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ZFAUV is a novel portable modular AUV. There are four fixed thrusters at tail, and two tunnel thrusters are set at front. The maneuverability of ZFAUV is relatively high. It can turn around in situ, move lateral or move up/down vertical. The yaw and pitch can be controlled by tunnel thrusters or differential control of tail thrusters, but differential control will reduce the forward force. Different from propeller-rudder AUVs, the turning radius is related to speed forward: the smaller the speed forward, the smaller the turning radius. The minimum turning radius tends to be zero. The mathematical model is built first; then CFD is used to predict the thrust and torque of tail thrusters and tunnel thrusters. Through numerical simulation, zigzag maneuver analysis in horizontal plane, and trapezoidal steering maneuver analysis in vertical plane, the maneuverability of ZFAUV is obtained. The maneuverability of ZFAUV becomes worse with the increase of speed. The maneuverability of differential control is better than that of tunnel control. In the case of specific thrust distribution of tail thrusters and tunnel thrusters, ZFAUV can turn around in situ (the maximum angular velocity is about 24.1°/s), move lateral or move up/down vertical (the maximum velocity is about 0.4m/s). Finally, an example, PID parameters tuning, is given to illustrate the application of maneuverability analysis. The dynamic performance of ZFAUV can be quickly and accurately analyzed by mathematical method, which has important guiding significance for the choice of control strategy and experiments and also has reference value for the later development of AUVs.

Autonomous Underwater Vehicle (AUV) is defined as a vehicle that can perform underwater tasks and missions autonomously, using onboard navigation, guidance, and control systems [

Due to the limitation of weight and economic cost, most AUVs use fewer thrusters to achieve multi-degree-of-freedom coupling motion control, which makes them typical underactuated system. The shape of these AUVs is generally streamlined, most are torpedo-shaped (e.g., the NERC Autosub6000 AUV and REMUS-100 AUV). The traditional propeller-rudder control mode is adopted by most AUVs that is a main propeller and control surface is arranged at the tail. At present, the control systems adopted by underactuated AUVs are generally classified into the following categories: X rudder[

We developed a small portable modular AUV with four fixed thrusters at tail, named ZFAUV; it weighs about 20kg. Meanwhile, in order to improve the maneuverability of ZFAUV, two tunnel thrusters are set at front: one is horizontal and the other is vertical, as shown in Figure

Modular ZFAUV.

To the best of our knowledge, the AUVs who have tunnel thrusters both at front and at rear are both conventional propeller-rudder AUVs. They are equipped with two tunnel thrusters (vertical usually) [

Researchers model the dynamic behavior of AUVs to evaluate their performances [

Mathematical model is needed to predict the maneuverability; although numerous underwater vehicle models have been presented, there is a little study of the model of this novel AUV with four fixed thrusters and two tunnel thrusters. Thus, there is an urgent need to build dynamics model for this novel AUV. Meanwhile, there are many hydrodynamic coefficients in the mathematical model. Generally, there are two different approaches to obtain the hydrodynamic coefficients. The conventional method is towing tank experiment, but it takes a long time and costs a lot. The other method is using of CFD. In the field of AUVs, the use of CFD has increased in recent years due to the increasing availability of powerful computers and user-friendly CFD software, which has become an almost completely necessary tool for predicting the hydrodynamic coefficients used in maneuverability predictions [

The purpose of this study is to simulate the maneuverability of ZFAUV. So the mathematical model is built first; then the CFD is used to predict the thrust and torque of thrusters. And the mathematical model is simplified in horizontal plane and vertical plane, respectively. The motion characteristics of ZFAUV are studied in detail, including steady linear maneuver, maneuverability in vertical plane, maneuverability in horizontal plane, etc.

The rest of this paper is organized as follows. Section

As shown in Figure

Thruster arrangement.

The forces and torques acting on ZFAUV are complex during the survey task, including gravity, buoyancy, thruster's thrust, and water resistance. In order to study the motion of ZFAUV, we only analyze the influence of thruster's thrust on the attitude of ZFAUV.

Figures

Motion of ZFAUV in horizontal plane.

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In the case of differential control, Figures

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Motion of ZFAUV in vertical plane.

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In the case of differential control, Figures

In order to model ZFAUV, two reference frames are defined to describe motion of ZFAUV: the body-fixed reference frame (

Reference frame of ZFAUV.

According to reference (Fossen, 2011; Xiaoming Wang, 2009)[

Some physical parameters in the model, such as

For ZFAUV, the power comes from each thruster. Therefore, the performance of thruster is very important for the dynamic performance and maneuverability of ZFAUV. Therefore, the thrust of tail thruster and tunnel thruster under different conditions needs to be analyzed in detail.

Thrust performance analysis of thruster is relatively complex. There are two main ways at present. First, physical test, the results are accurate, but the cycle is long, which requires a huge amount of manpower and material resources. Second, CFD simulation, with low cost and short cycle, can obtain complete data and can set up computational flow field according to research needs, which is sometimes difficult to achieve in physical experiments. Therefore, CFD method is used to predict the thrust.

In the numerical calculation, the propeller is assumed to be located deep below the water surface, without considering the influence of surface effects such as surface waves. In order to simulate the infinite flow field, the size of computational domain should be much larger than 10 times the diameter of the propeller to ensure that the boundary layer has no effect on the motion of the propeller. According to the characteristics of propeller when it rotates, the computational domain is divided into two parts: the dynamic region near the propeller and the static region far from the propeller, as shown in Figures

Physical model of tail thruster in CFD.

CFD result of tail propeller.

Three-dimensional solver in Fluent is used to solve the physical model, and unsteady uncoupled invisible algorithm is used. Modified RNG k-

According to the characteristics of ZFAUV, the speed forward is about 0-5 kn. The physical model used in CFD is shown in Figure

Figure

The thrust and torque at different flow velocity was calculated and analyzed, as shown in Table

Calculated thrust and torque of tail thruster at different flow velocity.

T/N Q/N•m | n/% | |||||
---|---|---|---|---|---|---|

10 | 20 | 40 | 60 | 80 | 100 | |

| ||||||

0.26 | 1.47 | 3.31 | 5.89 | 12.95 | 22.82 | 35.41 |

0.0173 | 0.0403 | 0.0714 | 0.1596 | 0.2815 | 0.4391 | |

0.56 | 1.17 | 2.78 | 5.06 | 11.59 | 20.71 | 32.43 |

0.0161 | 0.0381 | 0.0674 | 0.1538 | 0.2725 | 0.4249 | |

1.14 | 0.82 | 2.24 | 4.40 | 10.26 | 18.51 | 25.76 |

0.0119 | 0.0316 | 0.0605 | 0.1401 | 0.2519 | 0.3935 | |

1.79 | 0.48 | 1.69 | 3.68 | 8.93 | 16.46 | 20.39 |

0.0060 | 0.0223 | 0.0481 | 0.1198 | 0.2194 | 0.3560 | |

2.22 | 0.13 | 1.13 | 3.03 | 7.60 | 14.26 | 14.42 |

0.0018 | 0.0159 | 0.0412 | 0.1061 | 0.1988 | 0.3267 | |

2.87 | -0.22 | 0.57 | 2.37 | 6.27 | 12.06 | 8.46 |

-0.0024 | 0.0094 | 0.0343 | 0.0923 | 0.1782 | 0.2975 |

At rated speed, the thrust coefficient and torque coefficient can be obtained by the following equations:

The advance coefficient J and the efficiency of the propeller can be calculated by the following equations:

So the open water performance of tail thruster at rated speed can be obtained, as shown in Figure

Open water performance at rated speed.

Tunnel thruster is mainly used to change the heading and pitch angle of ZFAUV. The performance of tunnel thruster is deeply influenced when the forward velocity changes, accordingly influencing the turning radius and maneuverability. Therefore, it is necessary to analyze the thrust at different forward velocity.

Due to the existence of tunnel, and the fact that there's no duct, the physical model of tunnel thruster in CFD is different from that of tail thruster. Tunnel needs to be added to the physical model, as shown in Figure

Physical model of tunnel thruster in CFD.

The direction of flow is perpendicular to the axis of tunnel, so the angle between the direction of inflow and the axis of tunnel is set as 90 degrees. The physical model used in CFD is shown in Figure

Figure

CFD result of tunnel thruster.

The hydrodynamic performance of tunnel thruster is calculated and analyzed for different forward velocity, the thrust and torque is shown in Table

Calculated thrust and torque of tunnel thruster at different forward velocity.

T/N Q/N•m | n/% | |||||
---|---|---|---|---|---|---|

10 | 20 | 40 | 60 | 80 | 100 | |

| ||||||

0.26 | 0.23 | 0.51 | 1.81 | 4.19 | 8.06 | 13.04 |

0.003 | 0.007 | 0.026 | 0.058 | 0.110 | 0.176 | |

0.56 | 0.25 | 0.81 | 2.22 | 4.43 | 7.74 | 11.87 |

0.003 | 0.010 | 0.030 | 0.061 | 0.106 | 0.163 | |

1.14 | 0.23 | 1.06 | 3.55 | 6.49 | 8.02 | 13.27 |

0.003 | 0.013 | 0.042 | 0.082 | 0.110 | 0.178 | |

1.79 | 0.51 | 1.14 | 3.34 | 7.11 | 15.07 | 17.51 |

0.007 | 0.014 | 0.040 | 0.086 | 0.179 | 0.221 | |

2.22 | 0.075 | 0.82 | 3.63 | 6.85 | 14.21 | 23.6 |

0.002 | 0.011 | 0.042 | 0.083 | 0.167 | 0.279 | |

2.87 | 0.075 | 0.74 | 5.30 | 7.64 | 13.41 | 21.41 |

0.001 | 0.010 | 0.066 | 0.088 | 0.159 | 0.254 |

The thrust variation of tunnel thruster at different forward velocity can be drawn, as shown in Figure

The thrust of tunnel thruster.

In order to make the simulation result more real and reliable, the thrust and torque coefficients of the tunnel thruster should be corrected in real time according to the calculated speed.

On the basis of 6-DOF mathematical model in Section

In order to study the dynamical behavior of ZFAUV in horizontal plane, including steady maneuver and zigzag maneuver, the mathematical model has to be simplified in

When ZFAUV moves in horizontal plane,

In steady maneuver, the motion parameters remain unchanged and the acceleration parameters are zero:

where subscript

where

In steady linear maneuver,

So, ZFAUV is only possible to keep steady linear maneuver under the condition of

When ZFAUV is moving forward, if the tunnel thruster is executed, lateral force and yaw moment will be produced. Under the action of lateral force and yaw moment, ZFAUV will turn left or turn right.

According to (

where

Therefore,

The turning radius is as follows:

The relationship between the turning radius and the speed of tunnel thruster is shown in Table

Turning radius of ZFAUV at different speed.

Forward | Tunnel | ||||
---|---|---|---|---|---|

20% | 40% | 60% | 80% | 100% | |

20%(0.56m/s) | 6.1 | 3.3 | 2.8 | 2.5 | 2 |

40%(1.14m/s) | 13.2 | 9.5 | 8 | 5.8 | 4.4 |

60%(1.79m/s) | 29 | 20 | 14 | 10.5 | 8.2 |

80%(2.22m/s) | 52 | 30 | 20 | 15 | 12 |

100%(2.87m/s) | 72 | 38.5 | 26 | 21 | 17 |

As can be seen from Table

Heading zigzag parameters-tunnel control (50%).

| 10 | 20 | 40 | 60 | 80 | 100 |
---|---|---|---|---|---|---|

| 0.26 | 0.56 | 1.14 | 1.79 | 2.22 | 2.87 |

| 8.50 | 6.31 | 6.38 | 8.19 | 9.69 | 12.00 |

| 2.21 | 1.77 | 3.64 | 7.33 | 10.76 | 17.22 |

| 6.50 | 3.25 | 1.63 | 1.03 | 0.83 | 0.63 |

| 38.6 | 11.8 | 2.9 | 1.2 | 0.8 | 0.5 |

Turning radius at different speed.

Zigzag maneuver is widely used to evaluate the maneuverability of ships and torpedoes, so this can be used to evaluate the maneuverability of ZFAUV.

In a general zigzag maneuver, the vehicle is moving forward at constant speed and the rudder is executed to a specified maximum rudder angle in one direction at maximum rudder rate. The vehicle responds by turning in that direction. When the vehicle heading angle reaches a specified check heading angle, the rudder is turned at maximum rudder rate in the opposite direction until it reaches the maximum rudder angle specified. The vehicle reacts by turning in the opposite direction, and the procedure is repeated when the check heading angle in the opposite direction is reached. This results in a zigzag response that is used to assess the maneuverability of the vehicle. The maximum rudder angle and the check heading angle characterize the maneuver type; for instance, a 20/10 zigzag maneuver turns the rudders to 20° and changes direction when the check heading angle of 10° is reached [

For ZFAUV, the direction is controlled by tunnel thruster or differential control of tail thrusters; behind, tunnel thruster or differential control of tail thrusters is replaced by ‘rudder’. So the 50%/15° zigzag maneuver is adopted in this study;, that is, when ZFAUV is moving forward at constant speed, the ‘rudder’ is executed to 50% (the first execute), ZFAUV starts to turn right. Once the check heading angle of -15° is reached for the first execute, the ‘rudder’ is executed to -50% (the second execute). At this time, ZFAUV will continue to turn right because of inertia, but the turning rate gradually decreases. When

Performance of heading zigzag maneuver.

The characteristic parameters of zigzag maneuver include the initial turning period, the overshoot time, the overshoot heading angle, and the full cycle. The smaller the characteristic parameters, the better the heading changing ability of ZFAUV.

The initial turning period (

The overshoot time (

After the second execute, ZFAUV keeps turning in the original direction. The overshoot heading angle (

The characteristic parameters of ZFAUV at different speed can be obtained, as shown in Tables

Heading zigzag parameters-differential control (50%).

| 10 | 20 | 40 | 60 | 80 | 100 |
---|---|---|---|---|---|---|

| 0.16 | 0.39 | 0.81 | 1.27 | 1.57 | 2.03 |

| 6.25 | 5.31 | 4.13 | 4.00 | 4.25 | 4.94 |

| 1.00 | 1.04 | 1.67 | 2.54 | 3.34 | 5.01 |

| 5.85 | 3.88 | 1.98 | 1.23 | 0.98 | 0.75 |

| 60.6 | 31.1 | 9.5 | 4.2 | 2.4 | 1.7 |

From Figures

As shown in Figures

Then, the force acting on ZFAUV is shown in Figure

Force analysis when turning around in situ.

The equation of rotating along

Figure

Simulation result of turning around in situ.

The relationship between turning angular velocity and

Angular velocity of turning around in situ.

| 10 | 20 | 40 | 60 | 80 | 100 |

| ||||||

| 0.2 | 0.9 | 3.8 | 8.6 | 15.3 | 24.1 |

Angular velocity of turning around in situ.

From Figure _{5}, and the maximum turning angular velocity is about 24.1°/s.

As shown in Figures

Force analysis when moving lateral.

The equation along

Figure

Simulation result of moving lateral.

The maximum velocity of moving lateral is about 0.39m/s.

In order to study the dynamical behavior of ZFAUV in vertical plane, including steady maneuver and trapezoidal steering maneuver, the mathematical model has to be simplified in

When ZFAUV moves in vertical plane,

In steady maneuver of ZFAUV, the motion parameters remain unchanged and the acceleration parameters are zero:

where subscript

After simplification, the following can be obtained:

where

In order to ensure the safety of ZFAUV, the range of pitch angle is limited as

According to (

If

Trapezoidal steering maneuver is the most typical test to verify the maneuverability in vertical plane; it is of great significance to the study of deep maneuvering motion [

In a general trapezoidal steering maneuver, the vehicle is moving forward at constant speed and the ‘rudder’ is executed to a specified ‘rudder’ angle (

Figure

Simulation result of trapezoidal steering.

The characteristic parameters of trapezoidal steering maneuver include the execution time, the overshoot pitch angle, and the overshoot depth. The smaller the characteristic parameters, the better the depth changing ability of ZFAUV.

The execution time (

After the ‘rudder’ is executed to zero, the pitch angle will continue to increase and gradually return to zero. The overshoot pitch angle (

The overshoot depth (

The maximum ‘rudder’ angle and the check pitch angle characterize the maneuver type; for instance, a 10/7 trapezoidal steering maneuver turns the ‘rudder’ to 10° and set as 0 when the check pitch angle of 7° is reached. For ZFAUV, 30%/7° trapezoidal steering maneuver is adopted. The characteristic parameters at different speed can be obtained, as shown in Tables

Trapezoidal steering parameters-tunnel control (30%).

| 10 | 20 | 40 | 60 | 80 | 100 |
---|---|---|---|---|---|---|

| 0.33 | 0.56 | 1.14 | 1.79 | 2.22 | 2.87 |

| 3.62 | 3.62 | 4.12 | 4.87 | 5.62 | 6.75 |

| 12.8 | 7.8 | 3.4 | 1.6 | 1.2 | 0.8 |

| 0.21 | 0.98 | 2.29 | 4.20 | 5.96 | 9.13 |

Trapezoidal steering parameters-differential control (30%).

| 10 | 20 | 40 | 60 | 80 | 100 |
---|---|---|---|---|---|---|

| 0.25 | 0.39 | 0.81 | 1.27 | 1.57 | 2.03 |

| 3.75 | 3.75 | 4.12 | 5.00 | 5.87 | 7.5 |

| 3.0 | 2.5 | 1.1 | 0.4 | 0.2 | 0.1 |

| -0.22 | 0.03 | 0.24 | 0.45 | 0.63 | 0.96 |

From Figures

_{e} becomes smaller with the increase of velocity),

As shown in Figures

Force analysis when moving down vertical.

The equation along

Figure

Simulation result of moving down vertical.

The maximum velocity in the vertical direction is about 0.41m/s, which is slightly larger than that in the horizontal direction. For the influence of antenna, the area of ZFAUV in horizontal direction is little larger than that in vertical direction, so the steady moving velocity in horizontal direction is relatively low.

Path tracking is the basis of motion control of AUVs, and path tracking strategy and control algorithm are needed. Line-of-Sight (LOS) guidance is the most widely used guidance strategy, and PID is the most widely used controller. For PID controller, it is necessary to adjust the parameters. If the parameters of PID are not adjusted properly, the final effect will be greatly affected.

By using the mathematical model and LOS path tracking algorithm, the effect of path tracking is optimized by modifying the PID parameters. Figure

Tracking effect with different PID parameters.

The same parameters (P=0.8, I=0.05, D=1) are adopted in water experiment, as shown in Figure

Experimental result of path tracking with the same parameters.

These PID parameters can be tuned in one hour in the simulation system. If the parameters are tuned by water experiment, it will take about 5-7 days, which will waste a lot of time and money. This further highlights the importance of dynamic analysis and simulation.

For ZFAUV, there are four fixed thrusters at tail, and two tunnel thrusters are set at front. The maneuverability of ZFAUV is relatively high. Through the establishment of mathematical model and maneuverability analysis, the following conclusions can be drawn:

The research results of this paper have guiding significance for the future development of AUVs.

Some data used to support the findings of this study are included within the article; others are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This research is supported by the National Natural and Science Foundation of Hebei (No. E2018202259) and Scientific Research Project of Tianjin Education Commission (No. 2017KJ022).