# Control Allocation Design for Torpedo-Like Underwater Vehicles with Multiple Actuators

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and the Desired Control Law for Torpedo-Like Underwater Vehicles

#### 2.1. Problem Formulation

#### 2.1.1. Dynamics of Underwater Vehicle and Robust Control Law

**η**is the position and Euler angle vector of the underwater vehicle with respect to the earth frame, the vector

**v**stands for the linear and angular velocity vector of the underwater vehicle with respect to the body-fixed frame, and the transition matrix between

**η**= [x, y, z, φ, θ, ψ]

^{T}and

**v**= [u, υ, w, p, q, r]

^{T}as listed in Table 1 is

**J**(

**η**). The matrix

**M**represents the mass and inertia matrix of the underwater vehicle,

**C**(

**v**) is the Coriolis and centripetal matrix,

**D(v)**is the hydrodynamic damping matrix, and

**g**(

**η**) is the gravitational and buoyancy vector.

**τ**(

**e**)∈

**R**

^{6}is the desired H

_{∞}control law applied to control the torpedo-like underwater vehicle to track a desired trajectory, and

**τ**(

**e**) is also the desired target that the control allocation seeks to achieve.

**τ**

_{d}is the ocean environmental disturbances.

_{∞}control law

**τ**(

**e**), which is derived using the robust control concept, is chosen as below for the control allocation design of the torpedo-like underwater vehicle in Figure 1.

**e**is the trajectory tracking error vector between the torpedo-like underwater vehicle and the desired trajectory ${\eta}_{d}$ and defined as the following:

**τ**

**in Equation (2) is designed to eliminate the external disturbance, i.e., ocean current. As to the fuzzy logic term**

_{e}**ζ**(

**e**)

**Θ**

**, it is developed to deal with the perturbed uncertainties of the torpedo-like underwater vehicle.**

_{f}**λ**

**,**

_{1}**λ**

**,**

_{2}**Γ**, and

**Z**are designable positive parameters or positive definite vectors, and

**ζ**(

**e**) is the fuzzy architecture matrix.

#### 2.1.2. Control Allocation Design

_{∞}control law $\mathbf{\tau}(\mathbf{e})=\mathbf{J}(\mathbf{\eta}{)}^{-1}({\tau}_{e}+\mathbf{\zeta}(\mathbf{e}){\Theta}_{f})$ to drive the torpedo-like underwater vehicle to precisely track desired trajectory ${\eta}_{d}$.

#### 2.2. Actuator Models

**τ**(

**e**) and the output command vector

**f**of the installed actuators, including turning angles of fins, turning angles of rudders, and thrust of the thruster of the controlled underwater vehicle, could be presented as follows.

**T**represents the thrust configuration matrix of the actuators, which indicates how the overall actuator control commands perform on the underwater vehicle and can be expressed as

_{ypb}, d

_{ysb}, …, and d

_{yT}are the distances from the positions of each single installed actuator to the center of gravity of the torpedo-like underwater vehicle.

## 3. Control Allocation Methods for Torpedo-Like Underwater Vehicles

**τ**(

**e**) and output commands

**f**of the installed actuators in Equation (6), an analytically optimal searching method is developed to find out the optimal output command vector

**f**which can precisely convert the output commands of the installed actuators into the desired control command

^{*}**τ**(

**e**), the method is shown in the following.

#### Least Squares Optimization Method

**τ**(

**e**) to the installed actuators of the torpedo-like underwater vehicles and meanwhile minimizes the overall energy consumption of all the actuators. The mathematical representation of the control allocation problem of the torpedo-like underwater vehicles could be stated as follows:

**P**∈

**R**

^{9}

^{×}

^{9}is the weighting matrix of power consumption which is positive definite. This kind of problem is called least squares optimization, and it can be solved by using the Lagrange multiplier.

**λ**is a vector of Lagrange multipliers, which is introduced to convert the constrained problem into an unconstrained problem. Differentiating Equation (10) with respect to

**f**and

**λ**, it yields

**λ**can be derived from Equation (14):

**f**could be solved as follows:

**f**

^{*}, which is the output vector of the installed actuators, can be analytically found based on Equation (16). For practical applications, the transformation between outputs and inputs of the installed actuators should be further discussed.

**Remark**

**1:**

**I**∈

**R**

^{9}

^{×}

^{9}because the efficiency of all installed actuators is equal practically. However, for specific control purposes, which intend to emphasize stern control or bow control of torpedo-like underwater vehicles, adjustments of P are required. The guideline for selecting P is suggested as follows:

**K**∈

**R**

^{9}

^{×}

^{9}is the force coefficient matrix of the installed actuators and

**u**∈

**R**

^{9×1}is the actuator input command vector. The force coefficient matrix

**K**and the actuator input command vector

**u**will be detailed below.

**K**depends on configuration and types of installed actuators. The main thruster, which is non-rotatable, is usually mounted on the aft of the controlled underwater vehicle to produce propulsive forces along the x axis and pushes the torpedo-like underwater vehicles forward in this study; hence, the relationship between the thruster’s input and output can be described by ${F}_{T}={k}_{T}{u}_{T}$. Furthermore, rudders and fins are the key apparatus to steer the torpedo-like underwater vehicles. Rudders provide forces in the y axis to control sway and roll of the torpedo-like underwater vehicles; meanwhile, fins produce forces in the z axis to manipulate heave and roll of the underwater vehicles. Forces generated by rudders and fins are related to the traveling speed of the torpedo-like underwater vehicles, and the relationships between generated forces, traveling velocity, and turning angle of rudders and fins are as follows:

_{f}is the flow velocity relative to fins or rudders, and in the underwater vehicle case, it usually means the traveling speed of the vehicle, or linear velocity along the x-direction. It is clear that the fins and rudders need flow velocity or traveling speed to generate forces. C

_{L}and C

_{D}stand for the lift coefficient and drag coefficient, which could be measured through experiments, and δ is the attack angle of fins and rudders. The schematic diagrams of the fin and rudder operations are shown in Figure 3 and Figure 4 as follows.

**K**can be expressed as below:

## 4. Implementation for a Torpedo-Like Underwater Vehicle

#### 4.1. The Specifications of the Underwater Vehicle

_{x}, moment of inertia I

_{y}, and moment of inertia I

_{z}, listed in Table 2, during the trajectory tracking period.

#### 4.2. The Configuration Matrix of the Torpedo-Like Underwater Vehicle

#### 4.3. The Actuator Model of the Underwater Vehicle

**K**could be obtained by gathering input control commands and measuring generated control forces through simulations and practical tests.

#### 4.3.1. The Hydrodynamics Model of Fins and Rudders

^{2}, and the lift coefficient C

_{L}and drag coefficient C

_{D}are related to the shape of the fin and rudder. The fins and rudders of the discussed torpedo-like underwater vehicle was designed based on the fin shape NACA−0012 type, which had a chord length of 8 cm. The lift force and the drag force of these adopted fins and rudders could be predicted precisely using a software program called Xfoil. The Xfoil software program calculation needs to input the shape of the fin and rudder along with the Reynolds number, and the Reynolds number is determined by the kinematic viscosity of the fluid, velocity of the fluid, and the chord length.

_{f}is the fluid velocity, c is chord length, and υ is the kinematic viscosity. The desired traveling speed of the underwater vehicle was between 1 and 3 m/s and the kinematic viscosity of the water was 9.7937 × 10

^{−7}m

^{2}/s.

**K**in Equation (21) can be all set up as 0.1090.

#### 4.3.2. The Actuator Model of the Thruster

_{T}could be roughly approximated based on the relation in Figure 9:

#### 4.4. Control Allocation Verification for a Torpedo-Like Underwater Vehicle

- Step 1.
- Solve the control allocation of the fins and rudders.
- Step 2.
- Calculate the angle of fins and rudders.
- Step 3.
- Calculate the drag forces of the fins and rudders by using Equation (26).
- Step 4.
- Calculate the thruster force by the desired control law and compensation of drag forces.
- Step 5.
- Calculate the PWM signal of the thruster by using the radial basis function network.

_{∞}control law in Equation (2) and a sliding mode control [19] were adopted for guiding the developed torpedo-like underwater vehicle, and here one trajectory tracking case was selected to test the proposed control allocation method.

#### Scenario 1

**λ**

_{f1,}**λ**

_{f2,}**k**, and

_{sc}**k**are listed in Table 6.

_{s}_{∞}control law in Equation (2) and a sliding mode control, respectively. The tracking histories are illustrated in the following picture. The red dashed line is the desired trajectory, the green line is the tracking result of the nonlinear H

_{∞}control law, and the blue line is the tracking result of the sliding mode control law. From Figure 11, the controlled torpedo-like underwater vehicle precisely tracks the desired trajectory with a steady state velocity 2 m/s.

_{∞}control law outperforms the sliding mode control in position tracking ability. In addition, attitude tracking errors with respect to pitch and yaw directions of the controlled torpedo-like underwater vehicle converged quickly and bounded within ±0.5° in a steady state for two control laws, even as it was directly affected by induced forces and torques of time-varying ocean currents, and the sliding mode control law has better tracking performances in pitch and yaw directions.

_{∞}control law. It is easy to find out that the sliding mode control law possesses a quick convergence rate in the transient period.

_{∞}control law are illustrated as Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20. These figures reveal the facts that control forces and torques generated by the sliding mode control law to guide the torpedo-like underwater vehicle are larger than those of the nonlinear H

_{∞}control law during the trajectory tracking period.

**u**. For achieving this design target, the proposed control allocation method was adopted to make the transformation of

**τ**(

**e**), which was generated via two adopted control laws and

**u**for the controlled torpedo-like underwater vehicle. Based on the design procedures of the proposed optimization method, because efficiency of all installed actuators are assumed be equal, the weighting matrix

**P**for the optimization method in Equation (22) is selected as

**τ**(

**e**) could be carried out by all installed actuators properly. The reason for vertically long results near 0 s of Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28 is: “The initial tracking errors, especially for x axis, and pitch and yaw directions in Figure 12 and Figure 13 are larger than others when the guided torpedo-like underwater vehicle tracks the first waypoint”.

## 5. Conclusions

_{∞}control law and a sliding mode control law into input commands of actuators, including one thruster in the aft part, four rudders at the stern, and four fins at the forward part of the controlled torpedo-like underwater vehicle. The closed-form solutions of this control allocation problem were solved analytically. Simulations for the proposed control allocation method were examined based on a real torpedo-like underwater vehicle named “AUV Lab 611”. Simulation results reveal the fact that control commands generated by two control laws for precisely achieving the given trajectory tracking mission can be optimally and properly converted into all installed actuators of the controlled torpedo-like underwater vehicle.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Derivation of Radial Basis Function Network

_{i}, y

_{i}), i = 1, 2, …, N}, a function could be solved to represent the unknown system by interpolation. The unknown system could be expressed as

_{c}is the center of the radial basis function, and ||●|| represents the Euclidean 2-norm. In other words, the radial basis function stands for the relativity between the data point and the center of the radial basis function, and a common choice of the radial basis function is Gaussian function:

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**Figure 1.**The proposed torpedo-like underwater vehicle with 4 fins, 4 rudders, and 1 non-rotatable thruster.

**Figure 2.**The flow chart of the control allocation procedure for the torpedo-like underwater vehicle with multiple actuators.

**Figure 10.**The generated force with respect to time interval (PWM) using radial basis function network approximation.

**Figure 11.**Trajectory tracking history of the controlled torpedo-like underwater vehicle based on sliding mode control law (blue line) and the proposed nonlinear H

_{∞}control law (green line) for Scenario 1.

**Figure 12.**Histories of the position tracking errors of the controlled torpedo-like underwater vehicle: (

**a**) x-axis, (

**b**) y-axis, and (

**c**) z-axis.

**Figure 13.**Histories of the attitude tracking errors of the torpedo-like underwater vehicle: (

**a**) roll direction, (

**b**) pitch direction, and (

**c**) yaw direction.

**Figure 14.**The velocities of the controlled torpedo-like underwater vehicle with respect to sliding mode control and the nonlinear H

_{∞}control law, respectively.

**Figure 15.**Surge forces of the developed torpedo-like underwater vehicle with respect to the sliding mode control law and the nonlinear H

_{∞}control law.

**Figure 16.**Sway forces of the developed torpedo-like underwater vehicle with respect to the sliding mode control law and the nonlinear H

_{∞}control law.

**Figure 17.**Heave forces of the developed torpedo-like underwater vehicle with respect to the sliding mode control law and the nonlinear H

_{∞}control law.

**Figure 18.**Roll moments of the developed torpedo-like underwater vehicle with respect to the sliding mode control law and the nonlinear H

_{∞}control law.

**Figure 19.**Pitch moments of the developed torpedo-like underwater vehicle with respect to the sliding mode control law and the nonlinear H

_{∞}control law.

**Figure 20.**Yaw moments of the developed torpedo-like underwater vehicle with respect to the sliding mode control law and the nonlinear H

_{∞}control law.

**Figure 21.**The port stern fin angles of the controlled torpedo-like underwater vehicle for two control laws, respectively.

**Figure 22.**The starboard stern fin angles of the controlled torpedo-like underwater vehicle for two control laws, respectively.

**Figure 23.**The upward stern rudder angles of the controlled torpedo-like underwater vehicle for two control laws, respectively.

**Figure 24.**The downward stern rudder angles of the controlled torpedo-like underwater vehicle for two control laws, respectively.

**Figure 25.**The port bow fin angles of the controlled torpedo-like underwater vehicle for two control laws, respectively.

**Figure 26.**The starboard bow fin angles of the controlled torpedo-like underwater vehicle for two control laws, respectively.

**Figure 27.**The upward bow rudder angles of the controlled torpedo-like underwater vehicle for two control laws, respectively.

**Figure 28.**The downward bow rudder angles of the controlled torpedo-like underwater vehicle for two control laws, respectively.

DOF | Linear and Angular Velocities | Positions and Euler Angles | Forces and Moments | |
---|---|---|---|---|

1 | Motions in the x-direction (surge) | u | x | X |

2 | Motions in the y-direction (sway) | υ | y | Y |

3 | Motions in the z-direction (heave) | w | z | Z |

4 | Rotations about the x-axis (roll) | p | ϕ | K |

5 | Rotations about the y-axis (pitch) | q | θ | M |

6 | Rotations about the z-axis (yaw) | r | ψ | N |

Parameter | Value |
---|---|

Total length L | 1.54 m |

Nose length a | 0.15 m |

Midbody length b | 1.14 m |

Tail length c | 0.25 m |

Hull diameter d | 0.16 m |

Mass m | 14.56 kg |

Buoyancy B | 27.13 kg |

Moment of inertia I_{x} | 0.072 kg·m^{2} |

Moment of inertia I_{y} | 12.02 kg·m^{2} |

Moment of inertia I_{z} | 12.02 kg·m^{2} |

**Table 3.**The related specifications of fins and rudders of the developed torpedo-like underwater vehicle.

Actuator | Position (cm) | Generated Force | Control Command |
---|---|---|---|

Port bow fin | (39, −16, 0) | [0 0 −F_{pb}] | δ_{pb} (degree) |

Starboard bow fin | (39, 16, 0) | [0 0 −F_{sb}] | δ_{sb} (degree) |

Upward bow rudder | (39, 0, −16) | [0 F_{ub} 0] | δ_{ub} (degree) |

Downward bow rudder | (39, 0, 16) | [0 F_{db} 0] | δ_{db} (degree) |

Port stern fin | (−57, −16, 0) | [0 0 −F_{ps}] | δ_{ps} (degree) |

Starboard stern fin | (−57, 16, 0) | [0 0 −F_{ss}] | δ_{ss} (degree) |

Upward stern rudder | (−57, 0, −16) | [0 F_{us} 0] | δ_{us} (degree) |

Downward stern rudder | (−57, 0, 16) | [0 F_{ds} 0] | δ_{ds} (degree) |

Thruster | (−80, 0, 0) | [F_{T} 0 0] | u_{T} (PWM) |

No. | Waypoints (x, y, z) (m) | No. | Waypoints (x, y, z) (m) |
---|---|---|---|

1 | (4, 0, 5) | 6 | (60, 0, 9.8) |

2 | (20, 0.75, 5.3) | 7 | (70, −5, 9) |

3 | (30, 5, 7) | 8 | (80, −8, 7) |

4 | (40, 8, 9) | 9 | (90, −5, 5.4) |

5 | (50, 5, 9.8) | 10 | (100, −0.85, 5.5) |

Parameter | Value |
---|---|

λ_{1} | 4 |

λ_{2} | 4 |

Γ | $\left[\begin{array}{cccccc}0.3103& 0& 0& 0& 0& 0\\ 0& 0.3103& 0& 0& 0& 0\\ 0& 0& 0.3103& 0& 0& 0\\ 0& 0& 0& 0.32& 0& 0\\ 0& 0& 0& 0& 0.0441& 0\\ 0& 0& 0& 0& 0& 0.0441\end{array}\right]$ |

Z | $\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$ |

Parameter | Value |
---|---|

λ_{f1} | 4 |

λ_{f2} | 4 |

k_{sc} | 7 |

k_{s} | 4 |

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**MDPI and ACS Style**

Chen, Y.-Y.; Lee, C.-Y.; Huang, Y.-X.; Yu, T.-T.
Control Allocation Design for Torpedo-Like Underwater Vehicles with Multiple Actuators. *Actuators* **2022**, *11*, 104.
https://doi.org/10.3390/act11040104

**AMA Style**

Chen Y-Y, Lee C-Y, Huang Y-X, Yu T-T.
Control Allocation Design for Torpedo-Like Underwater Vehicles with Multiple Actuators. *Actuators*. 2022; 11(4):104.
https://doi.org/10.3390/act11040104

**Chicago/Turabian Style**

Chen, Yung-Yue, Chun-Yen Lee, Ya-Xuan Huang, and Tsung-Tso Yu.
2022. "Control Allocation Design for Torpedo-Like Underwater Vehicles with Multiple Actuators" *Actuators* 11, no. 4: 104.
https://doi.org/10.3390/act11040104