theory, every single part of HUG can be analyzed. In this article, the multibody dynamic method proposed by Huston is adopted. ^19,20

Lower body array

Petrel II with controllable wings can be divided into six components, including the glider body, battery pack, hydraulic oil, propeller, the left wing, and the right wing. The six components are orderly represented by B_K (K = 1 − 6)

$\{\begin{array}{c}K=\left(1,2,3,4,5,6\right)\\ L\left(K\right)=\left(0,1,1,1,1,1\right)\end{array}$

1

where K is body number and L(K) is lower body array (LBA).

Coordinate establishment

To satisfy multibody analysis, seven coordinates are established. One is the inertial coordinate and the others are body-fixed coordinates. As shown in Figure 6, the inertial coordinate E-XYZ is assigned by standard definition. The origin of E is on the surface of sea, axis EY vertically points upward and axis EX horizontally points north. The body-fixed coordinate of B ₁, also called moving coordinate B-xyz, is established in such a way that the origin is the buoyancy center of neutrally buoyant glider, axis Bx is parallel to the long axis of the glider body, and axis Bz lies along the right side of the glider body. Other body-fixed coordinates are $\left\{e^{\left(i\right)}\right\}={\left\{x^{\left(i\right)},y^{\left(i\right)},z^{\left(i\right)}\right\}}^T,\left(i=2\sim 6\right)$ , and established like B-xyz. The glider’s attitude relative to the inertial coordinate can be described by Euler angle (φ, ψ, θ), which represents roll about axis Bx, yaw about axis By, and pitch about axis Bz.

Figure 6.

Coordinates of multibody system.

Petrel II has 16 DoFs. The glider body has six DoFs (φ, ψ, θ, x, y, z). The battery pack has two DoFs, including translation of L_b and rotation of θ_b . The propeller has one DoF θ_p . Hydraulic oil, out and in, has one DoF L_o . Each of the two wings has three DoFs. The right wing has translation Lrw and rotations θ_rwy and θ_rwz .

Transformation matrix

$C_B^E$ is transformation matrix from the inertial coordinate to body-fixed coordinate; $C_E^B$ is reverse and $C_B^E={\left(C_E^B\right)}^T$ . The inertial coordinate can coincide with body-fixed coordinate through three rotations ²¹

$C_B^E=\left[\begin{array}{ccc}\text{cos}\theta \text{cos}\psi & \text{sin}\theta & -\text{cos}\theta \text{sin}\psi \\ -\text{sin}\theta \text{cos}\psi \text{cos}\phi +\text{sin}\psi \text{sin}\phi & \text{cos}\theta \text{cos}\phi & \text{sin}\theta \text{sin}\psi \text{cos}\phi +\text{cos}\psi \text{sin}\phi \\ \text{sin}\theta \text{cos}\psi \text{sin}\phi +\text{sin}\psi \text{cos}\phi & -\text{cos}\theta \text{sin}\phi & -\text{sin}\theta \text{sin}\psi \text{sin}\phi +\text{cos}\psi \text{cos}\phi \end{array}\right]$

2

In multibody dynamic analysis, Euler parameters (ε₁, ε₂, ε₃, ε₄, ε₅) are used to represent objective’s attitude to avoid singular points

$C_E^B=\left[S_{ij}\right]=\left[\begin{array}{ccc}{\epsilon }_1^2-{\epsilon }_2^2-{\epsilon }_3^2+{\epsilon }_4^2& 2\left({\epsilon }_1{\epsilon }_2-{\epsilon }_3{\epsilon }_4\right)& 2\left({\epsilon }_1{\epsilon }_3+{\epsilon }_2{\epsilon }_4\right)\\ 2\left({\epsilon }_1{\epsilon }_2+{\epsilon }_3{\epsilon }_4\right)& -{\epsilon }_1^2+{\epsilon }_2^2-{\epsilon }_3^2+{\epsilon }_4^2& 2\left({\epsilon }_2{\epsilon }_3-{\epsilon }_1{\epsilon }_4\right)\\ 2\left({\epsilon }_1{\epsilon }_3-{\epsilon }_2{\epsilon }_4\right)& 2\left({\epsilon }_2{\epsilon }_3+{\epsilon }_1{\epsilon }_4\right)& -{\epsilon }_1^2-{\epsilon }_2^2+{\epsilon }_3^2+{\epsilon }_4^2\end{array}\right]$

3

where S_ij represents the element of transformation matrix $C_E^B$ . ε_i = λ_i sin(θ₁/2) (i=1, 2, 3), ε₄ = cos(θ₁/2), and ${\text{ε}}_1^2+{\text{ε}}_2^2+{\text{ε}}_3^2+{\text{ε}}_4^2=1$ . λ_i is the unit vector. In general, it’s defined that θ₁ ≤ π. Euler parameters can be derived from equation (3)

$\{\begin{array}{c}\left|{\text{ε}}_1\right|=\sqrt{\left({\text{S}}_{11}-{\text{S}}_{22}-{\text{S}}_{33}+1\right)/4}\\ \left|{\text{ε}}_2\right|=\sqrt{\left(-{\text{S}}_{11}+{\text{S}}_{22}-{\text{S}}_{33}+1\right)/4}\\ \left|{\text{ε}}_3\right|=\sqrt{\left(-{\text{S}}_{11}-{\text{S}}_{22}+{\text{S}}_{33}+1\right)/4}\\ \left|{\text{ε}}_4\right|=\sqrt{\left({\text{S}}_{11}+{\text{S}}_{22}+{\text{S}}_{33}+1\right)/4}\end{array}$

4

Then Euler parameters of B₁ can be determined

$\{\begin{array}{c}{\epsilon }_1^{\left(1\right)}=\text{sin}\frac{\theta }{2}\text{sin}\frac{\psi }{2}\text{cos}\frac{\phi }{2}+\text{cos}\frac{\theta }{2}\text{cos}\frac{\psi }{2}\text{sin}\frac{\phi }{2}\\ {\epsilon }_2^{\left(1\right)}=\text{cos}\frac{\theta }{2}\text{sin}\frac{\psi }{2}\text{cos}\frac{\phi }{2}+\text{sin}\frac{\theta }{2}\text{cos}\frac{\psi }{2}\text{sin}\frac{\phi }{2}\\ {\epsilon }_3^{\left(1\right)}=\text{sin}\frac{\theta }{2}\text{cos}\frac{\psi }{2}\text{cos}\frac{\phi }{2}-\text{cos}\frac{\theta }{2}\text{sin}\frac{\psi }{2}\text{sin}\frac{\phi }{2}\\ {\epsilon }_4^{\left(1\right)}=\text{cos}\frac{\theta }{2}\text{cos}\frac{\psi }{2}\text{cos}\frac{\phi }{2}-\text{sin}\frac{\theta }{2}\text{sin}\frac{\psi }{2}\text{sin}\frac{\phi }{2}\end{array}$

5

Euler parameters of body B ₂, B ₃, and B ₆ with rotational DoFs are shown

$\{\begin{array}{c}{\epsilon }_1^{(2)}=\text{sin}\frac{{\theta }_b}{2}\\ {\epsilon }_2^{(2)}=0\\ {\epsilon }_3^{(2)}=0\\ {\epsilon }_4^{(2)}=\cos \frac{{\theta }_b}{2}\end{array}\{\begin{array}{c}{\epsilon }_1^{(3)}=\text{sin}\frac{{\theta }_p}{2}\\ {\epsilon }_2^{(3)}=0\\ {\epsilon }_3^{(3)}=0\\ {\epsilon }_4^{(4)}=\cos \frac{{\theta }_p}{2}\end{array}\{\begin{array}{c}{\epsilon }_1^{(6)}=\text{sin}\frac{{\theta }_{rwz}}{2}\text{sin}\frac{{\theta }_{rwy}}{2}\\ {\epsilon }_2^{(6)}=\cos \frac{{\theta }_{rwz}}{2}\text{sin}\frac{{\theta }_{rwy}}{2}\\ {\epsilon }_3^{(6)}=\text{sin}\frac{{\theta }_{rwz}}{2}\cos \frac{{\theta }_{rwy}}{2}\\ {\epsilon }_4^{(6)}=\cos \frac{{\theta }_{rwz}}{2}\cos \frac{{\theta }_{rwy}}{2}\end{array}$

6

Generalized speed

The whole system has 16 DoFs and 16 generalized coordinates. Sixteen generalized speeds q_l (l = 1∼16) are used to represent 16 DoFs

$\{\begin{array}{c}{q}_l=\left[q_1, q_2, q_3, \cdots \cdots , q_{15},q_{16}\right]\\ q_l^T=C_E^{B_i}{V}^T \left(i=1\sim 6\right)\\ V=\left[p_i,q_i,{\text{ r}}_i,u_i,v_i,w_i\right] \left(i=1\sim 6\right)\end{array}$

7

where V means velocity and angular velocity component in the body coordinate system. It is zero if there is no rotational or translational component. [p_i, q_i, r _i ] is angular velocity of glider components. [u_i, v_i, w _i ] is linear velocity of glider components.

Centroid velocity, centroid acceleration, angular velocity, and angular acceleration

Centroid velocity, centroid acceleration, angular velocity, and angular acceleration of multibody system can be described as below ^19,20

$\{\begin{array}{c}{\omega }_k={\omega }_{klm}{\dot{q}}_l{n}_{oe}\\ {\alpha }_k=\left({\omega }_{klm}{\ddot{q}}_l+{\dot{\omega }}_{klm}{\dot{q}}_l\right)n_{oe}\\ {\upsilon }_k={\upsilon }_{klm}{\dot{q}}_l{n}_{oe}\\ a_k=\left({\upsilon }_{klm}{\ddot{q}}_l+{\dot{\upsilon }}_{klm}{\dot{q}}_l\right)n_{oe}\end{array}$

8

where n_oe is the orthonormal unit vector in the inertial coordinate. ω_klm is the partial angular vector scalar component about n_oe, and υ_klm is the partial velocity vector scalar component about n_oe. The subscript k (1∼6), l (1∼16), and m (1∼3) represent six body components, 16 generalized speed, and three coordinate components, respectively.

Multibody dynamics

Forces and moments analysis

Petrel II with controllable wings is subjected to the following forces: gravity, buoyancy, propeller thrust, and hydrodynamic forces.

Gravity and buoyancy

The rotation and movement of the battery pack change the gravity center and produce moment. The oil distribution changes the buoyancy and makes the buoyancy center move along axis Bx, thus producing moment. The gravity and buoyancy forces are expressed as ²¹

$\{\begin{array}{c}{F}_{GB}={\left[f_x{f}_y{f}_z{m}_x{m}_y{m}_z\right]}^T\\ f_x=-\left(G-B\right)\text{sin}\theta \\ f_y=-\left(G-B\right)\text{cos}\theta \text{cos}\phi \\ f_z=\left(G-B\right)\text{cos}\theta \text{sin}\phi \\ m_x=y_{G}G\text{cos}\theta \text{sin}\phi +z_{c}G\text{cos}\theta \text{cos}\phi \\ m_y=-\left(x_{G}G-x_{B}B\right)\text{cos}\theta \text{sin}\phi -z_{G}G\text{sin}\theta \\ m_z=-\left(x_{G}G-x_{B}B\right)\text{cos}\theta \text{cos}\phi +y_{G}G\text{sin}\theta \end{array}$

9

where G is the magnitude of gravity and B is the magnitude of buoyancy. (x_G, y_G, z_G) and (x_B, 0, 0) are the coordinates of gravity center and buoyancy center of the glider body in B-xyz, respectively.

Propeller thrust

The propeller generates thrust and moment along an axis which deflects 2° about axis Bx in plane Bxz to overcome the deflection moment. The force and moment can be expressed as ²²

$F_p=\left[f_p m_p\right]={\left[K_T\rho n^2{D}^4 0 0 K_Q\rho n^2{D}^5 0 0\right]}^{\text{T}}$

10

where K_T is the thrust coefficient, K_Q is the moment coefficient, ρ is the density of sea water, n is the rotating speed of the propeller, and D is the diameter of the propeller.

Hydrodynamic forces

The hydrodynamic forces include inertial hydrodynamic forces and viscous hydrodynamic forces. Inertial hydrodynamic force is generated when the glider operates in an unsteady state and is linearly proportional to the acceleration and angular acceleration of the glider. The inertial hydrodynamic force of HUG is the sum of forces acting on the glider body and controllable wings.

The glider body has 6 DoFs and 36 inertial forces, but with two symmetrical planes, the 26 acceleration coefficients are zero. The inertial hydrodynamic forces acting on the glider body and controllable wings can be described as ²³

$F_{IB}=M_{IB}\dot{V}=\left[\begin{array}{cccccc}{\lambda }_{11}& 0& 0& 0& 0& 0\\ 0& {\lambda }_{22}& 0& 0& 0& {\lambda }_{26}\\ 0& 0& {\lambda }_{33}& 0& {\lambda }_{35}& 0\\ 0& 0& 0& {\lambda }_{44}& 0& 0\\ 0& 0& {\lambda }_{53}& 0& {\lambda }_{55}& 0\\ 0& {\lambda }_{62}& 0& 0& 0& {\lambda }_{66}\end{array}\right]\left[\begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\\ \dot{p}\\ \dot{q}\\ \dot{r}\end{array}\right]$

11

where λ is acceleration coefficient, λ₆₂ = λ₂₆ and λ₅₃ = λ₃₅.

As forces acting on the wings are along Bz, λ_w33 = λ_w35 = λ_w53 = 0. The acceleration coefficients of a single wing are approximate to those of a flat plate with finite length and width as below ²⁴

$\{\begin{array}{c}{\lambda }_{w11}=\frac{1}{4}\pi \rho c^{2}b\text{sin}\delta \\ {\lambda }_{w22}=\frac{1}{4}\pi \rho c^{2}b\text{cos}\delta \\ {\lambda }_{w26}={\lambda }_{w22}{x}_w\\ {\lambda }_{w44}=\frac{1}{32}\pi \rho c^{4}b\\ {\lambda }_{w66}={\lambda }_{w22}{x}_w^2\end{array}$

12

where c is the chord, b is half wingspan, δ is the flip angle of wing, and x_w is the distance between action spot of hydrodynamic force and the gravity center along axis Bx.

Viscous hydrodynamic force is the viscous drag when the glider is shearing the fluid. The viscous hydrodynamic force acting on the body can be described as ²⁵

$\{\begin{array}{l}{F}_{VB}={\left[f_x{f}_y{f}_z{m}_x{m}_y{m}_z\right]}^T\\ f_x=\frac{1}{2}\rho S{V}^2{C}_X\left(0\right)\\ f_y=\frac{1}{2}\rho S{V}^2\left(C_Y^{\alpha }\alpha +C_Y^r\dot{r}\right)\\ f_z=\frac{1}{2}\rho S{V}^2\left(C_Z^{\beta }\beta +C_Z^p\dot{p}+C_Z^q\dot{q}\right)\\ m_x=\frac{1}{2}\rho S{V}^{2}L\left(T_X^{\beta }\beta +T_X^p\dot{p}+T_X^q\dot{q}\right)\\ m_y=\frac{1}{2}\rho S{V}^{2}L\left(T_Y^{\beta }\beta +T_Y^p\dot{p}+T_Y^q\dot{q}\right)\\ m_z=\frac{1}{2}\rho S{V}^{2}L\left(T_Z^{\alpha }\alpha +T_Z^r\dot{r}\right)\end{array}$

13

where S is the maximum cross-sectional area of the glider body and V is the magnitude of the vehicle velocity in the body-fixed coordinate. α is the AoA and β is slide angle. C and T are the linear hydrodynamic coefficient calculated by FLUENT. $\dot{p}$ , $\dot{q}$ , and $\dot{r}$ are dimensionless angular speeds. C_X(0) is dimensionless axial and vertical coefficient and $C_Y^{\alpha },C_Z^{\beta },T_X^{\beta },T_Y^{\beta }\text{,\hspace{0.17em}and} T_Z^{\alpha }$ are position derivatives. $C_Y^r,C_Z^p,C_Z^q,T_X^p,T_X^q,T_Y^p,T_Y^q\text{,\hspace{0.17em}and} T_Z^r$ are orientation derivatives.

As to controllable wings, whose shape and attitude always change, viscous hydrodynamic force coefficients of the wings can be approximately estimated ²⁶

$\{\begin{array}{c}{C}_{xw}=C_{DO}+\frac{C_{yw}^2}{0.9\pi \lambda }\\ C_{yw}=\frac{1.8\pi \lambda }{\text{cos}\eta \sqrt{{\lambda }^2/{\text{cos}}^4\eta +4}+1.8}{\alpha }_w+\frac{C_{DC}}{\lambda }{\alpha }_w^2\\ C_{zw}=0\\ M_{xw}=z_w\left[C_{yw}\left(\text{cos}{\alpha }_{wl}-\text{cos}{\alpha }_{wr}\right)+C_{xw}\left(\text{sin}{\alpha }_{wl}-\text{sin}{\alpha }_{wr}\right)\right]\\ M_{yw}=0\\ M_{zw}=x_w\left[C_{yw}\left(\text{cos}{\alpha }_{wl}+\text{cos}{\alpha }_{wr}\right)+C_{xw}\left(\text{sin}{\alpha }_{wl}+\text{sin}{\alpha }_{wr}\right)\right]\end{array}$

14

where C_xw is coefficient of longitudinal force, C_yw is coefficient of vertical force, C_zw is coefficient of lateral force, M_xw is coefficient of roll moment, M_yw is coefficient of yaw moment, M_zw is coefficient of pitch moment, λ is aspect ratio, η is sweep angle, C_DO is 2-dimension profile coefficient of minimum drag, C_DC is coefficient of lateral drag, and α_w is whole attack angle of wings, which can be expressed as

${\alpha }_w=\alpha +\delta$

15

where α is the AOA of the glider and δ is the flip angle of the wings.

Coordinate transformation of forces and moments

The forces above are conducted in the body-fixed coordinate. When converted to the inertial coordinate system numbered zero, force F_k and moment M_k are

$\{\begin{array}{c}{F}_{km}^{\left(0\right)}=C_E^B{F}_{km}=C_E^B\left(f_{GB}+f_p+f_{IB}+f_{IW}+f_{VB}+f_{VW}\right)\\ M_{km}^{\left(0\right)}=C_E^B{M}_{km}=C_E^B\left(m_{GB}+m_P+m_{IB}+m_{IW}+m_{VB}+m_{VW}\right)\end{array}\left(m=1\sim 3\right)$

16

The generalized force corresponding to the generalized active speed is ¹⁹

$F_l={\upsilon }_{klm}{F}_{klm}+{\omega }_{klm}{M}_{klm}$

17

where F_klm and M_klm are scalar components of F_k and M_k along n_oe, respectively.

Kane equation

Based on multibody system dynamics, the Kane equation is used to describe the system ²⁷

$F_l+F_l^{*}=0 \left(l=1\sim 16\right)$

18

where F_l and $F_l^{*}$ are generalized active force and generalized inertial force of the multibody system, respectively. The generalized inertial force can be expressed as

$F_l^{*}=-a_{lp}{\dot{q}}_l-h_l$

19

where α_lp is generalized inertial coefficient and h_l is generalized inertial force coefficient.

Motion simulation

HUG integrates the motion modes of AUG and autonomous underwater vehicle (AUV). When working in AUG mode, Petrel II has the advantage of long endurance and operates in glide motion and spiral motion. When working in AUV mode, Petrel II gets the high speed and horizontal turning capacity. Glide motion emphasizes flight efficiency and static stability. Spiral motion and horizontal turning emphasize maneuverability. All motions need static stability. Former study shows that sweep angle affects the static stability of Petrel II most. However, the static stability is always below zero. ²⁸ For Petrel II with controllable wings, the static stability changes slightly when the sweep angle changes as shown in Table 1.

Table 1.

Static stability changes with the sweep angle.

Sweep angle (°)	l′ α
0	−0.0052
20	−0.0169
40	−0.0253
60	−0.0383

There are many researches about effect of AOA on glide motion. ^{11 –14} By varying the AoA, the glider can optimize the flight efficiency while maintaining stable. The effect of aspect ratio and sweep angle on flight efficiency in glide motion should be considered. When Petrel II operates in spiral motion, opposite AOA and aspect ratio have the most effect on maneuverability. When Petrel II operates in horizontal turning motion, opposite AOA has the most effect on maneuverability.

Flight performance evaluation index

Flight efficiency is evaluated by lift-drag ratio. There is such a relation between lift-drag ratio and glide angle as shown in equation (20). The smaller glide angle means the better flight efficiency

$\text{cot}\xi =\frac{L}{D}=\chi$

20

where ξ is the glide angle, L is the lift force along axis By, D is the drag force along axis Bx, and χ is the lift-drag ratio.

For glider’s low speed, static stability is taken into consideration. Dimensionless hydrodynamic center arm $l_{\alpha }^{\prime }$ is used to evaluate static stability. When $l_{\alpha }^{\prime }<0$ , the glider is in static stability.

Maneuverability is evaluated by turning radius. The smaller turning radius means the better maneuverability.

Motion modes, flight performance, and wing configurations

Based on multibody dynamic models, motion simulation is conducted using fourth-order Runge-kutta with variable time step. ²⁹ The motion parameters of initial state are set as zero as in equation (21). u₀ is set as 0.01 m/s to avoid singular values ³⁰

$\{\begin{array}{l}{X}_0=0,Y_0=0,Z_0=0, u_0=0.01,v_0=0,w_0=0 \hfill \\ p_0=0,q_0=0,r_0=0,{\psi }_0=0,{\theta }_0=0,{\varphi }_0=0\hfill \end{array}$

21

Table 2 shows the inputs of five different motion simulations.

Table 2.

Variables’ values in simulation groups.

Group	1	2	3	4	5
Battery moving (mm)	40	40	40	40	0
Battery rotating (°)	0	0	0	40	0
Oil mass (%)	55	55	55	55	12
Propeller velocity (r/min)	0	0	0	0	1000
Aspect ratio	6, 7, 8	6	6	6, 7, 8	6
Sweep angle (°)	30	0, 30,45	30	30	30
AOA (°)	0	0	±5, ±10, ±15	0	±5, ±10, ±15

AOA: angle of attack.

Glide motion, flight efficiency, and aspect ratio

Figure 7 shows the displacement of Petrel II in the E-XY. Slope of the curves represents glide angle. Figure 8 shows the glide angle, which agrees with Figure 7. When the glider is in stable glide motion, the pitch angle will hold a certain nonzero value which is different from the initial setting of zero. Transient fluctuations of angle occur before the system tends to be stable in the simulation. The displacement curves are smooth, because the start point has no impact on relative displacement. Figures 7 and 8 show that larger aspect causes smaller glide angle and has better flight efficiency.

Figure 7.

Glide motion simulation with different aspect ratios.

Figure 8.

Glide angle simulation with different aspect ratios.

Glide motion, flight efficiency, and sweep angle

Figures 9 and 10 show that there is an optimal sweep angle to make the glide angle smallest. Figure 11 is the glide angle with different sweep angles. Figure 11 shows that when the sweep angle is 26°, the glide angle is the smallest and the glider has the best flight efficiency.

Figure 9.

Glide motion simulation with different sweep angles.

Figure 10.

Glide angle simulation with different sweep angles.

Figure 11.

Relation between sweep angles and glide angles.

Spiral motion, maneuverability, and opposite AOA

Petrel II with controllable wings can change directions by rotating the two wings in opposite directions. Figure 12 shows that the turning diameters are 50 m, 80 m, and 130 m with different opposite AOAs of ±5°, ±10°, and ±15°, respectively. Figure 13 shows that the sideslip angles are 26°, 19°, and 9°, respectively, which agrees with Figure 12. It can be concluded that the opposite AOA of ±5° causes the smallest turning radius and best maneuverability.

Figure 12.

Spiral motion simulation with opposite AOA. AOA: angle of attack.

Figure 13.

Sideslip angle simulation with opposite AOA. AOA: angle of attack.

Spiral motion, maneuverability, and aspect ratio

Figure 14 shows that the turning diameters are 90 m, 105 m, and 120 m with different aspect ratios of 6, 7, and 8, respectively. Figure 15 shows that the sideslip angles are 16°, 17°, and 19°, respectively, which agrees with Figure 14. It can be concluded that smaller aspect ratio causes smaller radius and better maneuverability.

Figure 14.

Spiral motion simulation with different aspect ratios.

Figure 15.

Sideslip angle simulation with different aspect ratios.

Horizontal turning motion, maneuverability, and opposite AOA

Horizontal turning is completed by the propeller. Figure 16 shows that the turning diameters are 43 m, 79 m, and 134 m with different opposite AOAs of ±5°, ±10°, and ±15°, respectively. It can be concluded that the opposite AOA of ±5° causes the smallest turning radius and best maneuverability.

Figure 16.

Horizontal turning simulation with different opposite AOAs. AOA: angle of attack.

Field trial

Field trials were conducted in Danjiangkou Reservoir, China. The radius of the trial area is over 1 km and the average depth is 40 m. Figure 17 shows that the HUG is changing direction in thrust mode. The pressure sensor gathers depth information. Electronic compass TCM3 gathers attitude information. Altimeter sensor guarantees that the glider won’t strike the bottom. The displacement of glider along X and Y axis can be derived from depth and attitude information. With the limitation of shallow water, the glider can’t finish complete spiral motion. The antenna mask can’t locate GPS location under water. Instead of turning radius, the change amount of course angle in the same time is used to evaluate the maneuverability. Figures 18 to 22 show the trial results.

Figure 17.

Field trials in Danjiangkou Reservoir.

Figure 18.

Glide motion trial with different aspect ratios.

Figure 19.

Glide motion trial with different sweep angles.

Figure 20.

Spiral motion trial with different opposite AOAs. AOA: angle of attack.

Figure 21.

Spiral motion trial with different aspect ratios.

Figure 22.

Horizontal turning trial with different opposite AOAs. AOA: angle of attack.

Glide motion

Figures 18 and 19 show that the trial results agree with those of the simulation. In glide motion, lager aspect ratio and the sweep angle of 26° can get better flight efficiency.

Spiral motion

Figure 20 shows that glider with opposite AOA of 5° has the largest course angle change of 179° in 160 s in spiral motion and better maneuverability. Figure 21 shows the glider with aspect ratio of 6 has the largest course angle change of 81° in 160 s and better maneuverability.

One thing to note here is that glider with opposite AOA of 15° loses the turning capacity at the depth of 18 m in Figure 20. The reason is that AOA of ±15 is beyond the stalling angle that affects the turning capacity of the glider. In this case, the hydrodynamic forces acting on the glider change. Theoretical equations cannot explain the stalling phenomenon. Therefore, simulation results do not reflect this phenomenon.

Horizontal turning motion

Figure 22 shows that the glider with opposite AOA of 5° turns a circle with the least time of 137 s and has better maneuverability.

Conclusions and future work

HUG Petrel II has three motion modes, including buoyancy driven glide, buoyancy driven spiral turning, and propeller driven level-flight, which require different wing configurations to achieve optimal flight performance. In this article, controllable wings are designed based on linkage synthesis to change AOA, sweep angle, and aspect ratio. Kinematics and dynamics models are established based on multibody dynamics, which treats components separately and is good for analysis of controllable wings. Finally, the impact of wing configurations on flight efficiency in glide motion and maneuverability in spiral motion and horizontal turning motion is investigated by Matlab (Matlab2010a) simulation and field trials. The conclusions are listed as follows:

Glide motion, which puts long endurance and duration in the first place, values flight efficiency. Larger aspect ratio of 8 and the sweep angle of 26° should be realized to get smaller glide angle and better efficiency.

Spiral motion values maneuverability. The opposite AOA of 5° and smaller aspect ratio of 6 should be realized to get smaller turning radius and better maneuverability.

Horizontal turning focuses on maneuverability. The opposite AOA should be realized to get smaller turning radius and better maneuverability.

There is some work to be done in the future. In simulations and field trials of both spiral motion and horizontal turning motion, the opposite AoA of 5° shows the best maneuverability compared with 10° and 15°. The optimal opposite AoA between 1° and 10° should be determined in the future. Moreover, this article only explores the effect of each single parameter of the wings on flight performance. Orthogonal test of all factors, including AoA, sweep angle, and aspect ratio, should be conducted to determine the optimal wing configurations for various motion modes. In addition, the mechanism and cabin increase the complexity of system, so field experiment of long duration and distance should be conducted to justify the advantage of adding controllable wings.