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IEEE_Journal

Hydrodynamic Coefficients and Motion Simulations of

Underwater Glider for Virtual Mooring

Masahiko Nakamura

Research Institute for Applied Mechanics, Kyushu University 6-1 Kasuga-koen, Kasuga, Fukuoka, 816-8580, JAPAN

Kenichi Asakawa, Tadahiro Hyakudome

Japan Agency for Marine-Earth Science and Technology 2-15 Natsushima-cho, Yokosuka, Kanagawa, 237-0061, JAPAN

Satoru Kishima, Hiroki Matsuoka, Takuya Minami

Interdisciplinary Graduate School of Engineering Sciences, Kyushu University 6-1 Kasuga-koen, Kasuga, Fukuoka, 816-8580, JAPAN

Abstract

We are now developing a prototype of a 3000m-class underwater glider for virtual

mooring. The vehicle glides back and forth between the sea surface and the seabed

collecting ocean data at a specific point. Hydrodynamic forces acting on the half-size

model were measured to determine the optimal wing shape. Next, in order to obtain

the dynamical-hydrodynamic coefficients, forced oscillation tests were carried out using

the optimal shaped model. Finally, the motions of the glider were simulated using the

hydrodynamic coefficients obtained from these model experiments. The experimental

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 38, NO. 3, JULY 2013 pp.581-597

Manuscript received September 01, 2011; revised July 26, 2012 and September 28, 2012; accepted December 16, 2012. Date of publication April 22, 2013; date of current version July 10, 2013. Guest Editor: A. Chave. M. Nakamura is with the Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka 816-8580, Japan (e-mail: naka@riam. kyushu-u.ac.jp). K. Asakawa and T. Hyakudome are with the Japan Agency for Marine-Earth Science and Technology, Yokosuka, Kanagawa 237-0061, Japan. S. Kishima, H. Matsuoka, and T.Minami are with the Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan. Digital Object Identifier 10.1109/JOE.2012.2236152

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and calculated results are shown in this paper.

Index Terms Underwater glider, virtual mooring, model experiments, motion

simulation

I. INTRODUCTION

The ocean is well known to strongly influence global climate. Its heat capacity is

a thousand times greater than that of the atmosphere. It absorbs about 30 % of the

emitted carbon dioxide [1]. To understand the nature of global warming, the ocean

environment has been monitored using many means including profiling floats, moored

buoys, ships, and satellites. However, because of its vastness, it is difficult to gather

sufficient data even using all of these methods.

The Argo project is a breakthrough in oceanography. This international project,

to which many countries currently contribute, has about 3000 Argo floats [2] distributed

worldwide. These floats monitor the ocean environment down to 2000 m depth over

four years. Nevertheless, it is difficult to increase their number to cover all oceans

with adequate density because of the vastness of the world’s oceans. They cannot

remain in a designated area where data are needed because they float with seawater.

In addition, the change of seawater temperature has been observed even in waters

deeper than 2000 m where seawater temperatures were previously believed to be stable

[1].

Other methods such as artificial satellites, moored buoys, and research vessels are

used. These methods have their respective limitations. Artificial satellites are suited

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for gathering wide-range data, but they cannot monitor the underwater environment.

Moored buoy systems [3, 4, 5] can carry out long-term monitoring at a fixed point, but

traditional types cannot monitor depths from the seabed to the ocean surface. It is also

difficult to increase their number because of the costs of construction and maintenance.

Research vessels can only provide a limited range of data.

Underwater gliders [6] such as Seaglider [7, 8], Spray [9], and Slocum [10] have

drawn attention and have been used widely. Osse et al. [11] reported the development

of Deepglider, the objective maximum depth of which was 6000 m. Kawaguchi et al.

[12] developed the underwater glider ALBAC in 1995. At present in Japan, Arima

[13], Kato [14] and Yamaguchi [15] have engaged in research related to underwater

gliders. They can travel autonomously over long distances gathering ocean data.

However, their operating duration is shorter than one year as they cannot “sleep” like

Argo floats. They cannot provide long-term data as Argo floats or moored buoys can.

Fig. 1. Concept of virtual mooring by underwater glider

To solve these problems, we propose a virtual mooring system [16, 17] using an

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underwater glider, and in collaboration with Kyushu University, JAMSTEC (the Japan

Agency for Marine-Earth Science and Technology) is now developing a prototype of a

3000m-class vehicle given the name “Tsukuyomi” [18]. The name comes from the

ancient traditional Japanese God of the Moon.

The concept of a virtual mooring using an underwater vehicle is shown in Fig.1.

The vehicle houses various pieces of observation equipment and glides back and forth

between the sea surface and the seabed collecting ocean data at a specific point (virtual

mooring area). When the vehicle returns to the sea surface, the measured data are

transmitted to a research base by an Iridium communications system. The vehicle

then automatically checks its current position by GPS. If the position is outside the

sea area of the virtual mooring because of currents etc., the vehicle is controlled so that

it returns to the correct area during its next dive. Diving and surfacing are repeated

periodically. On the seabed, the vehicle sits and sleeps for a predetermined period and

power other than control equipment is shut off in order to reduce battery consumption.

Since the current speed near the seabed in the deep ocean is very small, a vehicle can

stay on the seafloor by increasing its weight in water with buoyancy control equipment.

Horizontal movement of the vehicle is carried out by gliding; therefore, the vehicle has

no thrusters. The gliding ratio and the course of the vehicle are controlled by moving

the position of the center of gravity, and this position is changed by moving a built-in

weight (battery). Depth is controlled by buoyancy control equipment [19]. The

prototype glider “Tsukuyomi” has been built based on this study, and operation tests are

being carried out. We will introduce the details of the prototype glider in another

paper.

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Table 1 Nomenclature

As a first step, we carried out tank tests using a half-size model to evaluate its

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hydrodynamic characteristics (static model experiments). The definition of the

symbols used in this paper is shown in Table 1. The vehicle shape is like a torpedo

with wings. Hydrodynamic forces acting on the body with various wing forms were

measured to determine the optimal body shape. Performance of the static stability of

the vehicle can be determined from attack and sideslip angle change tests.

Furthermore, the difficulty of fabrication of the body was also taken into consideration,

and the shape of the vehicle was modified.

In the next step, forced oscillation tests (dynamic model experiments) were

conducted for the optimally shaped model in order to obtain the

dynamical-hydrodynamic coefficients such as the added mass coefficients. In fluid

mechanics, an accelerating body moves some volume of surrounding fluid, and

therefore added mass can be modeled as some volume of fluid moving with the object

[20].

A lot of work has been done on underwater vehicle dynamics. A mathematical

model of an ROV by parameter identification using free-motion experimental data [21],

a mathematical model of a torpedo-shaped AUV by system identification using data

collected during a sea-trial mission [22], a plant-model of an underwater glider obtained

by parameter identification using flight data [23], a method to get an AUV model by

neural network [24] and so on [25, 26, 27] were presented. However, there is very

little research in which hydrodynamic forces acting on a model vehicle were measured

in a water tank and the hydrodynamic coefficients in motion equations were obtained

[28, 29, 30].

In the final step, the motions of the glider were simulated using the hydrodynamic

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coefficients obtained from the model experiments. The gliding performance, the

gliding performance in the current, the simulation of braking to make a soft landing on

the seabed and the circling motion depending on placement of the vehicle’s

center-of-gravity are shown. Since it is thought that the influence of the

compressibility of the materials to the buoyancy of the glider is large, that is taken into

consideration in the design of the buoyancy control equipment, but that is disregarded

in the motion simulations. The simulations in which the compressibility of the

materials are considered are future subjects.

The contents from Section 2 to Section 4 of this paper are based on the report

presented at UT’11+SSC’11 [31].

II. STATIC EXPERIMENTS

We had no choice but to adopt the torpedo type body for the following reasons:

(1) the buoyancy control equipment (Fig.2) [19] developed for the 3000m-class Argo

float is used to shorten the period of development of the glider, and the length of the

equipment is not small, (2) a fairing is not attached as much as possible in order to

reduce the weight and cost, (3) motion control by movable wings whose parts

penetrate a pressure vessel is not adopted in order to assure the reliability of prolonged

operation under high pressure.

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Fig. 2. Prototype of buoyancy control equipment

A. Model for Experiments

In the design of an underwater glider, the physical relationship between the

hydrodynamic center and the center of gravity is very important for stable gliding.

Moreover, the longitudinal and lateral distance-of-movement of the center of gravity by

shifting the weight (battery) should be minimal. Therefore, the effect of the main

wing form and vertical wing form on the performance of gliding and turning is very

large. Thus, the hydrodynamic forces acting on the body (Fig.3) with three different

kinds of main wings (Fig.4) and two different kinds of vertical wings (Fig.5) were

measured to evaluate the optimal body shape. The model vehicle shown in Fig.3 has

Main wing A and Vertical tail wing A. The scale of the model is 1/2, and the length L

is 1150 mm, the diameter D of the barrel is 159 mm.

The hydrodynamic forces acting on the body are measured by a 6-component

watertight load cell attached to the model (Fig.6). The moment center of the cell is in

agreement with the center of gravity of the vehicle, and its weight in water is almost

zero. This method of measurement increases accuracy because there is no necessity to

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subtract the force acting on the strut or change the measured moment around the

moment center of the cell to the moment around the center of gravity. The opening is

covered after installation of the load cell.

Fig. 3. Model of glider for virtual mooring

Fig. 4. Main wings

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Fig. 5. Vertical tail wings

Fig. 6. Load cell set in model

B. Experimental Condition

Three tests comprise the experimental conditions: resistance measurement test

(Fig.7), attack angle change test (Fig.8) and side slip angle change test (Fig.9).

The model shown in Fig.7 has Main wing B and Vertical tail wing A; the model shown

in Fig.8 has Main wing C and Vertical tail wing A, and the model shown in Fig.9 has

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Main wing A and Vertical tail wing A. The strut is connected to the towing carriage,

and the model is towed at speed U to measure hydrodynamic forces such as resistance,

lift force, drag force and moment around the center of gravity.

Fig. 7. Resistance measurement test (Main wing B, Vertical tail wing A)

Fig. 8. Attack angle change test (Main wing C, Vertical tail wing A)

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Fig. 9. Side slip angle change test (Main wing A, Vertical tail wing A)

C. Coordinate System

Fig. 10. Coordinate system

The coordinate system used in a motion simulation of the vehicle is shown in

Fig.10. The motion of the vehicle is described by a moving coordinate system, and

the position is expressed by a space -fixed coordinate system. Since the load cell is

housed in the model and its axis is compatible with the moving coordinate system, it is

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possible to obtain the hydrodynamic coefficients in the moving coordinate system

directly.

III. RESULTS OF STATIC EXPERIMENTS

A. Effect of Main Wing Form on Hydrodynamic Characteristics

The effect of main wing form on the hydrodynamic characteristics of the vehicle is

shown in Fig.11 and Fig.12. According to the general technique, the forces Fx’ and Fz’

and the moment My’ are nondimensionalized by 0.5LDU2 and 0.5L2DU2, respectively

[28], where is density of water. In Fig.11, the average values are calculated by using

the data between U = 0.3 m/s and U = 1.0 m/s. In Fig.12, fitting by the least squares

method is carried out using the data between = -12 deg and = 12 deg.

Fig. 11. Result of resistance measurement test (Damping coefficient Fx’ ≡ Xuu’ of surge)

Figure 11 shows the resistance of the vehicle when = 0 deg. The minimum

speed 0.3 m/s and maximum speed 1.0 m/s in the model tests are equivalent to 0.42 m/s

and 1.42 m/s of a full scale vehicle, respectively. The effect of the main wing form on

the resistance is quite small.

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Figure 12 shows the results of attack angle change tests. It is found that the force

acting on the vehicle in x-direction is proportional to the second power of the attack

angle. Under a small angle approximation, the attack angle satisfies = w / U with

222 wvuU and so attack angle is equivalent to non-dimensional velocity

w’ = w / U in the z-direction [28] when the vehicle has forward speed. Therefore, the

coefficient is written by Xww’ instead of X’. The Main wing A that has a

wing-shaped section has a very good performance. According to the increase in the

attack angle , the force to the plus direction of the x-axis increases greatly by the effect

of lift. It is found that the lift force is proportional to the attack angle; the coefficient

is written by Zw’ in the same manner as the induced drag force. Although the wing

area differs, the coefficient of Main wing A and Main wing C is almost the same. The

effect of aspect ratio on the coefficient seems large because the lift

force vvL SUSUC 22 5.0)2/2(5.0)/( [32]. Since the

coefficient of Main wing B is about 1/2 the value of Main wing A, the lift force of Main

wing B is about 1/2 the value of Main wing A. Considering that the wing area is about

1/2, the performance is not too bad although the aspect ratio is small [33]. The

moment around the y-axis is greatly affected by the shape of the main wing, and the

sign of the coefficient of Main wing C differs from that of Main wings A and B. In

Main wings A and B the static stability of pitching motion of the vehicle cannot be

obtained because the coefficient Mw’ has a plus sign in the small attack angle region

(Since the moment around the y-axis also increases when the attack angle is increased

by disturbance, influence of disturbance on pitching motion cannot be prevented.). On

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the other hand, Main wing C can provide static stability because the coefficient has a

minus sign. If Main wing A or B is adopted, the center of gravity of the vehicle should

be moved forward. Main wing C thus seems optimal because its position is difficult to

shift.

Fig. 12. Result of attack angle change test (Hydrodynamic coefficient Xww’ of surge, damping coefficient Zw’ of heave and damping coefficient Mw’ of pitch)

B. Effect of Vertical Tail Wing Form on Hydrodynamic Characteristics

The effect of vertical tail wing form on the hydrodynamic characteristics of the

vehicle is shown in Fig.13. The forces Fx’ and Fy’ and the moment Mz’ are

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nondimensionalized by 0.5LDU2 and 0.5L2DU2, respectively. In the figure, fitting

by the least squares method is carried out using the data between = -12 deg and =

12 deg. The hydrodynamic coefficients are obtained by excluding the stall-region.

Fig. 13. Result of side slip angle change test (Hydrodynamic coefficient Xvv’ of surge, damping coefficient Yv’ of sway and damping coefficient Nv’ of yaw)

In the case of =0 deg, the resistance of Vertical tail wing B is almost the same as

that of Vertical tail wing A. The latter has a wing-shaped section with a good

performance like Main wing A. According to the increase in the side slip angle , the

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force to the plus direction of the x-axis increases. The lift force of Vertical tail wing B

is very small as compared with that of Vertical tail wing A because of its small wing

area. Therefore, the sign of the inclination of the moment Mz’ is plus. Vertical tail

wing B cannot guarantee the static stability of the yawing motion. Although Vertical

tail wing A is considered to be desirable, this large wing may enlarge the radius of

turning. After the prototype glider is built, we want to resume the study in light of the

results of field experiments.

IV. MODIFICATION OF WING SHAPE DEPENDING ON FABRICATION

Based on experiments, it is found that Main wing C and Vertical tail wing A (Fig.4,

Fig.5 and Fig.8) are optimal for our underwater glider. However, a vertical tail wing

attached at the end of the main wing is expected to be much easier to fabricate. If the

main wing and the vertical tail wing are separately attached to the body, the cost of a

prototype vehicle will increase. An end-plate effect is also expected.

The modified vertical tail wing and the vehicle are shown in Fig.14 and Fig.15.

The wing area is 1.67 times that of Vertical tail wing A, and the aspect ratio is the same

as Vertical tail wing A. The wing is attached so that the back end might be even with

that of the main wing. Since the distance between the hydrodynamic center of the

vertical tail wing and the center of gravity of the vehicle is not changed but the wing

area is increased, it makes it easy to assure static stability.

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Fig. 14. Vertical tail wing D

Fig. 15. Modified model (Main wing C, Vertical tail wing D)

Fig. 16. Result of resistance measurement test (Main wing C, Vertical tail wing D) (Damping coefficient Fx’ ≡ Xuu’ of surge)

Figure 16 shows the resistance of the vehicle in the case of = 0 deg. Although

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the resistance is slightly large in the low speed region, the average value in the high

speed region is the same as that of Vertical tail wing A.

Fig. 17. Result of attack angle change test (Main wing C, Vertical tail wing D) (Hydrodynamic coefficient Xww’ of surge, damping coefficient Zw’ of

heave and damping coefficient Mw’ of pitch)

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Fig. 18. Result of side slip angle change test (Main wing C, Vertical tail wing D) (Hydrodynamic coefficient Xvv’ of surge, damping coefficient Yv’ of sway

and damping coefficient Nv’ of yaw)

The results of attack angle change tests are shown in Fig.17. It is found that the

inclination of the lift coefficient is slightly larger than that of the vehicle to which Main

wing C and Vertical tail wing A are attached. The inclination of the moment

coefficient is also large, and the static stability of the pitching motion is improved

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greatly. The end-plate effect thus appears to be great.

Finally, the effect of Vertical tail wing D on the hydrodynamic characteristics of

the vehicle is shown in Fig.18. Although the aspect ratio is the same as Vertical tail

wing A and the wing area is increased 1.67-fold, the lift is not large compared with that

of Vertical tail wing A. Therefore, the performance of the static stability of the yawing

motion is almost the same as that of Vertical tail wing A. However, the performance

in the large side slip angle region is greatly improved.

Considering the overall experimental results, the vehicle to which Main wing C

and Vertical tail wing D (Fig.15) are attached is considered to be optimal.

V. DYNAMIC EXPERIMENTS

After the optimal shape of the vehicle was decided, forced oscillation tests

(dynamic experiments) were carried out to obtain the dynamic-hydrodynamic

coefficients such as added mass coefficients.

Motion equations are necessary to determine the dynamical-hydrodynamic

coefficients by forced oscillation tests and to simulate the motions of the vehicle. The

coordinate system used to describe the motion is shown in Fig.10. Motion equations

of the glider and the analysis method to obtain the hydrodynamic coefficients are

recounted in an appendix. A lot of work has been done on the dynamics of

torpedo-shaped AUVs [22, 25, 26, 34, 35]. However, since the dynamics greatly

differs from an underwater glider, the results about hydrodynamic coefficients are

inapplicable. In the case of “Tsukuyomi”, the lift force of the main wing has dominant

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influence to the motion because of its large wing area compared with the body.

Five kinds of forced oscillation tests were carried out: forced surging test (Fig.19),

forced swaying (Fig.20), forced heaving (Fig.21), forced pure-pitching (Fig.22) and

forced pure-yawing (Fig.23). The strut is connected to the forced oscillation apparatus

put on the towing carriage, and the model is towed at speed U during the forced

oscillation. In the pure-pitching tests, it is necessary to add a translational motion in

the z-direction (heaving motion) which counterbalances the speed caused by the forced

pitching to the pitching motion in order to measure only the pitching moment. If the

translatory motion is not added, pitching moment and heaving force are measured

simultaneously. The same method is used in the pure-yawing tests.

Fig. 19. Dynamic experiment (Forced surge)

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Fig. 20. Dynamic experiment (Forced sway)

Fig. 21. Dynamic experiment (Forced heave)

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Fig. 22. Dynamic experiment (Forced pure-pitch)

Fig. 23. Dynamic experiment (Forced pure-yaw)

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VI. RESULTS OF DYNAMIC EXPERIMENTS

The main results of the forced surging tests, swaying tests, heaving tests,

pure-pitching tests and pure-yawing tests are shown in Fig.24 through Fig.28. The

towing speed U = 0.6m /s was determined from the nominal speed of the full scale

vehicle. Although it was thought that the response of the glider to the actuators and

disturbance was slow, the forced oscillation tests were carried out in the wide frequency

region. The solid lines in the figures represent the mean lines. The average values

are calculated using the data between = 1.5 rad/s and = 4.0 rad/s because the

measured force and moment in the low frequency region are very small and low

accuracy is suggested. In addition, in the case of the added mass and added moment

of inertia, average values are calculated using the results of towing tests.

Fig. 24. Hydrodynamic coefficients due to surge (Added mass coefficient A11’ and damping coefficient Xuu’ of surge caused by surging)

From Fig.24, it is found that the amplitude-effect is not measurable. Moreover, a

frequency-effect is not measurable because the data (marks) are located in a line

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parallel to the horizontal axis. In addition, the velocity-effect on the added mass is

very small. Xuu’ obtained from the static experiments indicated by a double circle is

well in agreement with the result obtained from the dynamic experiments. Figure 24

has suggested that constant hydrodynamic coefficients of surge can be used in the

motion equations.

Fig. 25. Hydrodynamic coefficients due to sway (Added mass coefficient A22’ and damping coefficient Yv’, Yvv’ of sway and damping coefficient Nv’ of yaw

caused by swaying)

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From Fig.25, it is found that the amplitude-effect and frequency-effect on the

added mass are not measurable and the velocity-effect is very small. When the

forward speed is not zero, the amplitude-effect on Yv’ (damping coefficient of sway

resulting from a lift force acting on the vertical tail wing and body) is not measurable,

Yvv’ (damping coefficient of sway resulting from viscosity) may be ignored because the

mean line is parallel to a horizontal axis and Yv’ obtained from the static experiments

indicated by a double circle is in agreement with the result obtained from the dynamic

experiments. When the forward speed is zero, the damping force in the y-direction is

proportional to the second power of yawing speed and the amplitude-effect and

frequency-effect on Yvv’ are minimal. When the forward speed is not zero, the

amplitude-effect and frequency-effect on Nv’ (damping coefficient of yaw resulting

from a lift force acting on the vertical tail wing and body) are also minimal and the

value obtained from the static experiments is in agreement with that obtained from the

dynamic experiments. Figure 25 has suggested that constant hydrodynamic

coefficients of sway and yaw can be used in the motion equations.

It is found from Fig.26 that the amplitude-effect and frequency-effect on the added

mass are negligible and the velocity-effect is slightly measurable. When the forward

speed is not zero, the amplitude-effect on Zw’ (damping coefficient of heave resulting

from a lift force acting on the main wing and body) is not measurable, Zww’ (damping

coefficient of heave resulting from viscosity) may be ignored and Zw’ obtained from the

static experiments is well in agreement with the result obtained from the dynamic

experiments. When the forward speed is zero, the damping force in the z-direction is

proportional to the second power of heaving speed and the amplitude-effect and

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frequency-effect on Zww’ are small. When the forward speed is not zero, the

amplitude-effect and frequency-effect on Mw’ (damping coefficient of pitch resulting

from a lift force acting on the main wing and body) are minimal and the value obtained

from the static experiments is well in agreement with that obtained from the dynamic

experiments. Figure 26 has suggested that constant hydrodynamic coefficients of

heave and pitch can be used in the motion equations.

Fig. 26. Hydrodynamic coefficients due to heave (Added mass coefficient A33’ and damping coefficient Zw’, Zww’ of heave and damping coefficient Mw’ of

pitch caused by heaving)

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Fig. 27. Hydrodynamic coefficients due to pitch (Added moment of inertia coefficient A55’ and damping coefficient Mq’, Mqq’ of pitch and damping coefficient Zq’

of heave caused by pitching)

From Fig.27, it is found that the amplitude-effect, frequency-effect and

velocity-effect on the added moment of inertia are not measurable. When the forward

speed is not zero, the amplitude-effect on Mq’ (damping coefficient of pitch resulting

from a lift force acting on the main wing) is not measurable, and Mqq’ (damping

coefficient of pitch resulting from viscosity) may be ignored. When the forward speed

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is zero, the damping moment around the y-axis is proportional to the second power of

pitching angular velocity and the amplitude-effect and frequency-effect on Mqq’ are not

measurable. When the forward speed is not zero, the amplitude-effect and

frequency-effect on Zq’ (damping coefficient of heave resulting from a lift force acting

on the main wing) are not measurable. Figure 27 has suggested that constant

hydrodynamic coefficients of pitch and heave can be used in the motion equations.

Fig. 28. Hydrodynamic coefficients due to yaw (Added moment of inertia coefficient A66’ and damping coefficient Nr’, Nrr’ of yaw and damping coefficient Yr’ of sway caused by yawing)

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It is found from Fig.28, that the amplitude-effect, frequency-effect and

velocity-effect on the added moment of inertia are not measurable. When the forward

speed is not zero, the amplitude-effect on Nr’ (damping coefficient of yaw resulting

from a lift force acting on the vertical tail wing) is negligible, and Nrr’ (damping

coefficient of yaw resulting from viscosity) may be ignored. When the forward speed

is zero, the damping moment around the z-axis is proportional to the second power of

yaw angular velocity and the amplitude-effect and frequency-effect on Nrr’ are

negligible. When the forward speed is not zero, the amplitude-effect and

frequency-effect on Yr’ (damping coefficient of sway resulting from a lift force acting

on the vertical tail wing) are not measurable. Figure 28 has suggested that constant

hydrodynamic coefficients of yaw and sway can be used in the motion equations.

Table 2 Hydrodynamic coefficients of glider

The values of the hydrodynamic coefficients are collected in Table 2. A forced

rolling test could not be performed because we had no forced rolling apparatus,

therefore, at this stage, the values of the coefficients for rolling Kp ’( = -0.05) are

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estimated values obtained using commercial CFD (Computational Fluid Dynamics)

software.

From the results of these experiments, it seems that the damping force and moment

in the low speed region differ from those in the high speed region. This should greatly

influence transient-motion simulation in which the vehicle begins diving or surfacing.

The damping forces calculated for different forward speed and motion speed (For

example, Fy = Yv ’ · (1/2 U L D) · va, or Fy = Yvv ’ · (1/2 L D) · |va| va .) are shown in

Fig. 29 through Fig.32, where, the forces and moments (Yv va, Zw wa, Mq q, Nr r) are

larger than the forces and moments (Yvv |va| va, Zww |wa| wa, Mqq |q| q, Nrr |r| r) in the low

speed region. The motion simulation after submerging begins is shown in Fig.33.

The center of gravity is moved ahead 2mm and buoyancy is reduced by decreasing the

volume by 0.0005 m3. The dynamics of the actuators are approximated to be first

order systems with time constants of 2 sec and 10 sec, respectively. The solid line

shows the simulated result in which the values of damping coefficients are changed

according to the velocity (Yvv |va| va, Zww |wa| wa, Mqq |q| q, Nrr |r| r (low speed region) >>>

Yv va, Zw wa, Mq q, Nr r (high speed region)). This curve is very smooth because the

damping forces and moments are changing continuously (see Fig.29 through Fig.32).

The dotted line shows the results using constant values of damping coefficients (Yv va,

Zw wa, Mq q, Nr r). If the values of the damping coefficients are not changed

according to the speed, the transient motion is estimated to be large.

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Fig. 29. Damping force due to sway

Fig. 30. Damping force due to heave

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Fig. 31. Damping moment due to pitch

Fig. 32. Damping moment due to yaw

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Fig. 33. Transient motion of glider

VII. RESULTS OF MOTION SIMULATION

Motion simulations were carried out using the hydrodynamic coefficients shown in

Table2. The principal dimensions of the full scale glider are shown in Table 3. The

vehicle has neutral buoyancy when the buoyancy control equipment is in the start

condition.

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Table 3 Principal dimension of glider

Fig. 34. Gliding motion of glider

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Fig. 35. Gliding performance of glider

Figure 34 shows the time series of the gliding motion of the vehicle. The center

of gravity is moved ahead 2mm and the volume is reduced by 0.0005 m3. The vehicle

is gliding stably at a speed of 0.51 m/s, and the gliding ratio is about 1.5. Figure 35

shows the gliding performance. The speed u, trim angle and gliding ratio in the

steady state and the time required until the vehicle reaches the maximum depth 3000 m

are obtained from the motion simulations carried out changing the positions of the

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center-of-gravity xG and the amount of volume adjustments . The maximum value

of xG was estimated from the weight and stroke of the built-in weight. From this

figure, it is found that the limit value of the gliding ratio is about 4 when the time to

reach a depth of 3000 m and the speed required for stable gliding are taken into

consideration because the vehicle must ordinarily submerge and surface once per day

for data transmission by the Iridium communications system.

Fig. 36. Gliding motion of glider when brake is operated

Figure 36 shows the gliding motion when the vehicle applies the brake for a soft

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landing on the seabed. The center of gravity is moved to the neutral position

(Submerging velocity of the vehicle is decreased by increasing the trim angle) at a

depth of 2677 m and the volume is increased (submerging velocity is decreased by

increasing the buoyancy) by 0.001 m3. The rate of increase of the volume is 0.00001

m3 per minute based on the capability of the buoyancy control equipment. The gliding

performance of the glider when the brake is operating is shown in Fig.37. It must be

recognized that the depth at which the brake must be applied is not related to the

amount of displacement of the center of gravity.

Fig. 37. Gliding performance of glider when brake is operated

40

Figure 38 shows the gliding performance in a current; the vehicle can glide against

a 0.36 m/s current. The vehicle can move forward 3 km by 1 cycle diving against the

current, and the excursion takes about 2.3 hours. In contrast, the vehicle can fly 24 km

within 30 hours in still water (see Fig.35). The excursion time in the current is shorter

because of the faster diving speed. Faster speed is required in order to fly against the

current.

Fig. 38. Gliding performance in current

The motion simulation when the vehicle repeats submerging and surfacing is

shown in Fig. 39 through Fig.41. When the vehicle reaches a depth of 2000 m, the

center of gravity is returned to the neutral position, and buoyancy volume is increased

simultaneously; the rate of increase of the volume is 0.00001 m3 per minute. Then,

when the trim angle becomes zero, the center-of-gravity is moved back. When the

vehicle reaches a depth of 1000 m, the center of gravity is again returned to the neutral

position, and volume is reduced at the same time; the rate of volume reduction is 0.0002

m3 per second. Then, when the trim angle becomes zero, the center-of-gravity is

moved ahead. The vehicle can continue moving forward by repeating the above

41

operations. As shown, the overshoot is small when the volume adjustment is small.

Fig. 39. Gliding motion of glider while cruising (a) (xG : ±2mm, : ±0.001m3)

42

Fig. 40. Gliding motion of glider while cruising (b) (xG : ±1mm, : ±0.001m3)

43

Fig. 41. Gliding motion of glider while cruising (c) (xG : ±2mm, : ±0.0005m3)

44

Fig. 42. Circling motion of glider

Finally, simulation of circling motion is shown in Fig.42. At first, the center of

gravity is moved ahead 2mm and the volume is reduced by 0.0005 m3. Thereafter, at t

= 1000 sec, the center of gravity is moved 5 mm in the starboard direction. The figure

shows that the vehicle performs a left turn when the center of gravity is moved in this

direction. This results from the left turn moment induced by the component of the

underwater-weight in the x-direction caused by the trim being larger than the right turn

moment induced by the component of the underwater-weight in the y-direction caused

45

by the heel. It is also found that the altitude is greatly reduced by this turnaround. In

a virtual mooring in a shallow sea region, it seems that a nondirectional disk type body

[17] will be advantageous because the vehicle can glide in any direction direct.

VIII. CONCLUSIONS

Tank tests were carried out using half-size models of an underwater glider intended

for use as a virtual mooring to measure the hydrodynamic coefficients and to get an

optimal body shape. The vehicle that has Main wing C and Vertical tail wing D shown

in Fig.15 was optimal to guarantee static stability of pitching and yawing motion.

Afterward, forced oscillation tests were carried out to obtain the

dynamic-hydrodynamic coefficients such as added mass coefficients. The

experimental results have suggested that the constant added mass coefficients for surge,

sway and heave and constant added moment of inertia coefficients for pitch and yaw

can be used in the motion equations (see Fig.24 through Fig.28). However, although a

frequency-effect on the damping coefficients was negligible (see Fig.24 through Fig.28),

it was found that the values of the coefficients have to be changed according to the

vehicle-velocity (see Fig.29 through Fig.32) to achieve the high-precision results of

motion simulations (see Fig.33).

The following results were drawn from the motion simulations using the obtained

hydrodynamic coefficients.

1. A limit value of the gliding ratio of “Tsukuyomi” is about 4 taking into

consideration the time to reach a depth of 3000 m and the speed required for stable

46

gliding, because the vehicle has to submerge and surface once per day for data

transmission using the Iridium communications system (see Fig.35).

2. The depth at which braking must be applied is not related to the amount of

displacement of the center of gravity (see the top figure of Fig.37).

3. The vehicle can glide against a 0.36m/sec current (see Fig.38).

4. The vehicle can continue moving forward by repeatedly submerging and surfacing

(see Fig.39 through Fig.41).

5. The vehicle performs a left turn when the center of gravity is moved in the starboard

direction (see Fig.42).

The prototype vehicle “Tsukuyomi” has been almost completed on August 2012,

and the field experiments are continued. These early results confirm its fundamental

functions [36].

APPENDIX

A. Motion Equations of Glider

The coordinate system used to describe the motion is shown in Fig.10. The

origin of the moving coordinate system is the center-of-gravity of the vehicle where the

built-in weight is located in the initial position. Equation 2 shows the relationship

between space-fixed coordinate X, Y, Z, Eulerian angle , , , velocity u, v, w, and

angular velocity p, q, r.

47

r

q

p

w

v

u

Z

Y

X

T

000

000

000

000

000

000

2

1

E

E

(1)

coscoscossinsinsincossinsincossincos

cossincoscossinsinsinsincoscossinsin

sinsincoscoscos

1E

(2)

seccossecsin0

sincos0

tancostansin1

2E (3)

where E1T represents the linear velocity transformation and E2 is the angular velocity

transformation [28, 37].

Motions of the glider are expressed as follows based on the research of a towed

vehicle [28, 37] carried out at Kyushu University. Fossen et al. also provide similar

equations [38, 39].

z

y

x

z

y

x

zzGG

yyGG

xxGG

GG

GG

GG

M

M

M

F

F

F

r

q

p

w

v

u

AIAxmym

AIAxmzm

AIymzm

AxmymAm

AxmzmAm

ymzmAm

6662

5553

44

3533

2622

11

000

000

000

000

000

000

(4)

48

where,

222

11

226

2352233

sin)(

)()(

)(

)()()(

awwavvuuu

cc

GGG

Gaax

wXvXuX

θgρm

rvqwAm

rpzmqpymrAxm

qAxmrvAm+qwAmF

(5)

aavvrav

cca

GGGay

vvYrYvY

θgρm

pwruAmruAm

rpymrqzmqpAxmpwAmF

cossin)(

)()()(

)()()(

2211

223533

(6)

aawwqaw

ccaG

GGaz

wwZqZwZ

gm

qupvAmquAmrqym

rpAxmqpzmpvAmF

coscos)(

)()()(

)()()(

3311

2622

22

(7)

ppKpK

gzzm

gyym

pwruzmqupvym

qupvymwvAA

rwAAqvAA

rqAIAIpwruzmM

ppp

BG

BG

ccGccG

aaGaa

aa

yyzzaaGx

cossin)(

coscos)(

)()(

)()(

)()(

)()()(

2233

53263562

5566

(8)

qqMqMwM

gxxm

gzzm

rvqwzmqupvAxm

rpAIAI

quAxmpvAxmrvqwzmM

qqqaw

BG

BG

ccGccG

zzxx

aGaGaaGy

coscos)(

sin)(

)()()(

)()()(

53

6644

3562

(9)

49

rrNrNwN

gyym

gxxm

rvqwympwruAxm

qpAIAIrvqwym

pwAxmruAxmM

rrrav

BG

BG

ccGccG

xxyyaaG

aGaGz

sin)(

cossin)(

)()()(

)(

)()(

62

4455

5326

(10)

ca

ca

ca

vvv

vvv

uuu

(11)

When a current velocity is given in the space-fixed coordinate system, the current

velocity component in the moving coordinate is calculated by the following equation:

Z

Y

X

c

c

c

V

V

V

w

v

u

E (12)

B. Analysis Method

The method of analyzing the forced heaving test is shown as an example of the

method used for all five types of forced oscillation tests.

When the forced heaving ( tzz a sin , see Fig.21 ) is carried out, the velocity

and acceleration are

tzw a cos (13)

tzw a sin2 (14)

Therefore, from the motion equations (Eq.4 ~ Eq.10), the hydrodynamic forces acting

on the vehicle are inferred from the following equations:

50

2wXF wwx (15)

wwZwZwAmF wwwz )( 33 (16)

wMgxxmwAxmM wBGGy )()( 53 (17)

By substituting Eq.13 and 14 for Eq.15 ~ Eq.17, the hydrodynamic forces acting on the

vehicle are written as follows:

tzXF awwx 222 cos (18)

tzZ

tzZtzAm

ttzZ

tzZtzAmF

aww

awa

aww

awaz

cos3

8

cossin)(

coscos

cossin)(

22

233

22

233

(19)

tzMgxxmtzAxmM awBGaGy cos)(sin)( 253 (20)

On the other hand, since Fx, Fz, and My are measured by the load cell, the following

equations are obtained by a Fourier series expansion:

tFtFF FxxaFxxax cossinsincos (21)

tFtFF FzzaFzzaz cossinsincos (22)

tMtMM MyyaMyyay cossinsincos (23)

The following equations are obtained by comparing Eq.18 - Eq.20 with Eq.21 - Eq.23.

Fzzaa FzAm cos)( 233 (24)

Fzzaawwaw FzZzZ

sin3

8 22 (25)

MyyaaG MZAxm cos)( 253 (26)

51

Myyaaw MzM sin (27)

Therefore,

mZ

FA

a

Fzza 233

cos

(28)

a

Fzzaawww z

FzZZ

sin

3

8 (29)

Ga

yMya xmz

MA

253

cos

(30)

a

Myyaw z

MM

sin (31)

Since Eq.29 is a linear equation for , Zww can be found from the inclination and Zw can

be found from the y-intercept. Since the damping effect of a lift force does not work

when speed U is zero, Zw is zero. The hydrodynamic coefficients are

nondimensionalized as follows:

DLAA 233

'33

2

1/ (32)

DLUZZ ww 2

1/' (33)

DLZZ wwww 2

1/' (34)

DLAA 353

'53

2

1/ (35)

DLUMM ww2'

2

1/ (36)

The other forced oscillation tests can be analyzed using the same technique.

52

ACKNOWLEDGMENTS

The authors are deeply indebted to Mr. Yuzuru Ito (Ocean Engineering Research,

Inc.) and Dr. Junichi Kojima (KDDI Laboratories) for their kind help during the study.

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