1. Introduction
With artificial intelligence at the helm, the advent of unmanned surface vehicles (USVs) has garnered significant attention, fueled by their potential to revolutionize maritime operations by enhancing safety and efficiency [
1,
2,
3,
4,
5,
6]. However, the successful deployment of USVs depends on the development of autonomous technology, which refers to the ability of these vehicles to plan and execute their missions in complex environments without human intervention, thereby enabling safe and efficient navigation [
7,
8]. Generally speaking, the most common approaches that contribute to the autonomous level of USVs are perception, localization and mapping, path planning and decision-making, and control system design. Central to achieving autonomy in USVs is the challenge of path planning, which involves determining an optimal path for the vehicle to traverse in order to accomplish its mission objectives while adhering to a set of predetermined rules and regulations. Compared to other types of autonomous vehicles, such as unmanned ground vehicles (UGVs), USV path planning may incorporate specialized techniques to handle challenges, such as wave prediction models, collision avoidance strategies for vessels, or algorithms, that account for hydrodynamic effects on the vehicle’s motion. This task is particularly challenging due to the dynamic nature of maritime environments, which are subject to constantly changing weather conditions, currents, and other environmental factors that can impact navigation [
9]. Achieving this goal requires the development of sophisticated path planning algorithms that enable these vehicles to navigate complex environments with minimal human intervention, paving the way for a future of safe and efficient maritime navigation [
10].
The field of path planning for USVs has been an active area of research in recent years, with numerous studies investigating the development of effective planning strategies for USVs. In general, two primary categories of path planning algorithms have been proposed: global approaches and local approaches [
7]. Global approaches involve the generation of a complete path for the USV based on prior knowledge of the environment, usually represented as a map. Such methods typically employ high-level planning techniques that treat the USV as a point object, neglecting its maneuverability and physical constraints. These methods are, therefore, more suitable for planning routes for long-distance voyages, where the emphasis is on efficient and safe navigation over extended periods. In contrast, local approaches generate a path by utilizing local information collected during the mission, enabling the USV to adapt to unexpected obstacles or changes in the environment. These methods fully consider the physical bounds of the USV’s mechanical system, leading to more precise tracking performance for the low-level controller. Although the design of such methods is generally more complex, as it requires the integration of high-level planning and low-level control techniques to ensure effective operation, it is more applicable in practice.
Presently, there is a growing affinity for deterministic approaches in path planning, with various methods, such as A* and D* lite, basking in the limelight of scientific popularity. In particular, Yu and Wang [
11] have put forward a hybrid algorithm that fuses artificial potential field (APF) and D* lite to navigate complex environments. This approach not only minimizes time cost but also enhances path safety through the APF. Nonetheless, it overlooks disturbances and energy consumption. Similarly, Yu et al. [
12] have proposed an improved D* lite that reduces expanded nodes, validated via simulations. However, the simulation fails to consider ship dynamics, smoothness, and safety. Meanwhile, Song et al. [
13] have utilized various smoothing techniques to mitigate the jagged effect in A*, which has been demonstrated to be effective through both experiments and simulations, making it a practical choice. Furthermore, Shah and Gupta [
14] have presented a quadtree representation of the marine environment, which accelerates the A* algorithm without significantly sacrificing solution optimality, as shown in simulations. To facilitate path planning for working ships in offshore wind farms, Xie et al. [
15] devised a multi-direction A* algorithm modified by an artificial potential field. Compared with the real-case trajectory, the minimum distance to the wind turbines has increased, and the path length outside the wind farm decreased dramatically. To solve the path planning problem under changing environments with multiple dynamic obstacles, Yao et al. [
16] proposed an Improved D* lite algorithm, which has demonstrated its efficacy in real-time path planning through simulation results. Although deterministic approaches have emerged as popular and reliable methods for path planning, these methods can be computationally expensive, particularly when operating in large and complex environments. This can have a significant impact on their performance, making them less suitable for real-time applications.
As a result, the meta-heuristic algorithms, including ant colony optimization (ACO) [
17], particle swarm optimization (PSO) [
18], and genetic algorithms (GA) [
19], have emerged as promising alternatives for path planning in marine robotics. These algorithms offer a set of high-level strategies to search for solutions, allowing them to optimize paths while considering multiple objectives with a comparatively low computational burden. Considering the effects of currents, Krell et al. [
20] devised an improved PSO method implemented in visibility graphs. For the safe navigation of ships, a quasi-reflection-based PSO was proposed by Xue [
21]. Incorporating the environmental loads, a hierarchical path planning framework based on GA is developed by Wang and Xu [
10]. For rapid path generation, a leader-vertex ant colony optimization algorithm is proposed by Liang et al. [
22], which ensures a leader of the ant colony and optimizes the route by vertex method. For both global and local path planning, a series of studies on artificial fish swarm algorithms have been conducted by Zhao et al. [
7,
8,
23]. Under different current distribution, a comprehensive study has been conducted by Ma et al. [
24] using multi-objective dynamic augmented PSO. Organically bridging the planning and tracking, Wang et al. [
25] devised an elite-duplication GA (EGA) strategy to optimally generate sparse waypoints in a constrained space. However, meta-heuristic algorithms commonly encounter a significant hurdle with regard to their global searching capabilities, as they are prone to be trapped in local minima or suboptimum, thereby impeding the identification of the global optimum necessary for producing high-quality trajectories. Additionally, the computational efficiency of existing methods is not satisfactory enough to facilitate efficient path generation within high-dimensional configuration spaces [
26]. Therefore, there is an imperative need for innovative techniques that can enhance the global searching capability and convergence rate of meta-heuristic algorithms for path planning.
In addition, a noteworthy limitation of most existing methods is the neglect of the non-holonomic constraints of the vehicle, which can lead to paths that are potentially infeasible. Specifically, the USV, being a non-holonomic robot, often functions as an underactuated system during its missions, resulting in limited maneuverability and motion flexibility [
27,
28]. Analogous to unmanned ground vehicles (UGVs), this restricts the USV’s motion to forward velocity and manipulation of the heading angle to attain its desired position, thus precluding lateral movement. Consequently, it is of paramount importance to ensure smooth and continuous transitions of yaw and curvatures at turning points in order to devise an effective trajectory for a USV. For instance, sharp turns may be deemed unfeasible for a USV due to the significant sideslip that ensues, deviating from the planned path. Thus, the motion dynamics of the USV should be meticulously accounted for in the path planning process [
10].
Inspired by the aforementioned literature review, this paper proposes a novel GA-variant meta-heuristic algorithm in combination with a fast-discrete Clothoid curve to optimize path generation. The main contributions of this paper are illustrated as follows:
By capturing the non-holonomic nature of USVs and the intricate ocean dynamics, a sophisticated optimization model is carefully devised for the path planning problem, whereby the effects of currents, increments of curvatures, and constraints of physical system are addressed jointly.
Introducing the random testing initialization algorithm and the adaptive design in the selection procedure, the proposed GA-variant facilitates strong global searching capabilities and a fast convergence rate, thereby contributing to the optimal generation of waypoint sequence.
Accommodating the non-holonomic constraints, the fast-discrete Clothoid curve is able to preserve and enhance the continuity of the path curve, resulting in robust coordination between the planning and control module.
This paper is organized in the following structures:
Section 2 presents the detailed modeling of the environment, USV, and the optimization model.
Section 3 introduces the methodology. Illustrative simulation results are shown in
Section 4. The conclusion is drawn in
Section 5.
3. Solver Design
3.1. Adaptive-Elite Genetic Algorithm
The genetic algorithm (GA) was initially proposed by Professor J. Holland in 1973 as a meta-heuristic optimization method. By simulating the evolutionary process of an artificial population, the GA manipulates each individual in the population through genetic operations, such as selection, crossover, and mutation. The process generates a new population with the best-performing individuals from the previous generation as the parents. The population evolves through several generations, and the individuals with the best fitness values are selected as the optimal solutions.
The GA’s strength lies in its ability to search a large solution space using stochastic searches and evolutionary operations, such as crossover and mutation, making it effective in handling non-linear and non-convex optimization problems. Moreover, the GA’s population size enables it to mitigate the impact of hyperparameter selection by allowing the algorithm to sample from a diverse set of solutions. Given that the optimization problem presented in this paper is an NP-hard nonlinear problem, we choose the GA as the primary framework.
3.1.1. Chromosome Representation
In evolutionary algorithms, chromosomes can be represented in various ways, such as binary-coded, real-coded, and decimal-coded. In our paper, we utilized the real-coded chromosome to directly represent the USV’s path. Specifically, we use a sequence of points that begins at an origin position and ends at a destination point. Each point, denoted as
, is saved along with its x and y coordinates and a pointer to the next point in the path.
Figure 2 illustrates this representation.
3.1.2. Initialization
The original population for the algorithm is obtained through population initialization. In the case of the original GA, a certain number of solutions are randomly generated in the solution space without the use of a heuristic function. This can lead to a random and unfocused solving process, resulting in a high proportion of poor solutions and low-quality genes in the population. This, in turn, requires a long convergence time during subsequent evolution and makes the solving process prone to being trapped in a local optimum.
To address this issue, we have developed a modified initialization method for GA inspired by failure analysis techniques used in software systems. Specifically, we have incorporated a candidate set adaptive random testing (ART) approach to improve the diversity of the initial population. By enhancing the initial diversity, the ART-based initialization method allows the GA to explore a broader range of potential solutions. This exploration can improve the algorithm’s ability to escape local optima and discover better solutions in the search space. Consequently, it enhances the chances of finding high-quality solutions and can potentially accelerate the convergence toward optimal or near-optimal solutions. In summary, compared to standard random initialization, the ART-based initialization method in the GA offers the advantage of generating an initial population that is more diverse and better distributed throughout the search space. This increased diversity can facilitate improved exploration of the solution space and potentially lead to better overall performance and convergence in the GA.
The main steps of the initialization process are illustrated as follows:
Step 1: candidate individuals are randomly generated.
Step 2: The objective distances between each candidate individual and the current individuals in the population set are calculated.
Step 3: The shortest distance between each candidate individual and the population set is identified.
Step 4: The candidate individual with the maximum distance value is selected and added to the population set .
3.1.3. Selection Operator
In a genetic algorithm, the selection operator is responsible for choosing individuals from a population for the crossover operator. This selection process is carried out based on a predefined regulation called the Roulette Wheel Selection (RWS) method. To perform Roulette Wheel Selection (RWS) in a genetic algorithm (GA), first, we compute the fitness values for each individual in the population and normalize them to obtain probabilities. Then, calculate the cumulative probabilities by summing up the normalized fitness values. After that, generate a random number between 0 and 1, and select individuals whose cumulative probability exceeds this random number. Repeat the selection process as needed to obtain the desired number of parents. Use the selected parents for crossover or recombination to generate offspring for the next generation. This iterative process allows individuals with higher fitness to have a greater chance of being selected, promoting the propagation of favorable traits in the GA. The RWS method ensures that the fitter individuals have a higher chance of being selected for the crossover, thus improving the overall quality of the population in the subsequent generation. The selection probability of each individual can be expressed as follows:
where
denotes the individual and
is the corresponding fitness value. As can be seen from Equation (15), better individuals have more chances to be selected by RWS, which leads to better solutions.
3.1.4. Hybrid Crossover
Crossover operators are utilized to combine two solutions and generate a new offspring with better performance in terms of a predefined objective. These operators can be applied to solutions with the same or different number of waypoints. The first crossover operator involves calculating the mean of the two parent solutions to produce a new offspring.
where two parents have gene coordinates denoted by
,
, and
,
, respectively. The gene coordinates of the offspring are represented by
and
. First, two parent chromosomes are selected according to the selection operator. Second, we select one of the parents as a reference chromosome. In this procedure, if the number of waypoints in the parents are the same, we choose the reference randomly. Otherwise, the one with smaller waypoint number is chosen. Then, waypoints of the offspring
are calculated by taking the mean of each waypoint of the reference chromosome and the nearest waypoint of the other parent. To determine the gene coordinates of the offspring
, the average of each gene in the selected parent and the closest gene in the other parent are computed.
To enhance the variability of the population and explore the entire available space, the second crossover operator is utilized in which the two parents are randomly merged:
where vector
consists of random numbers ranging from −1 to 1. When the number of genes differs between the parents, a similar approach to the first operator is used to combine genes with minimal distance. The primary aim of the first operator is to escape local optima, while the second one explores the environment randomly, preventing premature convergence.
3.1.5. Mutation Operator
The mutation operator in a Genetic Algorithm has a critical function in maintaining diversity within the population. The primary goal of the mutation operator is to randomly modify the value(s) of one or more genes within an individual’s chromosome. By introducing such changes, the mutation operator assists in preventing the GA from becoming trapped in a local optimum, which would hinder the search for the global optimum. In the absence of the mutation operator, the GA may converge to a suboptimal solution that is in close proximity to the initial population. Hence, mutation serves as a crucial component of GA by promoting exploration of the search space and preventing premature convergence to suboptimal solutions. This paper introduces two mutation operators to facilitate the genetic process:
The first operator is a random mutation that selects one position on chromosomes and changes the value in the free space as shown in the following figure:
A different mutation operator is utilized to enhance the path’s smoothness and length by adjusting the position of a gene. The operator integrates the present position (
) of a gene with the directions towards the genes located on either side,
and
, using the subsequent expressions:
where
and
are random positive coefficients from 0 to 1. As illustrated in
Figure 3b, this mutation operator leads to the creation of paths with shorter lengths and better smoothness. Combining both mutation operators result in a powerful tool that enhances both the searching and convergence capabilities of the algorithm.
3.1.6. Fitness Design
In this paper, a multi-objective fitness function is devised. For this purpose, a weighted linear combination of the mentioned objectives is considered:
The formula involves several parameters, including (cruising time), (smoothness objective), and (safety level), where , , and represent weight values, and their sum is equal to 1. To maintain consistency in the indicators’ magnitudes, coefficients , , and have been set to 0.1, 1, and 100, respectively, as shown in the equation.
Selecting appropriate weight values is a vital aspect of the algorithm’s performance. However, relying solely on empirical methods can be subjective. In an effort to achieve more balanced results, the Delphi weighting method [
19] was employed to determine the weight of each indicator. As a result, the weight coefficients for cruising time, smoothness, and safety are 0.395, 0.275, and 0.330, respectively.
3.1.7. Determination of and
The conventional Genetic Algorithm for USV path planning relies on two essential parameters, namely, the crossover rate and mutation rate, to regulate and of individuals in each iteration. However, using fixed values for these parameters may pose certain challenges. For instance, employing a large crossover and mutation probability can make it difficult to retain the best individuals, slow down population convergence, and consequently, delay the generation of the inspection path, thereby impacting operational efficiency. Conversely, a small and can negatively affect the searching process, leading to the local optimum. This, in turn, causes the USV to travel longer distances, reducing its efficiency.
To tackle the aforementioned challenges, a modified approach is suggested. This approach involves adjusting and during the algorithm execution. Specifically, in the early stages of the algorithm, and are increased to improve the global search ability, while in the final stage, the probabilities are decreased to facilitate good convergence. Adaptive probabilities allow the GA to dynamically adjust the rates of crossover and mutation based on the progress of the algorithm. Initially, higher probabilities promote exploration by encouraging diverse offspring. As the GA progresses, the probabilities can be reduced, shifting the focus towards exploitation of promising solutions. This balance between exploration and exploitation helps the GA efficiently search for optimal or near-optimal solutions.
To achieve this, adaptive functions are formulated as follows:
where
represents the initial
,
is the scaling coefficient, and
here is the average fitness of the population. Similarly, the mutation probability can be obtained with same structure as follows:
where
represents the initial
and
is the scaling coefficient. These functions dynamically adjust the crossover probability based on the mean fitness degree of the population at each generation. As a result, the USV can achieve a balance between exploration and exploitation, leading to faster convergence and better results.
3.2. Fast-Discrete Clothoid Curve
To ensure real-time performance and accommodate the USV’s kinematic constraints, we introduce a Fast-Discrete Clothoid Path (FDCP) to construct and connect the path. The FDCP employs a sequence of control points, referred to as waypoints, which are linked together using Clothoid segments. However, accurately generating Clothoids can be difficult due to their non-linear nature and multiple solutions. Thus, instead of directly computing the parameters of the Clothoid segments, our algorithm utilizes a variational approach that produces a polyline with linear discrete curvature, which approximates the Clothoid segment. This approach allows for efficient and precise path planning for the USV, while taking into account the vehicle’s non-holonomic features.
To determine the position of intersection points, the following conditions must be met when inserting or updating point C between neighboring points B and D, as shown in
Figure 4. To simplify the calculations, a normalized configuration is used where point B is located at (−1, 0) and point D at (1, 0). For each of the five control points denoted as P, its left and right neighbors are identified as
and
, respectively. The insertion point C must satisfy the following conditions to approximate the Clothoids accurately between these control points:
From the abovementioned conditions, we have the following equation:
where
denote the curvature at each point, it can be approximated by:
where
is the angle between
and
, and
is the scalar value of the length between
and
. According to the geometric relations in
Figure 4, we obtain:
Substituting
in previous equation using Equation (16), we have:
As more and more points are inserted, the polyline gets refined and the angles between segments approach
. Therefore, with a large number of sample points, we can approximate
. Solving Equation (14),
can be obtained by:
Point C is now inserted on the perpendicular bisector between B and D in distance . By iteratively inserting the intersection points (such as point C), we can approximate the Clothoid path with satisfactory computational performance.
Clothoid curves provide a continuous change in curvature, resulting in a smooth transition between straight segments and curved segments of a path. This helps reduce abrupt changes in the path and improves the vehicle’s stability and comfort. Moreover, by gradually changing the curvature, Clothoid curves minimize lateral acceleration during turns. This reduces the forces acting on the USV, enhancing safety and stability during maneuvering. Additionally, Clothoid curves enable more precise and controlled maneuvering. They allow for gradual changes in heading angle, facilitating smooth turns, and transitions between different paths or waypoints.
The flowchart of the methodology is illustrated by
Figure 5.