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Abstract

Formation control is one of the essential problems in multi-unmanned underwater vehicle (UUV) coordination. In this article, a practically oriented UUV formation control structure and method are proposed for the problem of large communication in leader–follower approach. To solve the problem of large communication in multi-UUVs, local sensing means of acoustic positioning is used to provide the real relative distance and angle information for the follower UUV. So, only a small amount of state information of the leader UUV needs to be sent to the follower UUV by acoustic communication. Then, the formation control structure in absence of follower position information is proposed. In this control structure, only the relative distance and angle, as well as velocity and heading of the leader UUV, are used for the formation controller design of the follower UUV. Backstepping and Lyapunov methods are used to design the formation controller without position information of the follower UUV. Two formation configurations of rectangle and triangle with five UUVs are simulated to verify the effectiveness of the method proposed. The simulation results show that the follower UUV can successfully constitute and maintain the desired formation by controlling each real relative distance and angle.

Keywords

Multi-UUVs formation control local sensing means follower position information leader–follower

Introduction

As an unmanned, mobile, and low-cost underwater implementation, unmanned underwater vehicles (UUVs) are rapidly developed and employed in a variety of applications in ocean scientific research, marine engineering, and military affairs. Compared with a single UUV, multi-UUV system can improve work efficiency and complete complex tasks by cooperation, which is becoming a hot research field. Usually, multi-UUVs need to maintain a certain formation in the process of moving and task executing, which is called formation control problem. It can be described as a configuration of multi-UUVs in which the vehicles maintain a desired formation structure while moving as a whole and fulfilling assigned tasks.¹ Formation control problem is so important for multi-UUVs because of potentially broad applications in ocean engineering and science.² More and more formation control research have been carried out and some formation control approaches have been proposed. In general, there are mainly four formation control approaches for multiple vehicles and robots: virtual structure approach,^3
–5 behavior-based approach,^6,7 leader–follower approach,^8
–10 and artificial potential field approach.^11
–13 Each of the above approaches has its own advantages and disadvantages.

The basic idea of virtual structure approach is to treat the UUV formation as a virtual rigid body structure.¹⁴ Each UUV occupies a fixed and unique position point on the virtual rigid body structure. Each UUV only needs to track its own position point to maintain the desired formation. The virtual structure approach is easy to describe the formation behaviors, which is profitable for formation keeping. However, the rigid structure constraint and large communication requirements limit its wide applications. Meanwhile, the robustness is not well guaranteed when the formation motion is time varying.¹⁵ For behavior-based approach, the formation control task is decomposed into several basic behaviors. These basic behaviors may include cohesion, collision avoidance, obstacle avoidance, moving to goal, and so on.¹⁶ Then, each UUV moves as a total behavior weighted of each basic behavior according to the formation control goal. Behavior-based approach is easy to realize distributed control and has strong adaptability to the environment. The disadvantage is that the whole behavior of the formation is difficult to design and formalize mathematically.¹⁷ Two roles of leader UUV and follower UUVs are assigned for the leader–follower approach. The leader UUV tracks a predefined path and the follower UUVs maintain a desired formation with the leader UUV.^18,19 The follower UUVs keep formation based on information received from the leader UUV.²⁰ The basic idea is to transform the formation control problem into the problem that follower UUVs track the position and direction with respect to the leader UUV. Leader–follower approach is easy to understand and implement. However, the failure of the leader UUV can lead to the failure of the whole formation team. Once the leader UUV is invalid, all the follower UUVs will lose their formation performance.^21,22 For the artificial potential field approach, an abstract artificial potential field is constructed in the working environment of UUVs. Then, the corresponding gravitational potential field and repulsive potential field are used to control each UUV to form a specific formation structure. This approach is suitable for formation control in collision avoidance problem. However, the main disadvantage is that it is difficult for potential field function design.²³

Considering the advantages of simplicity and extensibility, leader–follower approach will be adopted and researched in this article. However, there are two main limitations when the traditional leader–follower approach is applied to UUV formation control in practice.

For traditional leader–follower approach, the follower UUVs need to obtain its own position information in real time and obtain the position information of the leader UUV at the same time. Then, the follower UUVs calculate the relative distance and angle with respect to the leader UUV for formation control. In addition, most of the leader–follower approach is based on the first-order model to design formation controller for the follower UUVs. That is, the follower UUVs adopt position control to maintain the formation. Based on the above analysis, it can be known that the follower UUVs must equip high precision but expensive navigation equipment when using the traditional leader–follower approach, which increases the cost of the multi-UUV system. So, developing a leader–follower formation control method without follower position information is very beneficial to reduce the cost of multi-UUV system because the follower UUVs do not need to equip expensive navigation equipment.

For traditional leader–follower approach, the position of the leader UUV is needed for the follower UUVs to calculate the relative distance and angle. Therefore, the information sent by the leader UUV to the follower UUVs through acoustic communication includes not only its own heading, linear velocity in surge, and angular velocity in yaw but also position information. As we all know, compared with other information, the communication quantity of position information is relatively large. However, the large communication quantity may easily cause the problem of packet dropouts due to the low data rate, long propagation delay, and instability of the acoustic communication.²⁴ Therefore, developing a leader–follower formation control method with low communication quantity is very meaningful for the practical application of UUV formation control based on acoustic communication.

Motivated by the above discussion, this article aims to develop a leader–follower formation control method of multi-UUVs that can reduce acoustic communication quantity and do not need position information of the follower UUV. The main contributions of this article are given below:

A leader–follower formation control method of multi-UUVs in absence of follower position information is proposed. First, the follower UUVs do not need the position information of self and the leader UUV to calculate the relative distance and angle. Instead, the follower UUVs can obtain the relative distance and angle directly through acoustic positioning means. Second, the formation controller is designed based on the second-order model to control the heading and speed of the follower UUVs. Based on these two aspects, the follower UUVs do not need to obtain its own position information during formation control process. That is to say, there is no need to equip high precision but expensive navigation equipment for the follower UUVs, which can greatly reduce the cost of multi-UUV system.

An efficient and practical acoustic local sensing information interaction means is designed for multi-UUV system. In the local sensing information interaction means, the method of combining acoustic communication with acoustic positioning is adopted. The leader UUV sends a small amount of information of its heading, linear velocity in surge, and angular velocity in yaw to follower UUVs by acoustic communication. Meanwhile, the follower UUVs obtain the relative distance and angle by acoustic positioning. By this, the position information with large communication quantity does not need to be sent by the leader UUV through acoustic communication. The reduced communication quantity is very suitable for low data rate, long propagation delay, and instability acoustic communication for formation control of multi-UUV system.

Considering the practical feasibility and the current development of underwater acoustic engineering technology, the integrated communication and positioning equipment is proposed to be used for formation control of multi-UUV system to realize the above two objectives. Because the integrated communication and positioning equipment has the characteristics of dual function, high integration and low cost, it is very suitable for UUVs.

The remainder of the article is organized as follows: Problem statement for leader–follower formation control of multi-UUV system is presented in the second section. In the third section, models of single UUV and leader–follower relative motion are established. Next, the formation control structure and controller design are introduced in the fourth section. Simulation results are illustrated in the fifth section. Then, the rationality and limitation of simulation are analyzed. Conclusion is given in the last section.

Problem statement

The scenario considered in this article is shown in Figure 1. There is only a leader and some followers for a group of UUVs. The whole leader–follower UUVs need to maintain a fixed geometrical formation while navigating to mission waypoints.²⁵ The leader UUV has high precision underwater navigation ability to obtain complete position information. However, the follower UUVs do not equip navigation instrument in absence of position information. In the leader–follower formation, the leader UUV guides and navigates the whole formation to all mission waypoints. Each follower UUV is responsible for generating a suitable actuation signal to track the leader UUV with a desired relative distance and relative angle for the formation maintenance. The introduction of the relative distance and relative angle between the follower UUV and leader UUV will be introduced in the third section.

Figure 1.

Problem statement of leader–follower formation of UUVs. UUV: unmanned underwater vehicle.

To keep formation, information interaction using acoustic means is necessary in the multi-UUV system, such as acoustic communication and acoustic positioning. For the purpose of reducing communication quantity, a local sensing means combined with acoustic communication and acoustic positioning is adopted in this article. All the follower UUVs can obtain the real relative distance and relative angle with respect to the leader UUV by local acoustic positioning. In addition, all the follower UUVs can also obtain a small amount of state information of the leader UUV by acoustic communication. In this way, the information needed for formation control of the follower UUVs is shared by acoustic positioning and acoustic communication means, respectively. So, the communication quantity in the multi-UUV system is reduced, which can reduce packet dropouts.

The acoustic information interaction between multi-UUVs proposed in this article has two features:

The acoustic communication is unidirectional, only from the leader UUV to the follower UUVs as a timing broadcast form. In normal formation control process, each follower UUV does not need active communication to leader UUV or other follower UUVs. By this, the timing-unidirectional-broadcast communication of leader UUV greatly reduces the amount and frequency of information interaction between multi-UUVs and is very suitable for UUV formation control based on underwater acoustic communication. In practical applications, each follower UUV will communicate actively only in case of emergency to inform the leader UUV. However, the emergency active communication of follower UUVs is not the focus of this article. So, active communication of follower UUVs will not be mentioned here.

The function of underwater acoustic positioning is realized synchronously at the same time of underwater acoustic communication. In practical underwater acoustic engineering, the integrated communication and positioning equipment can realize this function.

The integrated communication and positioning equipment is a kind of underwater acoustic equipment with communication and positioning functions. Usually, it adds the acoustic communication function to the acoustic positioning system, which can realize the acoustic positioning and acoustic communication at the same time. The integrated communication and positioning equipment is not only a simple superposition of positioning and communication functions but also an organic combination and complement of acoustic positioning technology and acoustic communication technology. Using the integrated communication and positioning equipment, the leader UUV can periodically send positioning signal with a small amount of communication information to all the follower UUVs as a broadcast form. Then, each follower UUV can receive the communication information and calculate the relative distance and relative angle with respect to the leader UUV through the positioning signal. However, this article does not describe how it is actually realized in detail here. Only the flow of the information interaction from leader UUV to follower UUVs is presented.

Some assumptions are made for the development of a control scheme suitable for achieving the above requirements.

Assumption 1. There is no obstacle in the working space, and the information of the mission waypoints and desired path is known only to the leader UUV.

Assumption 2. There is a delay of the information between the leader UUV and the follower UUVs. Because the processing delay and propagating delay are necessary for underwater acoustic communication and positioning. The delay time of the underwater acoustic information is usually 3–8 s.

Assumption 3. The position information for the follower UUVs is unknown.

Remark 1. Assumption 3 means that the proposed approach does not require the position information to design the formation controller of follower UUVs. In this article, the relative distance and angle information by local acoustic positioning means will be used.

Remark 2. A classic waypoints following control method is adopted for the leader UUV to guide and navigate the whole formation to all mission waypoints. So, waypoints following control method will not be introduced in this article.

Modeling of leader–follower unmanned underwater vehicles

In this section, the simplified kinematics of UUV is introduced, and the leader–follower relative motion model of multi-UUVs is formulated.

Simplified kinematics of unmanned underwater vehicles

To describe the kinematics of UUV, two reference frames are defined first. They are the Earth reference frame $\{E\}$ and the body-fixed frame $\{B\}$ , as shown in Figure 2. The body-fixed frame is chosen so as to coincide with the center of buoyancy and the center of gravity.

Figure 2.

The Earth reference frame and body-fixed frame.

In this article, the UUV formation is only considered in horizon plane with constant depth. Without loss of generality, the motions in heave, roll, and pitch are neglected. Meanwhile, it is assumed that the leader UUV and the follower UUVs have same kinematics. Then, the simplified kinematics of the UUV is described as follows^26,27

\begin{matrix} \dot{η} = J (η) v \end{matrix}

where $η = {[x, y, ψ]}^{T}$ are the position $(x, y)$ and the heading angle $ψ$ of the UUV in the Earth-fixed frame. $v = {[u, v, r]}^{T}$ are the linear and angular velocities of the UUV in the body-fixed frame. u and v are the linear velocities in surge and sway. r is the angular velocity in yaw. $J (η)$ is the transformation between the two reference frames, as mentioned above.

Leader–follower relative motion model

For leader–follower formation control approach, there are different types of leader–follower relative motion systems. The method used in this article is an “ $L - ψ$ ” type, as shown in Figure 3. $L_{l f}$ is defined as the real relative distance between the follower UUV and the leader UUV. $ψ_{l f}$ is the real relative angle between the follower UUV and the leader UUV, as depicted in Figure 3. Then, the real relative distance $L_{l f}$ and real relative angle $ψ_{l f}$ are used to design the formation controller for the follower UUVs.

Figure 3.

The leader–follower relative motion.

For the convenience of analysis and controller design, the linear velocity in sway is neglected here. So, based on equation (1), the simplified kinematics of the leader UUV can be defined as

\{\begin{matrix} {\dot{x}}_{l} = u_{l} cos ψ_{l} \\ {\dot{y}}_{l} = u_{l} sin ψ_{l} \\ {\dot{ψ}}_{l} = r_{l} \end{matrix}

The simplified kinematics of the follower UUV can be defined as

\{\begin{matrix} {\dot{x}}_{f} = u_{f} cos ψ_{f} \\ {\dot{y}}_{f} = u_{f} sin ψ_{f} \\ {\dot{ψ}}_{f} = r_{f} \end{matrix}

In equation (2) and equation (3), $(x_{l}, y_{l})$ , $ψ_{l}$ , u_l , and r_l are the position, heading angle, linear velocity in surge, and angular velocity in yaw of the leader UUV, respectively. $(x_{f}, y_{f})$ , $ψ_{f}$ , u_f , and r_f are the position, heading angle, linear velocity in surge, and angular velocity in yaw of the follower UUV, respectively.

Then, the leader–follower relative motion model expressed with $L_{l f}$ and $ψ_{l f}$ can be obtained as

{\dot{L}}_{l f} = u_{l} cos (ψ_{l} - α) - u_{f} cos (ψ_{f} - α) = - u_{l} cos ψ_{l f} + u_{f} cos (ψ_{l f} + ψ_{l} - ψ_{f})

{\dot{ψ}}_{l f} = \frac{1}{L_{l f}} (u_{l} sin (ψ_{l} - α) - u_{f} sin (ψ_{f} - α)) - r_{l} = \frac{1}{L_{l f}} (u_{l} sin ψ_{l f} - u_{f} sin (ψ_{l f} + ψ_{l} - ψ_{f})) - r_{l}

Remark 3. However, only a leader UUV and a follower UUV are shown in Figure 3. Extending to formation problem of a group of three UUVs or more, the model is also applied and studied.

Assumption 4. The system states u_l , $ψ_{l}$ , r_l , u_f , $ψ_{f}$ , r_f of the leader UUV and the follower UUV are available for feedback and $0 \leq ψ_{l}, ψ_{f} < 2 π$ , $0 \leq r_{l}, r_{f} < π / 2$ from physical constraints.

Assumption 5. The leader UUV and the follower UUV can move forward and stop, that is, $u_{l} \geq 0$ and $u_{f} \geq 0$ , for all $t \geq 0$ .

Formation controller design

In this section, a formation control structure in absence of position information of the follower UUVs is proposed based on local sensing means mentioned above. Then, the formation controller is designed using backstepping and Lyapunov methods for the follower UUVs.

Formation control structure

The formation control structure designed in this article is shown in Figure 4. Because the leader UUV is responsible for guiding and navigating to mission waypoints, the formation control design is mainly applied to the follower UUV. According to the principle of leader–follower formation control, as long as the desired relative distance and angle between the follower UUV and the leader UUV are set, the desired position and angle of follower UUV are also uniquely determined. So, for the follower UUV, the control reference inputs are the desired relative distance L_D and angle $ψ_{D}$ . It can be found in Figure 4 that there are two-layer controllers: a formation controller and a proportional-integral-derivative (PID) controller. The formation controller is responsible for generating the desired commands of velocity and heading for the follower UUV to keep the formation. The classical PID controller is responsible for generating the desired commands of thrust and rudder angle to realize velocity and heading control for the follower UUV.

Figure 4.

The formation control structure of leader–follower UUVs. UUV: unmanned underwater vehicle.

In Figure 4, it can also be found that the formation controller does not require the position information of the follower UUV. Because the local sensing means is adopted in this article, the information of relative distance $L_{l f}$ and relative angle $ψ_{l f}$ obtained by local acoustic positioning is used for the formation controller. Other information used for the formation controller include u_l , $ψ_{l}$ , r_l of the leader UUV sent by acoustic communication and u_f , $ψ_{f}$ fed by the follower UUV self. It should be noticed that r_l of the leader UUV is used to design formation controller of the follower UUV. Usually, for leader–follower formation control approach, the follower UUV needs to track the heading angle $ψ_{l}$ of the leader UUV. However, the r_l can denote the changing trend of the leader UUV, which is beneficial for tracking the heading angle for the follower UUV. So, r_l of the leader UUV is used as an input of the formation controller for the follower UUV to produce heading angle command $ψ_{f_com}$ . As mentioned above, the integrated communication and positioning technology is assumed to use in the multi-UUV system. So, the information of $L_{l f}$ , $ψ_{l f}$ , u_l , $ψ_{l}$ , and r_l is sent together from the leader UUV to all follower UUVs periodically.

Controller design

In this section, the formation controller of the follower UUV in absence of position information is designed. First, the leader–follower formation control error model is established based on the desired relative distance L_D and desired relative angle $ψ_{D}$ . Considering the complexity of directly establishing the error model using $L_{l f} \to L_{D}$ and $ψ_{l f} \to ψ_{D}$ , the real relative distance $L_{l f}$ is decomposed as $L_{l f x}$ and $L_{l f y}$ corresponding to real relative angle $ψ_{l f}$ , which is shown in Figure 3. Then, the formation control errors can be established with $L_{l f x} \to L_{l f x D}$ and $L_{l f y} \to L_{l f y D}$ as

\{\begin{matrix} e_{1} = L_{l f x D} - L_{l f x} \\ e_{2} = L_{l f y D} - L_{l f y} \\ e_{3} = ψ_{l} - ψ_{f} \end{matrix}

According to geometric relationship shown in Figure 3 and let $β = ψ_{l f} + ψ_{l} - ψ_{f} - \frac{π}{2}$ , then

\{\begin{matrix} e_{1} = L_{D} sin β_{D} - L_{l f} sin β \\ e_{2} = L_{D} cos β_{D} - L_{l f} cos β \\ e_{3} = ψ_{l} - ψ_{f} \end{matrix}

Then, let $β_{D} = ψ_{D} + ψ_{l} - ψ_{f} - \frac{π}{2}$ , the derivative of error dynamics along equation (7) can be written as

\{\begin{cases} {\dot{e}}_{1} = {\dot{L}}_{D} sin β_{D} + L_{D} {\dot{ψ}}_{D} cos β_{D} + L_{D} {\dot{e}}_{3} cos β_{D} - {\dot{L}}_{l f} sin β - L_{l f} {\dot{ψ}}_{l f} cos β - L_{l f} {\dot{e}}_{3} cos β \\ {\dot{e}}_{2} = {\dot{L}}_{D} cos β_{D} - L_{D} {\dot{ψ}}_{D} sin β_{D} - L_{D} {\dot{e}}_{3} sin β_{D} - {\dot{L}}_{l f} cos β + L_{l f} {\dot{ψ}}_{l f} sin β + L_{l f} {\dot{e}}_{3} s i n β \\ {\dot{e}}_{3} = r_{l} - r_{f} \end{cases}

According to the leader–follower relative motion model described in the third section, equation (9) is further obtained

\{\begin{matrix} {\dot{e}}_{1} = {\dot{L}}_{D} sin β_{D} + L_{D} cos β_{D} ({\dot{ψ}}_{D} + {\dot{e}}_{3}) - u_{l} cos e_{3} + u_{f} + r_{l} L_{l f} cos β - {\dot{e}}_{3} L_{l f} cos β \\ {\dot{e}}_{2} = {\dot{L}}_{D} cos β_{D} - L_{D} sin β_{D} ({\dot{ψ}}_{D} + {\dot{e}}_{3}) + u_{l} sin e_{3} - r_{l} L_{l f} sin β + {\dot{e}}_{3} L_{l f} sin β \end{matrix}

So, the final error model is

\{\begin{cases} {\dot{e}}_{1} = {\dot{L}}_{D} sin β_{D} + L_{D} cos β_{D} ({\dot{ψ}}_{D} + {\dot{e}}_{3}) - u_{l} cos e_{3} + u_{f} + r_{l} L_{l f} cos β - {\dot{e}}_{3} L_{l f} cos β \\ {\dot{e}}_{2} = {\dot{L}}_{D} cos β_{D} - L_{D} sin β_{D} ({\dot{ψ}}_{D} + {\dot{e}}_{3}) + u_{l} sin e_{3} - r_{l} L_{l f} sin β + {\dot{e}}_{3} L_{l f} sin β \\ {\dot{e}}_{3} = r_{l} - r_{f} \end{cases}

To obtain stable formation control to keep the desired relative distance and angle, backstepping and Lyapunov methods are used to design the formation controller. From equation (10), it can be found that the error model is not only dependent on the real relative distance $L_{l f}$ and relative angle $ψ_{l f}$ but also dependent on u_l and r_l of the leader UUV.

Consider the Lyapunov function candidate as follows

V = \frac{e_{1}^{2} + e_{2}^{2}}{2} + k_{1} (1 - cos e_{3}), k_{1} > 0

where k ₁ is the control parameter and $k_{1} > 0$ .

It can be clearly seen from the above Lyapunov function that $V \geq 0$ , and only when $e_{1} = e_{2} = e_{3} = 0$ , $V = 0$ .

The derivative of V can be described as

\dot{V} = e_{1} {\dot{e}}_{1} + e_{2} {\dot{e}}_{2} + k_{1} {\dot{e}}_{3} sin e_{3}

Substituting equation (8) into equation (12), we obtain

\begin{array}{l} \dot{V} = e_{1} ({\dot{L}}_{D} sin β_{D} + L_{D} {\dot{ψ}}_{D} cos β_{D} - u_{l} cos e_{3} + u_{f} + r_{l} L_{l f} cos β) \\ + e_{2} ({\dot{L}}_{D} cos β_{D} - L_{D} {\dot{ψ}}_{D} sin β_{D} + u_{l} sin e_{3} - r_{l} L_{l f} sin β) + (r_{l} - r_{f}) \\ (e_{1} L_{D} cos β_{D} - e_{1} L_{l f} cos β - e_{2} L_{D} sin β_{D} + e_{2} L_{l f} sin β + k_{1} sin e_{3}) \end{array}

Substituting $\{\begin{matrix} L_{D} sin β_{D} = e_{1} + L_{l f} sin β \\ L_{D} cos β_{D} = e_{2} + L_{l f} cos β \end{matrix}$ into equation (13), we obtain

\dot{V} = e_{1} ({\dot{L}}_{D} sin β_{D} + L_{D} {\dot{ψ}}_{D} cos β_{D} - u_{l} cos e_{3} + u_{f} + r_{l} L_{l f} cos β) + e_{2} ({\dot{L}}_{D} cos β_{D} - L_{D} {\dot{ψ}}_{D} sin β_{D} + u_{l} sin e_{3} - r_{l} L_{l f} sin β) + (r_{l} - r_{f}) k_{1} sin e_{3}

The designed control law of u_f is

u_{f} = - k_{2} e_{1} - {\dot{L}}_{D} sin β_{D} - L_{D} {\dot{ψ}}_{D} cos β_{D} + u_{l} cos e_{3} - r_{l} L_{l f} cos β

where k ₂ is control parameter and $k_{2} > 0$ .

Substituting equation (15) into equation (14), we obtain

\dot{V} = - k_{2} e_{1}^{2} + e_{2} ({\dot{L}}_{D} cos β_{D} - L_{D} {\dot{ψ}}_{D} sin β_{D} + u_{l} sin e_{3} - r_{l} L_{l f} sin β) + (r_{l} - r_{f}) k_{1} sin e_{3}

The designed control law of r_f is

r_{f} = k_{3} ({\dot{L}}_{D} cos β_{D} - L_{D} {\dot{ψ}}_{D} sin β_{D} + u_{l} sin e_{3} - r_{l} L_{l f} sin β) + C

where k ₃ is control parameter, $k_{3} > 0$ , and C is the additional item.

Substituting equation (17) into equation (16), we obtain

\dot{V} = - k_{2} e_{1}^{2} + e_{2} ({\dot{L}}_{D} cos β_{D} - L_{D} {\dot{ψ}}_{D} sin β_{D} + u_{l} sin e_{3} - r_{l} L_{l f} sin β) + (r_{l} - k_{3} ({\dot{L}}_{D} cos β_{D} - L_{D} {\dot{ψ}}_{D} sin β_{D} + u_{l} sin e_{3} - r_{l} L_{l f} sin β) - C) k_{1} sin e_{3}

Simplify and obtain

\dot{V} = - k_{2} e_{1}^{2} + (e_{2} - k_{1} k_{3} sin e_{3}) ({\dot{L}}_{D} cos β_{D} - L_{D} {\dot{ψ}}_{D} sin β_{D} + u_{l} sin e_{3} - r_{l} L_{l f} sin β) + (r_{l} - C) k_{1} sin e_{3}

It can be found that $- k_{2} e_{1}^{2} \leq 0$ . So, define W

W = (e_{2} - k_{1} k_{3} sin e_{3}) ({\dot{L}}_{D} cos β_{D} - L_{D} {\dot{ψ}}_{D} sin β_{D} + u_{l} sin e_{3} - r_{l} L_{l f} sin β) + (r_{l} - C) k_{1} sin e_{3}

where

W \leq (e_{2} + k_{1} k_{3}) (|{\dot{L}}_{D}| + L_{D} |{\dot{ψ}}_{D}| + u_{l} + r_{l} L_{l f}) + k_{1} (r_{l} + |C|) = D

In equation (21), $e_{2} \to 0$ , C is bounded. Then, W must be bounded. So, $\dot{V} \leq - k_{2} e_{1}^{2} + D$ . Furthermore, the formation control errors can be stable by setting the values of k ₃ and C in the controller. Here, we select $k_{3} = \frac{e_{2}}{k_{1} sin e_{3}}$ , $C = r_{l}$ . Then, $\dot{V} = - k_{2} e_{1}^{2} < 0$ .

Finally, the control law is

\{\begin{matrix} u_{f_com} = - k_{2} e_{1} - {\dot{L}}_{D} sin β_{D} - L_{D} {\dot{ψ}}_{D} cos β_{D} + u_{l} cos e_{3} - r_{l} L_{l f} cos β \\ r_{f_com} = \frac{e_{2}}{k_{1} sin e_{3}} ({\dot{L}}_{D} cos β_{D} - L_{D} {\dot{ψ}}_{D} sin β_{D} + u_{l} sin e_{3} - r_{l} L_{l f} sin β) + r_{l} \end{matrix}

It should be noted that equation (22) gives the $r_{f_com}$ command. However, in this article, the $ψ_{f_com}$ command is needed for the follower UUV control, as shown in Figure 4. In fact, the $r_{f_com}$ command and the $ψ_{f_com}$ command can both be used for formation control. The two commands can be mutually converted according to the kinematics of UUV. The reason of using $ψ_{f_com}$ command is considering practical application to reduce the cost of the follower UUV. This article assumes that the follower UUV is not equipped with navigation equipment. So, the angular velocity in yaw r_f of the follower UUV cannot be obtained. However, compass is usually equipped for the follower UUV. That is to say, the heading angle $ψ_{f}$ can be obtained. So, in our control method, the $r_{f_com}$ command is converted to the $ψ_{f_com}$ command for the follower UUV control. According to equation (3), the $ψ_{f_com}$ command of the follower UUV is easy to be obtained using the real $ψ_{f}$ and $r_{f_com}$ commands.

Simulation results

In this section, the simulation results of formation control are presented to demonstrate the performance of the proposed control structure and method. There are five UUVs for simulation: one leader and four followers. Meanwhile, two configurations of rectangle and triangle are designed for the leader–follower formation. In addition, different communication delays are set in the two formation configurations to verify the effect when suffering information delays from the leader UUV to the follower UUVs.

For rectangle formation simulation, the initial information of positions $(x, y)$ and heading angles $ψ$ of all UUVs and the desired relative distances L_D and desired relative angles $ψ_{D}$ for the follower UUVs are presented in Table 1. In the simulation, velocity command in surge of the leader UUV is set as 2 m/s. The information delays from the leader UUV to all follower UUVs are set as 3 s. The two control parameters k ₁ and k ₂ in the final control law of equation (22) are set as 0.05 and 0.02, respectively. Figure 5 shows the trajectories of all UUVs from the initial formation to the desired formation. Figure 6 shows the real relative distances and angles of the follower UUVs in the formation. Figure 7 shows the linear velocities in surge and heading angles of all UUVs.

Table 1.

Initial information, desired relative distance and angle for rectangle formation.

UUV	Initial x (m)	Initial y (m)	Initial $ψ$ (°)	L_D (m)	$ψ_{D}$ (°)
Leader	40	40	45	—	—
Follower 1	45	−7.7	55	37.7	135
Follower 2	2.3	40	35	37.7	225
Follower 3	−5.7	−43.3	55	84.3	162
Follower 4	−45.3	12.3	35	84.3	198

UUV: unmanned underwater vehicle.

Figure 5.

The trajectories of the leader UUV and follower UUVs for rectangle formation. UUV: unmanned underwater vehicle.

Figure 6.

(a) The relative angles and (b) relative distances of the follower UUVs for rectangle formation. UUV: unmanned underwater vehicle.

Figure 7.

(a) The heading angles and (b) velocities of the leader UUV and follower UUVs for rectangle formation. UUV: unmanned underwater vehicle.

From Figure 5, it can be found that the leader UUV is responsible for navigating to waypoints and all follower UUVs maintain desired relative distances L_D and desired relative angles $ψ_{D}$ with respect to the leader UUV. It can also be seen in Figure 5 that the initial formation is not the desired rectangle. The reason is that the positions of all follower UUVs are randomly placed in the simulation. The results show that the follower UUVs can successfully constitute and maintain the desired rectangle formation after about 65 s, which can be shown in Figure 6. For the follower UUVs, the initial velocities are set as 0 and the initial heading angles are not accordant with the leader UUV. So, the velocities and heading angles of the follower UUVs are adjusted according to the velocity and heading angle of the leader UUV under the formation constraint. This can be distinctly found in Figure 7.

From Figure 6, it can be found that the follower UUVs well maintain the desired relative distance and angle and track the velocity and heading of the leader UUV. However, when the leader UUV makes a turn, the real relative distance and angle of the follower UUVs are deflected with the desired value, which made the maintenance effect of the formation a little poor. This is because when the leader UUV turns, its heading angle and angular velocity in yaw change greatly, which increase the control difficulty of the follower UUVs. It also can be analyzed by the final control law of equation (22). The control commands of linear velocity in surge and angular velocity in yaw of the follower UUVs are affected by the r_l of the leader UUV. When the leader UUV turns, the large change of r_l results in the fluctuation of the $u_{f_com}$ and $r_{f_com}$ control command of the follower UUVs. So, with the control command fluctuation of the follower UUVs, the whole formation control effect is a little poor. Moreover, the control effects of the follower UUVs are different according to the different position in the desired formation configuration.

For triangle formation simulation, the initial information of positions $(x, y)$ and heading angles $ψ$ of all UUVs, the desired relative distances L_D , and desired relative angles $ψ_{D}$ for the follower UUVs are presented in Table 2. Velocity command in surge of the leader UUV is also set as 2 m/s. The information delays from the leader UUV to all follower UUVs are set as 6 s. The two control parameters of k ₁ and k ₂ in triangle formation simulation are same as the rectangle formation simulation. Figure 8 shows the trajectories of all UUVs in the formation. Figure 9 shows the real relative distances and angles of the follower UUVs. Figure 10 shows linear velocities in surge and heading angles of all UUVs. It can be found that, although the information delay time increases, the follower UUVs can also successfully constitute and maintain the desired triangle formation by controlling each real relative distance and angle to the desired values. The time from the initial formation to the desired triangle formation is about 85 s, which can be shown in Figure 9. The velocities and heading angles of the follower UUVs are also tracked well with the leader UUV. Similarly, when the leader UUV turns, the maintenance effect of the formation is also a little poor. The reason is the same as the rectangle formation simulation.

Table 2.

Initial information, desired relative distance and angle for triangle formation.

UUV	Initial x (m)	Initial y (m)	Initial $ψ$ (°)	L_D (m)	$ψ_{D}$ (°)
Leader	40	40	45	—	—
Follower 1	45	−15	55	40	135
Follower 2	0	30	35	40	225
Follower 3	30	−65	55	80	135
Follower 4	−55	−20	35	80	225

UUV: unmanned underwater vehicle.

Figure 8.

The trajectories of the leader UUV and follower UUVs for triangle formation. UUV: unmanned underwater vehicle.

Figure 9.

(a) The relative angles and (b) relative distances of the follower UUVs for triangle formation. UUV: unmanned underwater vehicle.

Figure 10.

(a) The heading angles and (b) velocities of the leader UUV and follower UUVs for triangle formation. UUV: unmanned underwater vehicle.

Analysis on rationality and limitation of simulation

First, through the analysis of four aspects, the rationality of the simulation in the article is explained.

The formation control structure shown in Figure 4 is designed according to the method proposed in this article, considering the feasibility of engineering implementation. The simulation system is strictly constructed based on the control structure, including the leader and follower UUV model, actuator model, controller, underwater acoustic communication model, underwater acoustic positioning model, and so on. Therefore, it is reasonable to use the simulation system to verify the designed formation control structure and control method.

To conform to the actual situation, not only the kinematics model of UUV but also the dynamics model of UUV are established in simulation. In addition, the settings of sampling time and control period are consistent with the actual UUV system.

In simulation, the stated information obtained by the leader UUV and the follower UUVs and the acoustic interaction information sent by the leader UUV to follower UUVs is designed according to the actual situation. The specific design is as follows: the leader UUV can obtain its own position, heading angle, and linear and angular velocities. For follower UUVs, only their own heading angles, and linear and angular velocities can be obtained. For follower UUVs, the relative distance and angle with respect to the leader UUV, as well as the heading angle, and the linear and angular velocities of the leader UUV can be obtained through acoustic communication and acoustic positioning.

In simulation, the delay of underwater acoustic information interaction is set and simulated according to the actual characteristics of information processing and propagation in underwater acoustic communication and positioning.

However, the simulation has the following two limitations:

In simulation, the kinematics and dynamics model of UUV are appropriately simplified. The simplification is beneficial to the analysis of the problem and the design of formation controller but will cause the motion response of single UUV to be inconsistent with the actual situation. However, because all UUV models are simplified in the same way, it will not affect the effect of the whole formation, nor affect the verification for the formation control method proposed.

In simulation, there is no ocean environment model, mainly the ocean current model. Therefore, there is no ocean current interference in the simulation results of formation control. In fact, formation control under ocean current interference is an important and special research problem of UUV formation control. And, some solutions have been proposed in related research. However, the focus of this article is not on the ocean current interference.

Conclusion

This article researches the problems of reducing the communication quantity and not using position information of the follower UUVs in the leader–follower UUV formation control approach. So, local sensing means is proposed for the multi-UUVs to solve the above problems. Then, the information needed for the follower UUVs to track the leader UUV can be obtained both by acoustic positioning and acoustic communication. The acoustic positioning provides the relative distance and relative angle to the follower UUVs. The acoustic communication only provides a small amount of information, including linear velocity in surge, heading angle, and angular velocity in yaw of the leader UUV to the follower UUVs. Then, the formation control structure in absence of follower position information is proposed and the formation controller is designed. The effectiveness of the method is simulated by two formation configurations of rectangle and triangle with five UUVs. The simulation results show that the proposed control structure and controller are effective for the leader–follower UUV formation control. Under the condition of random position place, the follower UUVs can successfully constitute and maintain the desired formation by controlling the real relative distance and angle. Also, there are two information delays set in the simulations. The simulation results show that the method can be well applied to relatively large information delays for UUV formation control using acoustic communication. However, the formation control structure and method proposed in this article have specific application area which is that the multi-UUVs need to adopt integrated acoustic communication and positioning means for information interaction.

Footnotes

Acknowledgments

The authors would like to thank all the editors and anonymous reviewers for improving this article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the National Nature Science Foundation of China under grant no. 51809060, 51679057, and 51609048 and by the Natural Science Foundation of Heilongjiang Province of China under grant no. E2017014.

ORCID iD

Tao Chen

References

Emrani

Dirafzoon

Talebi

. Leader-follower formation control of autonomous underwater vehicles with limited communications. In: International conference on control applications, Denver, CO, USA, 28–30 September 2011, pp. 921–926. IEEE.

Yang

Wang

Zhang

. A decoupled controller design approach for formation control of autonomous underwater vehicles with time delays. IET Control Theory Appl 2013; 7(15): 1950–1958.

Mehrjerdi

Ghomman

Saan

. Nonlinear coordination control for a group of mobile robots using a virtual structure. Mechatronics 2011; 21(7): 1147–1155.

Askari

Mortazavi

Talebi

. UAV formation control via the virtual structure approach. J Aerosp Eng 2015; 28(1): 04014047.

Xue

Zhang

Multi-UUV formation coordination control based on combination of virtual structure and leader. In: International conference on mechatronics and automation, Changchun, China, 5–8 August 2018, pp. 1574–1579. IEEE.

Hou

Allen

. Intelligent behaviour-based team UUVs cooperation and navigation in a water flow environment. Ocean Eng 2008; 35 (3): 400–416.

Lee

Chwa

. Decentralized behavior-based formation control of multiple robots considering obstacle avoidance. Intel Serv Robot 2018; 11: 127–138.

Belleter

DJW

Pettersen

KY.

Underactuated leader-follower synchronisation for multi-agent systems with rejection of unknown disturbances. In: American control conference, Chicago, IL, USA, 1–3 July 2015, pp. 3094–3100. IEEE.

Edwards

Bean

Odell

, et al. A leader-follower algorithm for multiple AUV formations. In: IEEE/OES autonomous underwater vehicles conference, Sebasco, ME, USA, 17–18 June 2004, pp. 40–46. IEEE.

10.

Yan

Chen

, et al. Leader-follower formation control of UUVs with model uncertainties, current disturbances, and unstable communication. Sensors 2018; 18: 662.

11.

Jia

Formation control and obstacle avoidance algorithm of multiple autonomous underwater vehicles(AUVs) based on potential function and behavior rules. In: International conference on automation and logistics, Jinan, China, 18–21 August 2007, pp. 569–573. IEEE.

12.

Yan

Song

, et al. Dynamic formation control for autonomous underwater vehicles. J Cent South Univ 2014; 21: 113–123.

13.

Liang

Wang

, et al. Swarm control with collision avoidance for multiple underactuated surface vehicles. Ocean Eng 2019; 191: 1–10.

14.

Cheng

Cao

. Survey of cooperative path planning for multiple unmanned aerial vehicles. In: Asian pacific conference on mechanical components and control engineering, Tianjin, China, 20–21 September 2014, pp. 388–393.

15.

Yoo

Park

. Formation tracking control for a class of multiple mobile robots in the presence of unknown skidding and slipping. IET Control Theory Appl 2013; 7(5): 635–645.

16.

K-K

Park

M-C

Ahn

H-S

. A survey of multi-agent formation control. Automatica 2015; 53: 424–440.

17.

Park

. Adaptive formation control of underactuated autonomous underwater vehicle. Ocean Eng 2015; 96: 1–7.

18.

Gao

Guo

. Adaptive formation control of autonomous underwater vehicles with model uncertainties. Int J Adapt Control Signal Process 2018; 32: 1067–1080.

19.

Cui

How

BVE

, et al. Leader-follower formation control of underactuated autonomous underwater vehicles. Ocean Eng 2010; 37: 1491–1502.

20.

Das

Subudhi

Pati

. Cooperative formation control of autonomous underwater vehicles: an overview. Int J Autom Comput 2016; 13: 199–225.

21.

Liang

Wang

, et al. A novel distributed and self-organized swarm control framework for underactuated unmanned marine vehicles. IEEE Access 2019; 7: 112703–112712.

22.

Liang

Hou

, et al. Distributed coordinated tracking control of multiple unmanned surface vehicles under complex marine environments. Ocean Eng 2020; 205: 1–9.

23.

Zhu

Qian

. A survey on formation control algorithms for multi-AUV system. Unmanned Syst 2014; 2(4): 351–359.

24.

Das

Subudhi

Pati

. Co-operative control coordination of a team of underwater vehicles with communication constraints. Trans Inst Meas Control 2016; 38(4): 463–481.

25.

Rashidi

Mohammadloo

. Simultaneous cooperative localization for AUVs using range-only sensors. Int J Inform Acquis 2011; 8(2): 117–132.

26.

Chen

Yan

. Globally finite-time stable three-dimensional trajectory-tracking control of underactuated UUVs. Ocean Eng 2019; 189: 106329.

27.

Fossen

. Guidance and control of ocean vehicle. New York: Wiley, 1994, pp. 53–157.

Formation control of unmanned underwater vehicles using local sensing means in absence of follower position information

Abstract

Keywords

Introduction

Problem statement

Modeling of leader–follower unmanned underwater vehicles

Simplified kinematics of unmanned underwater vehicles

Leader–follower relative motion model

Formation controller design

Formation control structure

Controller design

Simulation results

Analysis on rationality and limitation of simulation

Conclusion

Footnotes

Acknowledgments

Declaration of conflicting interests

Funding

ORCID iD

References