Local Path Planning of Unmanned Surface Vehicles’ Formation Based on Vector Field and Flow Field Traction

Abstract

Formation obstacle avoidance is an essential attribute of the cooperative task in unmanned surface vehicle (USV) formation. In real-world scenarios involving multiple USVs, both formation obstacle avoidance and formation recovery after obstacle avoidance play a critical role in ensuring the success of collaborative missions. In this study, an Interfered Fluid Dynamic System (IFDS) algorithm was used for obstacle avoidance due to its excellent robustness, high computational efficiency and path smoothness. The algorithm can provide good local path planning for USVs. However, the use of the IFDS on USVs still has the defect of local extreme values, which has been effectively modified to obtain an enhanced IFDS (EIFDS). In formation, based on the leader–follower method, the virtual leader was used to determine the desired position of USVs in formation, and the streamlines generated by the EIFDS guided the USVs. In order to make the formation converge to the desired formation better, the vector and scalar of the EIFDS algorithm were uncoupled, and different designs were made to achieve convergence to the desired formation. The interfered residue of the IFDS is not suitable for addressing collision avoidance between USVs in practice. Therefore, the vector field method was employed to tackle the issue, with some enhancements made to optimize its performance. Subsequently, a weighted separation method was applied to combine the vector field and EIFDS, resulting in a composite field solution. Finally, the formation obstacle avoidance strategy based on composite fields was formed. The feasibility of this scheme was verified by simulation, and compared with the single IFDS formation method, the pairwise spacing of USVs behind obstacles could be increased, and the reliability of formation obstacle avoidance was increased.

1. Introduction

With the development of automation and intelligence, unmanned surface vehicles (USVs) have emerged as a pivotal component of marine intelligence. In comparison to conventional surface vessels, USVs offer distinct advantages such as compactness, agility, cost-effectiveness, efficiency, and autonomy in operation under challenging conditions [1]. They can collaborate with manned ships to execute tasks in intricate environments and are capable of independent task execution. However, for highly complex missions, a single USV may not suffice to meet the requirements. Consequently, research on multi-USV formations has gained prominence as a means to enhance mission effectiveness [2,3,4].

In the context of formation obstacle avoidance, the following practical scenario can be considered: Suppose there is a formation composed of multiple USVs executing a mission. The formation advances along a predetermined path, but obstacles are present on the path. These obstacles may be static (such as buoys or reefs) or dynamic (such as other vessels or floating objects). To successfully complete the mission, the formation needs to reasonably avoid these obstacles. During the obstacle avoidance process, each USV must maintain a safe distance from others to prevent collisions. After successfully avoiding the obstacles, the formation needs to readjust its configuration and return to the predetermined formation to continue the mission. The above description can be summarized into three key issues in formation obstacle avoidance: (1) the obstacle avoidance capability of individual members; (2) the need to maintain safe distances between members; and (3) the ability to restore the formation after obstacle avoidance (i.e., the stability of the formation structure).

2. Related Work

At present, the majority of scholarly research is focused on single USV obstacle avoidance, with relatively less exploration and fewer research findings in the area of obstacle avoidance for USV formations. Yoo S J et al. proposed a dynamic error model to achieve obstacle avoidance while maintaining communication function [5]. Park B S et al. proposed a novel route control method that utilizes heading control to navigate around obstacles while maintaining the original relative distance [6]. Dong Z et al. employed MPC to achieve coordinated formation tracking and collision avoidance [7].

The artificial potential field (APF) method is widely used for path planning and obstacle avoidance. Song L F et al. proposed a dual-layer obstacle avoidance scheme integrating the velocity obstacle (VO) and artificial potential field (APF) methods. The velocity obstacle (VO) method is employed to achieve stable and controllable control beyond the predetermined distance. In cases where the distance between the unmanned surface vehicle (USV) and the agent falls below the predetermined value, artificial potential fields (APFs) are utilized for emergency obstacle avoidance [8]. Furthermore, the consistency algorithm has been used in combination with APFs to adjust the formation distance between USVs to ensure collision avoidance [9,10]. He X Y et al. established a collision avoidance algorithm based on potential field function by adjusting motion parameters [11]. Furthermore, Wu W et al. proposed a tracking control method based on the virtual target point strategy and used APFs to refine the control method [12].

Han X et al. adjusted the trajectory parameters of a tracking system by integrating obstacle information, thereby achieving a seamless integration of the obstacle avoidance path and the tracking path [13]. To address the limitations of conventional VO in certain environments, Yuan X L and Huang Y et al. proposed a nonlinear VO-based collision avoidance method for unmanned boats [14,15]. In response to unknown external disturbances in VO avoidance, Zhang G Q et al. introduced a virtual first-order guidance approach. By tracking and evading the guidance body, it provides effective reference information for practical obstacle avoidance while dynamically adjusting compensation based on tracking error. The advantage of using a virtual guidance body lies in its ability to obtain a simplified ideal model, thus reducing computational complexity in parameter calculation [16].

Due to the incomplete motion of USVs, Zhang Z et al. employed the linked incomplete model to develop an obstacle avoidance control method, initially addressing the orientation-free characteristic of the model [17,18]. Compared with a single-obstacle environment, navigating multiple obstacles poses greater challenges. Lv G et al. established obstacle avoidance constraint conditions based on barrier functions and formulated a quadratic programming model [19,20].

The aforementioned studies either focused on a singular collision avoidance issue or solely addressed the avoidance of an individual obstacle. In obstacle avoidance methods, the artificial potential field (APF) suffers from the drawback of challenging parameter tuning. Meanwhile, model predictive control (MPC) not only relies on a high-precision motion model but also faces difficulties in determining the prediction step size and fails to consider the impact of external disturbances and obstacle maneuvers on the model. Given the intricate dynamic characteristics and substantial uncertainty associated with unmanned boats, addressing obstacle avoidance in trajectory tracking for unmanned boat formations represents a valuable research endeavor. Presently, research on obstacle avoidance methods predominantly centers around algorithm fusion and strategic innovation [21]. The advantages and disadvantages of the above mainstream schemes are shown in

Table 1. Current mainstream solutions to the obstacle avoidance problem.

Solution	Instance	Disadvantage
Based on geometric analysis methods	Velocity obstacle method	Unable to adapt quickly to changes in dynamic environments
Model predictive control (MPC) methods	Nonlinear MPC method	High computational requirements and workload; low computational efficiency
Stochastic planning methods	Rapidly exploring random tree algorithm	Low calculation efficiency in complex environments
Methods of simulating force fields	Artificial potential field (APF) method	Many local minima problems
Intelligent optimization algorithms	Neural networks and reinforcement learning	Sample inefficiency and generalizability problems; poor practicality

.

The study of obstacle avoidance for formations of unmanned vessels necessitates a direct and reliable approach that can be readily implemented in practical engineering. Drawing inspiration from the natural phenomenon of water flowing around obstacles, we developed a ship capable of seamless navigation and obstacle avoidance, ensuring consistent arrival at its destination. While simulation methods for fluid flow around obstacles are available, including those based on stream function, the interfered fluid dynamical system (IFDS) proposed by Wang Hong et al. emerges as the most valuable approach in research terms [22,23,24,25,26]. This fundamental method, grounded in dynamical systems (DSs) [27] with lower computational costs, compared to alternative obstacle avoidance methods, the IFDS exhibits superior robustness, reduced computational requirements, seamless path generation, and high adaptability. In complex and unfamiliar environments, particularly for local route planning purposes, the IFDS retains significant advantages in terms of obstacle avoidance performance, real-time capabilities, and path quality. Having a smooth, high-quality path is critical for USVs, as it greatly improves the effectiveness of control during subsequent path tracking. The IFDS method, in particular, presents a contrast to the aforementioned reference methods. It exhibits a reduced number of local extremes compared to APFs and demonstrates a smoother path and higher computational efficiency when compared to VO. The aforementioned basis enables the provision of effective path planning for formation obstacle avoidance, thereby enhancing the robustness of formation obstacle avoidance.

Therefore, this study will use the IFDS algorithm as the basis to build an obstacle avoidance framework as follows:

(1)

In Section 4 an enhanced IFDS is described. The traditional IFDS has problems of stagnation points and trap areas in two-dimensional environments. A modulation matrix is added on the basis of the IFDS to deal with the stagnation point problem. The problem of trap areas is solved by using the virtual target point method, and the process of determining a virtual target point is described.

(2)

In Section 5, the formation obstacle avoidance structure based on streamlined traction is constructed. The traditional virtual leader–follower formation structure is enhanced by incorporating streamlined traction and velocity amplitude constraints. The expected position of USVs in formation is determined by using the virtual leader–follower method. The EIFDS is used to connect the desired position to the USVs so that the USVs can reach the desired position and have obstacle avoidance capabilities. In order to maintain the formation structure, the velocity vector and velocity scalar of the EIFDS are decoupled, and the velocity scalar and vector are adjusted according to the distance between USVs and the desired position, so as to better maintain the formation.

(3)

In Section 6, in the obstacle avoidance between members of the formation, the IFDS interferes with the residual defects and is difficult to change due to the existence of the problem. In the member collision avoidance problem, the IFDS algorithm is abandoned, the vector field method is used, and this scheme is combined with the EIFDS by weight allocation, and a composite field formation obstacle avoidance strategy is formed.

3. Problem Formulation

The scenario assumes the presence of J USVs tasked with performing formation obstacle avoidance missions.

The kinematics equation of the j-th USV is considered as follows:

$$\left\{\begin{array}{c}{\dot{x}}_j=V_j\cos {\psi }_j\\ {\dot{y}}_j=V_j\sin {\psi }_j\\ {\dot{\psi }}_j={\varphi }_j+{\phi }_j\end{array}\right.$$

(1)

where $x_j,y_j\in R$ represents the position of the j-th USV in the geodetic coordinate system; ${\psi }_j$ indicates the heading angle of the USV; $V_j=\sqrt{{\mu }_j^2+v_j^2}$ represents the combined velocity amplitude of the USV, where ${\mu }_j$ and $v_j$ represent the longitudinal velocity amplitude and transverse velocity amplitude of the USV, respectively. ${\varphi }_j$ is the yaw angular velocity; ${\phi }_j=\arctan (v,\mu )\in (-\pi ,\pi ]$ is the sideslip angle.

The matrix $\Delta$ of $J\times 3$ is used to represent the positions of J USVs, as follows:

$$\Delta ={\left[\begin{array}{ccccc}{\eta }_1& {\eta }_2& {\eta }_3& \dots & {\eta }_J\end{array}\right]}^T$$

(2)

The designated position of J USVs in the formation is ${\Delta }_F$, as follows:

$${\Delta }_F={\left[\begin{array}{ccccc}{\eta }_{1F}& {\eta }_{2F}& {\eta }_{3F}& \dots & {\eta }_{JF}\end{array}\right]}^T$$

(3)

In order to form a complete formation, the USVs must converge towards the desired position (even if $\Delta$ converges to ${\Delta }_F$).

During this process, each USV must ensure a safe distance $R_k$ from the obstacle to avoid collision, as follows:

$$‖{\xi }_j-{\xi }_k‖>R_k$$

(4)

where ${\xi }_k=(x_k,y_k)$ is the position of the center point of the k-th obstacle; ${\xi }_j=(x_j,y_j)$.

At the same time, the formation members should also maintain an appropriate distance ${\hslash }_j^{jj}={\hslash }_j+{\hslash }_{jj}$ to prevent collision, as follows:

$$‖{\xi }_j-{\xi }_{\mathrm{jj}}‖>{\hslash }_j^{jj}$$

(5)

where ${\xi }_{jj}=(x_{jj},y_{jj})$ is the position of the adjacent USV.

4. Obstacle Avoidance Strategy

This section describes the EIFDS method, which enables a USV to have local path planning capability, and the USV can avoid obstacles and reach the desired position under the traction of the generated streamlines of the EIFDS.

4.1. Obstacle Modeling

Initially, a model of the obstacles is constructed, taking into account the high precision of radars and other sensors in target detection. To enhance the obstacle avoidance efficiency, we simplify the representation of obstacles (such as islands and port facilities) as circles, ellipses, rectangles, polygons, or combinations thereof, while boats and small vessels are represented as ellipses or circles at time $t$ within the terrestrial coordinate system. In order to standardize the representation of these obstacles, the obstacle modeling is represented by the following Equation (1) [28].

$${\wp }_k({\xi }_j)={({\displaystyle \frac{x_j-x_k}{a_1}})}^{2s_1}+{({\displaystyle \frac{y_j-y_k}{a_2}})}^{2s_2}$$

(6)

In the formula, parameters $a_1$ and $a_2$ can be adjusted to modify the axial length of the obstacle, while parameters $s_1$ and $s_2$ can be tuned to alter the shape of the obstacle. By manipulating these parameters, it is possible to adapt the obstacle’s shape to correspond with real-world boundary features. The aforementioned model is capable of accommodating most obstacles; in cases involving very large obstacles, multiple units can be combined to match the boundary volume.

4.2. Enhanced IFDS Algorithm

To ensure continuous navigation of the unmanned surface vehicle (USV) towards the designated target point, an initial flow field is established. The target point is set as ${\xi }_g=(x_g,y_g)$, and the initial smooth flow formula is expressed as follows:

$$U_j={\left[\begin{array}{cc}{\displaystyle \frac{(x_g-x_j)}{\sqrt{{(x_g-x_j)}^2+{(y_g-y_j)}^2}}}& {\displaystyle \frac{(y_g-y_j)}{\sqrt{{(x_g-x_j)}^2+{(y_g-y_j)}^2}}}\end{array}\right]}^T$$

(7)

From Equation (7), it can be seen that at any time t, the velocity vector of the USV always points toward the target point, regardless of the position of the target point in the coordinate system.

Based on Equation (7), we add K obstacles to the initial flow field to induce perturbations, and the expression is as follows:

$${\stackrel{•}{U}}_j=\overline{F}{U}_j$$

(8)

$$\overline{F}={\displaystyle \sum _{k=1}^K{W}_k{F}_k}$$

(9)

In this Equation, ${\overline{U}}_j$ denotes the overall flow field, $\overline{F}$ represents the total disturbance, and $F_k$ and $W_k$ denote the disturbance and weight of the kth obstacle, respectively. The formula is expressed as follows:

$$F_k=I-{\displaystyle \frac{n_k{n}_k^T}{{\left|{\wp }_k\right|}^{1/{\eta }_k}{n}_k^T{n}_k}}+{\displaystyle \frac{i{\tau }_k{n}_k^T}{{\left|{\wp }_k\right|}^{1/{\varphi }_k}{n}_k^T{n}_k{C}_k}}$$

(10)

$$W_k=\left\{\begin{array}{cc}\begin{array}{c}1\\ {\displaystyle {\prod }_{i=1,i\ne k}^K\frac{({\wp }_i-1)}{({\wp }_i-1)+({\wp }_k-1)}}\end{array}& \begin{array}{c}K=1\\ K\ne 1\end{array}\end{array}\right.$$

(11)

where $I$ represents the identity matrix. The disturbance matrix of the kth obstacle, denoted as $-{\displaystyle \frac{n_k{n}_k^T}{{\left|{\wp }_k\right|}^{1/{\eta }_k}{n}_k^T{n}_k}}$, signifies the perturbation caused by the kth obstacle on the initial flow field. ${\eta }_k$ denotes the disturbance coefficient, indicating that an increase in its value corresponds to a higher degree of disturbance. $n^k={\left[\begin{array}{cc}{\displaystyle \frac{\partial {\wp }_k}{\partial x}}& {\displaystyle \frac{\partial {\wp }_k}{\partial y}}\end{array}\right]}^T$ represents the normal vector associated with the surface of the kth obstacle. ${\displaystyle \frac{i{\tau }_k{n}_k^T}{{\left|{\wp }_k\right|}^{1/{\phi }_k}{n}_k^T{n}_k{C}_k}}$ is the new modulation matrix, which is introduced to address the stagnation point problem in the two-dimensional plane of the IFDS, as discussed in our previous work [29]. ${\tau }^k={\left[\begin{array}{cc}-{\displaystyle \frac{\partial {\wp }_k}{\partial y}}& {\displaystyle \frac{\partial {\wp }_k}{\partial x}}\end{array}\right]}^T$ represents a vector perpendicular to the normal vector $n_k$, while ${\phi }_k$ denotes the modulation coefficient. Moreover, there exists a positive correlation between the degree of deviation from the stagnation point and the modulation coefficient, with a typical value usually around 2. $i\in \left[1,-1\right]$ has the capability to adjust the direction of deviation.

Upon the foundation of static obstacles, we have integrated a design to accommodate dynamic obstacles, resulting in modifications to the synthetic flow field ${\stackrel{•}{U}}_j$. The expression for the total flow field is as follows:

$${\stackrel{•}{U}}_j=\overline{F}(U_j-q_{\mathsf{\Gamma }})+q_{\mathsf{\Gamma }}$$

(12)

where $q_{\mathsf{\Gamma }}$ is the collective velocity vector of K obstacles.

$$q_{\mathsf{\Gamma }}={\displaystyle \sum _{k=1}^K{W}_k{e}^{\frac{1-{\wp }_k}{{\rho }_k}}{q}_k}$$

(13)

where ${\rho }_k$ represents a constant parameter, while $q_k$ denotes the velocity vector of the obstacle.

The velocity vector of the unmanned surface vehicle (USV) at time t can be derived from Equation (12), and integrating the observation time yields the corresponding trajectory path of the USV.

The new modulation matrix is added to verify the feasibility of this method, which proves that the addition of the modulation matrix does not affect the feasibility of the original IFDS method, as follows:

(1)

Reach the target point by following the path.

Evidence: As the USV moves further away from the obstacle, the disturbance matrix $-{\displaystyle \frac{n_k{n}_k^T}{{\left|{\wp }_k\right|}^{1/{\eta }_k}{n}_k^T{n}_k}}$ and modulation matrix ${\displaystyle \frac{i{\tau }_k{n}_k^T}{{\left|{\wp }_k\right|}^{1/{\varphi }_k}{n}_k^T{n}_k}}$ in Equation (10) gradually approach 0 with increasing distance from the obstacle (${\wp }_k=\infty$), thereby yielding the result of ${\stackrel{•}{U}}_j=\overline{F}{U}_j=I{U}_j=U_j$ for the overall flow field. Consequently, the trajectory will ultimately converge toward the target point.

(2)

To satisfy the requirements of obstacle avoidance along the path, i.e., to demonstrate the impenetrability of obstacles.

Evidence: When the unmanned surface vehicle (USV) encounters an obstacle, Equation (6) is applied to determine ${\wp }_k({\xi }_j)=1$, followed by the disturbance matrix $F_k=I-{\displaystyle \frac{n_k{n}_k^T}{n_k^T{n}_k}}+{\displaystyle \frac{{\tau }_k{n}_k^T}{n_k^T{n}_k}}$. The initial flow field $U_j=C_1{s}_k+C_2{n}_k$ is then established and decomposed into the normal vector $n_k$ and obstacle vector $s_k$. $F_k{U}_j=C(F_k{s}_k+F_k{n}_k)$ and $F_k{n}_k=n_k-{\displaystyle \frac{n_k{n}_k^T{n}_k}{n_k^T{n}_k}}+{\displaystyle \frac{{\tau }_k{n}_k^T{n}_k}{n_k^T{n}_k}}={\tau }_k$. Vector ${\tau }_k$ is tangential to the surface of the obstacle and it will not penetrate barriers. Thus, it can be inferred that the normal velocity of the obstacle on its surface is 0, satisfying the impenetrability condition and meeting the requirement for obstacle avoidance.

In the practical implementation of the aforementioned method, it becomes evident that there may be an overlap of obstacle models. In such instances, the area enclosed by the two obstacles and the common tangent line is referred to as the trap area (TA), as illustrated in Figure 1. The inherent algorithmic limitations of the IFDS can result in a scenario where the USV enters a dead zone while still within TA, particularly when the target point is obstructed by the two obstacles. Within this dead zone, both existence ${\wp }_i$ and ${\wp }_k$ are equal to 1, resulting in a breakdown of the weight formula and subsequent failure in obstacle avoidance.

In the two-dimensional environment, the original IFDS does not consider the TA problem. In order to solve the TA problem, the obstacle filling method or the virtual target point method can be used. The introduction of the obstacle filling method has been explained in our previous work [29], and only the virtual target point method is described in detail here.

The process of generating virtual target points is depicted in Figure 2a.

(1)

Determine the presence of overlapping obstacles and guide the unmanned surface vehicle (USV) to navigate towards TA.

(2)

When the buffering distance is reached, a sector sailing range is determined based on the existing velocity amplitude and kinematic constraints, resembling the Dynamic Window Approaches. The ultimate objective of all navigational efforts is to obtain the virtual target point set at $Vi{r}_{1,2,3,4\dots N}\in \overline{Vir}$ (depicted as the red arc in Figure 2b).

(3)

Assess the viability of virtual target points within the specified set.

The entire set of points is initially assessed as follows:

To expedite the computation of assessing virtual target points, a preliminary filtration of non-compliant virtual target points can be implemented. Initially, an assessment is made to ascertain the potential for evading local trap regions, drawing inspiration from our previous radar research [30,31], followed by a determination of the point’s spatial positioning. In Figure 2b, a straight line $l_{1,2}$, passing through the tangent point as a vertical tangent line, originates from the virtual target point to form rays in the directions of $l_1$ and $l_2$. If the ray intersects only one of $l_1$ and $l_2$, the point is deemed unable to escape from the trap area. If the ray can intersect both $l_1$ and $l_2$, it is determined that the point can escape from the trap area.

After the initial assessment, a cost analysis is conducted for the remaining viable points.

Utilize cost function $\Im$, with $\gamma$ as the collision cost (to assess potential obstacle collisions), $\chi$ as the consumption cost (to evaluate additional sailing distance resulting from this virtual target point), and $\mu$ as the escape cost (ascertain if you are approaching another trap areas) to compute the optimal virtual target point.

$$\Im ={\lambda }_1\gamma +{\lambda }_2\chi +{\lambda }_3\mu$$

(14)

$$\gamma =\left\{\begin{array}{cc}{\displaystyle \prod _{k=1}^{K}1/{\wp }_k(Vi{r}_n)}& {\wp }_k(Vi{r}_n)>1\\ +\infty & {\wp }_k(Vi{r}_n)\le 1\end{array}\right.$$

$$\chi ={\displaystyle \frac{d\left(Vi{r}_n,{\xi }_g\right)-d(n_{\min },g)}{d(n_{\max },g)-d(n_{\min },g)}}$$

$$\mu =\left\{\begin{array}{cc}0& \mathrm{Do}\mathrm{not}\mathrm{sail}\mathrm{to}\mathrm{new}\mathrm{trap}\mathrm{areas}\\ +\infty & Otherwise\end{array}\right.$$

where $Vi{r}_n$ denotes the coordinates of the virtual target point, $d(Vi{r}_n,{\xi }_g)$ represents the distance between the virtual target point and the actual target point, and $d(n_{\max },g)$ signifies the maximum distance between the virtual target point in the set and the actual target point, while $d(n_{\min },g)$ indicates the minimum distance. Following computation, we select a virtual target point with a minimal substitute value as our navigation target for escaping the trap area.

When all the virtual target points have been assessed and found to not meet the requirements (with a high cost), the speed amplitude is decreased, and a new sectorial navigation domain is generated.

5. Formation Strategy

Under the design of the second section of the EIFDS, this section focuses on the design of the formation structure, which mainly consists of the combination of EIFDS calculation and the virtual leader–follower method, which is described as follows.

Formation Strategy of Streamlined Traction

The virtual leader formation structure is adopted in this paper to enhance the plasticity and robustness of the formation, as compared to the leader–follower formation structure. The position $P_O^F$ of the virtual leader is adopted as the origin of the formation coordinate system, with the track angle ${\psi }_t^{virl}$ of the virtual leader serving as the X-axis. The j-th member’s position in the formation is denoted as $P_j^F$ with respect to the formation coordinate system and as $P_j$ with respect to the geodetic coordinate system.

$$P_j=P_O^F+\left[\begin{array}{cc}\cos {\psi }_t^{virl}& -\sin {\psi }_t^{virl}\\ \sin {\psi }_t^{virl}& \cos {\psi }_t^{virl}\end{array}\right]P_j^F$$

(15)

The movement trajectory of the virtual leader to reach the target point will be determined by the aforementioned enhanced IFDS algorithm. The virtual leader solely takes into account static obstacles, while maintaining a fixed value for velocity amplitude $V_{virl}$. Furthermore, the virtual leader enables us to acquire the anticipated positions of the actual team members, which can then be utilized as target points for each member’s IFDS algorithm.

Due to the influence of the enhanced IFDS algorithm, the course of the follower will eventually point to the desired position in the formation, while still having the obstacle avoidance ability. In order to converge to the desired position of the formation, we adjust the velocity amplitude $V$ to maintain the formation.

The adjustment of the velocity amplitude is determined based on the distance and orientation of the follower relative to the desired position. After taking into account the significant amount of external noise, it becomes challenging in practical scenarios for the actuator to precisely control the followers’ positions. Therefore, the follower is deemed to have reached the corresponding area once it enters the radius $R_j$ of the desired location. At the front or back of the desired position of the follower, we will design different speed adjustment schemes.

The displacement vector between the desired position of the follower and the actual position can be calculated using Equation (15). By considering the vector angle $S_j^{virl}$ formed between this vector and the heading of the virtual leader, we can determine whether the follower is positioned ahead of or behind the desired location, as shown in Figure 3.

When $S_j^{virl}\ge {\displaystyle \frac{\pi }{2}}$, the j-th follower is judged to be positioned behind the desired location. The velocity amplitude $V_j$ of the j-th follower is determined by the following formula:

$$V_j=\left\{\begin{array}{c}\begin{array}{cc}{V}_{\max }& D_j^{Ex}>R_j^{dec}\\ \frac{V_{\max }+V_{virl}}{2}+\frac{V_{virl}-V_{\max }}{2}\cos (\frac{\pi }{R_j^{dec}-R_j}(D_j^{Ex}-R_j))& R_j\le D_j^{Ex}\le R_j^{dec}\\ V_{virl}& D_j^{Ex}<R_j\end{array}\end{array}\right.$$

(16)

where $R_j^{dec}$ is the deceleration range. Equation (16) enables follower to track the desired position when it lags behind and ensures that the velocity amplitude remains consistent with $V_{virl}$ upon reaching the interior of $R_j$. Consequently, the follower converges precisely at the desired position.

When $S_j^{virl}<{\displaystyle \frac{\pi }{2}}$, the j-th follower is judged to be positioned ahead of the desired location. We adjust the follower not only to reduce the velocity amplitude but also to fine-tune the velocity vector.

$${Vj}_{t+1}=\left\{\begin{array}{cc}{V}_{j_t}-C_{dec}\Delta T& {Vj}_{t+1}\ge V_{\min }\\ V_{\min }& {Vj}_{t+1}<V_{\min }\end{array}\right.$$

(17)

where $V_{j_t}$ and $V_{j_{t+1}}$ are the velocity amplitudes at time t, and at the next time, $\Delta T$ is the observation time, $C_{dec}$ is the deceleration constant that can change according to $D_j^{Ex}$, and $V_{\min }$ is the minimum velocity amplitude.

The target point of the follower is always the desired position, as mentioned in the preceding discussion. However, since the follower is positioned ahead of the desired position, it tends to take a significant detour at a large angle towards the desired position. This behavior should be avoided for USVs, as shown in Figure 4. Therefore, we adjust the velocity vector of follower located ahead of the desired position by changing its target point from the desired position itself to a point in the direction of the velocity vector of the desired position (the velocity vector of the desired position in the leader–follower method aligns with that of the virtual leader). The aforementioned issue of a significant detour can thus be effectively circumvented, and the modification of the target point does not impact the obstacle avoidance effectiveness of followers.

In the above formation decision, the enhanced IFDS is used to determine the navigation route of the virtual leader in the known environment, and the virtual leader determines the desired position and the navigation trajectory of the desired position of the follower in the formation, and the desired position is used as the target point of the enhanced IFDS algorithm for the follower. Since the enhanced IFDS algorithm will always make the follower flow to the desired position, the velocity decoupling method is used here to maintain the follower within the desired position range by controlling the velocity amplitude. Finally, the formation structure of the leader–follower method based on streamline traction is ultimately established, as shown in Figure 5.

6. Inter-Vehicle Collision Avoidance

From the formation structure combined with the EIFDS and a virtual leader–follower system in the above section, it can be seen that in the formation structure led by the streamline, each member of the formation has the ability to avoid obstacles and maintain formation. However, it is easy to see that there is still a possibility of collision between members, which is risky for formation navigation. In order to improve the reliability of formation navigation, this chapter will expound the obstacle avoidance between vehicles.

6.1. Interference Residual Problem of IFDS

The IFDS algorithm has excellent obstacle avoidance properties; at first, we also used the IFDS as a means of collision avoidance between vehicles. However, practical implementation revealed significant drawbacks in using it for this purpose. When a USV sails in formation, the direction of each member’s velocity vector is roughly consistent. After collision avoidance between vehicles, there are still residual effects of the flow field that can make the streamlines converge unnecessarily. Nevertheless, real-world scenarios may introduce issues such as communication and actuator delays that pose unnecessary collision risks due to inherent limitations of the algorithm itself, as shown in Figure 6.

The problem at hand can indeed be effectively resolved by adjusting parameters such as reaction coefficients. However, the incorporation of time-varying reaction coefficients and model predictive control would significantly escalate the computational demands, which contradicts our initial intention. Hence, we derive inspiration from vector field methods.

6.2. A Vector Field Method

The vector field method, as mentioned in [27], forms the foundation of our approach for ensuring collision avoidance among formation vehicles.

The vector field method employed is as follows:

$$f(r)={\left[\begin{array}{cc}{f}_x(r)& f_y(r)\end{array}\right]}^T=\kappa ({\alpha }^{T}r)r-a(r^{T}r)$$

(18)

where $r={\left[\begin{array}{cc}{r}_x& r_y\end{array}\right]}^T$, $\alpha ={\left[\begin{array}{cc}{\alpha }_x& {\alpha }_y\end{array}\right]}^T$, $\alpha \ne 0$. The vector field components $f_x$, $f_y$ read as follows:

$$\begin{array}{l}{f}_x=(\kappa -1) {\alpha }_x{r}_x^2+\kappa {\alpha }_y{r}_x{r}_y-{\alpha }_x{r}_y^2\\ f_y=(\kappa -1) {\alpha }_y{r}_y^2+\kappa {\alpha }_x{r}_x{r}_y-{\alpha }_y{r}_x^2\end{array}$$

(19)

The singularity of the vector field is exclusively at $r=0$ if and only if $\kappa \ne 1$. The integral of $f(r)$ is as follows when $\alpha ={\left[\begin{array}{cc}1& 0\end{array}\right]}^T$:

$$(r_x^2+r_y^2{)}^{\frac{\kappa }{2}}=C{z}_y^{\kappa -1}$$

(20)

$C$ is a constant. For $\kappa =0$, (16) reduces to $r_y=C$, i.e., the integrals are straight lines parallel to $\alpha ={\left[\begin{array}{cc}1& 0\end{array}\right]}^T$.

For $\kappa =1$, (16) reduces to $r_x^2+r_y^2=C^2$, i.e., the integrals are circles of radius $C$, where $C>0$ centered at the origin is (0, 0).

6.3. Vector Fields Are Used to Avoid Obstacles between USVs

The collision avoidance vector field between ships was designed based on the aforementioned introduction of vector fields. Firstly, the priority of USVs was established, assuming a total of $J$ USVs and setting the priority order for inter-vehicle collision avoidance as $US{V}_1>US{V}_2>US{V}_3>\dots \dots >US{V}_J$; subordinate USVs should consistently avoid any superior USVs.

We will demonstrate the vector field method using an example of circumventing $US{V}_1$ through $US{V}_J$. At time t, let the position of $US{V}_1$ be denoted as $r_{1_t}={\left[\begin{array}{cc}{x}_{1_t}& y_{1_t}\end{array}\right]}^T$ and that of $US{V}_J$ as $r_{J_t}={\left[\begin{array}{cc}{x}_{J_t}& y_{J_t}\end{array}\right]}^T$, the safe distance between the two boats is set to ${\hslash }_1+{\hslash }_J+{\hslash }_0$, ${\hslash }_1$ is the safe range of $US{V}_1$, ${\hslash }_J$ is the safe range of $US{V}_J$, ${\hslash }_0$ stands for the minimum separation between ${\hslash }_1$ and ${\hslash }_J$, and ${\hslash }_{ob}$ indicates the collision avoidance range. When the separation between the two vessels reaches $‖\overrightarrow{r}‖=‖r_{1_t}-r_{J_t}‖\le {\hslash }_1+{\hslash }_J+{\hslash }_0+{\hslash }_{ob}$, $US{V}_J$ will maneuver to avoid $US{V}_1$. At this juncture, the formula for the component of the collision avoidance vector field is derived from Equation (19) as depicted below:

$$\begin{array}{l}{f_{1_a}^J}_x={\alpha }_y(x_{1_t}-x_{J_t})(y_{1_t}-y_{J_t})-{\alpha }_x{(y_{1_t}-y_{J_t})}^2\\ {f_{1_a}^J}_y={\alpha }_x(x_{1_t}-x_{J_t})(y_{1_t}-y_{J_t})-{\alpha }_y{(x_{1_t}-x_{J_t})}^2\end{array}$$

(21)

where $\kappa =1$, and the vector $\alpha$ is positioned along the line connecting $US{V}_1$ to the desired position of $US{V}_J$ (the vector extends from the desired position of $US{V}_J$ to the position of $US{V}_1$). Equation (21) expresses the circular vector field $f_{1_a}^J={\left[\begin{array}{cc}{f_{1_a}^J}_x& {f_{1_a}^J}_y\end{array}\right]}^T$ centered at $US{V}_1$ and with the reflecting axis being the line containing $\alpha$, as shown in Figure 7. In this vector field, $US{V}_J$ can evade $US{V}_1$, a residual effect that persists, as depicted in Figure 6. However, the vector field method can effectively circumvent such issues.

To correct for residual effects, the displacement vector from $US{V}_1$ toward $US{V}_J$ is defined as $\overrightarrow{r}=r_1-r_J$. Subsequently, the angle $S_{\alpha }^r$ between $\overrightarrow{r}$ and $-\alpha$ is to be calculated. When considering the angle $S_{\alpha }^r\le \frac{\pi }{2}$, $US{V}_J$ is positioned in front of $US{V}_1$. At this juncture, the vector field component can be derived from Equation (15), with the design outlined as follows:

$$\begin{array}{l}{f_{1b}^J}_x=-{\alpha }_x{(x_{1_t}-x_{J_t})}^2-{\alpha }_x{(y_{1_t}-y_{J_t})}^2\\ {f_{1b}^J}_y=-{\alpha }_y{(y_{1_t}-y_{J_t})}^2-{\alpha }_y{(x_{1_t}-x_{J_t})}^2\end{array}$$

(22)

where $\kappa =1$, and the vector $-\alpha$ is positioned along the line connecting $US{V}_1$ to the desired position of $US{V}_J$ (the vector extends from $US{V}_1$ to the desired position of $US{V}_J$).The vector field $f_{1b}^J={\left[\begin{array}{cc}{f_{1b}^J}_x& {f_{1b}^J}_y\end{array}\right]}^T$, as defined by Equation (18), consists of all vectors aligned with the direction of vector $\alpha$, as illustrated in Figure 8.

The angle $S_{\alpha }^r$ between $\overrightarrow{r}$ and vector $-\alpha$ is utilized for determining the relative positioning of the two USVs, and the value of vector field $\kappa$ is determined based on this relative positioning. A composite vector field for inter-boat collision avoidance has been developed, which not only achieves collision avoidance but also effectively mitigates the issues arising from residual effects of the vector field.

Ultimately, we integrate the velocity vector derived from the collision avoidance vector field with the velocity vector produced by the enhanced IFDS to formulate the ultimate velocity vector $f^J$ of USV. To enhance the integration of the vector field with the enhanced IFDS, normalization of the vector field is performed.

$$f_{1ab}^J=\left\{\begin{array}{cc}\frac{f_{1a}}{‖f_{1a}‖}& S_{\alpha }^r\ge \frac{\pi }{2}\\ \frac{f_{1b}}{‖f_{1b}‖}& S_{\alpha }^r<\frac{\pi }{2}\end{array}\right.$$

(23)

6.4. Obstacle Avoidance Framework Combining Vector Field and EIFDS

Under the influence of the enhanced IFDS method, the sum of the velocity vector $f_{IFDS}^J={\stackrel{•}{U}}_{t_y}/{\stackrel{•}{U}}_{t_x}$ generated by Equation (12) and the velocity vector generated by Equation (23) is as follows:

$$f^J=f_{IFDS}^J+{\displaystyle \sum _{j=1}^{J-1}(1-{\delta }_j)f_{jab}^J}$$

(24)

The expression of ${\delta }_j$ can be described as follows:

$${\delta }_{\mathrm{j}}=\left\{\begin{array}{ll}1\hfill & ‖\overrightarrow{r}‖<{\hslash }_1+{\hslash }_J+{\hslash }_0\hfill \\ a{{\beta }_{1J}}^3+b{{\beta }_{1J}}^2+c{\beta }_{1J}+d\hfill & {\hslash }_1+{\hslash }_J+{\hslash }_0\le ‖\overrightarrow{r}‖\le {\hslash }_1+{\hslash }_J+{\hslash }_0+{\hslash }_{ob}\hfill \\ 0\hfill & {\hslash }_1+{\hslash }_J+{\hslash }_0+{\hslash }_{ob}<‖\overrightarrow{r}‖\hfill \end{array}\right.$$

(25)

where ${\hslash }_{ob}$ represents the predetermined collision avoidance range, and $‖\overrightarrow{r}‖$ denotes the distance between the vessel and the obstacle boat.${\beta }_{1J}={{\hslash }_1}^2-{‖\overrightarrow{r}‖}^2$, $a=\frac{2}{{({\beta }_0-{\beta }_{ob})}^3}$, $b=-\frac{3({\beta }_0+{\beta }_{ob})}{{({\beta }_0-{\beta }_{ob})}^3}$, $c=\frac{6{\beta }_0{\beta }_{ob}}{{({\beta }_0-{\beta }_{ob})}^3}$, $d=\frac{{{\beta }_0}^2({\beta }_0-3{\beta }_{ob})}{{({\beta }_0-{\beta }_{ob})}^3}$. ${\beta }_o={{\hslash }_1}^2-{({\hslash }_1+{\hslash }_J+{\hslash }_0)}^2$, ${\beta }_{ob}={{\hslash }_1}^2-{({\hslash }_1+{\hslash }_J+{\hslash }_o+{\hslash }_{ob})}^2$.

Following the aforementioned procedure, $US{V}_J$’s total velocity vector is $f^J={\left[\begin{array}{cc}{f}_x^J& f_y^J\end{array}\right]}^T$ with a heading angle of ${\psi }_t={\tan }^{-1}\left(f_y^J/f_x^J\right)$.

The ultimate differential formula for the guidance of $US{V}_j$ is as follows:

$$\left\{\begin{array}{c}{\dot{x}}_j=V_j\cos ({\tan }^{-1}\left(f_y^j/f_x^j\right))\\ {\dot{y}}_j=V_j\sin ({\tan }^{-1}\left(f_y^j/f_x^j\right))\end{array}\right.$$

(26)

From this integral, the trajectory planning for the navigation of the USV fleet can be derived.

7. Simulation

Between Section 2 and Section 4, the USV formation obstacle avoidance guidance strategy is developed using the enhanced IFDS method and vector field method to adjust the velocity vector of USV. This ensures that the USV consistently reaches the target point and avoids obstacles and collisions in formation, thereby ensuring heading direction safety. Additionally, in order to maintain formation integrity, adjustments are made to the velocity amplitude so that the USV can converge toward its desired position within the formation. To validate feasibility, corresponding simulations have been conducted to demonstrate these principles. The simulation environment used in this paper has been designed using matlab2022B.

The advantages of the IFDS and the vector field method over APFs are compared in [22,23,24,25,26,27]; they are characterized by smoothness and fewer local extreme values. This paper solely demonstrates the prominence of the enhanced IFDS and the vector field method in formation navigation.

7.1. The Comparison between Enhanced IFDS and IFDS

The enhanced IFDS overcomes the issue of local minimum encountered by the IFDS in the two-dimensional plane. Here, the USV is designed to drive to the stagnation point and TA, to determine whether the two methods will stay at the stagnation point because of the TA obstacle avoidance failure.

The verification of the stationary point problem necessitates the design of an obstacle avoidance environment wherein the initial position of the USV, the center of the obstacle circle, and the target point are aligned in a collinear manner. The initial position of the USV is set to (10,10), the target point of the USV is set to (150,150), the center of the obstacle circle is set to (75,75), and the radius of the obstacle is 15 m. The velocity amplitude V of the USV is set to 8.5 m/s. The disturbance coefficient ${\eta }_k$ of the IFDS is set to 3. The disturbance coefficient ${\eta }_k$ of the enhanced IFDS is set to 3, and the modulation coefficient ${\phi }_k$ of the enhanced IFDS is set to 1.

The enhanced IFDS, as depicted In Figure 9a,b, effectively circumvents the Issue of stationary points, and the direction of deviation is correlated with the parameter $i$ (take a positive value or a negative value). However, the stationary point problem in the 2D plane still persists for the IFDS.

The trap area problem can be verified by setting overlapping obstacles and aligning the USV velocity direction towards the trap area. The scenario is designed as follows.

The initial position of the USV is set to (10,10), and the target point of the USV is set to (110,110). The central point of the first obstacle is located at (65,75), the central point of the second obstacle is located at (75,65), and the diameter of both obstacles is 20 m. The disturbance coefficient ${\eta }_k$ of the IFDS is set to 3. The disturbance coefficient ${\eta }_k$ of the enhanced IFDS is set to 3, and the modulation coefficient ${\phi }_k$ of the enhanced IFDS is set to 1.

The enhanced IFDS effectively addresses the TA problem through the utilization of virtual target points, as shown in Figure 10a. The inability of the IFDS to address the TA problem results in a collision between the USV and the obstacle, as well as data value confusion.

7.2. The Comparison of Inter-Vehicle Obstacle Avoidance within Formation Using the IFDS and Vector Field Methods

In this design scenario, the collision avoidance scheme between vehicles is compared to determine whether the vector field method and IFDS will cause collision avoidance between vehicles and the interference residual problem (whether the course will still be affected by obstacles after obstacle avoidance).

The following simulation environment is designed in order to compare the vector field method and IFDS method. The initial position of $US{V}_1$ is set to (20,15), and the initial position of $US{V}_2$ is set to (−10,0). The two vehicles navigate in a formation manner along the Y-axis direction, with their paths intersecting. $US{V}_1$ has a higher weight than $US{V}_2$ (i.e., $US{V}_1>US{V}_2$), and the collision avoidance between two ships is achieved through the utilization of either the vector field method or the IFDS algorithm, with a comparative analysis conducted to evaluate their respective advantages and disadvantages.

The comparison in Figure 11 demonstrates that the vector field method exhibits no residual effect on inter-boat collision avoidance within a formation, whereas the IFDS encounters this issue. So, the vector field method possesses a distinct advantage in inter-boat collision avoidance within a formation.

7.3. Comparison of Formation Obstacle Avoidance Using Single IFDS Algorithm and Formation Obstacle Avoidance Using Compound Field Algorithm

Under the two strategies, the superiority of the composite field formation strategy is verified by comparing the distance of the formation members.

The formation obstacle avoidance environment is designed to verify the feasibility of formation obstacle avoidance by the composite field. The initial position of $US{V}_1$ is set to (1,40), the maximum velocity amplitude is 25 m/s, the initial position of $US{V}_2$ is set to (30,0), the maximum velocity amplitude is 23 m/s, and the initial position of $US{V}_3$ is set to (60,10). The maximum velocity amplitude is 27 m/s. The priority of each vehicle is $US{V}_3>US{V}_2>US{V}_1$. The velocity amplitude of the virtual leader is 6 m/s. The formation path is obstructed by four consecutive obstacles, with their center points set at (110,75), (133,155), (200,160), and (80,120) and a radius of 13 m, 12 m, 11 m, and 10 m each. The safety range ${\hslash }_{1,2,3}$ of each vehicle is set to 5 m. The disturbance coefficient ${\eta }_k$ of the enhanced IFDS is set to 3, and the modulation coefficient ${\phi }_k$ of the enhanced IFDS is set to 1.

As can be seen from the comparison between Figure 12 and Figure 13, when the same parameters are consistent, the composite field can keep the distance away from 10 m. However, in the two-dimensional plane, the formation structure using only the IFDS will make it difficult to maintain a safe range when the formation members avoid obstacles because of the IFDS interference residue.

7.4. Compound Field Formation Obstacle Avoidance under Multiple Obstacles

We set the coordinates of four USVs in the geodetic coordinate system as (1,40), (30,0), (60,0), (0,0). The maximum velocity amplitude is 17 m/s,18 m/s,15 m/s, and 16 m/s, respectively. The obstacle coordinates, respectively, are (75,110), (130,155), (200,160), (80,120), (150,100), and (50, 60). The radius of obstacles is 13 m, 12 m, 11 m, 10 m, 10 m, and 5 m, respectively. There is another dynamic obstacle, starting at (50,160) and ending at (200,40), with a speed of 5 m/s, and the radius is 10 m. The disturbance coefficient ${\eta }_k$ of the enhanced IFDS is set to 3, and the modulation coefficient ${\phi }_k$ of the enhanced IFDS is set to 1. The priority of each vehicle is $US{V}_1>US{V}_2>US{V}_4>US{V}_3$. The safety range ${\hslash }_{1,2,3,4}$ of each vehicle is set to 5 m.

As can be seen from Figure 14, the USV formation successfully avoids obstacles and maintains the formation, which demonstrates the feasibility of the formation strategy presented in this paper. Additionally, the inter-vehicle distance is consistently maintained at over 10 m, and the distance between the vehicles and dynamic obstacles is also kept within a safe range.

7.5. Formation Obstacle Avoidance in Environments with Dense Obstacles

Based on reference [32,33], considering the characteristics of the ship’s navigational environment, a relatively dense group of obstacles is provided. We set the coordinates of four USVs in the geodetic coordinate system as (15,45), (25,35), (35,25), and (45,15). The disturbance coefficient ${\eta }_k$ of the enhanced IFDS is set to 3, and the modulation coefficient ${\phi }_k$ of the enhanced IFDS is set to 1. The priority of each vehicle is $US{V}_1>US{V}_2>US{V}_4>US{V}_3$. The safety range ${\hslash }_{1,2,3,4}$ of each vehicle is set to 5 m.

As can be seen from Figure 15, the USV formation successfully avoids obstacles and maintains the formation, which demonstrates the feasibility of the formation strategy presented in this paper. Additionally, the inter-vehicle distance is consistently maintained at over 10 m, and the distance between the vehicles and dynamic obstacles is also kept within a safe range.

8. Conclusions

In this paper, a composite field formation obstacle avoidance strategy based on a leader–follower structure was proposed. Aiming at solving the extreme value problem of the IFDS in a two-dimensional plane, the EIFDS was improved. The velocity decoupling method was used to combine the EIFDS with the virtual leader–follower formation. To solve the obstacle avoidance problem among formation members, a vector field method and an improved scheme were used, and a composite field obstacle avoidance strategy was formed by combining it with the EIFDS. The effectiveness of EIFDS improvement and the advantage of the vector field method in obstacle avoidance between boats were verified by simulation experiments. Finally, compared with the single IFDS formation strategy, the superiority of compound field formation obstacle avoidance was verified in terms of the distance between formation members.

Finally, this method presented a novel decision-making scheme for cooperative obstacle avoidance in USV formations. However, it is currently limited to ideal environmental conditions and does not account for environmental noise and actuator constraints. These challenges will be addressed in future research, where a control scheme tailored to low communication conditions in complex maritime environments will be proposed. Additionally, the parameter optimization problem will be tackled using model predictive control (MPC) to enhance its practical applicability.