Motion parameter optimization for gliding strategy analysis of underwater gliders

Ocean Engineering 191 (2019) 106502

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Motion parameter optimization for gliding strategy analysis of underwater gliders Ming Yang a, b, Yanhui Wang a, b, c, *, Shuxin Wang a, b, c, Shaoqiong Yang a, b, Yang Song a, b, Lianhong Zhang a, b a b c

Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, School of Mechanical Engineering, Tianjin University, Tianjin, 300350, China Qingdao Institute for Ocean Engineering of Tianjin University, Qingdao, Shandong, 266237, China The Joint Laboratory of Ocean Observing and Detection, Pilot National Laboratory for Marine Science and Technology (Qingdao), Shandong, 266237, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Motion parameter optimization Gliding strategy analysis Underwater glider

Underwater glider is one of the most promising autonomous observation platforms for long-term ocean obser­ vation, which can glide through water columns by adjusting its buoyancy and attitude. In this paper, gliding range of the Petrel-L underwater glider is increased by optimizing the motion parameters according to different observation missions. The optimization is established to maximize the gliding range and minimize the energy consumption for one gliding cycle, which is solved by the inner penalty function method (IPFM). Based on the nonlinear variation rule of the buoyancy loss, a hidden gliding strategy model (HGSM) is proposed and the optimization result indicates that it can increase the gliding range by 24.13% for Petrel-L when the power related with time is 0.2 W. The optimization is applicable to other types of underwater gliders.

1. Introduction Driven by its net buoyancy, underwater glider (UG) can glide through water columns by adjusting the buoyancy and the pitching angle (Bachmayer et al., 2004). Since the concept of UG was first demonstrated by Stommel (1989), various underwater gliders including the Slocum (Webb et al., 2001), the Seaglider (Eriksen et al., 2001) and the Spray (Sherman et al., 2001) have been developed and widely applied in ocean observation for their temporal and spatial cruising ability (Rudnick et al., 2004). New types of underwater gliders, such as the Deepglider (Osse and Eriksen, 2007) with 6000-m operating depth, the ZRAY (D Spain et al., 2005) with heavy payload capability, and the Sea-explorer (Claustre et al., 2014) with high gliding velocity, have been designed. With the development of marine science, a larger gliding range is required for the observation of some larger-scale and longer-lasting ocean phenomena (Stuntz et al., 2016). Though gliding range can be elongated by increasing the battery capacity in the limited internal space of the underwater glider, it seems to be great significance to increase the gliding range through reducing energy consumption, improving gliding strates and optimizing motion parameters. This paper focuses on the motion parameter optimization and gliding strategy

analysis of the underwater glider. Several research results have been reported in the literatures to in­ crease the cruising ability and reduce the energy consumption, which can be realized by optimized control and improved designs. In terms of optimized control, several typical researches are listed. Yu et al. (2013) optimized gliding motion parameters and sensor scheduling to increase the gliding range of underwater gliders, without considering the varia­ tion of net buoyancy in the gliding cycles. Xue et al. (2018) proposed a multi-layer coordinate control strategy of underwater gliders by opti­ mizing the motion parameters. Huang et al. (2018) improved the control and energy efficiency over the pitch angle adjustment by a self-searching optimal active disturbance rejection control theory to increase the endurance of the thermal glider. Sang et al. (2018) proposed a new hybrid heading tracking control algorithm to improve the adaptability and robustness of underwater glider’s heading control. Claus and Bachmayer (2016) presented an energy optimal depth controller design methodology for long range autonomous underwater vehicles with ap­ plications to a propeller driven hybrid UG during level flight. Yoon and Kim (2016) formulated an optimal control problem for an underwater glider by considering depth constraints into a nonlinear two-point boundary value problem with inequality constraints, which was solved

* Corresponding author. Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, School of Mechanical Engineering, Tianjin Uni­ versity, Tianjin, 300350, China. E-mail address: [email protected] (Y. Wang). https://doi.org/10.1016/j.oceaneng.2019.106502 Received 18 June 2019; Received in revised form 11 September 2019; Accepted 28 September 2019 Available online 7 October 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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main contributions of this paper are in two aspects. (1) An optimization method is established to increase the gliding range, which takes the variation of the net buoyancy into account and analyzes the effect of the power of the carried sensors on the optimized motion parameters based on the different observation missions. The optimization is constructed by the kinematic model in the vertical plane, two gliding strategy model and an energy consumption model. The objective of the optimization is maximizing the gliding range and minimizing the energy consumption which is solved by the IPFM, and the trend of the optimized motion parameters for the different temporal power is researched. (2) A hidden gliding strategy model (HGSM) is presented by analyzing the nonlinear variation trend of the buoyancy loss, the application effect of which on improving the gliding range is analyzed in comparison with the CGSM. This paper is organized as follows. Section 2 introduces the Petrel-L underwater glider for modeling. In Section 3, the mathematical model is derived based on the kinematical model of glider motion in vertical plane, the gliding strategy model and the energy consumption model. In Section 4, an optimization is established to analyze how to control the motion parameters for the largest gliding range. In Section 5, the opti­ mization results and discussions are introduced. In Section 6, a sea trail is carried out to verify the validity of the optimization. Finally, Section 7 concludes the paper.

Fig. 1. Petrel-L underwater glider. Table 1 Technical specifications of Petrel-L. Technical specification

Value

Size Weight Depth Battery Range Navigation Science sensor

Diameter 0.24 m, length 2.6 m, wing span 1.2 m 93 kg 1000 m Lithium primary batteries >3000 km GPS, pressure sensor, altimeter and compass CTD

2. The Petrel-L underwater glider The Petrel-L, shown in Fig. 1, is an underwater glider developed by Tianjin University, China. It consists of a cylindrical pressure hull to which the emergency device, fixed wings, two fairings and a trailing antenna are attached. The subsystems inside the pressure hull include the buoyancy regulating unit, pitch-regulating unit, roll-regulating unit, control unit, communication and navigation unit and battery pack. Table 1 shows the main technical specifications of Petrel-L. As a core part of the Petrel-L, the buoyancy regulating unit consists of an external bladder, an internal reservoir, some hydraulic valves, a displacement sensor, a pressurized gear pump and a high pressure plunger pump. The pumps and valves can adjust the oil volume of the external bladder to change the buoyancy of the glider and thus drive its vertical motion. The displacement sensor monitors the oil volume of the internal reservoir when the buoyancy regulating unit works. During gliding, the attitude angles of glider including the pitching angle, the roll angle and the heading are adjusted by an internal translational and rotational battery pack, which can change the position of mass center of the underwater glider. Hence, the glider dives and climbs in a zigzag trajectory through the seawater.

by numerical methods. Liu et al. (2017) developed an optimal three-dimensional path planning method for low energy consumption. Miller (2018) applied the bio-inspired energy utilization strates to the underwater vehicles to achieve a 10–40% range promotion. The following studies focused on the optimal design to increase the cruising ability and reduce the energy consumption. Fu et al. (2018) proposed a multi-objective shape optimization scheme of underwater gliders for low drag, high efficiency, low energy consumption and long voyage. Li et al. (2018) presented a simplified shape optimization strategy for blended-wing-body underwater gliders, by which the lift-to-drag ratio is increased by 4.2%. Tang et al. (2018) designed a neutrally buoyant pressure hull using bi-directional evolutionary topology method to reduce the buoyancy loss. He et al. (2017) carried out a multi-objective optimization of a multi-bubble pressure cabin in the underwater glider with blended-wing-body using Kriging and the non-dominated sorting genetic algorithm. Wang et al. (2017) adopted the design of experiment (DOE) theory to solve the multi-objective design optimization of the underwater glider, which improved the hydrodynamic performance by 9.1%. Although the above investigations have provided numerous results related to the optimization design, there exist the following problems. (1) The net buoyancy variation is not taken into account in the motion parameter optimization in the previous works, which affects the gliding motion state and is influenced by the deformation of the pressure hull and the density variation of the seawater (Yang et al., 2017). (2) Inte­ grated with the CTDs, turbulence sensors, acoustic sensors and biochemical sensors, underwater gliders have been widely applied in the mesoscale eddy observation, the submesoscale eddy observation, the internal wave observation and the turbulent microstructure observation (Rudnick, 2016). The power of the carried sensors is necessary to be analyzed in the optimization, which has a significant impact on the gliding range. (3) Since the physical parameters of the seawater including density and temperature show nonlinear variation with the depth, therefore, the conventional gliding strategy model (CGSM) in the previous optimization may not be the optimal gliding strategy for the largest gliding range, which needs to be analyzed. In this paper, an efficient method is developed to increase the gliding range of underwater gliders by optimizing the motion parameters and analyzing the gliding strategy. Compared with the existing work, the

3. Modeling of the Petrel-L underwater glider 3.1. Buoyancy loss of Petrel-L As the glider ascends higher in the water column and the pressure relaxes, it becomes less buoyant because the hull is less compressible than seawater (Jenkins et al., 2003). Therefore, the net buoyancy de­ creases gradually in the diving phase and the climbing phase. This phenomenon is obvious particularly when the glider goes across the thermocline where the seawater density changes sharply. In the gliding motion, the buoyancy loss △Bglider is decided by the buoyancy of the glider, which is mainly related to the volume of the pressure hull and the density of the seawater. The buoyancy loss of the glider can be calcu­ lated as ΔBglider ðZÞ ¼ ρZ VZ g

ρ 0 V0 g

(1)

Here, Vz is the volume of the glider under the hydrostatic pressure of the seawater, V0 is the volume of Petrel-L under the atmospheric pres­ sure which is 91 L, ρz is the density of seawater varying with the depth and ρ0 is 1021.8 kg/m3. According to the International Gravity Formula proposed by Moritz (1984), the acceleration of gravity g is considered as 2

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9.8 kg/m3 because it varies little with the latitude and the height rela­ tive to sea level. The seawater density varies with temperature, salinity and pressure. In this paper, the density is considered as a variable changing with the depth Z and can be obtained by the sea trial data in South China Sea. The quartic polynomial fitted curve is described in Fig. 2. The maximum value is 1031.98 kg/m3 at the depth of 1000 m and the minimum value ρ0 is 1021.8 kg/m3 at sea surface. The expression of the density can be described as

3

Density(kg/m )

1030 1028 1026

ρZ ðZÞ ¼ p1 Z 4 þ p2 Z 3 þ p3 Z 2 þ p4 Z þ p5

1024 1022 1020 0

200

400

where Z is the depth of the seawater and the fitted coefficients pi (i ¼ 1, …,5) are 4.8547 � 10 11, 1.1823 � 10 7, 1.0165 � 10 4, 0.0422 and 1021.6 respectively. When the glider glides through the water columns, the temperature and pressure change gradually, which leads to the thermal stress and hydrostatic stress applied on the pressure hull. The temperature varia­ tion with the depth observed by Petrel-L in the South China Sea can be described by a quartic polynomial analogously and the variation trend is shown in Fig. 3.

Sea trial data Fitted curve 800 1000

600

Depth(m)

Fig. 2. The fitted curve of density.

T ¼ q1 Z 4 þ q2 Z 3 þ q3 Z 2 þ q4 Z þ q5

30

Sea trial data Fitted curve

Temperature(

20 15 10 5 0 0

200

400

600

Depth(m)

800

1000

Fig. 3. Fitted temperature curve.

Vol ume var i et y( mL)

300 250 200

Coupling effect Fitted curve Pressure effect Temperature effect

150

ΔVZ ðZÞ ¼ r1 Z 2 þ r2 Z

(4)

VZ ðZÞ ¼ V0

(5)

ΔVZ

where △VZ is the volume variation of the pressure hull, and the co­ efficients r1 and r2 are 1.384 � 104 and 0.386675 respectively. Substitute Eqs. (2), (4) and (5) into Eq. (1) and the buoyancy loss can be obtained

100 50 0 0

(3)

where the fitted coefficients qi (i ¼ 1, …,5) are 1.3811 � 10 10, 3.4465 � 10 7, 3.0883 � 10 4, 0.1265 and 28.7196 respectively. In this paper, the finite element analysis code ANSYS Workbench is adopted to analyze the deformation and volume variation of the pres­ sure hull due to temperature and hydraulic pressure changes in the seawater. Firstly, a steady-state thermal model is adopted and the temperature difference is imported into the model. Then a static struc­ tural model is established and the analysis result of the steady-state thermal model is imported. What calls for special attention is that the pressure inputted in the static structural model needs to match the temperature inputted in the steady-state thermal model according to the ocean environment. Moreover, the effects of pressure and temperature on the volume variety are simulated to evaluate the influence of tem­ perature. Fig. 4 shows the effects of coupling, pressure and temperature on the volume variety of the pressure hull respectively. It can be seen from Fig. 4 that the effects of pressure and temperature are nonlinearly coupling. Moreover, it is necessary to take the temperature into account in the volume variety simulation because the volume variety caused by temperature is 21.7% of that by coupling when the pressure is 10 MPa. To validate the calculated result, a pressure test is carried out to measure the strain of the pressure. The strain gages are pasted on the internal face of the pressure hull and the environment temperature is 28 � C. During the test, the temperature of water is 12 � C. Fig. 5 shows the pressure test and the result comparison which indicates the consistency between the simulation result and the test result. The fitted quadratic can be calculated.

)

25

(2)

200

400

600

Dept h( m)

800

ΔBglider ðZÞ ¼ p1 r1 Z 6 g þ ðp1 r2 þ p2 r1 ÞZ 5 g þ ðp2 r2 þ p3 r1 ÞZ 4 g þðp3 r2 þ p4 r1 ÞZ 3 g þ ðp4 r2 þ p5 r1 ÞZ 2 g þ p5 r2 Zg ρ0 V0 g

1000

(6)

The buoyancy loss is described in Fig. 6. As shown in the figure, there exists nonlinear variation and the net buoyancy decreases faster in the shallow sea than in deep sea because of the existence of the thermocline.

Fig. 4. The volume variation of the glider.

3

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1800

Circumferential strain (Simulation) Circumferential strain (Test) Axial strain (Simulation) Axial strain (Test)

Mi cr os t r ai n( 1/ 106 )

1600 1400 1200 1000 800 600 400 200 0

(a) Pressure test

2

4

Pr es s ur e

6

MPa)

8

10

(b) Result comparison Fig. 5. Pressure test and result comparison.

control the attitude of the glider. After arriving at the surface, the communication phase bns during which the battery pack moves fully forward and the hydraulic oil is pumped to the external bladder. The trailing antenna emerges from the surface and the pitching angle is larger than 45� . In this phase, the GPS module and the iridium satellite communication module bn to work and we can get the position of the glider and control the glider by the iridium satellite communication.

7

Buoyancy loss(N)

6 5 4

3.2.2. The HGSM gliding strategy The gliding model adopted mostly by the legacy underwater gliders is the CGSM, the gliding efficiency of which is low because the net buoyancy changes sharply when the glider goes through the thermo­ cline. Thus, the buoyancy regulating unit has to pump extra oil to the external bladder to keep a certain gliding velocity (Jenkins et al., 2003). In this work, a HGSM shown in Fig. 8 is proposed, in which the glider does not need to go through the thermocline for a few sequent gliding cycles. This gliding strategy can be adopted when the glider is far from the sea area of the task. For example, the glider needs to move to the sea area of the next task when it has finished the current task. Moreover, it can also be adopted to save the cost of the observation task when the sea area of the task is far from the coast and the glider is launched near the shore. In the HGSM, the minimum gliding depth needs to be confirmed, below which the buoyancy loss rate is the lowest, as expressed by Eq. (7) and Fig. 9. The minimum depth is 331 m, below which the net buoyancy loss rate R (net buoyancy loss per unit depth) is 2.934 � 10 3 N/m.

3 2 1 0 0

200

400

600

Depth(m)

800

1000

Fig. 6. The buoyancy loss of Petrel-L.

3.2. Gliding strategy modeling 3.2.1. The CGSM gliding strategy As shown in Fig. 7, the CGSM can be divided into five phases: the diving preparation phase, the diving phase, the climbing preparation phase, the climbing phase and the communication phase. In the diving preparation phase, the solenoid valve turns on and the hydraulic oil flows from the external bladder to the internal reservoir until the oil volume in the internal reservoir reaches the set value. Then, the pitchregulating unit bns to adjust the pitching angle to a desired value and the diving phase bns during which the roll-regulating unit adjusts the heading when the glider deviates from the target heading. When the glider reaches the target depth, the climbing preparation phase bns and the hydraulic oil is pumped back to the external bladder until the oil volume in the internal reservoir reaches the set value. Then the glider will climb and enter the climbing phase during which the pitchregulating unit and the roll-regulating unit work interruptedly to

R¼

ΔBglider ðZ ¼ 1000Þ 1000 Z

ΔBglider

(7)

3.3. Mathematical modeling in vertical plane 3.3.1. Mass distribution The mass configuration of a glider is critical because it moves in the seawater by adjusting its buoyancy. Shown in Fig. 10, the mass of a glider m consists of mass of the hull mh, internal moving point mass mm and ballast mass mb. The component mh is a fixed, uniformly mass. The vector rm and rb describes the position of the moving mass mm and ballast mass mb in the body fixed frame at time, t. The ballast mass mb1 and mb2 of the glider are the mass of fluid taken in during its descent and given 4

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Ocean Engineering 191 (2019) 106502

Communication phase

Diving preparation phase

Communication phase

ng

gp ha se

i Div

Cl

ase

im bin

ph

Climbing preparation phase Fig. 7. The process of CGSM. Communication phase

Diving preparation phase

Communication phase

Thermocline

Cl

im

bin

gp

ase

ph

ha se

ng

vi Di ……

Climbing preparation phase

Fig. 8. The schematic diagram of HGSM. -3

8

x 10

7

mh

R(N/m)

6

rb

mb

mm

5

Fig. 10. Glider mass definitions.

4

ΔBðZÞ ¼ mg

100

200

300

400

Depth(m)

500

600

ρ0 ðZÞV0 g � ΔBglider ðZÞ

(9)

If the △B ¼ 0, then glider is neutrally buoyant. The glider descends if △B > 0 and ascends if △B < 0. In order to maintain a steady glide, the state variables of glider (velocity, gliding angle and angle of attack) need remain constant.

3 2 0

rm

CB

700

3.3.2. Kinematic and dynamic model of glider motion Petrel-L glider is designed to have a separate control algorithm and actuators for controlling the lateral dynamics and vertical plane dy­ namics. In this paper, the vertical plane dynamics are controlled using the movable mass. Here, the lateral dynamics can be ignored since a fixed back rudder is adopted to make the glider glide in a straight motion in a vertical plane and thus stabilize the dynamics. Experiments using the Slocum glider show that the vertical plane steady glides are quali­ tatively consistent with the result of the model (Leonard and Graver, 2001). Thus, the study of the vertical plane dynamics as a representative

Fig. 9. Net buoyancy loss per unit depth.

out during its ascent respectively. Therefore, the total mass of the diving phase and climbing phase is � mh þ mm þ mb1 m¼ (8) mh þ mm mb2 and the net buoyancy of the glider △B is given by 5

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Ocean Engineering 191 (2019) 106502

E-frame

X

L

Y

D

B-frame

u Yaw w

p x

Roll

s

U

z B Fig. 11. Kinematic model of the glider.

of actual glider operation is useful. Leonard and Graver (2001) derived the equation of motions of glider in the vertical plane and described the mathematical model of the steady state dynamics of glider in the vertical plane. The kinematics of the vehicle can be explained based on Fig. 9 (Singh et al., 2017). Two frames, the inertial frame (E-frame) and the body frame (B-frame), are required in the work of building dynamic model of the glider. Definition of the two frames is same as that of the common underwater vehicles (Wang et al., 2015; Sarkar et al., 2016). The origins of the B-frame coincide with the center of the buoyancy. The E-frame is fixed to a point on the sea surface. In reference to E-frame, the velocity of the glider along x, y and z-axis is represented by u, v and w respectively while the angular velocity of the glider along x, y and z-axis is represented by p, q and s, respectively, all in B-frame. In Fig. 11, θ is the pitching angle (angle between X-axis of E-frame and x-axis of B-frame), α is the angle of attack (angle between gliding velocity vector and x-axis of B-frame), γ is the gliding angle (angle between gliding velocity vector and X-axis of E-frame), U is the gliding velocity, L and D are vehicle lift and drag, respectively. When the glider descends, γ and θ are positive and α is negative. When the glider ascends, γ and θ are negative and α is positive. In order to ease the analysis process, some restrictions have to be made to simplify the mass distribution on the glider (Graver, 2005). Pm denotes momentum of the moving mass, in the B-frame. Be represents the position in E-frame while Ω represent the angular velocity vector in B-frame, respectively. R is the rotational matrix which defines the orientation of glider in B-frame. The control inputs to the system are the ballast rate ub, internal force um acting on the moving mass. umx represent the internal force acting on moving mass along z-direction. mfx and mfz represent the added mass along x and z-axis, respectively. KM0 and KM are the zero moment co­ efficient and induced moment coefficient, respectively. Iyy represents the mass moment of inertia along y-axis. The roll, yaw and sway velocity terms are taken to be zero. Assuming zero side-slip angles, the hydro­ dynamic shear force is neglected. The simplified parameters for motion are expressed in Eq. (10) as: T

Be ¼ ½x 0 z� ; Ω ¼ ½0 q rm ¼ ½rmx 0 rmz �; rb ¼ ½rbx Pm ¼ ½Pmx 0 Pmz � um ¼ ½umx 0 umz �

T

0� ; U ¼ ½u 0 rbz �;

0

w�

E-frame as 2 3 2 3 x_ u 4 0 5 ¼ ½R�4 0 5 z_ w

(11)

2 3 2 3 0 0 4 θ_ 5 ¼ ½R�4 q 5 0 0

(12)

where The modified equations of motion are represented in Eq. (13) as: x_ ¼ u cos θ þ w sin θ; z_ ¼ u sin θ þ w cos θ 1w α ¼ tan ; θ_ ¼ q u � 1 q_ ¼ ½ mfz mfx uw ðrmx Pmx þ rbx Pbx rmz Pmz Þq Iyy mm rmz g sin θ

ðmm rmx þ mb rbx Þg cos θ

rmz umx þ rmx umz �

þMp 1 � u_ ¼ mfx

mfz wq

ðPmz þ Pbz Þq

1 � mfz

mfx wq

ðPmx þ Pbx Þq þ ΔBg cos θ

w_ ¼

r_mx ¼

Pmx mm

P_ mx ¼ umx ;

u

rmz q;

r_mz ¼

P_ mz ¼ umz ;

ΔBg sin θ þ L sin α

Pmz mm

u

L cos α

D cos α D sin α

umx

�

umz

�

rmx q

m_ b ¼ ub

(13)

MP represents the pitching angle moment, the expression of which is: � MP ¼ ðKM0 þ KM αÞ u2 þ w2 (14) U is denoted as (15)

U 2 ¼ u2 þ w2

T

3.3.3. Mathematical model of the steady state gliding motion The kinematic and equilibrium relations follow from Graver (2005) as:

(10)

θ ¼ γ þ α;

The velocity and the angular velocity in B-frame can be expressed in 6

ΔB cos γ ¼

L;

ΔB sin γ ¼

D

(16)

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Ocean Engineering 191 (2019) 106502

1.65

30

Quant i t y

35

mL/ s

40

W

1.7

Power

45

25 20 0

2

4

6

Load( MPa)

8

10

1.6 1.55 1.5

0

2

4

6

Load( MPa)

8

10

Fig. 12. The power and quantity of the pump.

The hydrodynamic drag and lift can be defined as: � D ¼ KD0 þ KD α2 U 2 ; L ¼ ðKL0 þ KL αÞU 2

X_ ¼ U cosðθ þ αÞ Z_ ¼ U sinðθ þ αÞ

(17)

The mathematical model established above is widely used in the design of the glider and Graver (2005) illustrated the use of the equa­ tions of motion to simulate the sawtooth gliding in the vertical plane by making use of the parameters of a glider in size as Slocum. In this paper, the parameters used for simulation are: KD0 ¼ 9.4546 N (s/m)2, KD ¼ 553.1856 N (s/m)2, KL0 ¼ 0 N (s/m)2, KL ¼ 604.2496 N (s/m)2, (s/m)2, KM ¼ 311.53 N (s/m)2, rmz ¼ 0.02 m, KM0 ¼ 0 N Iyy ¼ 28.2689 kg m2, mh ¼ 77 kg, mb ¼ 16.5 kg, mfx ¼ 2.7374 kg, mfz ¼ 105.1181 kg, U0 ¼ 0.3 m/s, θ0 ¼ �17� .

where KD0 and KD are the zero (angle of attack) drag coefficient and induced drag coefficient respectively, KL0 and KL are the zero (angle of attack) lift coefficient and induced lift coefficient respectively. Eqs. (16) and (17) can be used to define the relation between α and γ as: tan γ ¼

KD0 þ KD α2 KL0 þ KL α

D ¼ L

(18)

For a solution in the real number field, one must have ðKL tan γÞ2

4KD ðKD0 þ KL0 tan γÞ � 0

(19)

3.4. Energy consumption modeling

When the equation is satisfied in Eq. (19), the solution of the tanγ is: 0 sffi� ffiffiffiffiffiffiffiffiffiffi� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2KD @KL0 KL0 KD0 A (20) tan γ ¼ � þ KL KL KL KD is:

The energy consumption reduction of each part plays an important role in increasing the gliding range of the glider, and therefore an energy consumption model needs to be established in the optimization. The subsystems in the energy consumption consist of the buoyancy regu­ lating unit, the attitude adjustment unit, the measuring unit, the control unit and the communication unit.

Therefore, the attainable value range of the gliding angle for a glider 2

0 0 2K D @KL0 ; tan 1 @ 2 KL KL

π

γ¼4

sffi� ffiffiffiffiffiffiffiffiffiffi� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 113 2 KL0 KD0 AA5 þ KL KD

2

0 0 sffi� ffiffiffiffiffiffiffiffiffiffi� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 3 2 KL0 KD0 AA π5 1 @2KD @KL0 4 [ tan ; þ þ KL KL KL KD 2 And the expression of α can be deduced by Eq. (18) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! K tan γ 4KD cot γðKD0 cot γ þ KL0 Þ 1� 1 αðγÞ ¼ L 2KD K 2L

3.4.1. The buoyancy regulating unit The energy consumption of the buoyancy regulating unit is directly related to the control parameters, including the oil volume mb1 pumped back to the internal reservoir and the oil mass mb2 pumped into the external bladder. In the diving preparation phase, the solenoid valve works, and the quantity Qvalve and the power Pvalve are measured to be 24 mL/s and 5 W respectively. The energy consumption of solenoid valve for this phase can be obtained by Eq. (27).

(21)

(22)

Wvalve ¼

The gliding velocity can be expressed as a function of gliding angle using Eqs. (16) and (17) as follows: ΔBðZÞcos γ ¼

U¼

ðKL0 þ KL αÞU 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jΔBðZÞjcos γ KL0 þ KL αðγÞ

(26)

Pvalve mb1 ρoil Qvalve

(27)

Where ρoil is the density of the hydraulic oil. In the climbing preparation phase, the oil is pumped into the external bladder, and the quantity Qpump and the power Ppump are related to the load, namely the pressure applied on the external bladder by the seawater, which can be fitted by the measured data in Fig. 12. � � � � ρ gZ 3 ρ gZ 2 ρ gZ Qpump ¼ j1 0 6 þ j2 0 6 þ j3 0 6 þ j4 (28) 10 10 10

(23) (24)

Substituting from αðγÞ, Eq. (22) in Eq. (24), one obtains vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2jΔBðZÞjKD u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � U ¼t � 2KL0 KD cot γ þ KL sin γ KL � K 2L 4KD KD0 cot2 γ 4KD KL0 cot γ

Ppump ¼

(25)

k1 ρ0 gZ þ k2 106

(29)

Here, the coefficients (j1, …, j4) are 0.0158, 0.2536, 0.3487 and 101.9852, k1 is 1.9168, and k2 is 22.7191 respectively. Hence, the energy consumption of the pump in the climbing prepa­ ration phase can be calculated.

which is the relationship between the gliding velocity and the gliding angle. And the horizontal velocity UH and vertical velocity Uv are 7

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Ocean Engineering 191 (2019) 106502

0.45

CGSM HGSM

Average gliding velocity (m/s)

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0

0.5

1

1.5

2

2.5

Power (W)

3

3.5

Fig. 15. The optimal average gliding velocity.

1W 0.6 W

0.2 W

1200 25

0.25 0.2 0.15 0.1 0.05 0 30

(°)

1200 25

1100 20 900 10 800

m b2

(g)

15

(°)

900 10 800

F (m/J)

F (m/J)

F (m/J)

1200

0.25 0.2 0.15 0.1 0.05 0 30

1100 20

1200 25

1000 900 10 800

m b2

(g)

(g)

1200 1100 20

15

900 10 800

m b2

(g)

1000 15

(°)

900 10 800

1200 25

F (m/J)

0.25 0.2 0.15 0.1 0.05 0 30

1200 25

1100 20 900 10 800

m b2

(g)

1200 25

1100 20

1000 15

(°)

(g)

0.25 0.2 0.15 0.1 0.05 0 30

1100 20

1000

m b2

3.4 W

3W

F (m/J)

F (m/J)

m b2

25

1000

(°)

0.25 0.2 0.15 0.1 0.05 0 30

15

900 10 800

0.25 0.2 0.15 0.1 0.05 0 30

1100 20

2.6 W

(°)

15

(°)

1.8 W

25

Fig. 13. Procedure of the optimization solution.

(g)

m b2

1000

2.2 W

0.25 0.2 0.15 0.1 0.05 0 30

15

1100 20

1000

1.4 W

(°)

1200 25

1100 20

1000 15

0.25 0.2 0.15 0.1 0.05 0 30

F (m/J)

F (m/J)

F (m/J)

0.25 0.2 0.15 0.1 0.05 0 30

900 10 800

m b2

(g)

1000 15

(°)

900 10 800

m b2

(g)

(a) The objective function in CGSM 0.6 W

0.2 W

20 15 700

800

(a) The CGSM

900 m b2

(g)

1100

1200

0 30

20

(°)

(°)

600

20 15

400

m b2

400

10

m b2

20

(g)

(°)

10

(b) The HGSM

(30)

m b2

800 20

(g)

(°)

F (m/J)

F (m/J)

0.25 0.2 0.15 0.1 0.05 0 30

800 20 (°)

600 15 10

10

10

400

m b2

400

m b2

(g)

20

(°)

600 15 10

400

m b2

(g)

(g)

800 20 (°)

600 15

Fig. 16. The objective function’s variation trend.

8

m b2

25

(b) The objective function in HGSM

3.4.2. The attitude adjustment unit The attitude adjustment unit includes the pitch-regulating unit and the roll-adjustment unit, which adjust the pitching angle and the roll angle. The major energy consuming components are the motors and the actuators. In this paper, the energy consumption of the attitude adjust­ ment unit for one gliding cycle is referred as a constant. Then energy

400

0.25 0.2 0.15 0.1 0.05 0 30

600 10

(g)

800 20

800

25 15

m b2

3.4 W

0.25 0.2 0.15 0.1 0.05 0 30

(°)

400

25

(g)

3W

25

(31)

15

0.25 0.2 0.15 0.1 0.05 0 30

600 15

2.6 W

In conclusion, the energy consumption of the buoyancy regulating unit for one gliding cycle can be calculated.

600

(°)

2.2 W

25

600 400

10

20

(g)

0.25 0.2 0.15 0.1 0.05 0 30

800 20 (°)

m b2

1.8 W

0.25 0.2 0.15 0.1 0.05 0 30

15

400

800

25

600 15

1.4 W

25

Wbuo ¼ Wvalve þ Wpump

800

25

600 15

(g)

Fig. 14. The average gliding velocity.

Ppump mb2 ρoil Qpump

F (m/J)

F (m/J)

800

25

800

25

0.25 0.2 0.15 0.1 0.05 0 30

F (m/J)

(°)

1000

0.25 0.2 0.15 0.1 0.05 0 30

F (m/J)

25

0.25 0.2 0.15 0.1 0.05 0 30

F (m/J)

0.2 30

0.2

F (m/J)

0.3

F (m/J)

U (m/s)

U (m/s)

0.4

Wpump ¼

1W

0.4

0.5

10

400

m b2

(g)

M. Yang et al.

Ocean Engineering 191 (2019) 106502 24

30

(°)

(°)

1

1

2

2

(°)

22

(°)

m (100g)

25

b1

20

m

m (100g)

m

b2

F(100m/J)

18

(100g)

b1

(100g)

b2

F(100m/J)

20

16

15

14 12

10

10 8

5

6 4 0

0.5

1

1.5

2

Power(W)

2.5

3

3.5

0 0

4

0.5

(a) The CGSM

1

1.5

2

Power(W)

2.5

3

3.5

4

(b) The HGSM

Fig. 17. The result of the optimization.

25

I ncr eas i ng r at e( %)

20 15 10 5 0 -5 -10 -15 0

1

2

Power ( W)

3

4

Fig. 18. The increasing rate of gliding range.

Table 2 The parameters set in the sea trail. Motion parameters

mb1

mb2

θ1

Group 1 Group 2

780 g 690 g

1060 g 980 g

17� 15�

θ2 17� 15�

consumption of this part can be achieved by adding up the energy consumption of the attitude adjustment unit and calculating the average value for one profile. A total of 356 1000-m-deep profiles are analyzed and the energy consumption of the attitude adjustment unit Watt for one gliding cycle is calculated as 3.08 kJ. 3.4.3. The measuring unit and the control unit The measuring unit consists of the task sensors integrated on the gliders such as the CTD, dissolved oxygen sensor and other types of sensors. The control unit keeps running when the glider works, and its

Fig. 19. The moving trajectory of Petrel-L.

9

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Ocean Engineering 191 (2019) 106502

-400

-400

-200

-200

0

0

200

200

400

400

600

600

800

800

1000 1200 0

0.5

1

Test depth(m) Test angle(0.1° Simulation depth(m) Simulation angle(0.1° 1.5 2

Test depth(m) Test angle(0.1° Simulation depth(m) Simulation angle(0.1° 1.5 2 2.5

1000

2.5

Time(s)

x 10

(a)

1200 0

3

4

0.5

1

Time(s)

(b)

3

3.5

4

x 10

4

Fig. 20. Gliding trajectories obtained from the sea trail and simulation ((a) is the gliding trajectories and gliding angles of Petrel-L with the motion parameters of group 1 and (b) is the gliding trajectories and gliding angles of Petrel-L with the motion parameters of group 2).

unit are selected as the optimized variables, including the ballast mass mb, the gliding angle γ 1 in the diving phase, and γ 2 in the climbing phase.

Table 3 The comparison of model calculation and trail results. Parameters

Group

Simulation

Sea trail

Error

Gliding time Gliding range Pitching angle Objective function Gliding time Gliding range Pitching angle Objective function

group 1 group 1 group 1 group 1 group 2 group 2 group 2 group 2

234 min 5146 m 17� 0.1204 287 min 5635 m 15� 0.1316

238 min 4930 m 16.8� 0.1149 291 min 5420 m 14.7� 0.126

1.7% 4.2% 1.1% 4.5% 1.4% 3.8% 2% 4.3%

4.1.2. The objective function In this study, the optimization aims to maximize the gliding range and minimize the energy consumption of one gliding profile. Thus, the objective function of the optimization is the ratio of the gliding range to the energy consumption for one profile, which expresses the gliding range of the glider per energy consumption unit. In fact, limited by the depth of diving preparation phase and climbing preparation phase, the Petrel-L glides from the depth of 50 m to the depth of 950 m in the diving phase and from 1000-m deep to 0-m deep in the diving phase actually. Then the time T and the range for one gliding cycle XCGSM can be calculated.

energy consumption includes that of the monitoring sensor. Hence, energy consumption of these parts Pmea and Pcon are determined by the power and the sample frequency of the sensors which can be seen as variables in the energy consumption model.

TCGSM ¼ t0 XCGSM ¼

3.4.4. The communication unit Similar to the attitude adjustment unit, energy consumption of the communication unit Wcom can also be seen as a constant, which is calculated as 1.244 kJ by the energy consumption statistics.

900 X0 1000

þ t1000 0 ðγ2 ; mb2 Þ

1000 ðγ 1 ; mb1 Þ

þ X1000 0 ðγ 2 ; mb2 Þ

(33) (34)

And the objective function FCGSM is: FCGSM ¼

3.4.5. The total energy consumption of one gliding cycle Based on the above discussions, the total energy consumption of one gliding cycle can be computed as EG ¼ Wbuo þ Watt þ Wcom þ ðPmea þ Pcon Þ⋅TG

1000 ðγ 1 ; mb1 Þ

XCGSM EG ðT ¼ TCGSM Þ

(35)

4.1.3. The constraint conditions To take some practical conditions into consideration, some con­ straints need to be satisfied, which are described below.

(32)

4.1.3.1. Side constraints. Each optimized variable needs to have a feasible ron in which the optimum value can be achieved. Taking the reality into account, the scope of the imported motion parameters needs to be wide enough to get a correct optimized result. Hence, the side constraints of the four imported motion parameters can be expressed.

where T is the time of one gliding cycle. 4. Optimal model for gliding range The optimizer is designed to maximize the gliding range and mini­ mize the energy consumption of one gliding profile. The optimization design is mathematically modeled as follows.

g1 : 660 g � mb1 � 1200 g

(36)

g2 : 760 g � mb2 � 1300 g

(37)

4.1. Optimal model with CGSM strategy

g3 : 10o � γ1 � 30o

(38)

4.1.1. The optimized variables In the optimization, the motion parameters imported into the control

g4 : 10o �

(39)

10

γ2 � 30o

M. Yang et al.

Ocean Engineering 191 (2019) 106502

results of the average gliding velocity with different glide angles γ and ballast mass mb2. It can be seen in Fig. 14 that the average gliding ve­ locity increases with the gliding angle and the oil volume pumped into the external bladder. Fig. 15 shows the variation trend of the optimal average gliding velocity in CGSM and HGSM. It can be seen that the gliding velocity increases with the power of the measurement unit and control unit and that the average gliding velocity in the CGSM is higher than that in the HGSM with the same power because the buoyancy loss makes the ballast masses larger in the CGSM. In the CGSM, the optimal gliding velocity reduces to 0.25 m/s and becomes stable when the power of the mea­ surement unit and control unit is lower than 0.4 W. For different observation missions of the underwater glider, the power of the measuring unit is different, which needs to be considered. Thus, in the optimization, the power of the measuring unit and the control unit is referred as a time-dependent variable. Fig. 16 shows the partial result of the objective function with different gliding angles γ and ballast mass mb2, in which the power range of the measuring unit and the control unit is 0.2–3.4 W. From Fig. 16, we can see the variation trend of the optimal point (the pink point in the hook face) and the optimal area (the red portion in the hook face) in CGSM and HGSM. The minimum oil volume and the minimum gliding angle cannot be the optimal point any more with the power increase of the control unit and measuring unit. The optimization results in CGSM and HGSM are described in Fig. 17, from which we can see that γ 1, γ2 and F decrease with the power increase in a nonlinear trend and γ 1 is larger than γ2 in CGSM and HGSM. Moreover, the ballast mass mb1 and mb2 increase with the power increase in a linear trend basically and the difference of mb1 and mb2 becomes larger when the power is larger than 3.5 W in CGSM and 1 W in HGSM. Fig. 18 shows the increasing rate of gliding range if the HGSM is applied to the Petrel-L. It can be seen in Fig. 18 that the HGSM is effective to increase the gliding range when the power is lower than 0.8 W and that the enhancing effect is more obvious when the power is lower. When the power decreases to 0.2 W, the gliding range can in­ crease by 24.13%. The reason is that the energy consumption of the buoyancy regulating unit plays a leading role in the entire energy con­ sumption when the power is less than 0.8 W. Thus, the HGSM has its advantages because the buoyancy loss in the CGSM is inevitable and the ballast masses cannot be smaller, which consume the extra energy. When the power turns larger, the optimized ballast masses become larger to shorten the gliding time and the increasing rate of the ballast mass mb2 in the HGSM is larger than that in the CGSM, as shown in Fig. 17. Moreover, the gliding depth for one profile in the CGSM is larger than that in the HGSM and the depth difference is changeless. Thus, with the increase of the power, the CGSM is more suitable than the HGSM.

4.1.3.2. Other constraints. In addition, for one gliding cycle, the parameter connected with the oil volume needs to meet the following relationship in order to keep a minimum net buoyancy. Jenkins et al. (2003) introduced that the legacy gliders are presently using a variable net buoyancy volume of 100–150 cc. Therefore, the minimum net buoyancy volume of Petrel-L is 1 N. g5 :

mb2 g þ ΔBglider ð1000Þ �

(40)

1 N

4.2. Optimal model with HGSM strategy 4.2.1. The optimized variables Similarly, the motion parameters imported into the control system are selected as the optimized variables, including the ballast mass mb. According to Eq. (16), the angle of attack can be expressed by the gliding angle. Thus, to reduce the difficulty of solving the optimization, the gliding angle γ1 and γ 2 are selected as the optimized variables instead of γ 1 and γ 2. 4.2.2. The objective function Similarly, the gliding time THGSM and gliding range RHGSM of the HGSM can be expressed by Eq. (41) and Eq. (42). THGSM ¼ t331 XHGSM ¼

1000 ðγ 1 ; mb1 Þ

569 X331 669

þ t1000

1000 ðγ 1 ; mb1 Þ

331 ðγ 2 ; mb2 Þ

þ X1000

331 ðγ 2 ; mb2 Þ

(41) (42)

The objective function FHGSM is shown in Eq. (43). It is worth noting that the communication unit does not consume energy in HGSM and the energy consumption of the attitude adjustment unit appears to scale with the gliding depth. FHGSM ¼

XHGSM EG ðT ¼ THGSM Þ

(43)

4.2.3. The constraint conditions 4.2.3.1. Side constraints. The side constraints in the HGSM is similar to those in the CGSM, which is shown as g1 : 290 g � mb1 � 600 g

(44)

g2 : 390 g � mb2 � 700 g

(45)

g3 : 10o � γ 1 � 30o

(46)

g4 : 10o �

(47)

γ2 � 30o

6. Verification of the model

4.2.3.2. The other constraint. The other constraint in the HGSM is same as that in the CGSM. � g5 : mb2 g þ ΔBglider ð1000Þ ΔBglider ð331Þ � 1 N (48)

In order to ensure the validity of the optimization described above, Petrel-L experienced the sea trial from June 30th, 2018 to November 17th, 2018 for performance test in South China Sea, as shown in Fig. 19, and 734 gliding cycles are counted in Fig. 18 including 692 1000-mdeep gliding cycles. The gliding range reaches 3619.6 km and the average range of 1000-m-depth gliding cycles is 4.93 km. The average gliding time for one gliding cycle is 263 min. The power of the measuring unit and the control unit is 0.75 W in the sea trail. Table 2 shows the two groups of motion parameters set in the sea trail to analyze the influence of motion parameters on the gliding range. To avoid the influence of the current on the gliding range, the target heading of the two sequent profile is opposing and the average gliding range is calculated as the gliding range for one cycle. By inputting the parameters in Table 2 into the model described above, the gliding tra­ jectories and pitching angle of the two gliding cycle can be described in Fig. 20 and the average gliding range, gliding time and pitching angle for one gliding cycle can be calculated as shown in Table 3. It can be seen that there is a good consistent between the results of the optimization

4.3. The solution algorithm In this paper, the optimization of motion parameters is a linear constraint optimization issue consisting of some inequality constraints. In this study, the IPFM is applied to solve this problem in consideration of its high precision and high efficiency, which is effective to solve the constrained optimization problem. Procedure of the optimization solu­ tion is shown in Fig. 13. 5. Results and discussions Before optimizing the model, we assume that Eq. (40) and Eq. (48) are the equalities in the optimal model and the angle of attack in the diving phase is equal to that in the climbing phase. Fig. 14 shows the 11

Ocean Engineering 191 (2019) 106502

M. Yang et al.

model and sea trail, which proves the validity of the optimization model.

Eriksen, C.C., Osse, T.J., Light, R.D., Wen, T., Lehman, T.W., Sabin, P.L., Ballard, J.W., Chiodi, A.M., 2001. Seaglider: a long-range autonomous underwater vehicle for oceanographic research. Ieee J. Oceanic Eng. 26 (4), 424. Fu, X., Lei, L., Yang, G., Li, B., 2018. Multi-objective shape optimization of autonomous underwater glider based on fast elitist non-dominated sorting genetic algorithm. Ocean. Eng. 157, 339–349. Graver, J.G., 2005. Underwater gliders: dynamics, control and design. J. Fluids Eng. 127 (3), 523–528. He, Y., Song, B., Dong, H., 2017. Multi-objective optimization design for the multibubble pressure cabin in bwb underwater glider. Int. J. Naval Architect. Ocean Eng. 10 (2018), 439–449. Huang, Z., Liu, Y., Zheng, H., Wang, S., Ma, J., Liu, Y., 2018. A self-searching optimal ADRC for the pitch angle control of an underwater thermal glider in the vertical plane motion. Ocean. Eng. 159, 98–111. Jenkins, S.A., Humphreys, D.E., Sherman, J., Osse, J., Jones, C., Leonard, N., Graver, J., Bachmayer, R., Clem, T., Carroll, P., 2003. Underwater Glider System Study. Scripps Institution of Oceanography. Leonard, N.E., Graver, J.G., 2001. Model-based feedback control of autonomous underwater gliders. Ieee J. Oceanic Eng. 26 (4), 633–645. Li, C., Wang, P., Dong, H., Wang, X., 2018. A simplified shape optimization strategy for blended-wing-body underwater gliders. Struct. Multidiscip. Optim. 58 (5), 2189–2202. Liu, Y., Ma, J., Ma, N., Zhang, G., 2017. Path planning for underwater glider under control constraint. Adv. Mech. Eng. 9 (8). Miller, T.F., 2018. A bio-inspired climb and glide energy utilization strategy for undersea vehicle transit. Ocean. Eng. 149, 78–94. Moritz, H., 1984. Geodetic reference system 1980. Bull. Geod. 3 (58), 388–398. Osse, T.J., Eriksen, C.C., 2007. The deepglider: a full ocean depth glider for oceanographic research. Oceans 2007, 4449125. Rudnick, D.L., 2016. Ocean research enabled by underwater gliders. Annu. Rev. Mar. Sci. 8 (1), 519–541. Rudnick, D.L., Davis, R.E., Eriksen, C.C., Fratantoni, D.M., Perry, M.J., 2004. Underwater gliders for ocean research. Mar. Technol. Soc. J. 38 (2), 73–84. Sang, H., Zhou, Y., Sun, X., Yang, S., 2018. Heading tracking control with an adaptive hybrid control for under actuated underwater glider. Isa T 80, 554–563. Sarkar, M., Nandy, S., Vadali, S.R.K., Roy, S., Shome, S.N., 2016. Modelling and simulation of a robust energy efficient auv controller. Math. Comput. Simulat. 121, 34–47. Sherman, J., Davis, R.E., Owens, W.B., Valdes, J., 2001. The autonomous underwater glider "Spray. Ieee J. Oceanic Eng. 26 (4), 437. Singh, Y., Bhattacharyya, S.K., Idichandy, V.G., 2017. CFD approach to modelling, hydrodynamic analysis and motion characteristics of a laboratory underwater glider with experimental results. J. Ocean Eng. Sci. 2 (2), 90–119. Stommel, H., 1989. The Slocum Mission. Stuntz, A., Kelly, J.S., Smith, R.N., 2016. Enabling Persistent Autonomy for Underwater Gliders with Ocean Model Predictions and Terrain-Based Navigation, vol 3. Tang, T., Li, B., Fu, X., Xi, Y., Yang, G., 2018. Bi-directional evolutionary topology optimization for designing a neutrally buoyant underwater glider. Eng. Optim. 50 (8), 1270–1286. Wang, C., Zhang, F., Schaefer, D., 2015. Dynamic modeling of an autonomous underwater vehicle. J. Mar. Sci. Technol. 20 (2), 199–212. Wang, Z., Yu, J., Zhang, A., Wang, Y., Zhao, W., 2017. Parametric geometric model and hydrodynamic shape optimization of a flying-wing structure underwater glider. China Ocean Eng. 31 (6), 709–715. Webb, D.C., Simonetti, P.J., Jones, C.P., 2001. Slocum: an underwater glider propelled by environmental energy. Ieee J. Oceanic Eng. 26 (4), 447. Xue, D., Wu, Z., Wang, Y., Wang, S., 2018. Coordinate control, motion optimization and sea experiment of a fleet of Petrel-II gliders. Chin. J. Mech. Eng. 31 (1), 17. Yang, Y., Liu, Y., Wang, Y., Zhang, H., Zhang, L., 2017. Dynamic modeling and motion control strategy for deep-sea hybrid-driven underwater gliders considering hull deformation and seawater density variation. Ocean. Eng. 143, 66–78. Yoon, S., Kim, J., 2016. Longitudinal Trajectory Optimization of an Underwater Glider in Finite Depth Water. Yu, J., Zhang, F., Zhang, A., Jin, W., Tian, Y., 2013. Motion parameter optimization and sensor scheduling for the sea-wing underwater glider. Ieee J. Oceanic Eng. 38 (2), 243–254.

7. Conclusions In this paper, we established an optimized mathematical model which includes the steady motion model, the gliding strategy model and the energy consumption model. Then, the motion parameters are opti­ mized to increase the gliding range, considering different powers related to the gliding time and different gliding strates. The two gliding strates, CGSM and HGSM, are analyzed base on the result of the optimization. The experimental study on the Petrel-L validates the effectiveness of the proposed methods and the following conclusions can be drawn. 1) The establishment of steady motion model, hydrodynamic model, gliding range model and energy consumption model is an effective way to obtain the optimal motion parameters of the underwater gliders for the largest gliding range based on the different observa­ tion mission. 2) With the augment of the power related to gliding time, the optimal gliding angles are decreased and the ballast masses are increased for the largest gliding range. 3) The HGSM is effective to increase the gliding range when the power of the control unit and the measuring unit is less than 1.2 W for Petrel-L and the enhancing effect of which is more obvious with the decrease of the power. Therefore, the HGSM can be applied when the target area is far from the current location of the glider. In the future, the optimization of the pressure hull and the hydro­ dynamic shape will be studied to enlarge the gliding range, and the ocean current will be considered in the optimization, which has a certain influence on the gliding range. Acknowledgments This work was jointly supported by National Key R&D Program of China (Grant Nos. 2016YFC0301100) and National Natural Science Foundation of China (Grant Nos. 51721003, 51722508 and 11902219). The authors also would like to express their sincere thanks to L. Ma for her to revise the grammar. References Bachmayer, R., N.E, L.J., 2004. Underwater Gliders: Recent Developments and Future Applications. Claus, B., Bachmayer, R., 2016. Energy optimal depth control for long range underwater vehicles with applications to a hybrid underwater glider. Aut. Robots 40 (7), 1307–1320. Claustre, H., Beguery, L., Patrice, P.L.A., 2014. Sea explorer glider breaks two world records multisensor uuv achieves global milestones for endurance, distance. Sea Technol. 55 (3), 19–22. D Spain, G.L., Jenkins, S.A., Zimmerman, R., Luby, J.C., Thode, A.M., 2005. Underwater acoustic measurements with the liberdade/X-ray flying wing glider. J. Acoust. Soc. Am. 117 (4), 2624.

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