Collision-free formation method for multi-unmanned surface vessels based on distributed multi-objective optimization under random perturbation

Abstract

This invention discloses a collision-free formation method for multiple unmanned surface vessels (USVs) under random disturbances based on distributed multi-objective optimization. The method includes applying a distributed multi-objective optimization algorithm based on zero gradient sums, determining optimal weights according to the Pareto front and utopia value of the multi-objective cost function, and obtaining a smooth formation trajectory satisfying the kinematic constraints of the USVs using cubic B-spline curves based on the optimal time-series waypoint sequence. An anti-disturbance trajectory tracking control law is constructed for the multiple USVs based on the smooth formation trajectory. This anti-disturbance trajectory tracking control law enables collision-free formation control of multiple USVs under random disturbances based on distributed multi-objective optimization, allowing the multiple USVs to accurately track the smooth optimal trajectory even in random disturbance environments, thus achieving collaborative formation tasks. This invention solves the problem that existing methods struggle to guarantee the safe, low-consumption, and reliable execution of formation tasks by multiple USVs in complex and confined waters.

Description

Technical Field

[0001] This invention relates to the fields of multi-objective optimization technology, unmanned swarm collaborative planning technology, and unmanned surface vessel (USV) disturbance resistance control technology, and particularly to a collision-free formation method for multiple USVs based on distributed multi-objective optimization under random disturbances. Background Technology

[0002] Multi-UAVs (Multi-Unmanned Surface Vessels) serve as crucial platforms for maritime transport and collaborative operations, offering a wide range of applications including maritime spatial layout maintenance, trajectory planning, marine environmental monitoring, and maritime rescue. Formation missions, in particular, require the synchronized deployment and precise formation of multiple UAVs to achieve efficient area occupation, thus laying the terrain foundation for subsequent collaborative tasks. However, collision avoidance and energy consumption are two critical factors to consider in formation missions. For example, when multiple UAVs navigate through narrow ocean areas cluttered with coral reefs and buoys, inadequate collision avoidance mechanisms can lead to collisions, resulting in equipment damage and mission failure. Furthermore, inappropriate trajectory planning can result in circuitous routes, leading to a sharp increase in energy consumption. Therefore, comprehensively considering these two indicators is essential for achieving efficient, reliable, and safe formation missions.

[0003] Traditional research often relies on human experience to set the weights of the objective function. This "trial and error" parameter tuning method is highly subjective and uncertain, making it difficult to guarantee optimal navigation of multiple unmanned surface vessels (USVs) in variable ocean environments. Therefore, exploring a scientific weighting decision-making mechanism is of great significance. Through systematic sampling and performance boundary analysis, the competitive relationships between various task indicators can be revealed mathematically. This not only eliminates the interference of subjective factors on formation performance but also ensures the theoretical rigor of the decision results. Furthermore, the transition from discrete decision points to physically feasible trajectories is crucial for the successful implementation of the planning layer. The original waypoints generated by distributed optimization algorithms are usually discrete. If used directly as reference signals, they can lead to frequent speed jumps and drastic heading adjustments during USV navigation. How to smooth the preprocessing, effectively eliminate the mechanical shock caused by command jumps to the propulsion system, protect the actuators from fatigue damage, and make the USV's trajectory smoother and more fluid while ensuring obstacle avoidance safety are also worthy of further research. Trajectory tracking is a crucial step in multi-UAV swarm operations. Wave disturbances must be considered because they directly exert forces and torques on the multiple UAVs, significantly impacting the controller's duty cycle design, thereby interfering with propeller speed and reducing tracking accuracy. Therefore, modeling wave disturbances as random disturbances acting on the control input and analyzing and modeling them in a more precise manner can provide a potential approach for designing multi-UAV tracking controllers.

[0004] Currently, pre-generating waypoints or trajectories in the planning layer of formation missions can significantly reduce the computational burden on multi-UAV controllers. Some studies have proposed a path search strategy based on trajectory units, implementing the A* algorithm through a designed cost function. Others utilize deconvolution operations in deep learning theory to generate the solution space of waypoints in multi-UAV anti-submarine search missions. Still other studies have proposed a predictive trajectory method integrating dynamic constraints and heading angle optimization to achieve safe formation navigation of multi-UAVs. However, the distribution of waypoints is directly affected by the trade-offs between mission indicators, and the above studies are difficult to simultaneously meet the multi-objective requirements of real-world scenarios. A potential solution is to formulate a multi-objective cost function and assign weights to sub-tasks. For such problems, centralized optimization algorithms compute the optimal solution through a central node, providing a method for multi-objective decision-making problems. However, such methods are prone to single points of failure, and as the number of UAVs increases, the computational and communication overhead of the central node increases rapidly, leading to reduced planning efficiency and failing to meet the requirements of large-scale cluster collaborative operations. Secondly, for disturbance rejection control, existing research typically characterizes wave effects as suppressable or compensable external signals and employs methods such as observer-based estimation, intelligent approximation algorithms, or modeling them as white noise using Iton's formula for analysis. However, wave disturbances are more similar to non-uniformly distributed non-white noise and stationary processes, making them difficult to analyze using methods characterized by white noise.

[0005] Therefore, there is an urgent need to provide a collision-free formation method for multiple unmanned surface vessels (USVs) based on distributed multi-objective optimization under random perturbation, so as to effectively enable multiple USVs to perform formation tasks safely, with low power consumption and reliably in complex and restricted waters. Summary of the Invention

[0006] This invention provides a collision-free formation method for multiple unmanned surface vessels based on distributed multi-objective optimization under random perturbation, in order to overcome the above-mentioned technical problems.

[0007] To achieve the above objectives, the technical solution of the present invention is as follows: A collision-free formation method for multiple unmanned surface vessels (USVs) based on distributed multi-objective optimization under random perturbation includes the following steps: S1: Establish a mathematical model for each unmanned surface vessel (USV) in the USV cluster, and obtain a second-order mathematical model for the USV based on the mathematical model. S2: Define the position decision variables of the unmanned surface vessel (USV) and design a multi-objective optimization model for each USV based on the position decision variables; The multi-objective optimization model is transformed into a single-objective optimization model for solving the optimal decision position of the unmanned surface vessel using a scalar method. S3: Based on the single-objective optimization model and location decision variables, construct a distributed multi-objective optimization algorithm based on zero gradient sum; use a discretized weight vector generation algorithm to obtain several sets of candidate weight vectors corresponding to the weights of each objective cost function in the distributed multi-objective optimization algorithm based on zero gradient sum. Substitute each group of candidate weight vectors into a distributed multi-objective optimization algorithm based on zero gradient sums for computation until the numerical matrix corresponding to the cost function values ​​of each objective after the algorithm converges is obtained. Use each numerical matrix as the initial solution set of the multi-objective optimization model, and perform non-dominated solution screening on the initial solution set to obtain the Pareto solution set. Obtain the Euclidean distance between the Pareto solution set after normalization and the utopian solution of each objective cost function, and select the candidate weight vector corresponding to the Pareto solution with the smallest Euclidean distance as the optimal weight. Based on the set sampling time period, the coordinate values ​​of the optimal position decision variables corresponding to each sampling time are recorded in real time according to the optimal weight and the distributed multi-objective optimization algorithm based on zero gradient sum, so as to obtain the optimal time-series waypoint sequence for multi-unmanned surface vessel collision-free formation. S4: Using cubic B-spline curves, a smooth formation trajectory that satisfies the kinematic constraints of the unmanned surface vessel is obtained based on the optimal time-series waypoint sequence; S5: Based on the smooth formation trajectory and the second-order mathematical model of the unmanned surface vessel, construct the anti-disturbance trajectory tracking control law for multiple unmanned surface vessels, and realize the collision-free formation control of multiple unmanned surface vessels based on distributed multi-objective optimization under random disturbances according to the anti-disturbance trajectory tracking control law.

[0008] Furthermore, S1 specifically includes the following steps: S11: The mathematical model for each unmanned surface vessel (USV) in the USV swarm is as follows:

[0009] In the formula: Represents a position vector and ; These represent the roll coordinate position, pitch coordinate position, and heading angle, respectively. Indicates the heading angle in the body coordinate system; Represent the space of real numbers; Describes the velocity vector and ; These represent the roll rate, pitch rate, and angular velocity of the heading angle, respectively. These represent the rotation matrix, inertia matrix, Coriolis matrix, and damping matrix, respectively. This represents a control input vector that includes force and torque; Represents a random disturbance acting on force and torque, which satisfies ; This represents the upper bound of the random perturbation; Indicates transpose; This represents the expectation operation; Denotes the noise intensity matrix, which satisfies ; This represents the upper bound of the noise intensity matrix; Represents an unmanned surface vessel swarm The number of unmanned surface vessels in the area; express The first derivative; express The first derivative; S12: The second-order mathematical model of the unmanned surface vessel is obtained based on the mathematical model described above:

[0010] In the formula: Represents the state vector and ; Represents the system matrix and ; express A zero matrix of dimension; express An identity matrix of 3D; Represents nonlinear terms and , Describes the newly defined matrix and .

[0011] Furthermore, S2 specifically includes the following steps: S21: Defining Unmanned Surface Vessels Location decision variables and The initial position decision variable is set as and Design a multi-objective optimization model for each unmanned surface vessel based on its location decision variables. for:

[0012] In the formula: unmanned surface vessel Distance from the initial position and ; unmanned surface vessel Distance from the preset formation and , Indicates collision avoidance and ; This indicates that the collision avoidance group includes unmanned vessels. The state vectors of the other unmanned surface vessels; Including unmanned surface vessels The collection of all unmanned surface vessels except those mentioned above; Indicates obstacle avoidance items and ; Indicates the coordinates of the o-th obstacle; Indicates the preset formation; These represent the coordinates of the formation's center; Indicates positive gain; Representing sets respectively Index of unmanned surface vessels and all obstacles; Indicates the total number of obstacles; The collision avoidance penalty function is represented as follows: , or ; Indicates the maximum navigation boundary distance of multiple unmanned surface vessels; Represents the power exponent and ; These represent the penalty magnitude constant and the trend constant, respectively. Represents a set of sailings and , In the set The second-order continuous differentiability and , Representing the cost function The number of corresponding sub-cost functions; S22: The multi-objective optimization model is transformed into a single-objective optimization model for solving the optimal decision position of the unmanned surface vessel using the scalar method:

[0013] In the formula: This represents the weights of each sub-objective function and , and , The single-objective optimization model is solvable, and its optimal solution is... satisfy .

[0014] Furthermore, S3 specifically includes the following steps: S31: Based on the single-objective optimization model and location decision variables, construct a distributed multi-objective optimization algorithm based on zero gradient sum as follows:

[0015] In the formula: express The first derivative; This represents the sliding manifold function used to ensure that the sum of gradients converges to zero by accelerating convergence; Indicates the preset time and ; The weighting coefficients representing the allocation time and ; Indicates nonlinear gain and ; Indicates morphological parameters; Indicates the first The neighbors of the unmanned boat gathered; Indicates the time quantity of integration; Let represent a binary variable used to characterize the communication relationship between unmanned surface vessels (USVs). If it exists, then... 1, otherwise 0; express The partial derivatives; unmanned surface vessel Location decision variables; Represents the inverse of the Hessian matrix; S32: Using a discretized weight vector generation algorithm, obtain several sets of candidate weight vectors corresponding to the weights of each objective cost function in a distributed multi-objective optimization algorithm based on zero gradient sum; S33: Substitute the candidate weight vectors of each group into the distributed multi-objective optimization algorithm based on zero gradient sum for calculation until the numerical matrix corresponding to the cost function values ​​of each objective is obtained after the algorithm converges. and Each numerical matrix is ​​used as the initial solution set for the multi-objective optimization model. S34: Perform non-dominated solution filtering on the initial solution set to obtain the Pareto solution set; The method for screening non-dominated solutions is as follows: For any two initial solutions in the initial solution set ; If the following conditions are met: Then it is called a solution. Dominant Solution ,in Numerical matrices The first in Group elements, Indicates the number of initial solutions; Remove all dominated initial solutions from the initial solution set, and retain the non-dominated initial solutions, which are denoted as the Pareto solution set; S35: Obtain the Pareto solution set after normalization and the corresponding utopian solutions for each objective cost function. Euclidean distance between for:

[0016] In the formula: This represents the first Pareto solution in the normalized solution set. One solution; express Transpose of; The candidate weight vector corresponding to the Pareto solution with the minimum Euclidean distance is selected as the optimal weight. S36: Based on the set sampling time period, and using the optimal weights combined with a distributed multi-objective optimization algorithm based on zero gradient sums, the optimal position decision variables corresponding to each sampling time are recorded in real time. The coordinate values ​​are used to obtain the optimal time-series waypoint sequence for collision-free formations of multiple unmanned surface vessels. ; , unmanned surface vessel The total number of waypoints corresponding to the optimal time-series waypoint sequence; Waypoints The location coordinates.

[0017] Furthermore, S4 specifically includes the following steps: S41: Based on the optimal time-series waypoint sequence The curvature of discrete waypoints is calculated as follows:

[0018] In the formula: Indicates the curvature of waypoints; Indicates the cross product operator symbol; S42: Set the curvature threshold for the minimum turning radius of the unmanned surface vessel. and retain the satisfaction In addition to the waypoints, the retained waypoints also include the coordinate extreme points and stationary points that ensure the range of the trajectory envelope: The extreme point of coordinates is defined as: ; The criteria for determining a station are: ;in, unmanned surface vessel The minimum linear velocity; Indicates the simulation step size; S43: Based on step S42, obtain the filtered waypoint sequence as follows: and the filtered waypoint sequence Using the basis functions of a cubic B-spline curve as control points, a smooth formation trajectory satisfying the kinematic constraints of the unmanned surface vessel is obtained. for:

[0019] In the formula: This represents the waypoints in the filtered waypoint sequence; The basis functions represent cubic B-spline curves; This indicates the preset total travel time.

[0020] Furthermore, S5 specifically includes the following steps: S51: Based on the smooth formation trajectory and the second-order mathematical model of the unmanned surface vessel, define the unmanned surface vessel... trajectory tracking error for:

[0021] In the formula: Represents the reference trajectory vector and ; express The first derivative; Indicates the desired heading angle and , Represents the trajectory components in the roll and pitch directions; This represents the derivative of the expected heading angle; express The abbreviated form; express The first derivative; S52: Based on trajectory tracking error The disturbance-resistant trajectory tracking control law for multiple unmanned surface vessels is constructed as follows:

[0022] In the formula: Indicates control gain and ; Represents the feedback gain matrix and . Linear matrix inequalities The solution; Indicate design parameters and ; Represents the optimal fuzzy weight matrix The estimate; Indicates a fuzzy compensation term; Represents a fuzzy basis function vector; Indicates IF The number of T HEN rules; The derivative of the expected reference trajectory and ; express The first derivative; Indicate design parameters and ; express The first derivative; Collision-free formation control of multiple unmanned surface vessels based on distributed multi-objective optimization under random disturbances is achieved by using an anti-disturbance trajectory tracking control law.

[0023] This invention provides a collision-free formation method for multiple unmanned surface vessels (USVs) based on distributed multi-objective optimization under random perturbation, with the following beneficial effects: (1) A multi-objective cost function, i.e., a multi-objective optimization model, was designed to consider the optimal trade-off between safety and energy consumption. The optimal weights were locked by using a distributed multi-objective optimization algorithm based on zero gradient sum and the Pareto solution set visualized by the parallel coordinate graph combined with the Utopian solution distance method. In addition, the inverse term of the Hessian matrix was introduced into the zero gradient sum framework to accelerate the second-order optimization. This ensures that each unmanned surface vessel node can accurately converge to the global optimal decision state within a preset time by communicating with its neighbors. This invention enables multiple unmanned surface vessels to autonomously achieve the optimal balance between obstacle avoidance safety and energy consumption efficiency within a preset time. It not only ensures the real-time performance and reliability of formation tasks in complex environments, but also significantly improves the overall efficiency of multi-unmanned surface vessel operations through scientific weight decision-making.

[0024] (2) By generating a smooth trajectory with second-order continuity using cubic B-splines, a feature control point selection mechanism based on discrete curvature and motion state is introduced. Combined with the construction of quasi-uniform node vectors with endpoint interpolation characteristics, the transformation from discrete waypoints to smooth analytical trajectories is achieved, ensuring accurate waypoints during the initial and convergence phases. This invention eliminates the problem of abrupt changes in control commands caused by discrete waypoints, effectively reducing the mechanical losses of the unmanned surface vessel's actuators and improving the energy efficiency and success rate of navigation.

[0025] (3) In the presence of wave disturbances, an anti-disturbance trajectory tracking control law that is more in line with the actual formation mission and tracks the well-fitted multi-unmanned surface vessel was effectively designed. Subsequently, by using stochastic differential equations combined with the stability theory of Lyapunov functions to analyze the random disturbances of forces and torques acting on the multi-unmanned surface vessel, the noise tolerance of the multi-unmanned surface vessel system under actual complex sea conditions was improved. Attached Figure Description

[0026] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0027] Figure 1 This is a flowchart of the collision-free formation method for multiple unmanned surface vessels based on distributed multi-objective optimization under random perturbation according to the present invention; Figure 2This is a communication topology diagram between multiple unmanned surface vessels in this embodiment; Figure 3 For this embodiment Map showing the waypoint planning results for multiple unmanned surface vessels in an environment with internal obstacles; Figure 4 This is a simulation diagram of the optimal weights and normalized cost function under the parallel coordinate graph in this embodiment; Figure 5 This is a simulation diagram showing the spacing between each unmanned surface vessel (USV) and the spacing between each USV and obstacles in this embodiment. Figure 6 This is a graph showing the error of the decision variables in this embodiment; Figure 7 This is a multi-unmanned surface vessel formation trajectory diagram generated based on the B-spline method in this embodiment; Figure 8 This is a graph showing the trajectory tracking error of the unmanned surface vessel (USV) swarm in this embodiment. Figure 9 This is a fuzzy adaptive weight curve diagram in this embodiment; Figure 10 This is a diagram showing the control input curves for multiple unmanned surface vessels in this embodiment; Figure 11 This is a graph showing the random disturbance in this embodiment. Detailed Implementation

[0028] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0029] This embodiment aims to successfully achieve formation missions for multiple Autonomous Surface Vehicles (ASVs) in complex environments. It provides a collision-free formation method for multiple ASVs under random perturbations based on distributed multi-objective optimization, enabling safe formation in obstacle-prone and random-perturbation environments. Specifically, at the planning layer, a multi-objective cost function is designed in the distributed zero-gradient optimization algorithm. Weights are scientifically generated using parallel coordinate graphs and the Utopian solution distance method, ensuring the system converges to the optimal waypoint within a preset time, balancing obstacle avoidance safety and energy efficiency. At the control layer, a cubic B-spline method is used to fit the waypoints to a smooth trajectory, and a trajectory tracking scheme resistant to random perturbations is proposed for accurate tracking. Finally, based on Lyapunov functions and a method combining random differential equations (RDEs) with Lyapunov functions, the stability of the distributed zero-gradient optimization algorithm at the planning layer and the noise-state probability-dependent practical stability of the multi-ASV system at the control layer are proven, theoretically ensuring the safety and efficiency of the multi-ASV formation mission. Figure 1 As shown, the specific steps include: S1: Establish a mathematical model for each unmanned surface vessel (USV) in the USV cluster, and obtain a second-order mathematical model for the USV based on the mathematical model. Specifically, in this embodiment, it is assumed that Multiple unmanned surface vessels jointly perform formation missions, including all real numbers, positive real numbers, and real numbers. The sets of matrices are represented as follows: ; Undirected graphs for internal communication of unmanned surface vessels Description, in which For the image The set of nodes, For the image The set of edges, Let be the adjacency matrix. Indicates the first The neighbors of the unmanned surface vessel gathered, and set up Including unmanned surface vessels The collection of all unmanned surface vessels except those mentioned above. (Definition diagram) The Laplace matrix is , among which when Sometimes, and , The degree matrix represents the graph; if the graph If they are connected, then define for eigenvalues; Assuming each unmanned surface vessel is affected by random perturbations, then: S11: Establish the mathematical model of each unmanned surface vessel (USV) in the USV swarm, i.e., the first... The kinematic and dynamic equations of the unmanned surface vessel are as follows: (1) In the formula: Represents a position vector and ; These represent the roll position, pitch position, and heading angle in the Earth coordinate system, respectively. Indicates the heading angle in the body coordinate system; Represent the space of real numbers; Describes the velocity vector and ; These represent the roll rate, pitch rate, and angular velocity of the heading angle, respectively. These represent the rotation matrix, inertia matrix, Coriolis matrix, and damping matrix, respectively. This represents a control input vector that includes force and torque; Represents a random disturbance acting on force and torque, which satisfies ; This represents the upper bound of the random perturbation; Indicates transpose; This represents the expectation operation; Denotes the noise intensity matrix, which satisfies ; This represents the upper bound of the noise intensity matrix; This indicates the number of unmanned surface vessels (USVs) in the USV swarm. express The first derivative; express The first derivative; S12: The second-order mathematical model of the unmanned surface vessel is obtained based on the mathematical model described above: (2) In the formula: Represents the state vector and ; Represents the system matrix and ; express A zero matrix of dimension; express An identity matrix of 3D; Represents nonlinear terms and , Describes the newly defined matrix and ; S2: Define the position decision variables of the unmanned surface vessel (USV), and design a multi-objective optimization model for each USV based on the position decision variables; use scalar methods to convert the multi-objective optimization model into a single-objective optimization model for solving the optimal decision position of the USV. Specifically, the following steps are included: S21: Defining Unmanned Surface Vessels Location decision variables and The initial position decision variable is set as and Design a multi-objective optimization model for each unmanned surface vessel based on its location decision variables. for: (3) In the formula: unmanned surface vessel Distance from the initial position and ; unmanned surface vessel Distance from the preset formation and , Indicates collision avoidance and ; This indicates that the collision avoidance group includes unmanned vessels. The state vectors of the other unmanned surface vessels; Including unmanned surface vessels The collection of all unmanned surface vessels except those mentioned above; Indicates obstacle avoidance items and ; Indicates the coordinates of the o-th obstacle; Indicates the preset formation; These represent the coordinates of the formation's center; Indicates positive gain; Representing sets respectively Index of unmanned surface vessels and all obstacles; Indicates the total number of obstacles; The collision avoidance penalty function is represented as follows: , or ; Indicates the maximum navigation boundary distance of multiple unmanned surface vessels; Represents the power exponent and ; These represent the penalty magnitude constant and the trend constant, respectively. Represents a set of sailings and , In the set The second-order continuous differentiability and , Representing the cost function The number of corresponding sub-cost functions; where exist yes Strongly convex , Denotes the strong convexity coefficients that are greater than zero. exist yes Strongly convex , Represents the strong convexity coefficients that are greater than zero; gradient and exist They are Lipschitz continuous sum Lipschitz is continuous, among which ; S22: The multi-objective optimization model is transformed into a single-objective optimization model for solving the optimal decision position of the unmanned surface vessel using the scalar method: (4) In the formula: This represents the weights of each sub-objective function and , and , Assuming the optimization problem of the single-objective optimization model is solvable, its optimal solution... satisfy ; S3: Obtain the optimal time-series waypoint sequence for collision-free formation of multiple unmanned surface vessels, specifically including the following steps: S31: Select parameters Based on the single-objective optimization model and location decision variables, a distributed multi-objective optimization algorithm based on zero gradient sum is constructed as follows: (5) In the formula: express The first derivative of is intended to achieve global consistency; This represents the sliding manifold function used to ensure that the sum of gradients converges to zero by accelerating convergence; Indicates the preset time; Indicates the weighting coefficient for the allocated time; Indicates nonlinear gain; Indicates morphological parameters; Indicates the first The neighbors of the unmanned boat gathered; Indicates the time quantity of integration; Let represent a binary variable used to characterize the communication relationship between unmanned surface vessels (USVs). If it exists, then... 1, otherwise 0; express The partial derivatives; unmanned surface vessel Location decision variables; Represents the inverse of the Hessian matrix; assuming a diagram It is an undirected and connected graph. Let... ,if , , , ,but It is the optimal waypoint in the single-objective optimization model, and it is within the predetermined time. All internal states Able to converge ; This embodiment proves the first based on the Lyapunov function. The optimal waypoints that can be obtained from the planning layer of an unmanned surface vessel are as follows: First, we prove that the sum of local gradients in time... The internal energy converges to zero when hour, The derivative with respect to time can be calculated as: (6) In the formula: Choose the Lyapunov function ,but The derivative with respect to time is ;when At that time, we can obtain: (7) Therefore at the specified time Inside, Approaching zero, and (8) According to equation (6), we can obtain Combining ,exist At that time, it is possible to obtain Next, prove the state. Able to be at the scheduled time Converging to the optimal solution ,when At that time, by What can be obtained: (9) From equation (9), we can obtain that Because when hour, ,therefore exist This holds true when the gradient is zero, which means that the zero gradient and manifold also hold true.

[0030] Choose the Lyapunov function Its satisfaction .definition ,in , . because and ,therefore .

[0031] right Taking the derivative and scaling it further, we get: (10) Design form parameters satisfy Equation (10) can be further written as: (11) If a system and Its Lyapunov function satisfy ,in and If the system is globally time-stable, the stable time will not exceed [a certain value]. Therefore, equation (11) can be used to appropriately preset the time as follows: hour, ,at this time The convergence time proposed in this embodiment can be arbitrarily specified by the user, independent of system parameters, thereby avoiding computation time consumption, and also providing a wider range of parameter selection. Therefore, at the preset time Internally, a distributed multi-objective optimization algorithm based on zero gradient sums can generate optimal waypoints. .

[0032] S32: Using a discretized weight vector generation algorithm, obtain several sets of candidate weight vectors corresponding to the weights of each objective cost function in a distributed multi-objective optimization algorithm based on zero gradient sum; Specifically, in order to fully traverse the solution space and extract the Pareto front, this embodiment employs a discretized weight vector generation algorithm: under constraints... Next, define the basic weight components. and weight increment step size ,Will One increment unit in The full combinatorial allocation is performed among the objective functions, and the calculation formula is as follows: The algorithm generated a total of The candidate weight vectors ensure the uniformity and completeness of the sampling space; S33: Substitute the candidate weight vectors of each group into the distributed multi-objective optimization algorithm based on zero gradient sum for calculation until the numerical matrix corresponding to the cost function values ​​of each objective is obtained after the algorithm converges. and Each numerical matrix is ​​used as the initial solution set of the multi-objective optimization model; where and This indicates the extraction of sub-cost function elements from the cost function matrix obtained one step before the Pareto front; S34: Perform non-dominated solution filtering on the initial solution set to obtain the Pareto solution set; The method for screening non-dominated solutions is as follows: For any two initial solutions in the initial solution set ; If the following conditions are met: Then it is called a solution. Dominant Solution ,in Numerical matrices The first in Group elements, Indicates the number of initial solutions; Remove all dominated initial solutions from the initial solution set, and retain the non-dominated initial solutions, which are denoted as the Pareto solution set; S35: Considering the differences in the dimensions of each objective function, the Pareto solution set is normalized using Min-Max. The normalized cost is... The calculation is as follows: ,in They are respectively The maximum and minimum elements in the vector, and the range of the target values ​​after mapping are: ,in This means that the indicator has reached its optimal level in the current Pareto frontier. Represents the worst; Define the utopian solution in the normalized space as It represents Each objective function simultaneously reaches its own minimum value. This embodiment uses parallel coordinate graph technology to visualize and analyze the Pareto solution set and observe the trade-offs between the objectives. It obtains the Pareto solution set after normalization and the utopian solutions of the corresponding objective cost functions. Euclidean distance between for: In the formula: This represents the first Pareto solution in the normalized solution set. One solution; express Transpose of; And select Euclidean distance The Pareto solution corresponding to the minimum value is used as the candidate weight vector corresponding to the cost function, i.e., the objective function, at this point, and the optimal weights are taken as the optimal weights. ; S36: In determining the optimal weight vector Then, these parameters are fed back as fixed parameters to a distributed multi-objective optimization algorithm based on zero gradient sums. Based on the set sampling time period, and according to the optimal weights combined with the distributed multi-objective optimization algorithm based on zero gradient sums, the optimal position decision variables corresponding to each sampling time are recorded in real time. The coordinate values ​​are used to obtain the optimal time-series waypoint sequence for collision-free formations of multiple unmanned surface vessels. unmanned surface vessel The total number of waypoints corresponding to the optimal time-series waypoint sequence; Waypoints The location coordinates.

[0033] S4: Using cubic B-spline curves based on the optimal time-series waypoint sequence, obtain a smooth formation trajectory that satisfies the kinematic constraints of the unmanned surface vessel. This includes the following steps: S41: Based on the optimal time-series waypoint sequence The curvature of discrete waypoints is calculated as follows: (12) In the formula: Indicates the curvature of waypoints; Indicates the cross product operator symbol; S42: Set the curvature threshold for the minimum turning radius of the unmanned surface vessel. and retain the satisfaction In addition to the waypoints, the retained waypoints also include the coordinate extreme points and stationary points that ensure the range of the trajectory envelope: The extreme point of coordinates is defined as: ; The criteria for determining a station are: ;in, unmanned surface vessel The minimum linear velocity; Indicates the simulation step size; S43: Based on step S42, obtain the filtered waypoint sequence as follows: And the filtered waypoint sequence is As control points, they are used for B-spline curve fitting of the navigation trajectory; This embodiment also includes (1) time variable parameterization and normalization: physical time Parameter space mapped to B-splines :at this time ,in Given a preset total sailing time, all subsequent basis function calculations are based on normalized parameters. (2) Constructing a quasi-uniform node vector: In order for the generated B-spline trajectory to accurately pass through the start and end points (i.e., to have endpoint interpolation properties), a quasi-uniform node vector is constructed. ;for The total number of nodal vectors in a cubic B-spline is: , The node vector is defined as: ,in For a node, the vector structure of this node forces the curve to... and The locations are respectively with control points and Tangent to satisfy the boundary conditions of the trajectory; (3) Trajectory fitting: basis functions of cubic B-splines First, we define the zeroth-order basis functions using the Cox-de-Boor recurrence formula: (13) Then calculate using the recursive formula. basis functions of order: (14) in, During the calculation process, if the denominator appears Then the value of this item is specified as . ; Using the basis functions of a cubic B-spline curve and the control points, a smooth formation trajectory satisfying the kinematic constraints of the unmanned surface vessel is obtained. for: (15) In the formula: This represents the waypoints in the filtered waypoint sequence; The basis functions represent cubic B-spline curves; This represents the preset total sailing time. The smooth formation trajectory generated in this embodiment has second-order continuity. Due to the use of cubic B-splines, the trajectory is continuous in terms of position and velocity (first derivative) and acceleration (second derivative). This means that the unmanned surface vessel will not experience sudden changes in commands during navigation, which can effectively protect the propulsion motor and reduce energy consumption. If a fine adjustment is needed to a certain obstacle avoidance point in the future, only a few adjacent control points need to be changed without causing drastic fluctuations in the entire trajectory.

[0034] S5: Based on the smooth formation trajectory and the second-order mathematical model of the unmanned surface vessel (USV), a disturbance-resistant trajectory tracking control law for multiple USVs is constructed. Based on this law, collision-free formation control of multiple USVs under random disturbances, using distributed multi-objective optimization, is achieved. Specifically, this includes the following steps: S51: Based on the smooth formation trajectory and the second-order mathematical model of the unmanned surface vessel, define the unmanned surface vessel... trajectory tracking error for: In the formula: Indicates the desired heading angle and , Represents the trajectory components in the roll and pitch directions; This represents the derivative of the expected heading angle; express The abbreviated form; express The first derivative; Represents the reference trajectory vector and ; express The first derivative; S52: Based on trajectory tracking error The disturbance-resistant trajectory tracking control law for multiple unmanned surface vessels is constructed as follows: (16) (17) In the formula: Indicates control gain and ; Represents the feedback gain matrix and . Linear matrix inequalities The solution; Indicate design parameters and ; Represents the optimal fuzzy weight matrix The estimate; Indicates a fuzzy compensation term; Represents a fuzzy basis function vector; Indicates IF The number of T HEN rules; The derivative of the expected reference trajectory and ; express The first derivative; Indicate design parameters and ; express The first derivative; Collision-free formation control of multiple unmanned surface vessels based on distributed multi-objective optimization under random disturbances is achieved by using an anti-disturbance trajectory tracking control law.

[0035] This embodiment also includes a method based on Lyapunov functions to prove that the trajectory tracking error closed-loop system of multiple unmanned surface vessels is noise-state probabilistically stable, i.e., the input of the first... Tracking error of an unmanned surface vessel Trajectory tracking control law Fuzzy weight update law Output the derivative form of the Lyapunov function. This demonstrates that all unmanned surface vessels (USVs) can track the desired reference trajectory, and that the closed-loop tracking error system of multiple USVs is noise-state probabilistically stable, specifically as follows: Define parameter quantity ; Define vectors and matrices Compact form:

[0036] Therefore, when designing the anti-disturbance trajectory tracking control law (16) and the fuzzy weight update law (17), the feedback gain matrix... and positive scalar When the mean square of the tracking error converges to the neighborhood of the zero point, the second-order mathematical model (2) of the unmanned surface vessel can realize the trajectory tracking control task.

[0037] According to equations (2) and (16), the trajectory tracking error The dynamics can be derived as follows: (18) The Lyapunov function is chosen as follows: Its time derivative can be written as: (19) Define error transformation And using the following Young's inequality: (20) in: The constant satisfies .

[0038] Therefore, equation (19) can be further scaled up to: (twenty one) in: , and Then, based on the conditions of random perturbation. , can be obtained .

[0039] Therefore, from equation (21), it can be concluded that the trajectory tracking error system is noise-state probabilistically stable. By selecting a suitable... Under the presence of random perturbations, all unmanned surface vessels were able to successfully track their reference trajectory. This completes the proof of the stability of the trajectory tracking error. Thus, this embodiment, by applying a distributed multi-objective optimization algorithm based on zero gradient sums, determining the optimal weights based on the Pareto front and utopia value of the multi-objective cost function, generating smooth formation trajectories for multiple unmanned surface vessels using cubic B-spline curves, and applying a multi-unmanned surface vessel anti-disturbance trajectory tracking control law, enables all unmanned surface vessels to successfully complete the formation task under random disturbance conditions.

[0040] This embodiment also includes the following application examples: This embodiment will utilize a distributed multi-objective optimization algorithm based on zero gradient sum, Pareto front of multi-objective cost function, cubic B-spline curve, and multi-unmanned surface vessel (USV) anti-disturbance trajectory tracking control law to complete the simulation of multi-USV formation strategy. The specific steps are as follows: Assumption Several unmanned surface vessels (USVs) jointly perform formation missions. The communication between multiple USVs is modeled as an undirected connected graph, such as... Figure 2 As shown. In the cost function, i.e., the single-objective optimization model (4), the collision avoidance gain is set to The formation center and obstacle positions are designed as follows: and In the penalty function, In the distributed multi-objective optimization model (5) based on zero gradient sum, let: and .

[0041] The matrix in the multi-unmanned surface vessel model is defined as follows:

[0042] ; In this embodiment, equations (16) and (17), the initial state of the unmanned surface vessel, and related parameters are set as follows: ; ; First, such as Figure 3 As shown in the example, this embodiment provides a simulation time of 20 seconds in an obstacle environment, such as... Figure 3 (a) is shown Waypoint results generated at each time; such as Figure 3 (b) shows The waypoint results generated at each moment. For example... Figure 3 (c) is shown The waypoint results generated at each moment. For example... Figure 3 (d) shows The waypoint results generated at each time point. Four unmanned surface vessels can achieve the desired results within a preset time using a zero-gradient sum distributed multi-objective optimization algorithm (5). Collision-free waypoints are generated internally. Then, based on the Pareto front non-dominated solution set generated by distributed iteration, a parallel coordinate graph is used to perform multi-dimensional visualization mapping and conflict trade-off analysis of the four normalized cost functions under the multi-objective cooperative task, such as... Figure 4 As shown. Subsequently, by calculating the minimum Euclidean distance between each candidate solution in the normalized solution space and the theoretical utopian solution, the compromise solution with the optimal comprehensive performance index is precisely identified from the solution set. The basic weight components are selected as follows. The weight increment step size is designed as follows: , Therefore, the optimal weight vector for this system under the current task environment is determined to be... ,but And when the collision weight Exceeding the formation mission completion weight Weighted by energy consumption At that time, all unmanned surface vessels (USVs) will complete the formation mission with safety as the highest priority, even if the final optimal waypoint may deviate from the formation center; the distance between each USV and the distance between each obstacle will be as follows: Figure 5 As shown, Figure 5 (a) The spacing between each unmanned surface vessel: ; Figure 5 (b) The spacing between each unmanned surface vessel and the obstacle: In the image, no unmanned surface vessels collided and all distances exceeded the safety threshold. , In the planning layer, there are no decision variables for each individual. and the optimal solution The error curve between them is shown in the figure. Figure 6 As shown, Figure 6 (a) is the decision variable and the optimal solution The error between; Figure 6 (b) is the decision variable and the optimal solution The error between them indicates that the distributed optimization algorithm is in the preset time. The convergence characteristics within the range are then observed. Subsequently, by filtering out unnecessary waypoints, the B-spline method generates smooth and kinematically constrained multi-unmanned surface vessel trajectories, such as... Figure 7 As shown, no collisions occurred on any of the trajectories, and a formation was formed. Finally, the small trajectory tracking errors of the multi-UAVs in three directions, the bounded fuzzy adaptive weights, and the control inputs of the multi-UAVs are as follows: Figures 8 to 10 As shown, random disturbances suffered by multiple unmanned surface vessels like Figure 11 As shown, it is generated through a second-order filter: The initial state satisfies The constant is chosen as . This represents a white noise vector. All simulation results demonstrate that the algorithm can achieve an effective trade-off between conflicting formation mission objectives. Thus, the method described in this embodiment, by applying a distributed multi-objective optimization algorithm based on zero gradient sums, determining optimal weights according to the Pareto front and utopia value of the multi-objective cost function, generating smooth formation trajectories using cubic B-spline curves, and combining this with an anti-disturbance trajectory tracking control law, enables multiple unmanned surface vessels (USVs) to accurately track smooth optimal trajectories even under random disturbance environments. While ensuring collision avoidance safety between vessels and obstacles throughout the entire voyage, it successfully achieves multi-USV cooperative formation missions, effectively mitigating the potential trade-off between mission completion and safety obstacle avoidance requirements.

[0043] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A collision-free formation method for multiple unmanned surface vessels (USVs) based on distributed multi-objective optimization under random perturbation, characterized in that, Specifically, the following steps are included: S1: Establish mathematical models for each unmanned surface vessel (USV) in the USV cluster, and obtain second-order mathematical models for the USVs based on the mathematical models. S2: Define the position decision variables of the unmanned surface vessel (USV), and design a multi-objective optimization model for each USV based on the position decision variables; The multi-objective optimization model is transformed into a single-objective optimization model for solving the optimal decision position of the unmanned surface vessel using a scalar method. S3: Based on the single-objective optimization model and location decision variables, construct a distributed multi-objective optimization algorithm based on zero gradient sum; use a discretized weight vector generation algorithm to obtain several sets of candidate weight vectors corresponding to the weights of each objective cost function in the distributed multi-objective optimization algorithm based on zero gradient sum. Substitute each group of candidate weight vectors into a distributed multi-objective optimization algorithm based on zero gradient sums for computation until the numerical matrix corresponding to the cost function values ​​of each objective after the algorithm converges is obtained. Use each numerical matrix as the initial solution set of the multi-objective optimization model, and perform non-dominated solution screening on the initial solution set to obtain the Pareto solution set. Obtain the Euclidean distance between the Pareto solution set after normalization and the utopian solution of each objective cost function, and select the candidate weight vector corresponding to the Pareto solution with the smallest Euclidean distance as the optimal weight. Based on the set sampling time period, the coordinate values ​​of the optimal position decision variables corresponding to each sampling time are recorded in real time according to the optimal weight and the distributed multi-objective optimization algorithm based on zero gradient sum, so as to obtain the optimal time-series waypoint sequence for multi-unmanned surface vessel collision-free formation. S4: Using cubic B-spline curves, a smooth formation trajectory that satisfies the kinematic constraints of the unmanned surface vessel is obtained based on the optimal time-series waypoint sequence; S5: Based on the smooth formation trajectory and the second-order mathematical model of the unmanned surface vessel, construct the anti-disturbance trajectory tracking control law for multiple unmanned surface vessels, and realize the collision-free formation control of multiple unmanned surface vessels based on distributed multi-objective optimization under random disturbances according to the anti-disturbance trajectory tracking control law.

2. The collision-free formation method for multiple unmanned surface vessels based on distributed multi-objective optimization under random perturbation as described in claim 1, characterized in that, S1 specifically includes the following steps: S11: The mathematical model for each unmanned surface vessel (USV) in the USV swarm is as follows: In the formula: Represents a position vector and ; These represent the roll coordinate position, pitch coordinate position, and heading angle, respectively. Indicates the heading angle in the body coordinate system; Represent the space of real numbers; Describes the velocity vector and ; These represent the roll rate, pitch rate, and angular velocity of the heading angle, respectively. These represent the rotation matrix, inertia matrix, Coriolis matrix, and damping matrix, respectively. This represents a control input vector that includes force and torque; Represents a random disturbance acting on force and torque, which satisfies ; This represents the upper bound of the random perturbation; Indicates transpose; This represents the expectation operation; Denotes the noise intensity matrix, which satisfies ; This represents the upper bound of the noise intensity matrix; Represents an unmanned surface vessel swarm The number of unmanned surface vessels in the area; express The first derivative; express The first derivative; S12: Based on the mathematical model described above, the second-order mathematical model of the unmanned surface vessel is obtained as follows: In the formula: Represents the state vector and ; Represents the system matrix and ; express A zero matrix of dimension; express An identity matrix of 3D; Represents nonlinear terms and , Describes the newly defined matrix and .

3. The collision-free formation method for multiple unmanned surface vessels based on distributed multi-objective optimization under random perturbation as described in claim 2, characterized in that, S2 specifically includes the following steps: S21: Defining Unmanned Surface Vessels Location decision variables and The initial position decision variable is set as and Design a multi-objective optimization model for each unmanned surface vessel based on its location decision variables. for: In the formula: unmanned surface vessel Distance from the initial position and ; unmanned surface vessel Distance from the preset formation and , Indicates collision avoidance and ; This indicates that the collision avoidance group includes unmanned vessels. The state vectors of the other unmanned surface vessels; Including unmanned surface vessels The collection of all unmanned surface vessels except those mentioned above; Indicates obstacle avoidance items and ; Indicates the coordinates of the o-th obstacle; Indicates the preset formation; These represent the coordinates of the formation's center; Indicates positive gain; Representing sets Index of unmanned surface vessels and all obstacles; Indicates the total number of obstacles; The collision avoidance penalty function is represented as follows: , or ; Indicates the maximum navigation boundary distance of multiple unmanned surface vessels; Represents the power exponent and ; These represent the penalty magnitude constant and the trend constant, respectively. Represents a set of sailings and , In the set The second-order continuous differentiability and , Representing the cost function The number of corresponding sub-cost functions; S22: The multi-objective optimization model is transformed into a single-objective optimization model for solving the optimal decision position of the unmanned surface vessel using the scalar method: In the formula: This represents the weights of each sub-objective function and , and , The single-objective optimization model is solvable, and its optimal solution is... satisfy .

4. The collision-free formation method for multiple unmanned surface vessels based on distributed multi-objective optimization under random perturbation as described in claim 3, characterized in that, S3 specifically includes the following steps: S31: Based on the single-objective optimization model and location decision variables, construct a distributed multi-objective optimization algorithm based on zero gradient sum as follows: In the formula: express The first derivative; This represents the sliding manifold function used to ensure that the sum of gradients converges to zero by accelerating convergence; Indicates the preset time and ; The weighting coefficients representing the allocation time and ; Indicates nonlinear gain and ; Indicates morphological parameters; Indicates the first The neighbors of the unmanned boat gathered; Indicates the time quantity of integration; Let represent a binary variable used to characterize the communication relationship between unmanned surface vessels (USVs). If it exists, then... 1, otherwise 0; express The partial derivatives; unmanned surface vessel Location decision variables; Represents the inverse of the Hessian matrix; S32: Using a discretized weight vector generation algorithm, obtain several sets of candidate weight vectors corresponding to the weights of each objective cost function in a distributed multi-objective optimization algorithm based on zero gradient sum; S33: Substitute the candidate weight vectors of each group into the distributed multi-objective optimization algorithm based on zero gradient sum for calculation until the numerical matrix corresponding to the cost function values ​​of each objective is obtained after the algorithm converges. and Each numerical matrix is ​​used as the initial solution set for the multi-objective optimization model. S34: Perform non-dominated solution filtering on the initial solution set to obtain the Pareto solution set; The method for screening non-dominated solutions is as follows: For any two initial solutions in the initial solution set ; If the following conditions are met: Then it is called a solution. Dominant Solution ,in Numerical matrices The first in Group elements, Indicates the number of initial solutions; Remove all dominated initial solutions from the initial solution set, and retain the non-dominated initial solutions, which are denoted as the Pareto solution set; S35: Obtain the Pareto solution set after normalization and the corresponding utopian solutions for each objective cost function. Euclidean distance between for: In the formula: This represents the first Pareto solution in the normalized solution set. One solution; express transpose; The candidate weight vector corresponding to the Pareto solution with the minimum Euclidean distance is selected as the optimal weight. S36: Based on the set sampling time period, and using the optimal weights combined with a distributed multi-objective optimization algorithm based on zero gradient sums, the optimal position decision variables corresponding to each sampling time are recorded in real time. The coordinate values ​​are used to obtain the optimal time-series waypoint sequence for collision-free formations of multiple unmanned surface vessels. ; , unmanned surface vessel The total number of waypoints corresponding to the optimal time-series waypoint sequence; Waypoints The location coordinates.

5. The collision-free formation method for multiple unmanned surface vessels based on distributed multi-objective optimization under random perturbation as described in claim 3, characterized in that, S4 specifically includes the following steps: S41: Based on the optimal time-series waypoint sequence The curvature of discrete waypoints is calculated as follows: In the formula: Indicates the curvature of waypoints; Indicates the cross product operator symbol; S42: Set the curvature threshold for the minimum turning radius of the unmanned surface vessel. and retain the satisfaction In addition to the waypoints, the retained waypoints also include the coordinate extreme points and stationary points that ensure the range of the trajectory envelope: The extreme point of coordinates is defined as: ; The criteria for determining a station are: ;in, unmanned surface vessel The minimum linear velocity; Indicates the simulation step size; S43: Based on step S42, obtain the filtered waypoint sequence as follows: and the filtered waypoint sequence Using the basis functions of a cubic B-spline curve as control points, a smooth formation trajectory satisfying the kinematic constraints of the unmanned surface vessel is obtained. for: In the formula: This represents the waypoints in the filtered waypoint sequence; The basis functions represent cubic B-spline curves; This indicates the preset total travel time.

6. The collision-free formation method for multiple unmanned surface vessels based on distributed multi-objective optimization under random perturbation as described in claim 5, characterized in that, S5 specifically includes the following steps: S51: Based on the smooth formation trajectory and the second-order mathematical model of the unmanned surface vessel, define the unmanned surface vessel... trajectory tracking error for: In the formula: Represents the reference trajectory vector and ; express The first derivative; Indicates the desired heading angle and , Represents the trajectory components in the roll and pitch directions; The derivative of the expected heading angle; express The abbreviated form; express The first derivative; S52: Based on trajectory tracking error The disturbance-resistant trajectory tracking control law for multiple unmanned surface vessels is constructed as follows: In the formula: Indicates control gain and ; Represents the feedback gain matrix and ; Linear matrix inequalities The solution; Indicate design parameters and ; Represents the optimal fuzzy weight matrix The estimate; Indicates a fuzzy compensation term; Represents a fuzzy basis function vector; Indicates IF The number of T HEN rules; The derivative of the expected reference trajectory and ; express The first derivative; Indicate design parameters and ; express The first derivative; Collision-free formation control of multiple unmanned surface vessels based on distributed multi-objective optimization under random disturbances is achieved by using an anti-disturbance trajectory tracking control law.

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