### 1. Introduction

### 2. Mathematical Model of the Towfish

**is (**

*ν**u*,

*v*,

*w*,

*p*,

*q*,

*r*)

*∈*

^{T}*R*

^{6}, and (

*u*,

*v*,

*w*) and (

*p*,

*q*,

*r*) represent the velocity and angular velocity, respectively, along the three axes of the coordinate system attached to the towfish.

*M*=

*M*+

_{RB}*M*∈

_{A}*R*

^{6×6}, where

*M*represents the rigid-body inertial matrix,

_{RB}*M*represents the added mass matrix from the hydrodynamic force, and

_{A}*C*(

**)=**

*ν**C*(

_{RB}**)+**

*ν**C*(

_{A}**) . Here,**

*ν**C*(

_{RB}**) represents the rigid-body Coriolis and centripetal force matrix.**

*ν**C*(

_{A}**) represents the Coriolis and centripetal force matrix of the hydrodynamic force. In addition,**

*ν**D*(

**)=**

*ν**D*+

_{l}*D*(

_{n}**)∈**

*ν**R*

^{6×6}, where

*D*represents the linear damping matrix of the hydrodynamic force, and

_{l}*D*(

_{n}**) represents a nonlinear damping matrix.**

*ν*

*f***,**

_{c}

*f***∈**

_{e}*R*

^{3}, and

*f***∈**

_{b}*R*

^{6}are the towing force from the cable, the elevator force, and the restoring force generated by the weight and buoyant force, respectively.

*r***and**

_{c}

*r***∈**

_{e}*R*

^{3}are the position vectors from the center of gravity to the towing point and the elevator center, respectively, and × represents the vector product. If

*S*(

**) is a skew-symmetric matrix, the relational expression**

*a***×**

*a***=**

*b**S*(

**)**

*a***,**

*b***,**

*a***∈**

*b**R*

^{3}, is satisfied if

**=(**

*a**a*,

_{x}*a*,

_{y}*a*)

_{z}*,*

^{T}*S*(

**) is given by**

*a**τ*,

*τ*is given by

##### (3)

*x*,

_{g}*y*,

_{g}*z*) and (

_{g}*x*,

_{b}*y*,

_{b}*z*) represent the center of gravity and center of buoyancy, respectively;

_{b}*c*and

*s*represent the cosine and sine, respectively. In addition,

*ϕ*and

*θ*represent the roll and pitch angle of the towfish, respectively, which are obtained as follows. where

**=(**

*η*_{2}*ϕ*,

*θ*,

*ψ*)

*∈*

^{T}*R*

^{3},

**=(**

*ν*_{2}*p*,

*q*,

*r*)

*∈*

^{T}*R*

^{3}, and

*R*

_{i}_{,}

*is a rotation matrix that represents rotation by an angle*

_{j}*j*about the

*i*axis.

### 3. Positions of the Towing Point and Center of Gravity

*xz*plane.

*f*

_{c}

_{(}_{xz}**=(**

_{)}*f*,

_{cx}*f*)

_{cz}*and*

^{T}

*f*_{e}

_{(}_{xz}**=(**

_{)}*f*,

_{ex}*f*)

_{ez}*are the towing forces and elevator forces in the*

^{T}*xz*plane, respectively.

*r*_{c}**=(**

_{(}_{xz}_{)}*r*,

_{cx}*r*)

_{cz}*and*

^{T}

*r*_{e}**= (**

_{(}_{xz}_{)}*r*, 0)

_{ex}*are the position vectors from the center of gravity to the towing point and to the elevator center, respectively. Using these variables, the pitching moment of the towfish generated by the towing force, restoring force, and elevator force in the*

^{T}*xz*plane can be calculated as

*f*

_{b}_{5}is the fifth component of

**in Eq. (3). In addition, ⊗ is the vector product in the plane, and satisfies interaction formulas such as**

*f*_{b}**⊗**

*a***=(**

*b**E*

**)**

*a*

^{T}**=(**

*b**E*

^{T}

*a**)*

^{T}**,**

*b***,**

*a***∈**

*b**R*

^{2}. Here

*E*is a rotation matrix that represents rotation by 90° in the counterclockwise direction in the plane and is given by

### 3.1 Cases Where Only the Towing Point Changes [Cases (a) and (b)]

*f*

_{b}_{5}, can be ignored because the moments caused by the weight and buoyant force are not generated.

*r*, which is in equilibrium with the maximum and minimum elevator forces in Eq. (5), is calculated using the Eq. (7) for cases (a) and (b). The maximum and minimum elevator forces,

_{cx}*f*

_{ez}_{max}and

*f*

_{ez}_{min}are defined as being generated at the maximum angles in the positive and negative directions. The positions of

*r*, which can controlled by the elevator force,

_{cx}*f*, according to Eq. (7), are shown in Fig. 4.

_{ez}*r*=

_{cx}*r*

_{cz}*f*/

_{cx}*f*in the area corresponding to case (a) becomes the optimal towing point for maintaining the attitude of the towfish without an additional elevator force because

_{cz}*r*is constant, even if

_{cz}f_{cx}*r*changes. The attitude becomes more difficult to control as the towing point,

_{cx}*r*, moves farther away from center of gravity because the elevator force required for attitude control increases.

_{cx}### 3.2 Cases in which the Center of Gravity and Towing point Change [Cases (c)–(h)]

*r*and

_{cx}*x*that can be controlled by the elevator force,

_{g}*f*are examined as follows.

_{ez}The maximum and minimum towing points (

*r*_{cx}_{max},*r*_{cx}_{min}) are calculated in the case where the center of gravity and center of buoyancy are at the same position;The maximum and minimum positions (

*x*_{g}_{max},*x*_{g}_{min}) of the center of gravity are calculated when the obtained maximum and minimum towing points and the center of gravity are on the same line of the*z*-axis;The maximum and minimum towing points (

*r*_{cxg}_{max},*r*_{cxg}_{min}) at the maximum and minimum centers of gravity obtained in procedure (2) are calculated.

*f*

_{b}_{5}, is −(

*z*

_{g}*W*−

*z*

_{b}*B*)

*sθ*−(

*x*

_{g}*W*−

*x*

_{b}*B*)

*cθcϕ*and can be simplified to −

*x*

_{g}*W*when the center of buoyancy is fixed at the origin and the towfish is stabilized (

*θ*,

*ϕ*≈0). The maximum and minimum towing points are obtained equally in Eq. (7) if procedure (1) is performed using Eq. (5) and the maximum and minimum positions

*x*

_{g}_{max},

*x*

_{g}_{min}are calculated as follows if procedure (2) is performed: The maximum and minimum towing points

*r*

_{cxg}_{max},

*r*

_{cxg}_{min}arecalculated as follows if procedure (3) is performed:

### 4. Simulations

*ρ*,

*s*,

*u*, and

*α*represent the density of water, the area of a single elevator, the velocity of the towfish, and the angle of attack, respectively.

*C*and

_{D}*C*are the drag and lift coefficients, respectively; their values were obtained according to the airfoil model of the National Advisory Committee for Aeronautics (NACA). Because the angle of attack is the same as the elevator angle,

_{L}*f*and

_{ex}*f*are expressed as functions of the elevator angle (Park et al., 2016). If

_{ez}*δ*and

_{l}*δ*are the angles of the left and right elevators, respectively, the pitch angle of the towfish can be controlled by following the value of the synchronous elevator angle,

_{r}*δ*= (

_{S}*δ*+

_{r}*δ*)/2. The simulations were performed using the following equations of motion obtained from Eqs. (1) and (10).

_{l}*K*,

_{p}*K*, and

_{d}*K*are the control gains in a proportional–integral–derivative [PID] controller and represent the proportional, derivative, and integral gains of the elevator angle, respectively.

_{i}*θ*and

_{d}*θ*represent the target pitch angle (reference input) and current pitch angle, respectively. The values 0.065 and 0.11 used in Eq. (11) are the slopes of the drag and lift coefficients plotted for various elevator angles, respectively, and were linearized by referring to the values for the NACA 0018 model.

*L*and

*D*, respectively.

*I*is the moment of inertia of the towfish with respect to the

_{y}*y*-axis. The other symbols represent hydrodynamic force derivatives (Fossen, 1994).

*X*,

_{u}*Z*, and

_{w}*M*were calculated using the towfish data, and

_{q}*X*,

_{u}*Z*,

_{w}*M*,

_{q}*X*

_{|}

_{u}_{|}

*,*

_{u}*Z*

_{|}

_{w}_{|}

*, and*

_{w}*M*

_{|}

_{q}_{|}

*were selected on the basis of the simulations.*

_{q}### 4.1 Simulation Conditions

*W*′ =

*W*−

*B*denotes the underwater mass. The control gains of the attitude-maintaining controller,

*K*,

_{p}*K*, and

_{d}*K*, were set to 3, 5, and 0.01, respectively, and the left and right elevator angles were maintained within ±30° to ensure the stability of the towfish.

_{i}### 4.2 Simulations of Cases (a) and (b)

*r*

_{cx}_{max}≈ 0.14 m and

*r*

_{cx}_{min}≈ −0.12 m are obtained from the conditions given in Table 2 and Eq. (5). The simulations were performed by selecting difficult-to-control cases for

*r*

_{cx}_{max}and

*r*

_{cx}_{min}and easily controlled cases with

*r*= 0.02 m and

_{cx}*r*= −0.02 m, as shown in Fig. 4.

_{cx}*r*

_{cx}_{max}and

*r*

_{cx}_{min}, respectively; they show that the target pitch angle (0°) of the towfish is reached very slowly, even at the maximum elevator angle (±30°). In these cases, even if the attitude of the towfish is controlled, it is difficult to address additional disturbances because there is little margin in the elevator angle. By contrast, the portions of the dotted line indicated by ● and ▴ are relatively easy to control, and the target pitch angle (0°) can be reached quickly. Moreover, the system can react immediately even when disturbances are applied because there is considerable margin in the elevator angle.

### 4.3 Simulations of Cases (e) and (f) and Cases (c) and (h)

*x*

_{g}_{max}≈ 0.1 m and

*x*

_{g}_{min}≈ −0.12, respectively. The maximum and minimum towing positions at this time are calculated as

*r*

_{cxg}_{min}≈ 0.27 m and

*r*

_{cxg}_{max}≈ −0.26 m. The simulations were performed by selecting cases (e) and (f), in which control is difficult, and cases (c) and (h), in which control is relatively easy. The values of

*r*and

_{cx}*x*for each case are listed in Table 3. Fig. 7 is a simulation result showing the pitch angle and synchronous elevator angle of the towfish when a pitch angle of 0° is applied as the reference input for the four positions shown in Table 3. The areas corresponding to cases (e) and (f) (positions ○ and ▵) have little margin in the elevator angle; therefore, the target pitch angle 0° is reached very slowly. In these cases, it is difficult to immediately address an additional disturbance. However, at the positions (▴ and ●) corresponding to cases (c) and (h), the target pitch angle is obtained quickly because there is considerable margin in the elevator angle. In these cases, it is possible to respond adequately to additional disturbances.

_{g}### 4.4 If Disturbance is Applied in Cases (c) and (f)

*r*and

_{cx}*x*were set to the values in Table 3, and disturbances were applied. Fig. 8 is a simulation result showing the pitch angle and synchronous elevator angle of the towfish when a pitching moment of 50

_{g}*NM*is applied as a disturbance at positions ● and ▵ among the areas corresponding to cases (c) and (f). The reference input was selected as a pitch angle of 0°, as in the previous simulations. The position indicated by ● has considerable margin in the elevator angle; therefore, even if the disturbance is applied, the target pitch angle 0° is reached quickly. However, the position indicated by ▵ has little margin in the elevator angle; thus, the target pitch angle 0° cannot be maintained even if an extreme elevator angle of −30° is continuously applied to both elevators. Theses results show that it is important to avoid positions at which there is little margin in the elevator angle. Moreover, if the towing point and center of gravity are located at positions where the elevator angle has considerable margin, the system can respond quickly to additional disturbances.