USV Search Mission Planning Methodology for Lost Target Rescue on Sea

Abstract

Quick and efficient mission planning is essential in maritime search and rescue (SAR). This includes defining the search area and developing an effective strategy. The task is fraught with challenges due to the difficulty of determining location information and the impact of complex meteorological environments. The primary objective of SAR mission planning is the rapid deployment of unmanned surface vehicles (USVs) to the incident area. While many planning algorithms prioritize the shortest route, there’s a lack of mission planning measures that maximize SAR effectiveness. In addition, the joint deployment of USVs increases the success rate compared to individual operations. Therefore, this paper presents a task assignment framework for USVs in SAR missions that considers the probability of success and time constraints. USVs are used to search for lost targets, and the framework consists of the following three modules: (1) a module for predicting the location of the overboard target to be rescued; (2) a module for modeling the probability of mission success; (3) a module for assigning search tasks to USVs. The framework first analyzes the search area. Then, it predicts the target location with a stochastic particle method, which incorporates marine environment forecast data to update the mission target location. To improve the scientific nature of USV search and rescue mission plans, an evaluation model is developed to assess mission capability. Simulation experiments and task scheme analysis validate its effectiveness.

1. Introduction

As the primary mode of international freight transportation, more than 90 percent of the world’s trade is carried by sea each year. However, maritime transportation is prone to frequent accidents due to the complex and dynamic marine environment influenced by both natural and human factors. According to data released by the China Maritime Search and Rescue Center (CMSARC) for the period from February to June 2023, a total of 546 SAR operations were conducted. During these operations, a total of 2847 search and rescue vessels and 66 search and rescue aircraft were deployed and 363 vessels in distress and 3481 people were successfully rescued. Remarkably, these efforts saved the lives of 3392 people. These statistics underscore the increasing frequency of maritime accidents, driven by the rapid expansion of the shipping industry. Therefore, improving China’s maritime search and rescue capabilities has become a matter of paramount research importance.

At present, emergency search and rescue at sea relies mainly on manually operated search vessels. This method is limited by the ability of search equipment and humans to coordinate, resulting in longer search times, lower efficiency, and an increased chance of missing search and rescue targets. However, the USV offers a promising solution. It has autonomous navigation, control, and communication capabilities. They possess characteristics such as high mobility, cost-effectiveness, and large search radius [1], and represent a novel technological approach to address the weaknesses in coordinated maritime SAR capabilities, increased search and rescue risks, and operational costs [2]. Compared to traditional maritime SAR methods, the use of USV offers significant advantages. It efficiently reduces operational costs and quickly and accurately locates maritime SAR targets. As a key tool in maritime search and rescue operations, the widespread use of USVs has completely changed the face of emergency response at sea. It has breathed new life into the field of maritime emergency management. However, an urgent challenge for current research is how to optimize the use of USVs in the case of missing or unknown targets in maritime search and rescue. Vessels that run aground, sink, burn, or explode at sea are subject to the influence of oceanic wind patterns, currents, and other variables once lost, posing a formidable challenge to rescue personnel. In the context of search missions for lost targets at sea, a pressing and complex challenge is to efficiently strategize the deployment and use of USV. The primary objectives are to minimize the expected mission duration and maximize the probability of mission success [3]. The complexity of this problem arises from many sources of uncertainty, including the uncertainty of the search area, the dynamic variability of the marine environment, and the inherent limitations of search resources.

Consequently, this research focuses on the process of planning USVs search missions for lost targets at sea. In the field of maritime emergency SAR missions, it seeks to formulate advanced task allocation algorithms and decision support systems. The overall goal is to optimize the effectiveness of USV search missions. At the same time, this research combines model-derived data to facilitate optimal delineation of SAR areas. Using the predictive models of target dynamic behaviour and advanced task planning technology, this study aims to intelligently configure search tasks for USVs. This effort focuses on improving the effectiveness of the rapid deployment of USVs in SAR missions, thereby maximizing the probability of successfully locating lost targets. It provides a more efficient and sustainable solution for responding to the emergency rescue needs of missing targets.

The paper makes significant contributions in the following areas:

1.

It presents a mission planning framework that facilitates collaboration among multiple unmanned boats for efficient rescue of lost targets.

2.

It uses Monte Carlo stochastic particle simulation to delineate the search area and estimates the initial mission point for unmanned boats using a Bayesian approach.

3.

The paper introduces an adaptive approach that leverages the search capabilities of unmanned aerial vehicles (UAVs) to maximize mission effectiveness in rapidly planning for lost targets within a constrained time frame.

This paper is organized as follows. Section 2 presents a literature review of relevant publications. In Section 3, we present a mission planning framework for collaborative search and rescue using unmanned vessels for lost targets at sea. In Section 4, we elaborate the methodological model for USVs search and rescue mission planning. Section 5 presents a case study of unmanned vessel mission planning. Section 6 discusses the importance and potential of collaborative unmanned vessel search and rescue for lost targets. Finally, Section 7 summarizes the main work and outlines future research directions.

2. Research on Related Methods

Maritime SAR, as defined by international conventions and relevant Chinese laws and regulations, refers to the coordinated efforts of maritime rescue forces to locate and assist distressed entities. This includes vessels in distress due to grounding, sinking, fire or explosion, forced landings, and aircraft crashes, as well as stranded, injured or missing persons [4]. In accordance with the guidelines provided by the National Maritime Search and Rescue Manual and the International Aeronautical and Maritime Search and Rescue Manual, the maritime search process includes four basic phases: (1) assessing the distress status of the target in the water; (2) determining the optimal search area; (3) formulating the search program and plan; (4) conducting the rescue operation immediately upon locating the target.

The first objective in recovering a lost target at sea is to accurately define the search area. This step is the basis for the development of a comprehensive search program and plan that allows efficient allocation of resources in a limited number of USVs. The emergency rescue process for lost-at-sea targets has received considerable attention from a wide range of experts and scientists. These scientists are actively pioneering methods and technologies to improve the efficiency of SAR operations. Their research focuses primarily on the analysis of the search area, including simulations and analyses that use factors such as ocean currents, wind direction, and other environmental variables to determine the drift path of the target. This, in turn, allows the search area to be effectively defined. Researchers are also investigating search task allocation and resource optimization, simplifying the allocation of search work through the use of relevant algorithms and decision support systems, and ultimately improving search efficiency.

Pan Wei [5] tackled the challenge of marine area prioritization using the Monte Carlo method. This approach involved simulating random particles and constructing a probability distribution density model based on probability distributions. Bourgault F et al. [6] introduced a framework for a Bayesian driven methodology. This framework used metrics such as average detection time and cumulative probability of detection as key control parameters. It was developed to address the problem of airborne vehicles searching for individual lost targets at sea. L. Lin et al. [7] addressed the challenge of prioritizing search regions in the context of search problems. Their investigation focused on mapping the probability distribution of search regions with prioritization. Burciu Z. [8] estimated the search region using a search target drift regression model. This estimation method involved the application of the Fokker–Planck equation to delineate the map of search probability regions. Reference [9] focused on the maritime search scenario for UAV. Within this scenario, they addressed communication and collision avoidance constraints, culminating in the development of a minimum time search (MTS) algorithm rooted in ant colony optimization.

Erol et al. [10] used a decision tree approach to identify the regular characteristics of SAR areas. Their analysis, based on 1247 marine casualties in Turkish waters, provided valuable insights into this area. Li Hao [11] conducted an in-depth investigation of the factors influencing the inclusion probability. Specifically, the study examined aspects such as region radius and drift error from the perspective of inclusion probability. Breivik et al. [12] conducted a comprehensive assessment of the wind-driven drift problem affecting object markers at sea. Their results emphasized the critical role of downwind conditions in this context. Reference [13] postulated that targets within the sea area conform to a uniform distribution. They also established an optimization model for ship selection. This model focused on the objective of efficiently covering the sea area in the shortest possible time, taking into account the ship’s search capabilities and the distance to be covered. Reference [14] proposed a holistic path-planning framework designed for surface autonomous ships (MASS). This framework was developed to address the challenge of maximizing the probability of successfully locating individuals in the water during SAR missions.

The aforementioned research efforts primarily revolve around the optimization of SAR task areas through algorithmic approaches. The overarching goal is to maximize the probability of inclusion of the rescue target within the search area, with the goal of near 100% inclusion. This tracking takes into account various constraints, including time and other related conditions, and the ultimate goal is to improve the efficiency and effectiveness of the search process.

The research advantage of this paper is that (1) the program framework can carry out the search for lost targets at sea, with strong adaptability, for the initial position analysis, plus, the article designed a prediction model that combines the position of the target under the environment of the ocean wind field and current field, and based on this, set up the SAR task point; (2) it can quickly generate a multi-target joint SAR program for the USVs, so as to effectively guide the sea search and rescue of the activity; (3) the SAR capability of the USV has been analyzed and taken into account in the generated program, which can make the program more valid and scientific.

3. Methodology

Despite notable progress in research on area search for lost targets at sea, the use of USVs in SAR missions continues to face key technical challenges. For example, there are still some unresolved issues, even with the establishment of the Maritime Search Area Delineation Program. These include the effective and judicious allocation of target areas for USV search missions, the achievement of efficient coordinated search and rescue operations by USVs, and the achievement of optimal alignment between USV and search missions. This alignment will be pursued while ensuring the success rate of USV search and rescue missions.

In this study, the Monte Carlo stochastic particle method is used to simulate and analyze the sea search area. In addition, we integrate the Bayesian method to predict the target in need of rescue [15], and then generate a probability distribution map for the target. This map is used as a basis for the planning of USV search missions. At the same time, we have developed a model to represent the detection probability of USVs in changing sea conditions. Building on this, we present a task assignment framework that emphasizes probability of success and time constraints in USV search and rescue missions. The framework is illustrated visually in Figure 1.

Based on the above research framework, we plan to conduct research in the following three areas.

(1)

Predicting location of the rescue target

Using positional data on the target prior to its loss, we used maximum likelihood estimation to determine the search area for the target. In parallel, we used the Monte Carlo random particle method to predict the probability of inclusion within the search area. This prediction involves incorporating marine environmental data and associating random particles with their respective motion characteristics. We then used time-step statistics to construct the probability distribution of the search target over a given time period. This distribution serves as the Bayesian prior for predicting the expected position of the rescuer relative to the target to be rescued.

(2)

Mission Success Probability Prediction

Using empirical data on the probability of finding drones in various marine environments, we have developed a model for predicting the probability of finding drones. Then, we developed a method to simulate the success probability of the search task. At the same time, we have established a framework for evaluating the search and rescue capabilities.

(3)

USV Mission Planning

Guided by the predicted location of the target to be rescued, we integrate a model that considers time and resource constraints. In addition, we have also included the evaluation results of the search and rescue capabilities of USVs. We then perform the allocation of search and rescue tasks to USV.

4. USV Collaborative Mission Planning for Targets Lost at Sea

4.1. Determining the Search Area and Predicting the Position

Under the influence of the marine environment, the target to be rescued experiences drift in the open sea. In our research methodology, the first step is to use maximum likelihood estimation to create a probability surface model for the inclusion of the target. This model effectively limits the search area. Then, we use the Monte Carlo random particle method to predict the potential position of the missing target.

4.1.1. Probabilistic Models for High Probability Estimation Target Inclusion

Given the complex and ever-changing nature of ocean climatic conditions, coupled with the limited information available about the lost target, we introduce a circular normal distribution [16] to characterize the initial probability distribution of the lost target. We then determine the search and rescue area by analyzing the probability density associated with the potential locations of the target. In addition, we collect critical information resources, including latitude, longitude, and bearing data for the distressed target. This information is obtained from available sources such as pre-loss alarm data. The probability distribution of the position of the distressed target is well represented by a two-dimensional normal distribution. To establish the SAR base point, we denote the coordinates of the SAR base point position as (x,y), taking into account the SAR base point position error. This error follows a two-dimensional normal distribution with parameters μ_x and μ_y, which allows us to derive the following:

$$f\left(x,y\right)=\frac{e^{-\frac{1}{2(1-{\xi }^2)}\left[\frac{{(x-{\mu }_x)}^2}{{\delta }_x^2}+\frac{2\xi (x-{\mu }_x)(y-{\mu }_y)}{{\delta }_x{\delta }_y}+\frac{{(y-{\mu }_y)}^2}{{\delta }_y^2}\right]}}{2\pi {\delta }_x{\delta }_y\sqrt{1-{\xi }^2}}$$

(1)

The above formula is noted as:

$$(x,y)\sim N({\mu }_x,{\mu }_y,{\delta }_x^2,{\delta }_y^2,\xi )$$

(2)

where μ_x is the mean of the horizontal coordinate x, μ_y is the mean of the vertical coordinate y, δ_x is the standard deviation of the horizontal coordinate x, δ_y is the standard deviation of the vertical coordinate y, and $\xi$ is the correlation coefficient of the variables x and y, which is usually less than 1. Taking the search and rescue base point as the origin (0, 0) so that μ_x = μ_y = 0, δ_x = δ_y = δ, and ξ = 0, the probability density function for the position of the distress target is shown below.

$$f\left(x,y\right)=\frac{1}{2\pi \delta }{e}^{-\frac{x^2+y^2}{2{\delta }^2}}$$

(3)

In order to better display the probability density distribution of the target location, according to the above equation, −2.5 ≤ x ≤ 2.5, −2.5 ≤ y ≤ 2.5, is chosen to display the probability density distribution of the initial location of the search base. See Figure 2 for the probability density distribution.

Examining the probability density distribution of the initial search base point, it is apparent that the position of the accidental ship follows a circular and regular probability density distribution. This distribution is centered on the last reported position, with the probability of inclusion increasing as the search radius increases. Therefore, the final position of the accident ship will be in the distribution area with the highest probability of the accident ship position. To calculate the inclusion probability of the outer tangent square, we use the following formula to determine the inclusion probability within the sea area:

$$P(A)=1-e^{-\frac{R^2}{2}}$$

(4)

where e is the base of the natural logarithm and R is the radius of the circle under the standard deviation. If the radius of the circle is one standard deviation E, then the length of the outer tangent square side is two standard deviations E.

$$E=\sqrt{D_{\mathrm{e}}^2+X^2+Y^2}$$

(5)

Here, X indicates the initial position error, generally using navigation positioning error data, fishing boat inertial navigation system positioning error is usually taken as 1 NM; Y indicates the position error of the USV search device, USV using the GNSS (Global Navigation Satellite System) and radar system, its positioning error two errors weighted at 1.3 NM; D_e is the trajectory projection error.

To facilitate the design of the search route, the search region is extended to the outer tangent square of the circular region, and the inclusion probability of this extended square is determined. Within this square, the probability associated with points on its horizontal and vertical coordinates represents the joint probability. In other words, both the horizontal coordinate x and the vertical coordinate y follow a normal distribution, and their joint probability is calculated using the standard normal distribution. This can be expressed as follows:

$$P(x)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{e}^{-\frac{x^2}{2}}}dx$$

(6)

$$P(y)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{e}^{-\frac{y^2}{2}}}dy$$

(7)

If the probability of containing (POC) is 50%, solve for R = 1.18, the probability of inclusion of a circle is 50% when the radius is 1.18 standard deviations. Similarly, calculate the inclusion probability of a square. If the probability of its horizontal coordinate x in the interval (−1.18,1.18) is 0.761, and the probability of its vertical coordinate y in the interval (−1.18,1.18) is 0.761, then the inclusion probability of its area is 0.761 × 0.761 = 0.579. According to the literature [17], it is known that if the radius of the circle searching in the search area is 3E, the inclusion probability is higher than 99.8%.

During the SAR planning process, the search area is typically divided into uniform units. The inclusion probability distribution is then determined using the probability distribution of the initial position. The sum of the containment probabilities for all units equals 100%. The containment probability distribution is shown in Figure 3.

The search area can be divided into distinct sub-regions by using any point along the two adjacent sides of the enlarged square sea as the boundary. The probability of the target being in each sub-region is determined by multiplying the respective intervals of the horizontal and vertical coordinates. The probability of each subregion getting selected is shown in

Table 1. The subregion contains the probability distribution.

Area Code	A1	A2	A3	A4	A5	A6	A7	A8	A9
POC	0.15%	3.65%	0.15%	3.65%	85.12%	3.65%	0.15%	3.65%	0.15%

.

Because the likelihood of targets being located in different subareas varies, those with higher probabilities require faster response times and higher search priorities. This requires the allocation of high-quality search resources. During actual search operations, it is not feasible to search all potential distress areas at sea due to manpower and resource constraints. In order to increase search efficiency, SAR coordinators typically divide the total search area encompassing distress targets, giving priority to sub-areas with higher probabilities.

4.1.2. Monte Carlo Stochastic Particle Method for Position Prediction

Monte Carlo methods are based on statistical mathematics, which makes them very suitable for dealing with complex systems and determining the solutions with the highest probability of occurrence by statistical methods. In the probabilistic prediction model presented in this paper, we employ the basic concept of the Monte Carlo method by treating the salvage target as a single particle and modeling its trajectory as a stochastic process. Assuming that the position of the floating object follows a Markov process and modeling the transition probability of the floating object from one position to the next, we replicate a large number of these particles following the same rule. Additionally, we assume that the motions of these particles are uncorrelated with each other. This enables us to analyze the probability region of the drifting target based on the statistical attributes of these particle distributions after a time interval ∆t. As the number of simulation iterations increases, the probability region of the target drift stabilizes. The basic steps of the Monte Carlo method used to generate the search region simulation solution are as follows:

(1)

Assume that the drift of the target in distress is modeled as a function g, and the search area is the integral of g:

$$I={\displaystyle {\int }_{A}g}(x)dx$$

(8)

(2)

Rewrite I in expected form:

$${\displaystyle {\int }_{A}g}(x)dx={\displaystyle {\int }_A\frac{g(x)}{f(x)}}f(x)dx=Q(h(X))$$

(9)

where h(X) = g(x)/f(x), the probability density function of the random variable x is f(x), and f(x) is an appropriately chosen function greater than 0 on the support of g(x).

(3)

Simulate the generation of random numbers X₁, X₂, …, X_N, which are independently and identically distributed and are drawn from the distribution corresponding to f(x).

(4)

${\stackrel{\wedge }{I}}_N$ is used to estimate I.

$$\widehat{I_N}=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}h}\left(X_i\right)$$

(10)

The Monte Carlo estimate of ${\stackrel{\wedge }{I}}_N$ converges probabilistically to I by satisfying Equation.

$$P\left(\underset{N\to \infty }{\lim }{\widehat{I}}_N=I\right)=1$$

(11)

Assuming that S particles exhibiting the drift characteristics of a specific maritime distress target move within the distressed sea area under identical sea state conditions, a computer is employed to stochastically simulate various drift speeds, denoted as V_N, for these S particles. Based on the initial positions and the assigned drift speeds, the AP98 model [18] is utilized to compute the resulting positions of these particles after undergoing downwind drift over a time interval of ∆t hours. This calculation incorporates considerations of wind effects and scenario-specific currents. The drift integral formula is expressed as follows:

$$x(t)=x(t_0)+{\displaystyle {\int }_{t_0}^{t}u(x,t)dt+{\delta }_x}$$

(12)

$$y(t)=y(t_0)+{\displaystyle {\int }_{t_0}^{t}v(x,t)dt}+{\delta }_y$$

(13)

$${\delta }_x=\xi \sqrt{2{\kappa }_x\Delta t},{\delta }_y=\varsigma \sqrt{2{\kappa }_y\Delta t}$$

(14)

where x(t), y(t) represent the position coordinates of east–west and north–south, respectively, u and v are the projections of the wind pressure velocity vectors in the east–west and north–south directions, respectively, ∆t is the simulated time, δ is the physical error correction, ξ and ζ are the two random numbers selected from a standard normal distribution, and κ_x and κ_y are the vortex diffusion coefficients on the sea surface.

At the current time t for the search region, which is further divided into subregions, we retrieved the probability of inclusion in the subregion using the following formula. S_Ai is the number of particles in the subregion and S is the total number of particles in the region: you can obtain the current probability of the search for the target inclusion.

$$PO{C}_t=\frac{S_{A_i}}{S}$$

(15)

4.2. USV Search Mission Success Probability Solution Modeling

The success of a maritime search operation hinges on a combination of two critical factors. Firstly, the planned search mission area must exhibit a high probability of containing the target, and secondly, the USV tasked with executing the search mission in that area must possess the requisite mission capabilities. The success of an USV search and rescue mission requires a comprehensive consideration of these two dimensions of probability, each representing the probability that the search operation will locate the target. This is expressed in terms of the search success rate, denoted as the probability of success (POS). These two likelihood dimensions are characterized as follows: the POC pertains to the current SAR mission area’s likelihood of encompassing the target, while the probability of detection (POD) is indicative of the USV’s ability to spot the lost target.

$$POS=POC\times POD$$

(16)

Based on the above equation, achieving a 100% success rate in UAS searches requires that both the probability of detection and the probability of containment reach 100%. In real-world search and rescue operations, however, it is difficult to achieve this ideal state due to various human and environmental factors. Regional probability of containment and probability of detection typically fluctuate between 0% and 100%. As a result, it becomes necessary to assign USVs to collaborate in the rescue mission simultaneously to compensate for these fluctuations and increase the chances of success. It is obvious that the probability of finding the target to be rescued in a single SAR mission planned by a single UAV is relatively low. The search process often requires the planning and coordination of multiple UAVs. Thus, the concept of cumulative probability of discovery (POD_cum) was born. In a search operation, assuming that Q is the set of subareas in the search and rescue area, Q = {1, 2, …, q} is the number of subareas, I is the set of unmanned boats in the search area I = {1, 2, …, i}, the cumulative probability of discovering the distressed target by multiple unmanned boats.

$$PO{D}_{cum}=1-{\displaystyle \prod _{i=1}^I(1-PO{D}_i)}$$

(17)

In this equation, I is the number of USVs to search and POD_i is the probability of finding the USV. Note that POD_cum is more representative of multiple USV search areas, which is also an important indicator for assessing the probability of success of USV search missions.

$$PO{S}_Q{}_{{}_{task}}=PO{C}_t\times PO{D}_{cum}$$

(18)

The success rate of an USV mission is determined by multiplying the probability that the USV will detect the target by the probability that the search area contains the target at the current time. If the mission’s POS is initially set to a high value, but no distress target is found during the mission, it is essential to assess whether there was an omission during the mission or if there are errors in the relevant data and information. In such cases, the search operation may need to be rescheduled.

Modeling the Probability of Target Detection for USVs in Distress at Sea

The USV search process is inherently uncertain, and search results can be affected by several factors, including USV performance and environmental conditions in the search area. The term “probability of discovery” for USVs refers to the likelihood of locating a lost target within the predetermined search area. Based on search experience, it is observed that when the search target is in close proximity to the USV, the probability of detecting the target is higher. Conversely, if the target is beyond the maximum detection range of the USV, it cannot be located.

Statistical data fitting and probability model analysis are used to determine the probability of detection of USVs in the search process. Since this paper mainly focuses on search and rescue of personnel, it refers to classical data on the probability of detection of small targets such as people in the water, life rafts, dinghies, and other medium-sized targets. Statistical data from reference [19] on the probability of detection of search and rescue personnel at sea is used, as well as reference to the probability of detection of rescue personnel under airborne optical conditions [20]. A Gaussian function-fitting method is used to construct the model and the model parameters are corrected accordingly. The Gaussian function is an exponential function model with the following formula:

$$y=a{e}^{-\frac{{(x-b)}^2}{c}}$$

(19)

To speed up the convergence of the algorithm and improve its performance and accuracy, the Gauss–Newton method is used to optimize the parameter solution step. The principle of minimizing the sum of squared errors is used for parameter selection. Observed data are then fitted to analyze the relationship between variables and to derive the function parameters.

The model was constructed using the Gaussian function. The formula for the relationship between the distance unit (km) and the probability of detection of unmanned boats for medium targets (life rafts, dinghies, etc.) under the condition of visibility ≥15 km was also constructed, as follows:

$$PO{D}_M=0.96e^{\frac{-{(x_t-1.27)}^2}{16.48}}$$

(20)

where POD_M is the probability of detecting a medium target, and x_t is the distance between the USV and the current mission point at the current time t. Substituting the data to obtain the correspondence between the probability of detecting a medium target and the distance is shown in

Table 2. Detection probability versus distance for medium targets.

Distance/NM	0.3	0.6	1.0	1.4	1.8	2.2	2.6	3.0	3.4	3.8	4.2
POD	0.93	0.95	0.93	0.86	0.73	0.59	0.45	0.31	0.20	0.12	0.07

.

Similarly, the distance and probability of detection equations for unmanned boats are obtained for small targets (life rafts, dinghies, etc.):

$$PO{D}_L=0.91e^{\frac{-{(x_t-1.28)}^2}{17.72}}$$

(21)

where POD_L is the probability of detecting small targets and x_t is the distance between the USV and the current mission point at the current time t. The substitution of the data to obtain the correspondence between the probability of detecting small targets and the distance is shown in

Table 3. Detection probability versus distance for small targets.

Distance/NM	0.3	0.6	1.0	1.4	1.8	2.2	2.6	3.0	3.4	3.8	4.2
POD	0.88	0.90	0.89	0.82	0.71	0.58	0.44	0.32	0.22	0.14	0.08

.

A probability of discovery model for a single USV searching for a lost sea target is formulated. In addition, the probability of discovery model for USVs searching for medium and small targets is fitted using statistical data from the literature. This model serves as the basis for the subsequent phase of USV search task allocation.

4.3. Multiple USV Collaborative Search Tasking

The maritime environment is known for its complexity and variability, and individual USVs have limited capacity. Consequently, large area search tasks require the cooperative efforts of multiple USVs. Within a given search area, we predict the probability distribution of positions using the Monte Carlo stochastic particle method. This method is based on the last reported position of the lost target and takes into account its motion characteristics. In addition, we use the Bayesian method to predict position points within the region that serve as initial task positions for USVs. Multiple USVs are then assigned search and rescue tasks with the goal of achieving fast, efficient, and cost-effective wide-area dragnet searches. Once the target is located, the boats can quickly reach it for rescue operations. This process is illustrated in Figure 4.

To increase the efficiency of search and rescue operations, the search area is typically divided into several grids. These grids are treated as a large number of independent bounded regions that do not overlap. The predictive analysis of the containment probability of the target to be rescued is then performed based on the approach described in Section 4.1.

4.3.1. Rescue Mission Bayesian Initial Point Prediction

In the Bayesian method, any unknown parameters can be replaced by probability distribution functions (PDFs). By dynamically updating the probability of inclusion in the search region using the random particle method and integrating it with the Bayesian approach, we can predict the distribution of lost targets within the region. The USV search operation is initiated using the predicted location of the lost target as the starting point for the search mission. The state of the target to be searched and recovered in region i at time k is represented as x_ik within each region. The Bayesian prediction of the starting point for the rescue mission involves three steps:

(1)

Determining the initial probability distribution function.

First, the a priori probability of the target’s location is estimated using a maximum likelihood approach, which quantifies the initial probability distribution within its search and rescue area. Therefore, the probability distribution of its different regional states at time t₀ is represented as a vector:

$$\left[p(x_{1t_0}\left|z_{1t_0}\right.),p(x_{2t_0}\left|z_{2t_0}\right.),\dots ,p(x_{n{t}_0}\left|z_{n{t}_0}\right.)\right]$$

(22)

Update the inclusion probability POC based on the Monte Carlo random particle method to obtain the probability distribution under k moments:

$$\left[p(x_{1t_k}\left|z_{1t_k}\right.),p(x_{2t_k}\left|z_{2t_k}\right.),\dots ,p(x_{n{t}_k}\left|z_{n{t}_k}\right.)\right]$$

(23)

(2)

Probabilistic updating the process in the subregion.

At time k, the latest observation information sequence z_i(1:tk) is obtained by combining the environmental information and the Monte Carlo stochastic particle simulation, which is combined with the probability distribution at time k − 1, and the probability distribution at time k is obtained by applying the marginal distribution with the conditional independence formula:

$$p(x_{i{t}_k}\left|z_{i(1:t_k)}\right.)=\frac{p(z_{i(1:t_k)}\left|x_{i{t}_k}\right.)p(x_{i{t}_k}\left|z_{i(1:t_k-1)}\right.)}{p(z_{i(1:t_k)}\left|z_{i(1:t_k-1)}\right.)}$$

(24)

It is worth noting that when k = 1, the update is obtained using the following equation:

$$p(x_{i{t}_k}\left|z_{i(1:t_k-1)}\right.)\equiv p(x_{i{t}_0})$$

(25)

(3)

Forecast of targets to be searched and rescued in the sub-region.

When analyzing the probability distribution of a lost target, it is crucial to consider the element of time. Assuming that we are at moment k, and we have knowledge of the probability distribution at moment k − 1, we can predict the potential probability distribution at moment k + 1 using the following Chapman–Kolmogorov equation.

$$p(x_{i{t}_{k+1}}\left|z_{i(1:t_k)}\right.)={\displaystyle \int p(x_{i{t}_{k+1}}\left|x_{i{t}_k}\right.)}p(x_{i{t}_k}\left|z_{i(1:t_k)}\right.)d{x}_{i{t}_k}$$

(26)

With the help of the above analysis, the probability model in the sub-region at the k + 1 moment and the final state model of the particle movement at the k + 1 moment, the prediction of the target to be searched and rescued in the sub-region is realized, which serves as the starting point of the next stage of the unmanned boat search task.

4.3.2. USV Configuration Program for Maritime Rescue

A Bayesian approach is used to predict the location of the lost target within the sub-region as a starting point for the search task at a future time. The prediction of the number of starting points of the search mission is based on a combination of the probabilities of inclusion of the region. The assessment of the search and rescue capability of USVs is an important prerequisite for mission planning. The search and rescue capability of USVs is quantified by E_q (E_q is a variable ranging from 0 to 1), and the higher its value, the stronger the capability. The USV search and rescue mission capability evaluation method is a fuzzy comprehensive judgment method. First, the comprehensive evaluation index system of search and rescue capability is constructed, and the task capability index system is constructed by combining the characteristics of the search task. The evaluation indexes are mainly centered around USV search and rescue scale, task satisfaction, environmental adaptability, task implementation timeliness, program economy, and other five dimensions to measure and build its index system as shown in Figure 5.

Capacity assessment indicators were constructed and hierarchical analysis was used to determine the weights of the indicators.

$$B=\{B_1,B_2,B_3,B_4,B_5\}$$

(27)

Create a formula representation of the corresponding indicator rubrics according to {good, better, fair, poor}:

$$V=\{v_1,v_2,v_3,v_4\}$$

(28)

In order to comprehensively assess and analyze the SAR capability for the task at hand, a multidisciplinary team of experts is employed to apply a comprehensive evaluation methodology. This method allows quantitative analysis of objective search and rescue environment and the subjective search and rescue ability, with the aim of achieving the best search and rescue results. Using the degree of affiliation to document the evaluations of various evaluation subjects on specific dimensional indicators, a fuzzy judgment matrix is constructed for the comprehensive evaluation of the aforementioned indicators to measure the search and rescue capability of USV.

$$R=\left[\begin{array}{l}{R}_1\hfill \\ R_2\hfill \\ R_3\hfill \\ R_4\hfill \\ R_5\hfill \end{array}\right]=\left[\begin{array}{llll}{r}_{11}\hfill & r_{12}\hfill & r_{13}\hfill & r_{14}\hfill \\ r_{21}\hfill & r_{22}\hfill & r_{23}\hfill & r_{24}\hfill \\ r_{31}\hfill & r_{32}\hfill & r_{33}\hfill & r_{34}\hfill \\ r_{41}\hfill & r_{42}\hfill & r_{43}\hfill & r_{44}\hfill \\ r_{51}\hfill & r_{52}\hfill & r_{53}\hfill & r_{54}\hfill \end{array}\right]$$

(29)

Here, R_n refers to the value of the set of rubrics for the evaluation indicator; r_n1, r_n2, r_n3, and r_n4 refer to the degree of affiliation of the four rubrics of the evaluation indicator, which are good, better, average, and poor, respectively. Their degree of affiliation must satisfy the following equation.

$${\displaystyle \sum _{j=1}^4{r}_{ij}=1}$$

(30)

Based on the determined weights of the USVs, the results of the search and rescue capability evaluation are derived:

$$C=B\times R=\left\{c_1,c_2,c_3,c_4\right\}$$

(31)

where c₁, c₂, c₃, and c₄ are the combined SAR capability indicator affiliations. The USV’s ability to search and rescue is expressed as follows:

$$E_q=w_i\times (c_1+c_2+c_3)$$

(32)

where w_i is the confidence coefficient. In combination with empirical data, it is set to 0.6.

The assessment of the search and rescue capability of USV can partially mitigate the idealization of the algorithmic allocation model. After comprehensively considering the capabilities of USV in search and rescue, a search plan for the lost target area is devised using the limited number of USVs to ensure the efficient execution of search and rescue missions. It is important to emphasize that the essence of mission planning lies in maximizing mission effectiveness, which is computed by formulating the exponential function equation of effectiveness:

$$P_q=1-e^{-E_q\ast u_q}·PO{S}_{q_{task}}$$

(33)

where u_q is the number of searches USVs in the target search and rescue subregion q. POS_qtask is the probability of success of the search and rescue task in the current subregion q.

When assigning tasks to USVs, it is very important not only to ensure the highest probability of successful search and rescue, but also to ensure that the number of times of Uses and the duration of the task remain within the prescribed limits. This leads to the formulation of the allocation model:

$$max{\displaystyle \sum _{q=1}^Q(1-e^{-E_q{u}_q}·PO{S}_{q_{task}})}$$

(34)

$$s.t.{\displaystyle \sum _{q\in Q}{t}_q\le T_Q,{\displaystyle \sum _{q\in Q}{u}_q\le I}}$$

(35)

where t_q is the time constraint for the arrival of the USV at the mission point, T_Q is the total time of the planning operation mission, and I is the number of unmanned vehicles mobilized for this mission.

5. Simulation Experiments

On 24 August 2022, at 13:25, the Ningbo Maritime Search and Rescue Center received a distress call from the “Zhejiang Fishery 3088”, which had gone missing in the XX sea area. The initial position of the accident was recorded as 123.55° E and 30.41° N. A total of four crew members and one observer were on board. After conducting a preliminary investigation and assessing the situation, it was determined that the vessel may have experienced a power failure in its batteries while returning to sea. This power failure prevented the main engine from starting and the vessel lost contact with its original location at sea awaiting rescue. It is important to note that the area where the vessel was lost was experiencing predominantly southwesterly winds of forces between 4 and 5, with intermittent gusts of precipitation due to the influence of tropical clouds.

5.1. Simulation Condition Settings

How long a person can survive in seawater after a fall depends on several factors, with water temperature being a key determinant. Individuals who accidentally fall overboard are often at risk of underwater hypothermia, anxiety, and hypoxia. A relationship has been established between seawater temperature and the survival time of individuals in such situations [21]. This relationship is illustrated in Figure 6. It is clear that rapid search and rescue efforts can significantly increase the chances of survival for individuals in the water. Therefore, the plan is to complete the rescue within 4 h.

According to the report, the lost sea area currently experiences light to moderate waves with a height of 1 m to 1.5 m, and the sea water temperature falls within the range of approximately 22 °C to 26 °C, which is conducive to human survival and maintaining physical strength. Based on the initial distress time and location of the target to be located, a search radius of 3E was determined, and then an outer square with a side length of 6E was constructed. Using the National Maritime Search and Rescue Handbook, the error in the projected course of an unmanned craft was estimated to be 5%. Using the standard deviation formula, this calculation resulted in an approximate search area of 100 nautical miles² in the accident area. The accident area is further subdivided into nine search sub-areas along the x and y directions using (−E,E) as the dividing point. The probability distribution within these sub-areas is determined based on the a priori probability model of the target estimation obtained by the great likelihood method in Section 4.1.1, as shown in Figure 7.

It is worth noting that at the time of the incident, a standby rescue vessel was positioned in the sea area at the coordinates 123.46° E, 29.93° N. Under the direction of the Maritime Search and Rescue Center (MSRC), this vessel was instructed to proceed to the site of the accident at a speed of 27 knots. As directed by the MSRC, the vessel was to reach the coordinates 123.55° E, 30.59° N, where unmanned boats would be launched to initiate the search and rescue mission. Based on data calculations, the distance to this designated area is approximately 39.87 nautical miles and it is estimated to take approximately 1.5 h to reach the mission area.

5.2. Results of Location Prediction

Meteorological observation data from the National Ocean Search and Rescue Platform was accessed to obtain information on the weather conditions in the vicinity of the lost target area. The wind field prediction report and sea time prediction data obtained are shown in Figure 8. Based on the analysis of this data, it was determined that the prevailing wind direction in the vicinity of the lost sea area was mainly from the south–southwest direction, with a wind speed of 5 to 6 m per second. As a result, the drift direction of the persons in the water and the life rafts is expected to be north–northeast under the influence of these meteorological conditions.

The direction of seawater flow is a critical element in understanding ocean currents. The direction of seawater flow is typically expressed in degrees, with calculations based on a reference point of due north as zero degrees, increasing clockwise, and due east as 90 degrees. To provide insight into the velocity of the ocean current and its directional changes over time, SAR platform forecast data was used to generate Figure 9. The image shows that the ocean current in the vicinity of the lost sea area is predominantly east–northeast, with current velocities ranging from 0.2 to 0.6 m per second. As a result of these oceanographic conditions, individuals in the water and life rafts are expected to drift in an east–northeast direction.

Based on the observation data, a simulation experiment was conducted using the Monte Carlo stochastic particle method. The experiment was designed to simulate the scenario in which the search and rescue vessel arrives at its designated position after 1.5 h and prepares to deploy the unmanned boat for the search mission.

At the start of the simulation, specific parameters were used for wind speed, wind direction, current speed, and current direction. The wind speed was set at 5.22 m per second with a wind direction of 209 degrees. Additionally, the flow velocity was set to 0.38 m per second and the flow direction was 93 degrees. The Monte Carlo random particle method was used with 10,000 random particles and a simulation step of 1 min. The step size was set to 400. The simulation results are shown in Figure 10.

To effectively illustrate the changes in inclusion probability within the search area, we analyzed the inclusion probability at coordinate location (0,−1.67E), which corresponds to the boundary point between regions A5 and A8. The simulation results for this specific coordinate location are presented in Figure 11.

Based on the inclusion probability analysis using the Monte Carlo particle method at 1.5 h, the inclusion probability within the subregions was calculated using joint generalization. In addition, the initial inclusion probability distribution map for each subregion was determined based on the great likelihood estimation of the target. The updated inclusion probability within the subregions is shown in Figure 12.

The inclusion probability updated in Figure 12 was used as the prior probability in the Bayesian network. Next, the wind speed, wind direction, flow velocity, and flow direction parameters at the 1.5 h time point were incorporated into the stochastic particle model to generate the probability distribution of particles within the region. The parameters selected for this time were a wind speed of 6.98 m/s with a wind direction of 199° and a flow velocity of 0.57 m/s with a flow direction of 59°. The Bayesian-based starting point prediction for the rescue mission was as shown in Figure 13, where the asterisk represents the starting point of the task and the number represents the serial number of the task.

Considering the current wind field and current information when the rescue boat arrives at the scene, the targets likely to be lost are in areas A2, A3, A5, and A6. Based on the Bayesian-based location prediction for these lost targets within this area, this prediction serves as the starting point for the USV search mission during this phase. Subsequently, the rescue mission planning for the USVs in this area is performed. Based on comprehensive analysis, No. 5, No. 6, and No. 8 may be medium targets, that is, people in distress use life rafts.

5.3. Multi-USV Tasking

Three types of USVs are available on the rescue vessel, each with specific parameters, as detailed in

Table 4. USV Basic Information.

Number	1	2	3	4	5	6	7	8	9
Velocity (kn)	6.7	9.2	8.2	6.7	8.2	8.2	6.7	9.2	9.2
Search Coverage Capacity (nm2/h)	12	20	15	12	15	15	12	20	20
Prep Time (min)	12	24	18	12	18	18	12	24	24

. It is worth noting that these weather and sea conditions do not impose any restrictions on the operation of the USVs. In addition, once an USV is assigned to its starting point within a sub-region, it focuses solely on searching that designated area and does not participate in searching other sub-regions once its assigned search is completed.

Combined with USV data, such as search task details for the lost target area, sea conditions, usage times, task allocation status, ocean wind and current field conditions, expected task duration, task distance, and other related information, various indicators usually show uncertainty to varying degrees. To evaluate the USV allocation program, a fuzzy comprehensive judgment method is applied. First, a hierarchical analysis method is applied to determine the weights of the five levels within the index system for assessing the comprehensive evaluation of USV search capability. The construction of the weight matrix is shown as follows:

$$B=[0.2389,0.4804,0.2114,0.0693]$$

It is expected that 1.5 h emergency rescue ship arrives at the designated location emergency rescue ship to launch the USVs. In the evaluation process, through the expert evaluation and scoring, the index comment of USV search capability is given, and the fuzzy comprehensive evaluation matrix of USV search and rescue capability is derived:

$$R=\left[\begin{array}{llll}0.36\hfill & 0.52\hfill & 0.08\hfill & 0.04\hfill \\ 0.08\hfill & 0.44\hfill & 0.32\hfill & 0.16\hfill \\ 0.16\hfill & 0.32\hfill & 0.48\hfill & 0.04\hfill \\ 0.36\hfill & 0.60\hfill & 0.04\hfill & 0\hfill \\ 0.32\hfill & 0.40\hfill & 0.24\hfill & 0.04\hfill \end{array}\right]$$

The results of the comprehensive evaluation of search and rescue capabilities were calculated based on the fuzzy comprehensive evaluation matrix and the weight vectors of the factors:

$$C=[0.2389,0.4804,0.2114,0.0693]$$

The calculation of the SAR mission effectiveness of USV is based on the comprehensive evaluation results of their SAR capability. The mathematical assignment model described in Section 4.3.2 and a dynamic planning algorithm are used to plan SAR missions for USVs. The starting point for the SAR mission is determined using Bayesian methods, and the positions of the USV in the vicinity of the position are generated for mission planning. The simulation results of the mission planning experiment are presented in Figure 14 and Figure 15.

Based on the above mission schedule, taking into account the change of USV speed parameters and the inclusion probability of the mission area, a comprehensive search and rescue mission schedule of USVs is obtained. The USVs are expected to reach the mission point within 1.2 h and can begin searching the area immediately. If USVs arrive in the sub-area, they can collaborate on the search task, significantly reducing the time required for emergency search operations. The collected analog simulation experimental data are used to construct the SAR mission planning information data, as shown in

Table 5. Search for mission program information.

Number	1	2	3	4	5	6	7	8	9
Task Point	5	2	3	8	1	6	9	4	7
Ditance/NM	7.39	9.00	10.02	5.78	8.83	6.65	5.51	10.65	10.37
POD	0.8291	0.8287	0.8287	0.8290	0.8708	0.8684	0.8287	0.8946	0.8299
POC	0.15	0.19	0.15	0.35	0.15	0.19	0.35	0.21	0.21
Sub-area	A2	A6	A2	A5	A2	A6	A5	A3	A3
pq	0.9083	0.8758	0.9077	0.7712	0.9030	0.8699	0.7713	0.8518	0.8625

.

From the data collected in the above simulation, it can be seen that the inclusion probability of A2, A3, A5, and A6 regions for the target reaches 90%, and at the same time for the A2 region, its task search task effectiveness is 90.63%, and at the same time for the A3 region, its task search task effectiveness is 93.06%, for the A5 region, its task search task effectiveness is 77.13%, and for the A6 region, its task search task effectiveness is 87.29%. For the A5 area, the task search performance is 77.13%, and for the A6 area, the task search performance is 87.29%. According to the judgment of sea wind field and sea current, the lost target is predicted to drift east–northeast, and the effectiveness analysis shows that the deployment of key forces to this position area is in line with the mission expectation.

In order to strengthen the justification of the framework model proposed in this paper, a comparative analysis with similar research methods was carried out. Both approaches were compared with references [22,23]. Reference [22] adopts a search theory perspective for maritime search and rescue operations, addressing the optimization of search and rescue efforts for multiple objectives. It formulates a multi-objective hierarchical nonlinear mathematical model for the allocation of search and rescue resources. Reference [23], on the other hand, focuses on task allocation in maritime search scenarios with multi-intelligence technology. It uses a traditional auction rule algorithm to construct a task allocation model for multi-intelligence sea search.

In our comparative experiments, we conducted simulations based on the parameters defined in this paper. These experiments include the current marine environmental conditions, the probability information in this area, and data related to unmanned search and rescue vessels and rescue missions. In the comparison experiment, the code execution time under the three schemes and the total distance of the generated schemes are shown in Figure 16.

As shown in Figure 16, it is evident that the approach presented in this paper has a notable advantage in terms of algorithm execution time. It efficiently assigns tasks to the maritime unmanned boat formation in a shorter time frame. Furthermore, the resulting scheme offers a technological advantage in terms of distance cost. The detailed program data generated from references [22,23] have been compiled and organized in

Table 6. Literature 22 Program Results.

Number	1	2	3	4	5	6	7	8	9
Task Point	1	2	3	4	5	6	7	8	9
Ditance/NM	7.71	9.01	10.03	4.78	8.52	6.56	5.51	11.66	10.39
Time/h	1.15	0.98	1.22	0.71	1.04	0.80	0.82	1.27	1.13

and

Table 7. Literature 23 Program Results.

Number	1	2	3	4	5	6	7	8	9
Task Point	4	7	6	1	2	5	3	9	8
Ditance/NM	6.93	8.40	9.71	5.60	8.08	7.01	6.84	10.78	11.29
Time/h	1.03	0.91	1.18	0.84	0.99	0.85	1.02	1.17	1.23

.

Analyzing the data in Table 6 and Table 7, it can be seen that the route length in the scheme proposed in this paper, 74.12, is better than the literature [22] with 74.16 and literature [23] with 74.64. In addition, considering the speed profiles of the USVs in conjunction with the three schemes, we can determine the longest subtask time among them, which serves as an estimate of the time required to complete the scheme. In this regard, the estimated completion time for the program in this paper is 1.42 h, while literature [22] requires 1.47 h and literature [23] requires 1.43 h. This suggests that the research framework presented in this paper is better equipped to provide decision makers with a higher quality unmanned boat search and rescue allocation plan in a shorter time frame.

6. Discussion

In this study, we presented a mission planning framework designed to facilitate the collaborative rescue of lost targets using USVs. This framework supports search operations for lost targets at sea using both manned vessels and USVs. The main features of the proposed framework are as follows:

(1)

We have used a prediction module to predict the position of an outboard target by using a high probability method. This module generates a search area with the inclusion probability of the location of the lost target. And the wind and current field data in the marine environment and Monte Carlo random particle method are used to perform simulations. This approach produces a probability distribution map for task feasibility.

(2)

The framework recognizes that the probability of success of USV search missions is determined not only by the probability of inclusion within sub-areas, but also by the probability of target detection by USV. Consequently, it conducts evaluations and analyses of both the search and rescue capabilities of USVs and the effectiveness of their missions. This comprehensive assessment of USV capabilities informs mission allocation decisions with the goal of increasing search and rescue effectiveness and decreasing overall time. The framework also provides a heuristic approach to searching for lost targets.

(3)

Unlike traditional mission planning approaches, this framework uses the predicted location of the lost target as the starting point for USV missions. It takes into account the influence of various variables, such as individual survival time, wind, and current patterns on target drift, and incorporates parameters related to USV. This overall consideration is helpful for establishing a search and rescue plan that takes into account the practicality and feasibility of the plan, thus improving the ease of carrying out search and rescue tasks for missing target.

Furthermore, the framework presented in this study can be seamlessly integrated into existing search and rescue procedures for lost targets. It provides regional predictions of lost target locations and facilitates USV task assignment and search coordination, serving as a valuable decision support tool for search and rescue personnel during operations. Certainly, in practical applications, the development of decision-making programs for USVs involved in maritime search and rescue operations often faces challenges related to condition monitoring errors and equipment malfunctions. Current research using a signal-based approach to fault detection [24], and even calibration of sensor fault data for troubleshooting [25], provides valuable inspiration for addressing these issues.

In the next phase of our research, we will look more closely at the construction of planning schemes that fuse data and planning models in the context of multi-intelligence maritime search and rescue operations. At the same time, we will discuss and analyze the joint search and rescue scene of UAV and USV maritime decision planning algorithm based on search and rescue theory. Our main focus will be refining the planning scheme framework for the search and rescue model, with particular emphasis on improving data accuracy and reliability.

7. Conclusions

In this paper, we have conducted a simulation study that addresses the challenge of coordinating USVs in a collaborative search for lost targets at sea. In view of the complexity of this problem, involving multiple parameters and limitations, we introduce a task allocation framework, which depends on the probability of success and time constraints related to the use of search and rescue tasks. In summary, our approach in this paper involved several key steps. First, we performed an analysis of the search area for the lost target using the large likelihood estimation method. Then, we predicted the location of the target based on marine environment forecast data by using the stochastic particle method to update the inclusion probability within the region. In addition, we evaluated and analyzed the search and rescue capability and mission effectiveness of USVs in improving the success rate of the task assignment scheme. We conducted simulation experiments and analyzed task assignment schemes to demonstrate the effectiveness of our proposed framework. It is worth noting that our focus in this paper was primarily on the SAR process for responding to a lost target at sea. We used a relatively simple model for path concatenation in the assignment process.

Indeed, the practical application of a synergistic framework for USVs working together to search for lost targets at sea presents significant challenges and requires further effort. This paper has primarily focused on a scenario where multiple USVs are launched by rescue vessels, which are typically part of a unified organizational system. However, in real-world situations, rescue forces responding to lost targets at sea often come from different organizations and systems and involve different rescue vessels. These forces can be viewed as discrete entities. The research proposal still has some limitations, which will be studied and explored in the next phase and needs to be combined with more maritime unmanned boat search and rescue cases to explore the applicability of our proposal under mature data and needs to be further explored for the framework modeling solution for the case of missing data and data errors.

In the future, the framework must be expanded to deal with joint search and rescue operations involving many different forces and ships responding to complex maritime accidents. Efficient and coordinated coverage of USVs in a given region is also a topic worthy of study and development for improving the effectiveness of maritime search and rescue missions.