**2.1.3 Mechanism of the water-jet propulsion system**

Fig.4 is the structure of one single water-jet propeller. It is composed of one water-jet thruster and two servo motors (above and side). The water-jet thruster is sealed inside a plastic box for waterproof. And we use waterproof glue on servo motors for waterproof. The thruster can be

Fig. 6. Electrical Schematics for Prototype System

**2.3 Power supply**

power system.

(a) S3C44B0X Board (b) ATmega2560 Board (c) Pressure Sensor (d) Gyro Sensor

Development of a Vectored Water-Jet-Based Spherical Underwater Vehicle 7

We adopt two power supply for the spherical underwater vehicle. The highest power consumption components in our vehicle are propellers. For each of them, the thruster has a working voltage of 7.2*V* and 3.5*A* current drain, servo motors can work under 5*V* with relatively small current. Therefore, we use two 2-cells LiPo batteries as the power supply for the propellers. The capacity of each battery is 5000*mAh* with parameter of 50*c* − 7.4*V*. Besides, we use 4 AA rechargeable batteries for the control boards. We carried out the power consumption test for one LiPo battery, and Fig.8 gives the battery discharge graph of the

Fig. 8. Power Consumption of the Whole System. Blue line – one propeller working; green

In this section, we will discuss about the working principles, modeling method and the identification experiment for the water-jet propeller. Many literatures have presented the computing formula for the torque and thrust exerted by a thruster. Most of them are base on

line – two propellers working; red line – three propellers working

**3. Principles and modeling of the propulsion system**

Fig. 7. Electrical Components for the Experimental Prototype Underwater Vehicle

rotated by these two servo motors, therefore, the direction of jetted water can be changed in *X*-*Y* plane and *X*-*Z* plane, respectively.

Fig. 4. Structure of a Water-jet Propeller

Three of the water-jet propellers are mounted on the metal support frame, as shown in Fig.5. Three of them are circumferentially 2*π*/3 apart from each other.

Fig. 5. Water-jet Propellers mounted on Support Frame

## **2.2 Electrical system design**

We adopt a minimal hardware configuration for the experimental prototype vehicle. For a single spherical underwater vehicle, there are three major electrical groups, sensor group, control group and actuator group. Fig.6 gives the electrical schematics. At present, we only use one pressure sensor for depth control and one gyro sensor for surge control. One ARM7 based control board is used as central control, data acquisition, algorithm implement and making strategic decisions. One AVR based board is used as the coprocessor unit for motor control. It receives the commands from ARM and translates the commands into driving signals for the water-jet propellers.

Fig.7 gives the main hardware for this vehicle. Fig.7(a) is the ARM7 based board with S3C44B0X on it, which can fulfill our requirement at present. Fig.7(b) is the AVR based board with ATmega2560 on it. RS232 bus is used for the communication between ARM7 and AVR. In Fig.7(c) is the set of pressure sensor with the sensor body(right) and its coder (left). It use RS422 bus for data transmission. Digital gyro sensor CRS10 is shown in Fig.7(d), we use the build in AD converter of S3C44B0X for data acquisition.

Fig. 7. Electrical Components for the Experimental Prototype Underwater Vehicle

#### **2.3 Power supply**

4 Will-be-set-by-IN-TECH

rotated by these two servo motors, therefore, the direction of jetted water can be changed in

(a) Design (b) Prototype

Three of the water-jet propellers are mounted on the metal support frame, as shown in Fig.5.

(a) Design (b) Prototype

We adopt a minimal hardware configuration for the experimental prototype vehicle. For a single spherical underwater vehicle, there are three major electrical groups, sensor group, control group and actuator group. Fig.6 gives the electrical schematics. At present, we only use one pressure sensor for depth control and one gyro sensor for surge control. One ARM7 based control board is used as central control, data acquisition, algorithm implement and making strategic decisions. One AVR based board is used as the coprocessor unit for motor control. It receives the commands from ARM and translates the commands into driving

Fig.7 gives the main hardware for this vehicle. Fig.7(a) is the ARM7 based board with S3C44B0X on it, which can fulfill our requirement at present. Fig.7(b) is the AVR based board with ATmega2560 on it. RS232 bus is used for the communication between ARM7 and AVR. In Fig.7(c) is the set of pressure sensor with the sensor body(right) and its coder (left). It use RS422 bus for data transmission. Digital gyro sensor CRS10 is shown in Fig.7(d), we use the

*X*-*Y* plane and *X*-*Z* plane, respectively.

Fig. 4. Structure of a Water-jet Propeller

Three of them are circumferentially 2*π*/3 apart from each other.

Fig. 5. Water-jet Propellers mounted on Support Frame

build in AD converter of S3C44B0X for data acquisition.

**2.2 Electrical system design**

signals for the water-jet propellers.

We adopt two power supply for the spherical underwater vehicle. The highest power consumption components in our vehicle are propellers. For each of them, the thruster has a working voltage of 7.2*V* and 3.5*A* current drain, servo motors can work under 5*V* with relatively small current. Therefore, we use two 2-cells LiPo batteries as the power supply for the propellers. The capacity of each battery is 5000*mAh* with parameter of 50*c* − 7.4*V*. Besides, we use 4 AA rechargeable batteries for the control boards. We carried out the power consumption test for one LiPo battery, and Fig.8 gives the battery discharge graph of the power system.

Fig. 8. Power Consumption of the Whole System. Blue line – one propeller working; green line – two propellers working; red line – three propellers working

#### **3. Principles and modeling of the propulsion system**

In this section, we will discuss about the working principles, modeling method and the identification experiment for the water-jet propeller. Many literatures have presented the computing formula for the torque and thrust exerted by a thruster. Most of them are base on

So, a general transform matrix can be obtained:

*<sup>p</sup>*1, **<sup>Φ</sup>***<sup>b</sup>*

*<sup>p</sup>*2, **<sup>Φ</sup>***<sup>b</sup>*

*<sup>p</sup>* = (**Φ***<sup>b</sup>*

demonstration of surge, heave and yaw.

Fig. 10. Orientation of Propellers

in vehicle-fixed coordinate as:

Fig. 11. Propulsion Forces for Surge, Heave and Yaw

⎧ ⎪⎪⎨

*pFxb* <sup>=</sup> **<sup>Φ</sup>***b<sup>T</sup>*

*pFyb* = 0 *pFzb* = 0 *p*1 3 ∑ *i*=1

⎪⎪⎩

coordinate, **Φ***<sup>b</sup>*

is a constant vector.

*<sup>p</sup>*P*<sup>b</sup>* = **Φ***<sup>b</sup>*

where *<sup>p</sup>*P*<sup>b</sup>* is the position vector of propeller-fixed coordinate expressed in vehicle-fixed

Development of a Vectored Water-Jet-Based Spherical Underwater Vehicle 9

to vehicle-fixed coordinate, *<sup>p</sup>*P*<sup>p</sup>* is the position vector in propeller-fixed coordinate and the C

Now, let us take a look at three motions, surge, heave and yaw. The definition of these three motions can be found in (Fossen, 1995). Before that, we define two angles which will be used for orientation of propellers. Fig.10 gives the definition of *α* and *β*. Fig.11 gives a

(a) Rotation in X-Y Plane (b) Rotation in X-Z Plane

(a) Surge (b) Heave (c) Yaw

The first case is surge. In this case, two of the water-jet propellers will work together, and the other one could be used for brake. So, from Fig.11(a), two water-jet propellers in the left will be used for propulsion, and if we want to stop the vehicle from moving, the third propeller can act as a braking propeller. From Equation 4, the resultant force for surge can be expressed

(*p***F***ip* <sup>+</sup> **<sup>e</sup>**1*Ci*) �<sup>=</sup> <sup>0</sup>

(5)

*<sup>p</sup>* · *<sup>p</sup>*P*<sup>p</sup>* <sup>+</sup> <sup>C</sup> (4)

*<sup>p</sup>*3)*<sup>T</sup>* is the transform matrix from propeller-fixed coordinate

the lift theory, and mainly focus on blades type propellers (Newman, 1977), (Fossen, 1995) and (Blanke et al., 2000). Our propellers are different with blades type propellers, therefore, we try to find another method for the modeling of water-jet propellers. In (Kim & Chung, 2006), the author presented a dynamic modeling method in which the flow velocity and incoming angle are taken into account. We will use this modeling method for our water-jet propellers.

#### **3.1 Working principles**

Before modeling of propulsion system, we want to give some basic working principles about the water-jet propellers. Fig.9(a) shows the top view of distribution of three propellers. They can work together to realize different motion, such as surge and yaw.

#### Fig. 9. Distribution and Coordination of Multiple Propellers

If we let *θ* be the interval angle of each water-jet propeller, as shown in Fig.9(b), then, for the purpose of kinematics transform, three propeller-fixed coordinates are introduced for propellers, which are fixed in the rotation center of the propellers. So we can see, these three propeller-fixed coordinates are actually transform results of vehicle-fixed coordinate reference frame. Meanwhile, it should be noted that, this transform only happens in *X*-*Y* plane. Let the matrix form of the coordinates transform be given as:

$$
\begin{pmatrix} X\_1 \\ Y\_1 \\ Z\_1 \end{pmatrix} = \begin{pmatrix} X \\ Y \\ Z \end{pmatrix} + \begin{pmatrix} -R \\ 0 \\ 0 \end{pmatrix} \tag{1}
$$

$$
\begin{pmatrix} X\_2 \\ Y\_2 \\ Z\_2 \end{pmatrix} = \begin{pmatrix} c\theta & s\theta & 0 \\ -s\theta & c\theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} X \\ Y \\ Z \end{pmatrix} + \begin{pmatrix} \frac{1}{2}Rc\theta - Rs\theta c\frac{\pi}{6} \\ -\frac{1}{2}Rc\theta - Rc\theta c\frac{\pi}{6} \\ 0 \end{pmatrix} \tag{2}
$$

$$
\begin{pmatrix} X\_3 \\ Y\_3 \\ Z\_3 \end{pmatrix} = \begin{pmatrix} c2\theta & s2\theta & 0 \\ -s2\theta & c2\theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} X \\ Y \\ Z \end{pmatrix} + \begin{pmatrix} \frac{1}{2}Rc2\theta + Rs2\theta c\frac{\pi}{6} \\ -\frac{1}{2}Rc2\theta + Rc2\theta c\frac{\pi}{6} \\ 0 \end{pmatrix} \tag{3}
$$

where *R* is the radius of the vehicle, *s*(·) ≡ sin(·) and *c*(·) ≡ cos(·).

So, a general transform matrix can be obtained:

6 Will-be-set-by-IN-TECH

the lift theory, and mainly focus on blades type propellers (Newman, 1977), (Fossen, 1995) and (Blanke et al., 2000). Our propellers are different with blades type propellers, therefore, we try to find another method for the modeling of water-jet propellers. In (Kim & Chung, 2006), the author presented a dynamic modeling method in which the flow velocity and incoming angle are taken into account. We will use this modeling method for our water-jet propellers.

Before modeling of propulsion system, we want to give some basic working principles about the water-jet propellers. Fig.9(a) shows the top view of distribution of three propellers. They

(a) Propeller Distribution (b) propeller-fixed Coordinates

If we let *θ* be the interval angle of each water-jet propeller, as shown in Fig.9(b), then, for the purpose of kinematics transform, three propeller-fixed coordinates are introduced for propellers, which are fixed in the rotation center of the propellers. So we can see, these three propeller-fixed coordinates are actually transform results of vehicle-fixed coordinate reference frame. Meanwhile, it should be noted that, this transform only happens in *X*-*Y* plane. Let the

can work together to realize different motion, such as surge and yaw.

Fig. 9. Distribution and Coordination of Multiple Propellers

matrix form of the coordinates transform be given as:

⎛ ⎝

*X*1 *Y*1 *Z*1

*cθ sθ* 0 −*sθ cθ* 0 0 01

*c*2*θ s*2*θ* 0 −*s*2*θ c*2*θ* 0 0 01

where *R* is the radius of the vehicle, *s*(·) ≡ sin(·) and *c*(·) ≡ cos(·).

⎞ ⎠ = ⎛ ⎝

> ⎞ ⎠

> ⎞ ⎠

⎛ ⎝

*X Y Z* ⎞ ⎠ +

⎛ ⎝

*X Y Z* ⎞ ⎠ +

*X Y Z* ⎞ ⎠ + ⎛ ⎝

−*R* 0 0

⎛

⎜⎜⎜⎝

⎛

⎜⎜⎜⎝

⎞

1 2

−1 2

> 1 2

−1 2

*Rcθ* − *Rsθc*

*Rcθ* − *Rcθc*

*Rc*2*θ* + *Rs*2*θc*

*Rc*2*θ* + *Rc*2*θc*

0

0

⎠ (1)

⎞

⎟⎟⎟⎠

⎞

⎟⎟⎟⎠

(2)

(3)

*π* 6

*π* 6

> *π* 6

*π* 6

**3.1 Working principles**

⎛ ⎝

⎛ ⎝

*X*3 *Y*3 *Z*3

⎞ ⎠ = ⎛ ⎝

*X*2 *Y*2 *Z*2 ⎞ ⎠ = ⎛ ⎝

$${}^{p}P\_{b} = \Phi\_{p}^{b} \cdot {}^{p}P\_{p} + \mathbf{C} \tag{4}$$

where *<sup>p</sup>*P*<sup>b</sup>* is the position vector of propeller-fixed coordinate expressed in vehicle-fixed coordinate, **Φ***<sup>b</sup> <sup>p</sup>* = (**Φ***<sup>b</sup> <sup>p</sup>*1, **<sup>Φ</sup>***<sup>b</sup> <sup>p</sup>*2, **<sup>Φ</sup>***<sup>b</sup> <sup>p</sup>*3)*<sup>T</sup>* is the transform matrix from propeller-fixed coordinate to vehicle-fixed coordinate, *<sup>p</sup>*P*<sup>p</sup>* is the position vector in propeller-fixed coordinate and the C is a constant vector.

Now, let us take a look at three motions, surge, heave and yaw. The definition of these three motions can be found in (Fossen, 1995). Before that, we define two angles which will be used for orientation of propellers. Fig.10 gives the definition of *α* and *β*. Fig.11 gives a demonstration of surge, heave and yaw.

Fig. 10. Orientation of Propellers

Fig. 11. Propulsion Forces for Surge, Heave and Yaw

The first case is surge. In this case, two of the water-jet propellers will work together, and the other one could be used for brake. So, from Fig.11(a), two water-jet propellers in the left will be used for propulsion, and if we want to stop the vehicle from moving, the third propeller can act as a braking propeller. From Equation 4, the resultant force for surge can be expressed in vehicle-fixed coordinate as:

$$\begin{cases} \ \ ^p F\_{xb} = \mathbf{0} \mathbf{e}\_{p1}^T \sum\_{i=1}^3 (\,^p \mathbf{F}\_{ip} + \mathbf{e}\_1 \mathbf{C}\_i) \neq \mathbf{0} \\\ \ ^p F\_{yb} = \mathbf{0} \\\ ^p F\_{zb} = \mathbf{0} \end{cases} \tag{5}$$

*D* is diameter of the nozzle *Vo* is velocity of outlet flow

and the central flow velocity,

Therefore, we can also get:

By rewriting Equation 12, we get:

where

*J*<sup>0</sup> is the advance ratio.

*γ* is incoming angle of ambient flow

Because the diameter of the nozzle is small, the velocity difference in the nozzle can be ignored, so we consider the axis flow velocity *Va* as a linear combine of incoming flow velocity

Development of a Vectored Water-Jet-Based Spherical Underwater Vehicle 11

By assuming that the flow is incompressible, therefore, from equation of continuity, we know

*ρaVaAa* = *ρoVo Ao* (9)

*Va* = *Vo* (10)

*<sup>a</sup>* (11)

<sup>2</sup>*D*2Ω2) (12)

<sup>2</sup>) (13)

<sup>2</sup>) (14)

*<sup>D</sup>*<sup>Ω</sup> (15)

(8)

*Va* = *k*1*Vi* + *k*2*Vc*

*Vc* <sup>=</sup> <sup>1</sup> 2 *D*Ω

that the volume of incoming flow must equal to the outlet flow, then we get:

where, *ρ<sup>a</sup>* = *ρ<sup>o</sup>* is density of flow, *Aa* = *Ao* is cross-section of the nozzle.

Meanwhile, we know that, the propulsive force of the water-jet thruster is:

<sup>1</sup>*Vi*

<sup>4</sup> (*k*<sup>2</sup> <sup>1</sup>( *Vi <sup>D</sup>*<sup>Ω</sup> )

<sup>4</sup> (*k*<sup>2</sup> <sup>1</sup>( *Vi*

*<sup>J</sup>*<sup>0</sup> <sup>=</sup> *Vi*

By substituting Equation 8 in Equation 11, we can get:

*Ft* <sup>=</sup> *<sup>π</sup>*

*Ft <sup>ρ</sup>D*4Ω<sup>2</sup> <sup>=</sup> *<sup>π</sup>*

Then, we can let the non-dimensional parameter be:

these parameters and find out their relationship.

*KT*(*J*0) = *<sup>π</sup>*

<sup>4</sup> *<sup>ρ</sup>D*2(*k*<sup>2</sup>

*Ft* = *ρAV*<sup>2</sup>

<sup>2</sup> + 2*k*1*k*2*ViD*Ω + *k*<sup>2</sup>

<sup>2</sup> + 2*k*1*k*<sup>2</sup>

*<sup>D</sup>*<sup>Ω</sup> )<sup>2</sup> <sup>+</sup> <sup>2</sup>*k*1*k*<sup>2</sup>

*<sup>D</sup>*<sup>Ω</sup> <sup>=</sup> *Vf cos<sup>γ</sup>*

Now, the modeling becomes measuring of three parameters, flow velocity, incoming angle and angular velocity of thruster. For this purpose, we designed an experiment to measure

*Vi <sup>D</sup>*<sup>Ω</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup>

*Vi <sup>D</sup>*<sup>Ω</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup>

*Vi* = *Vf cosγ*

where, **e**<sup>1</sup> = (1, 0, 0)*T*.

Then, for the heave case, all the three water-jet propellers will work and the side servo motor will rotate to an angle that *β* > *π*/2. Therefore, in this case, the resultant force for heave can be expressed in vehicle-fixed coordinate as:

$$\begin{cases} \ ^pF\_{xb} = 0\\ \ ^pF\_{yb} = 0\\ \ ^pF\_{zb} = \Phi\_{p3}^{b\top} \sum\_{i=1}^3 (\ ^p\mathbf{F}\_{ip} + \mathbf{e}\_3\mathbf{C}\_i) \neq 0 \end{cases} \tag{6}$$

where, **e**<sup>3</sup> = (0, 0, 1)*T*.

The third case is yaw which is rotating on z-axis. By denoting in propeller-fixed coordinates, *α* should have the same orientation, clockwise or counterclockwise, that means, *α<sup>i</sup>* > 0 or *α<sup>i</sup>* < 0. So in yaw, rotation moment will take effect. We can write the equation for yaw in vehicle-fixed coordinate as:

$$\begin{cases} \ ^pF\_{xb} = \mathbf{o} \mathbf{b}\_{p1}^T \sum\_{i=1}^3 (\,^p\mathbf{F}\_{ip} + \mathbf{e}\_1 \mathbf{C}\_i) \neq 0\\ \ ^pF\_{yb} = \mathbf{o} \mathbf{b}\_{p2}^T \sum\_{i=1}^3 (\,^p\mathbf{F}\_{ip} + \mathbf{e}\_2 \mathbf{C}\_i) \neq 0\\ \ ^pF\_{zb} = 0\\ \ ^pM\_{xb} + ^pM\_{yb} + ^pM\_{zb} \neq 0 \end{cases} \tag{7}$$

where, **e**<sup>2</sup> = (0, 1, 0)*T*, **e**<sup>3</sup> = (0, 0, 1)*T*.

#### **3.2 Modeling of single water-jet propeller**

In the author's previous research (Guo et al., 2009), the modeling for orientation of water-jet propeller is presented. Therefore, in this part, we will only discuss about the hydrodynamics modeling of the water-jet thruster. The method we refer to is presented in (Kim & Chung, 2006). For the purpose of dynamic modeling of water-jet propeller, we give the flow model of the water-jet thruster, which is shown in Fig.12. The shaft is perpendicular to the nozzle, and there are two blades.

Fig. 12. Flow Model of the Water-jet Thruster (top view)

where,


*D* is diameter of the nozzle

8 Will-be-set-by-IN-TECH

Then, for the heave case, all the three water-jet propellers will work and the side servo motor will rotate to an angle that *β* > *π*/2. Therefore, in this case, the resultant force for heave can

The third case is yaw which is rotating on z-axis. By denoting in propeller-fixed coordinates, *α* should have the same orientation, clockwise or counterclockwise, that means, *α<sup>i</sup>* > 0 or *α<sup>i</sup>* < 0. So in yaw, rotation moment will take effect. We can write the equation for yaw in

(*p***F***ip* <sup>+</sup> **<sup>e</sup>**3*Ci*) <sup>0</sup>

(*p***F***ip* <sup>+</sup> **<sup>e</sup>**1*Ci*) <sup>0</sup>

(*p***F***ip* <sup>+</sup> **<sup>e</sup>**2*Ci*) <sup>0</sup>

(6)

(7)

where, **e**<sup>1</sup> = (1, 0, 0)*T*.

where, **e**<sup>3</sup> = (0, 0, 1)*T*.

there are two blades.

where,

vehicle-fixed coordinate as:

where, **e**<sup>2</sup> = (0, 1, 0)*T*, **e**<sup>3</sup> = (0, 0, 1)*T*.

**3.2 Modeling of single water-jet propeller**

be expressed in vehicle-fixed coordinate as:

⎧ ⎪⎪⎨ *pFxb* = 0 *pFyb* = 0 *pFzb* = **<sup>Φ</sup>***b<sup>T</sup>*

*pFxb* <sup>=</sup> **<sup>Φ</sup>***b<sup>T</sup>*

*pFyb* <sup>=</sup> **<sup>Φ</sup>***b<sup>T</sup>*

*pFzb* = 0

*p*3 3 ∑ *i*=1

*p*1 3 ∑ *i*=1

*p*2 3 ∑ *i*=1

*pMxb* <sup>+</sup> *pMyb* <sup>+</sup> *pMzb* <sup>0</sup>

In the author's previous research (Guo et al., 2009), the modeling for orientation of water-jet propeller is presented. Therefore, in this part, we will only discuss about the hydrodynamics modeling of the water-jet thruster. The method we refer to is presented in (Kim & Chung, 2006). For the purpose of dynamic modeling of water-jet propeller, we give the flow model of the water-jet thruster, which is shown in Fig.12. The shaft is perpendicular to the nozzle, and

⎪⎪⎩

⎧ ⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

Fig. 12. Flow Model of the Water-jet Thruster (top view)

Ω is angular velocity of the thruster

*Vc* is central flow velocity in the nozzle

*Vi* is velocity of incoming flow


Because the diameter of the nozzle is small, the velocity difference in the nozzle can be ignored, so we consider the axis flow velocity *Va* as a linear combine of incoming flow velocity and the central flow velocity,

$$\begin{array}{l}V\_a = k\_1 V\_i + k\_2 V\_c\\V\_c = \frac{1}{2}D\Omega\\V\_i = V\_f \cos \gamma\end{array} \tag{8}$$

By assuming that the flow is incompressible, therefore, from equation of continuity, we know that the volume of incoming flow must equal to the outlet flow, then we get:

$$
\rho\_a V\_a A\_a = \rho\_o V\_o A\_o \tag{9}
$$

where, *ρ<sup>a</sup>* = *ρ<sup>o</sup>* is density of flow, *Aa* = *Ao* is cross-section of the nozzle. Therefore, we can also get:

$$V\_a = V\_o \tag{10}$$

Meanwhile, we know that, the propulsive force of the water-jet thruster is:

$$F\_l = \rho A V\_a^2 \tag{11}$$

By substituting Equation 8 in Equation 11, we can get:

$$F\_t = \frac{\pi}{4} \rho D^2 (k\_1^2 V\_l^2 + 2k\_1 k\_2 V\_l D \Omega + k\_2^2 D^2 \Omega^2) \tag{12}$$

By rewriting Equation 12, we get:

$$\frac{F\_t}{\rho D^4 \Omega^2} = \frac{\pi}{4} (k\_1^2 (\frac{V\_i}{D\Omega})^2 + 2k\_1 k\_2 \frac{V\_i}{D\Omega} + k\_2^2) \tag{13}$$

Then, we can let the non-dimensional parameter be:

$$K\_T(f\_0) = \frac{\pi}{4} (k\_1^2 (\frac{V\_i}{D\Omega})^2 + 2k\_1 k\_2 \frac{V\_i}{D\Omega} + k\_2^2) \tag{14}$$

where

$$J\_0 = \frac{V\_i}{D\Omega} = \frac{V\_f \cos \gamma}{D\Omega} \tag{15}$$

*J*<sup>0</sup> is the advance ratio.

Now, the modeling becomes measuring of three parameters, flow velocity, incoming angle and angular velocity of thruster. For this purpose, we designed an experiment to measure these parameters and find out their relationship.

3.3.2.1 Equivalent cross-section variation of propellers

cross-section II. So we try to find an equation to describe this variation.

Fig. 15. Variation of Equivalent Cross-section of Propellers

on the equivalent cross-section.

Fig. 16. Variation of Equivalent Cross-section

As a vectored water-jet-based propulsion system, it should be noted that both the propulsive force and its direction can be changed. Therefore, when the propeller changes its direction, actually, the incoming angle of flow is also changing, and the equivalent cross-section of the propeller is changing. From Equation 11 we know the propulsive force will change if cross-section *A* changes. Fig.15 gives a demonstration of this case. When the propeller rotate from position I to II, the equivalent cross-section will change from cross-section I to

Development of a Vectored Water-Jet-Based Spherical Underwater Vehicle 13

Considering that the measured force from stain gage is actually a resultant force of propulsive force and fluid force. And we also know that the fluid force acted on the propeller depends

So the first experiment is measurement of the equivalent cross-section variation. The propeller is submerged in the flow which has a speed of 0.2*m*/*s*, propeller is powered off. And we only change its orientation in *X* − *Y* plane. Because of experiment limits, we can not change the flow direction, so in the experiment, the incoming angle equals to the orientation angle of the propeller. Fig.15 gives a demonstration of the equivalent cross-section. We give some special angles, 0, *π*/6, *π*/3, 2*π*/3, 5*π*/6, *π*, for this experiment. Fig.16 gives the experiment data of equivalent cross-section. You may notice that, we did not adopt the orientation angle of *π*/2. Because, when the propeller rotate to *π*/2, which means that the measure surface of the strain

gage is parallel to the flow direction, the strain gage can not measure the flow force.


Table 1. Experiment Condition

### **3.3 Experiments for the dynamics modeling**

In this part, we try to identify the dynamics model of the water-jet propeller by experiment. What we are interested in is the relation of flow incoming angles, flow velocities and propulsive forces. Experiment condition is listed in Table 1.
