Multi-unmanned ship formation collision and obstacle avoidance control method with event triggering and signal quantization

Abstract

The invention provides a multi-unmanned ship formation collision and obstacle avoidance control method with event triggering and signal quantification, which comprises the following steps: acquiring sea condition information of surrounding environment and other surrounding ships, and establishing a trajectory control mathematical model of an under-actuated unmanned ship formation; a uniform quantizer is adopted to quantize control input and state variables in the control system; estimating quantization state information of each USV and uncertain items existing in the model by using an ESO technology; a linear model is introduced to describe a quantization process, a hierarchical design method is adopted, a kinematics guidance law and a dynamics quantization tracking control law are designed, a controller does not need priori information of quantization parameters, and an event-triggered formation control strategy is designed; and the stability of the multi-unmanned ship formation collision and obstacle avoidance control system with event triggering and signal quantization is proved by using an input state stability theory. According to the invention, while an event triggering mechanism and signal quantization are considered, formation collision avoidance and obstacle avoidance of multiple USVs are considered.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of artificial intelligence, in particular, especially relates to a multi-USV formation collision avoidance and obstacle avoidance control method with event triggering and signal quantization. BACKGROUND

[0002] With the rapid development of unmanned and automation technologies, unmanned systems are increasingly widely used in various fields. Among them, unmanned surface vehicles (USVs) have been widely used in ocean engineering, ocean monitoring, maritime search and rescue, and underwater detection due to their small size, flexibility, speed, low cost, easy control, and no risk of personnel casualties. USVs can easily and accurately navigate in complex marine environments, and their unmanned nature also effectively avoids personnel casualties in dangerous situations, significantly improving the safety of maritime operations. As the demand for efficient ocean solutions continues to grow, the deployment of USVs is becoming more common, and the development of their formation control systems represents a major shift in ocean operation patterns.

[0003] Compared with a single USV, a multi-USV formation exhibits excellent fault tolerance and adaptability through redundant communication architecture and distributed task allocation algorithms, which can effectively reduce the workload of operators and improve the sustainability, scalability, and intelligence of ocean operations. However, existing research has focused more on collision avoidance and obstacle avoidance for individual USVs, while collaborative collision avoidance and obstacle avoidance strategies for multi-USV formations are still limited. Although complex collision avoidance control algorithms have been developed, their practical application is fundamentally constrained by the bandwidth limitations of maritime communication systems. Solving these bandwidth limitations is crucial for achieving sustainable autonomous maritime operations, as it can enhance environmental perception, real-time collision avoidance, and energy-efficient path tracking capabilities, thereby reducing operational and environmental costs.

[0004] In maritime practice, information between sensor components needs to be quantized and encoded before being transmitted through channels. Existing research has focused more on analyzing the role of input or state quantization in multi-USV formation control systems. In fact, input state quantization can reduce the transmission and processing burden of state information while maintaining system accuracy. By quantizing state variables such as ship position and speed, continuous state space can be converted into a discrete set, thereby simplifying the computational complexity of collision avoidance decisions. In addition, the quantization method can enhance the robustness of the system under noise and uncertainty conditions. Therefore, it is particularly important to design a distributed formation control method that combines input quantization and state quantization.

[0005] Further, studying the multi-USV formation collision avoidance and obstacle avoidance control system containing event triggering mechanism, signal quantization and actuator fault tolerance is of great significance. This integrated approach not only optimizes the utilization of communication and computing resources, but also enhances the robustness and real-time response capability of the system. In complex marine environments, this strategy has great application potential and provides a solid theoretical and technical foundation for efficient cooperative operation of autonomous fleets. SUMMARY

[0006] The application provides a multi-unmanned ship formation collision avoidance and obstacle avoidance control method with event triggering and signal quantization.

[0007] The technical means adopted by the application are as follows:

[0008] A multi-unmanned ship formation collision avoidance and obstacle avoidance control method with event triggering and signal quantization comprises:

[0009] S1, obtain the sea state information of the surrounding environment and other ships, introduce an improved artificial potential field repulsion function, and establish a trajectory control mathematical model of the underactuated unmanned ship formation;

[0010] S2, use a uniform quantizer to quantize the control input and state variables in the control system;

[0011] S3, use ESO technology to estimate the quantized state information of each USV and the uncertain terms in the model;

[0012] S4, introduce a linear model to describe the quantization process, use a hierarchical design method to design the kinematic guidance law and dynamic quantization tracking control law, so that the designed controller no longer needs prior information of the quantization parameters, and design an event-triggered formation control strategy;

[0013] S5, use input-to-state stability theory to prove the stability of the multi-unmanned ship formation collision avoidance and obstacle avoidance control system with event triggering and signal quantization.

[0014] Further, step S1 specifically comprises:

[0015] Step S1 specifically comprises:

[0016] S11, given an underactuated multi-USV formation, a kinematic model of the motion of the i-th unmanned ship in the multi-USV formation system is established as follows:

[0017]

[0018] where x i , y i represent the ship's center of mass coordinates described in the geodetic coordinate system; denotes the ship's course angle; u i , v i and r i denote the surge velocity, sway velocity and rotational velocity of the ship, respectively;

[0019] S12, a nonlinear dynamic mathematical model of the underactuated unmanned ship is established as follows:

[0020]

[0021] wherein, both denote the mass of the USV; both denote the hydrodynamic derivative term; I z denotes the moment of inertia around the z-axis; the functions f iu (·), f iv (·) and f ir (·) all denote nonlinear uncertain terms; τ iuw , τ ivw and τ irw all denote disturbances caused by unknown ocean factors; Q(τ iu ) and Q(τ ir ) respectively denote the quantized values of the system control inputs τ iu and τ ir ;

[0022] S13, according to the artificial potential field theory, it is assumed that each follower in the formation has the same high potential field, and the repulsive potential function between the follower USVs in the formation is represented as:

[0023]

[0024] wherein, i, j = 1...N and i≠j, the Euclidean distance between any two USVs in the formation is given by ; and R are the upper and lower bounds of the collision avoidance region; the repulsive potential function is zero outside the detection region and tends to infinity within the lower bound of the detection region; if the distance between the two USVs satisfies

[0025] then the collision avoidance potential function is greater than zero and plays a role in the additional control input;

[0026] S14, the repulsive function is introduced, and the calculation formula is as follows:

[0027]

[0028] S15, the obstacle avoidance potential function between each USV and the obstacle is represented as:

[0029]

[0030] where k = 1,..., m represents the existence of m obstacles, the Euclidean distance between the multi-USV and the kth obstacle is denoted as and R 0 is the upper and lower bound of the collision avoidance region; the obstacle avoidance repulsion function is given by the following formula:

[0031]

[0032] Further, the step S2 specifically comprises:

[0033] S21, using a uniform quantizer to quantize the state variables and control inputs in the system, and the specific quantization process is expressed as:

[0034]

[0035] wherein and χ > 0 represents the quantization step, I1 = χ and I i+1 = I i + χ; therefore, the quantization error of the uniform quantizer is bounded and expressed as

[0036] S22, defining the ideal trajectory as where p0(t) represents a continuous and differentiable parameter trajectory; the ideal parameter trajectory p0(t) has a bounded first derivative and a bounded second derivative; it is shown that the ideal parameter trajectory is smooth and stable over time, i.e. there is a constant p m satisfying

[0037] S23, using the graph of Λ = {T, Φ} to represent the relationship between the N USVs and the virtual leader; wherein Φ = (i, j) ∈ Γ × Γ represents the set of edges, T = n0, n1,..., n N represent the set of points; in this case, n i , n j represent the communication flow from node j to node i; the adjacency matrix is expressed as If there is an edge (n j , n i ) ∈ Φ, then a ij = 1; otherwise, a ij = 0;

[0038] S24, setting the main control goal as: enabling each USV in the ship formation system to still track the time-varying ideal trajectory in the presence of communication bandwidth constraints and obstacles, i.e.

[0039]

[0040] ||p i (t)-p j (t)||≥ R ,||p i (t)-p k (t)||≥ R 0

[0041] where P i (t) = [x i , y i ] T denotes the actual position of each USV in the formation, P 0id (t) denotes the position deviation of each USV relative to the virtual leader parameter trajectory, and ζ denotes a constant greater than zero.

[0042] Further, the step S3 specifically comprises:

[0043] S31, according to the step S11 and the step S12, rewriting the mathematical model of the i-th USV as follows:

[0044]

[0045] where, ν i = [u i , v i , r i ] T , is a rotation matrix, and satisfies:

[0046]

[0047] S32, designing the ESO as follows:

[0048]

[0049] where σ i denotes a constant greater than 0, and are observer states, and respectively denote the quantized state variables and the observation values of Q(ν i ) = [Q(u i ), Q(v i ), Q(r i )] T ; denotes a system uncertainty term the observed value of denotes the rotation matrix based on state quantization, and satisfies:

[0050]

[0051] S33, define and satisfies:

[0052]

[0053] then we get:

[0054]

[0055] S34, because there is a constant greater than 0 and such that

[0056] S35, define the dynamic error of ESO as follows:

[0057]

[0058] where,

[0059] S36, for the designed ESO, there is a positive definite matrix satisfying the given conditions, so that the ESO error subsystem is input to state stable;

[0060] S37, let where, and derivation is obtained:

[0061]

[0062] where,

[0063] S38, according to there is a constant greater than 0 satisfying

[0064] S39, in order to analyze the stability of ESO error system, two inequalities are given as follows:

[0065]

[0066] where, ∈ i > 0, denotes the upper bound of the yaw angle velocity r i and satisfies

[0067] S310, design Lyapunov function, as follows:

[0068]

[0069] wherein, is a positive definite matrix, and the derivative of the above formula is:

[0070]

[0071] S311, when then:

[0072]

[0073] wherein, 0 i <1, and satisfies wherein and are the maximum eigenvalue and the minimum eigenvalue of the matrix , respectively;

[0074] S312, since then:

[0075]

[0076] Therefore, the error subsystem of ESO is input to state stable.

[0077] Further, step S4 specifically comprises:

[0078] S41, in the kinematics subsystem, a guidance law is designed based on the actuator to realize the tracking of the ideal trajectory of the unmanned ship formation;

[0079] S42, in the dynamics subsystem, a linear analytical model is used to describe the quantization process, while an event-triggered driving and an actuator fault-tolerant strategy are introduced into the system, and a system control law and an adaptive law are designed based on a sliding mode control strategy.

[0080] Further, step S41 specifically comprises:

[0081] S411, define a distributed formation tracking error, as follows:

[0082]

[0083] wherein, represents the estimated value of the actual position of the multiple USVs in the formation, and P0 represents the actual position of the virtual leader, represents a rotation matrix and satisfies:

[0084]

[0085] S412, derive the defined distributed formation tracking error, get:

[0086]

[0087] Wherein,

[0088] S413, in order to eliminate the influence of underactuated on USV formation control system, set error transformation, as follows:

[0089]

[0090] Wherein, δ0∈R is a constant greater than 0, combined with step S412, get:

[0091]

[0092] Wherein, h i =diag{l i ,δ0},

[0093] S414, define From step S3, it is known that the designed ESO is stable, and the observation error can converge to a small residual set, therefore, the kinematic guidance law based on ESO is designed, as follows:

[0094]

[0095] Wherein,

[0096] S415, according to step S414, the kinematic error is expressed as

[0097] Further, step S42, specifically includes:

[0098] S421, according to the nonlinear dynamic mathematical model of underactuated unmanned ship established in step S12, get:

[0099]

[0100] And the above formula is simplified as:

[0101]

[0102] Wherein,

[0103] S422, let Q(τ iu )=q 11iu (t)τ iu +q 12iu(t), Q(τ ir ) = q 11ir (t)τ ir + q 12ir (t), and:

[0104]

[0105] where q 11iu (t) and q 11ir (t) are unknown parameters; since the sign remains unchanged throughout the quantization process, from the above equation, q 11iu (t) > 0, q 11ir (t) > 0; in addition, if |τ iu (t)| < b and |τ ir (t)| < b, considering that Q(τ iu (t)) and Q(τ ir (t)) are bounded, q 12iu (t) and q 12ir (t) are also bounded, and satisfy

[0106] S423, considering the fault tolerance of the actuator, the controller is designed as follows:

[0107]

[0108] where q1(t) = ρ0q 11 (t), q2(t) = ρ0q 12 (t), 0 < ρ0 < 1;

[0109] S424, the control target of the kinetic subsystem is set as follows:

[0110]

[0111] where a1 and a2 both represent small positive integers;

[0112] S425, the integral sliding surface is defined as follows:

[0113]

[0114] where b iu > 0 and b ir > 0, and the derivative of the above equation is:

[0115]

[0116] S426, according to step S425, we have:

[0117]

[0118] wherein ρ iu , μ iu , ρ ir , μ ir all represent constants greater than 0, μ iu ≥ μ ud , μ ir ≥ μ rd and μ ud > 0, μ rd > 0;

[0119] S427, since q 1iu (t) and q 1ir (t) are unknown and time-varying, an adaptive method is used to estimate their boundaries, in order to prevent singular problems that occur when the estimated value tends to zero, the lower bound of q 1iu (t) and q 1ir (t) is used for estimation; define time-varying gain η iu = 1 / q 1iu (t) min and η ir = 1 / q 1ir (t) min , wherein q 1iu (t) min and q 1ir (t) min are the lower bounds of q 1iu (t) and q 1ir (t) respectively, therefore, the USV formation tracking control law is designed as follows:

[0120]

[0121] wherein γ1, γ2, c iu , c ir , ω iu , ω ir , ε iu , ε ir and all represent constants greater than 0;

[0122] S428, on the basis of considering quantization in the control system, a time-varying threshold event-triggered mechanism is introduced, defined as follows:

[0123]

[0124] wherein m represents a time constant, represents angle tracking, satisfying

[0125] S429, there is a continuous time-varying coefficient θ i (t) satisfying, θi (t k+1 ) = ±1, θ i (t k ) = 0, and |θ i (t)|≤1, it is obtained that i.e. Then the analysis is as follows:

[0126] When θ i (t) = 0, it is obtained that t = t k ,

[0127] When |θ i (t)|<1, it is obtained that t∈[t k ,t k+1 ];

[0128] When |θ i (t)|=1, the corresponding time is t = t k+1 , and it is obtained that For t∈[t k+1 ,t k+2 ], the analysis is the same as above, and thus, at any time, it satisfies where θ i (t) satisfies θ i (t k ) = 0, θ(t k+1 ) = ±1, and |θ i (t)|≤1;

[0129] S4210, based on s iu , s ir , kinematics and dynamics error system is expressed as:

[0130]

[0131] Further, step S5 specifically includes:

[0132] S51, considering the USV formation tracking collision avoidance obstacle avoidance control system with quantization, actuator fault tolerance and event triggering mechanism, combining the designed ESO, kinematic guidance rate, dynamic control rate and adaptive law, with the state s iu ,s ir , and input Ω iu , Ω irThe platoon tracking control system is input-to-state stable, the tracking error can converge to a small residual set, and all signals in the designed control system are uniformly ultimately bounded;

[0133] S52, define a Lyapunov function as follows:

[0134]

[0135] And take the derivative of the defined Lyapunov function:

[0136]

[0137] S53, combine step S426 and step S52 to obtain:

[0138]

[0139] S54, combine the designed control law and the adaptive law to obtain:

[0140]

[0141] Wherein, q 1i (t)>0 and

[0142] S55, since Then Let Thus, we obtain:

[0143]

[0144] S56, considering Then we obtain:

[0145]

[0146] S57, since And Then we obtain:

[0147]

[0148] Wherein,

[0149] S58, since Then we obtain:

[0150]

[0151] S59, let Then we obtain:

[0152]

[0153] where is a positive constant and satisfies

[0154] S510, define s u = [s 1u , s 2u ,..., s Nu ] T , s r = [s 1r , s 2r ,..., s Nr ] T ,

[0155] S511, outside the collision avoidance region Thus then we have

[0156]

[0157] S512, note that then we have

[0158]

[0159] where Thus, the designed error subsystem with event-triggered, actuator fault-tolerant, and input and state quantization is input-to-state stable, and satisfies

[0160]

[0161] where R c = diag{1, 1 / 2γ1η 1u ,..., 1 / 2γ1η Nu , 1 / 2γ2η 1r ,..., 1 / 2γ2η Nr};

[0162] S513, inside the collision avoidance region, then we have

[0163]

[0164] S514, note that then we have

[0165]

[0166] where Therefore, the designed error subsystem with event-triggering, actuator fault-tolerant and input and state quantization is input-to-state stable, and satisfies:

[0167]

[0168] S515, since s iu , s ir is bounded, thus is bounded; according to step S414, the kinematic guidance signal u ic and r ic are both bounded, thus it is proved that u i and r i are bounded; according to step S3, the ESO is input-to-state stable, according to the stability theory of the cascade system, combined with step S512, ||Y(t)|| is bounded and satisfies:

[0169]

[0170] Compared with the prior art, the present application has the following advantages:

[0171] 1. The multi-unmanned ship formation collision avoidance and obstacle avoidance control method with event triggering and signal quantization provided by the present application first considers the combination of event triggering, signal quantization and actuator fault-tolerance, solves the problem of USV formation control in a more complex marine environment with limited communication bandwidth and quantization, and further enhances the applicability of the proposed control scheme by adding the actuator fault-tolerance function. Specifically, the scheme surpasses the traditional method and takes into account the actuator fault-tolerance, so that the system can operate effectively even in the case of component failure or fault. The combination of the event triggering mechanism and the signal quantization strategy avoids unnecessary calculation and communication overhead, effectively reduces the communication burden, thereby improving the robustness and efficiency of the control system, and better meets the needs of maritime practice.

[0172] 2. The multi-unmanned ship formation collision avoidance and obstacle avoidance control method with event triggering and signal quantization provided by the present application is based on the characteristics of the marine environment with high uncertainty and strong disturbance, and aims to smooth the quantized signal and reduce the influence of quantization error by estimating the quantized state information of each USV and the uncertainty term in the system model. By introducing the ESO framework, the robustness of the system under quantization conditions is effectively improved, the adaptability to marine environment uncertainty and external disturbance is enhanced, and higher control performance is achieved.

[0173] 3. The multi-unmanned ship formation collision avoidance and obstacle avoidance control method with event triggering and signal quantization provided by the application uses a linear model to describe the quantization process, effectively eliminating the need for the controller to obtain prior parameter information of the quantization parameters. This method not only simplifies the design complexity of the control algorithm, but also reduces the dependence on accurate quantization parameters. Through this improvement, the control system can be more flexible in implementation and adjustment under different operating conditions, ensuring that the system remains efficient and stable in the face of various environmental changes. In addition, this solution also enhances the adaptability of the system, enabling it to work more reliably in uncertain marine environments and meet complex task requirements.

[0174] 4. The multi-unmanned ship formation collision avoidance and obstacle avoidance control method with event triggering and signal quantization provided by the application conducts comparative experiments before and after quantization of the unmanned ship formation control system based on the Matlab platform. At the same time, an adaptive quantization tracking controller is designed to achieve collision avoidance and obstacle avoidance control under the condition of unmanned ship formation trajectory tracking. After considering the event triggering mechanism and input and state quantization of the actuator fault tolerance, the execution frequency of the actuator is reduced, the control amplitude is reduced, and the system control input curve is more suitable for maritime engineering practice.

[0175] Based on the above reasons, the application can be widely promoted in the field of artificial intelligence. BRIEF DESCRIPTION OF DRAWINGS

[0176] In order to more clearly illustrate the technical solutions in the embodiments of the application or the prior art, the drawings needed to be used in the embodiment or prior art description will be briefly introduced below. Obviously, the drawings in the following description are some embodiments of the application, and other drawings can also be obtained by those skilled in the art without creative labor.

[0177] Figure 1 The method flowchart of the application.

[0178] Figure 2 The unmanned ship distributed formation collision avoidance and obstacle avoidance control result graph provided by the embodiment of the application.

[0179] Figure 3 The 5 unmanned ship lateral velocity and rudder angle velocity tracking graph provided by the embodiment of the application.

[0180] Figure 4 The 5 unmanned ship lateral and longitudinal tracking error graph provided by the embodiment of the application.

[0181] Figure 5 The 5 unmanned ship control input graph under different communication environments provided by the embodiment of the application.

[0182] Figure 6The control input diagram of the five unmanned ships in different communication environments is provided for the embodiments of the present application.

[0183] Figure 7 The event trigger diagram of the fourth unmanned ship in the five unmanned ships provided for the embodiments of the present application is taken as an example. DETAILED DESCRIPTION

[0184] In order for those skilled in the art to better understand the present application, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, but not all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor should belong to the protection scope of the present application.

[0185] It should be noted that the terms “include” and “have” and any variations thereof in the specification and claims of the present application and the above-mentioned drawings are intended to cover non-exclusive inclusion, for example, a process, method, system, product or device including a series of steps or units does not have to be limited to only those steps or units clearly listed, but can include other steps or units not clearly listed or inherent to the process, method, product or device.

[0186] As shown in Figure 1 The present application provides a multi-unmanned ship formation collision avoidance and obstacle avoidance control method with event triggering and signal quantization, which comprises:

[0187] S1, obtaining the sea state information of the surrounding environment and other ships, introducing an improved artificial potential field repulsion function, and establishing a trajectory control mathematical model of the under-actuated unmanned ship formation;

[0188] S2, using a uniform quantizer to quantize the control input and state variables in the control system;

[0189] S3, using the ESO technology to estimate the quantized state information of each USV and the uncertain terms existing in the model;

[0190] S4, introducing a linear model to describe the quantization process, using a hierarchical design method to design the kinematic guidance law and the dynamic quantization tracking control law, so that the designed controller no longer needs the prior information of the quantization parameters, and designing an event-triggered formation control strategy;

[0191] S5, using the input-to-state stability theory to prove the stability of the multi-unmanned ship formation collision avoidance and obstacle avoidance control system with event triggering and signal quantization.

[0192] In the implementation, as a preferred embodiment of the present application, step S1 specifically comprises:

[0193] S11. Given an underactuated multi-USV formation, establish the kinematic model of the motion of the i-th unmanned vessel in the multi-USV formation system as follows:

[0194]

[0195] Among them, x i 、y i Indicates the coordinates of the ship's center of mass described in the geodetic coordinate system; Indicates the ship's heading angle; u i 、v i and r i They represent the surge speed, sway speed and rotation speed of the ship respectively;

[0196] S12. Establish a nonlinear dynamic mathematical model of the underactuated unmanned vessel as follows:

[0197]

[0198] in, Both represent the quality of USV; Both represent hydrodynamic derivative terms; I z Represents the moment of inertia around the z-axis; function f iu (·), f iv (·), f ir (·) represents nonlinear uncertainties such as fluid dynamic damping and centripetal force; τ iuw , τ ivw , τ irw Both represent disturbances caused by unknown ocean factors; Q(τ iu ) and Q(τ ir ) represent the system control input τ iu and τ ir The quantitative value of

[0199] S13. According to the artificial potential field theory, assuming that each follower in the formation has the same high potential field, the repulsive field potential function between multiple USVs following in the formation is expressed as:

[0200]

[0201] Where i,j=1...N and i≠j, the Euclidean distance between any two USVs in the formation is given by given; and R are the upper and lower bounds of the collision avoidance region; the repulsive potential function is zero outside the detection region and tends to infinity within the lower bound of the detection region; if the distance between the two USVs satisfies

[0202] Then the collision avoidance potential function greater than zero and acts in the additional control input;

[0203] S14, repulsive force function is introduced, the calculation formula is as follows:

[0204]

[0205] S15, the obstacle avoidance potential function between each USV and the obstacle is represented as:

[0206]

[0207] wherein k = 1,..., m represents that there are m obstacles, and the Euclidean distance between the multi-USV and the kth obstacle is represented as and R0 is the upper and lower bound of the collision avoidance area; the obstacle avoidance repulsive force function is given by the following formula:

[0208]

[0209] In specific implementation, as a preferred embodiment of the present application, step S2 specifically comprises:

[0210] S21, a uniform quantizer is used to quantize the state variables and control inputs in the system, and the specific quantization process is expressed as:

[0211]

[0212] wherein and χ>0 represents the quantization step, I1=χ and I i+1 =I i +χ; therefore, the quantization error of the uniform quantizer is bounded and expressed as

[0213] S22, the ideal trajectory is defined as wherein p0(t) represents a continuous and derivable parameter trajectory; the ideal parameter trajectory p0(t) has differentiability, and the first derivative and the second derivative are bounded; it is indicated that the ideal parameter trajectory is smooth and stable over time, that is, there is a constant p m satisfying

[0214] S23, the relationship between the N USVs and the virtual leader is represented by the graph of Λ={T,Φ}; wherein Φ=(i,j)∈Γ×Γ represents the set of edges, and T=n0,n1,...,n N represent the set of points; in this case, n i ,n j represent the communication flow from node j to node i; the adjacency matrix is represented as If there is an edge (n j , n i )∈Φ, then a ij =1; otherwise, a ij =0;

[0215] S24, set the main control target as: enabling each USV in the ship formation system to still track the time-varying ideal trajectory in the presence of communication bandwidth constraints and obstacles, that is:

[0216]

[0217] ||p i (t)-p j (t)||≥ R ,||p i (t)-p k (t)||≥ R 0

[0218] Wherein, P i (t)=[x i ,y i ] T represents the actual position of each USV in the formation, P 0id (t) represents the position deviation of each USV relative to the virtual leader parameter trajectory, and ζ represents a constant greater than zero.

[0219] In specific implementation, as a preferred embodiment of the present application, step S3 specifically comprises:

[0220] S31, according to steps S11 and S12, rewrite the mathematical model of the i-th USV as follows:

[0221]

[0222] Wherein, ν i =[u i ,v i ,r i ] T , is a rotation matrix, and satisfies:

[0223]

[0224] S32, design ESO as follows:

[0225]

[0226] Wherein, σ iRepresents a constant greater than 0, and is the observer state, and Represent the quantized state variables and Q(ν i )=[Q(u i ),Q(v i ),Q(r i )] T Observed values ​​of Represents the system uncertainty term The observed value of Represents a rotation matrix based on state quantization, and satisfies:

[0227]

[0228] S33. Definition And satisfy:

[0229]

[0230] Then we get:

[0231]

[0232]

[0233] S34. Due to That is, there is a constant greater than 0 and Make

[0234] S35. Define the dynamic error of ESO as follows:

[0235]

[0236] in,

[0237] S36. For the designed ESO, there exists a positive definite matrix Satisfy given conditions so that the ESO error subsystem is input-state stable;

[0238] S37, Order in, And The derivative is:

[0239]

[0240] in,

[0241] S38、According to There is a constant greater than 0 Satisfy

[0242] S39, in order to analyze the stability of the ESO error system, two inequalities are given as follows:

[0243]

[0244] Wherein, ∈ i > 0, Indicates the upper bound of the yaw angle velocity r i And satisfies

[0245] S310, design Lyapunov function, as follows:

[0246]

[0247] Wherein, Indicates a positive definite matrix, and the derivative of the above formula is:

[0248]

[0249] S311, when Then get:

[0250]

[0251] Wherein, 0 i < 1, and satisfies Wherein And The maximum eigenvalue and the minimum eigenvalue of the matrix ;

[0252] S312, since Then get:

[0253]

[0254] Therefore, the error subsystem of ESO is input to state stable.

[0255] In this embodiment, there are two main purposes for introducing ESO in the control system. First, it helps to obtain the state information of the USV in the quantization environment and estimate the inherent uncertainty in the ship model. In addition, ESO also plays a filtering and smoothing role for the quantization curve, ensuring that the quantized state variable can be used as a continuous and differentiable variable, enhancing its applicability and feasibility in the design of the controller.

[0256] In specific implementation, as a preferred embodiment of the present application, step S4 specifically comprises:

[0257] S41, in the kinematics subsystem, a kinematic guidance law is designed based on the actuator to realize tracking of the ideal trajectory of the unmanned ship formation;

[0258] S42, in the dynamics subsystem, a linear analytical model is used to describe the quantization process, an event-triggered driving and an actuator fault-tolerant strategy are introduced into the system to save communication resources, and a system control law and an adaptive law are designed based on a sliding mode control strategy.

[0259] In specific implementation, as a preferred embodiment of the present application, step S41 specifically comprises:

[0260] S411, a distributed formation tracking error is defined as follows:

[0261]

[0262] wherein, represents an estimated value of the actual position of the multiple USVs in the formation, P0 represents an actual position of a virtual leader, represents a rotation matrix and satisfies:

[0263]

[0264] S412, the defined distributed formation tracking error is differentiated to obtain:

[0265]

[0266] wherein,

[0267] S413, in order to eliminate the influence of under-actuation on the USV formation control system, an error transformation is set as follows:

[0268]

[0269] wherein, δ0∈R is a constant greater than 0, and in combination with step S412, the following is obtained:

[0270]

[0271] wherein, h i = diag{ l i , δ0},

[0272] S414, the following is defined: It is known from step S3 that the designed ESO is stable, and the observation error can converge into a small residual set, therefore, a kinematic guidance law based on the ESO is designed as follows:

[0273]

[0274] wherein,

[0275] S415, according to step S414, the kinematic error is represented as

[0276] In particular implementation, as a preferred embodiment of the present application, step S42 specifically includes:

[0277] S421, according to the nonlinear dynamics mathematical model of the underactuated unmanned ship established in step S12, the following is obtained:

[0278]

[0279] And the above formula is simplified as:

[0280]

[0281] wherein,

[0282] S422, let Q(τ iu )=q 11iu (t)τ iu +q 12iu (t), Q(τ ir )=q 11ir (t)τ ir +q 12ir (t), and:

[0283]

[0284] wherein, q 11iu (t) and q 11ir (t) are unknown parameters; since the sign remains unchanged during the entire quantization process, from the above formula, q 11iu (t)>0, q 11ir (t)>0; in addition, if |τ iu (t)|<b and |τ ir (t)|<b, considering that Q(τ iu (t)) and Q(τ ir (t)) are bounded, q 12iu (t) and q 12ir (t) are also bounded, and satisfy

[0285] S423, considering the fault tolerance of the actuator, the controller is designed as follows:

[0286]

[0287] wherein, q1(t)=ρ0q11 (t), q2(t) = p0q 12 (t), 0 < p0 < 1;

[0288] S424, set the control target of the kinetic subsystem, as follows:

[0289]

[0290] wherein a1, a2 both represent a smaller positive integer;

[0291] S425, define the integral sliding surface, as follows:

[0292]

[0293] wherein b iu > 0 and b ir > 0, the derivative of the above equation is:

[0294]

[0295] S426, according to step S425, we have:

[0296]

[0297] wherein p iu , m iu , p ir , m ir all represent a constant greater than 0, m iu > m ud , m ir > m rd and m ud > 0, m rd > 0;

[0298] S427, since q 1iu (t) and q 1ir (t) are unknown and time-varying, an adaptive method is used to estimate their boundaries, in order to prevent singular problems that occur when the estimated value tends to zero, the lower bound of q 1iu (t) and q 1ir (t) is used for estimation; define time-varying gains η iu = 1 / q 1iu (t) min and η ir = 1 / q 1ir (t) min , wherein q 1iu (t) min and q 1ir (t) min are the lower bounds of q 1iu (t) and q 1ir (t) respectively.The lower bound of (t), thus, the USV formation tracking control law is designed as follows:

[0299]

[0300] where γ1, γ2, c iu , ω ir , ω iu , ε ir , ε iu , ε ir and are constants greater than 0;

[0301] S428, On the basis of considering quantization in the control system, an event-triggered mechanism of time-varying threshold is introduced, which further saves communication resources and reduces the burden of communication signal transmission, and is defined as follows:

[0302]

[0303] where m represents a time constant, represents angle tracking, and satisfies

[0304] S429, there is a continuous time-varying coefficient θ i (t) satisfies θ i (t k+1 ) = ±1, θ i (t k ) = 0, and |θ i (t)| ≤ 1, so that that is, Then the analysis is as follows:

[0305] When θ i (t) = 0, t = t k ,

[0306] When |θ i (t)| < 1, t ∈ [t k , t k+1 ];

[0307] When |θ i (t)| = 1, at this time, the corresponding time is t = t k+1 , and For t ∈ [t k+1 , t k+2 ], the analysis is the same as above, thus, at any time, it satisfies where θ i (t) satisfies θi (t k )=0,θ(t k+1 )=±1, and |θ i (t)|≤1;

[0308] S4210, based on s iu , s ir , The kinematic and dynamic error systems are expressed as:

[0309]

[0310] In specific implementation, as a preferred embodiment of the present invention, step S5 specifically includes:

[0311] S51, consider the USV formation tracking and collision avoidance control system with quantization, actuator fault tolerance and event triggering mechanism, combined with the designed ESO, kinematic guidance rate, dynamic control rate and adaptive law, with state s iu ,s ir , and input Ω iu ,Ω ir The formation tracking control system is stable in terms of input state, the tracking error can converge to a small residual set, and all signals in the designed control system are uniformly and ultimately bounded;

[0312] S52. Define the Lyapunov function as follows:

[0313]

[0314] And take the derivative of the defined Lyapunov function:

[0315]

[0316] S53, combining step S426 and step S52 to obtain:

[0317]

[0318] S54. Combining the designed control law and adaptive law, we get:

[0319]

[0320] Among them, q 1i (t)>0 and

[0321] S55. Due to but make Thus we obtain:

[0322]

[0323] S56, considering Thus we obtain:

[0324]

[0325] S57, since and Thus we obtain:

[0326]

[0327] where,

[0328] S58, since Thus we obtain:

[0329]

[0330] S59, let Thus we obtain:

[0331]

[0332] where, is a normal number and satisfies

[0333] S510, define s u = [s 1u , s 2u ,..., s Nu ] T , s r = [s 1r , s 2r ,..., s Nr ] T ,

[0334] S511, outside the collision avoidance area Therefore Thus we obtain:

[0335]

[0336] S512, note that Thus we obtain:

[0337]

[0338] where, Therefore, the error subsystem designed with event-triggering, actuator fault-tolerant and input and state quantization is input-to-state stable, and satisfies:

[0339]

[0340] wherein, R c = diag{1, 1 / 2γ1η 1u ,..., 1 / 2γ1η Nu , 1 / 2γ2η 1r ,..., 1 / 2γ2η Nr};

[0341] S513, in the collision avoidance area, then:

[0342]

[0343] S514, noting that then:

[0344]

[0345] wherein, Therefore, the error subsystem designed with event-triggering, actuator fault-tolerant and input and state quantization is input-to-state stable, and satisfies:

[0346]

[0347] S515, since s iu , s ir is bounded, therefore is bounded; according to step S414, the kinematic guidance signal u ic and r ic are both bounded, therefore u i and r i are bounded; according to step S3, the ESO is input-to-state stable, according to the stability theory of cascade systems, combined with step S512, ||Y(t)|| is bounded and satisfies:

[0348]

[0349] Embodiment 1

[0350] In order to verify the effectiveness of the multi-unmanned ship formation collision avoidance and obstacle avoidance control method and system with event triggering and signal quantization proposed in the present application, computer simulation research is carried out by using MATLAB / Simulink in this embodiment. The model parameters are selected as follows:

[0351] m u= 25.8 kg

[0352] m v = 33.8 kg

[0353] m r = 2.76 kg m 2

[0354]

[0355] f v = -36.5 |v|v - 0.8896v - 0.805v |r| - m u u i r

[0356] f r = -0.75 |r|r - 1.90r + 0.08 |v|r + (m u - m v )u i v - 1.0948u i r

[0357] The time-varying parameter trajectory of the virtual leader is designed as p0(t) = [8sin(0.02t), 1 - cos(0.02t)] T . The initial states of the five USVs in the formation are designed as P1 = [15, 16] T , P2 = [0, 21] T , P3 = [7, -7] T , P4 = [0, 28] T and P5 = [0, 7] T . The ideal formation initial positions are designed as P 1d = [15, 16] T , P 2d = [0, 21] T , P 3d = [0, 14] T , P 4d = [0, 28] T and P 5d = [0, 7] T . The controller design parameters take p0 = 0.80, b iu = b ir = 10, m iu = m ir = 2, c iu = c ir = 0.02, w iu = w ir = 0.20, g1 = 3, g2 = 2, e iu = e ir= 0.10, ε iu = ε ir = 0.10, χ = 0.1, the potential function parameter is taken as R = 2, R 0 = 1,

[0358] As Figures 2-7 Simulation results of the designed USV distributed formation tracking control strategy. Figure 2 The tracking results of five underactuated USVs on the time-varying ideal trajectory. It can be seen that these USVs can effectively track the predetermined route from the starting point in a relatively short time, while maintaining the specified inter-vehicle distance and formation even in the face of internal system failures and external disturbances. In autonomous ship formation, collision avoidance and obstacle avoidance are given priority over maintaining formation when there is a possibility of collision. In areas with low collision probability, the formation error converges to a small range around the equilibrium point. Figure 3 The longitudinal velocity and ROT of multiple USVs after signal quantization are shown. Figure 4 The lateral and longitudinal tracking errors of the formation under quantization, event-triggered control, and actuator fault tolerance conditions are shown respectively. The results show that the tracking error of the coordinated formation of multiple USVs based on event-triggered, signal quantization, and actuator fault tolerance can converge to a smaller residual set more quickly. Figure 5 The upper part of the graph in Figure 6 describes the ship control input without quantization and event-triggered mechanism. Figure 5 The lower part of the graph in Figure 6 describes the ship control input with quantization and event-triggered mechanism. Figure 7 The time interval between two consecutive instances of the fourth USV is shown.

[0359] The simulation results show that the addition of quantization and event-triggered mechanism greatly reduces the execution frequency of the controller. This reduction relieves the signal load of the communication channel and better addresses the limitations of limited bandwidth in maritime communications. After implementing the quantization optimization, the system execution frequency is significantly reduced. This modification effectively reduces redundant communication activities and reduces energy consumption. The optimized configuration improves system efficiency and enables effective coordination between multiple USVs while significantly reducing resource utilization during complex task execution. The proposed quantization optimization method has a dual advantage: it is an effective method to improve the performance of multiple USV systems, and it establishes a theoretical framework for the development of energy-aware autonomous systems. These findings provide valuable insights for designing resource-saving multiple USV architectures in practical applications.

[0360] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present application, and are not intended to limit the present application; although the present application has been described in detail with reference to the above embodiments, those skilled in the art should understand that the technical solutions recorded in the above embodiments can be modified, or some or all of the technical features can be replaced by equivalents; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application.

Claims

1. A collision and obstacle avoidance control method for a multi-unmanned vessel formation with event triggering and signal quantization, characterized in that: include: S1. Obtain information about the surrounding environment and sea conditions of other ships, introduce an improved artificial potential field repulsion function, and establish a mathematical model for trajectory control of the under-actuated unmanned ship formation; S2, using a uniform quantizer to quantize the control input and state variables in the control system; S3, using ESO technology to estimate the quantitative state information of each USV and the uncertainties in the model; S4. Introduce a linear model to describe the quantization process and adopt a hierarchical design method to design the kinematic guidance law and the dynamic quantized tracking control law. This makes the designed controller no longer require prior information on the quantized parameters, and designs an event-triggered formation control strategy. S5. Using the input-to-state stability theory, the stability of the multi-unmanned ship formation collision avoidance and obstacle avoidance control system with event triggering and signal quantization is demonstrated.

2. The method for collision and obstacle avoidance control of a multi-unmanned vessel formation with event triggering and signal quantization according to claim 1 is characterized in that: Step S1 specifically includes: S11. Given an underactuated multi-USV formation, establish the kinematic model of the motion of the i-th unmanned vessel in the multi-USV formation system as follows: Among them, x i 、y i Indicates the coordinates of the ship's center of mass described in the geodetic coordinate system; Indicates the ship's heading angle; u i 、v i and r i They represent the surge speed, sway speed and rotation speed of the ship respectively; S12. Establish a nonlinear dynamic mathematical model of the underactuated unmanned vessel as follows: in, Both represent the quality of USV; Both represent hydrodynamic derivative terms; I z Represents the moment of inertia around the z-axis; function f iu (·), f iv (·), f ir (·) denotes nonlinear uncertainty terms; τ iuw , τ ivw , τ irw Both represent disturbances caused by unknown ocean factors; Q(τ iu ) and Q(τ ir ) represent the system control input τ iu and τ ir The quantitative value of S13. According to the artificial potential field theory, assuming that each follower in the formation has the same high potential field, the repulsive field potential function between multiple USVs following in the formation is expressed as: Where i,j=1...N and i≠j, the Euclidean distance between any two USVs in the formation is given by given; and R are the upper and lower bounds of the collision avoidance region; the repulsive potential function is zero outside the detection region and tends to infinity within the lower bound of the detection region; if the distance between the two USVs satisfies Then the collision avoidance potential function is greater than zero and plays a role in additional control input; S14. Introduce the repulsion function, and the calculation formula is as follows: S15. The obstacle avoidance potential function between each USV and the obstacle is expressed as: Where k = 1, ..., m means there are m obstacles, and the Euclidean distance between multiple USVs and the kth obstacle is expressed as and R 0 is the upper and lower bounds of the collision avoidance area; the obstacle avoidance repulsion function is given by the following formula:

3. The method for collision and obstacle avoidance control of a multi-unmanned vessel formation with event triggering and signal quantization according to claim 1 is characterized in that: Step S2 specifically includes: S21. Use a uniform quantizer to quantize the state variables and control inputs in the system. The specific quantization process is expressed as: in, and χ>0 indicates quantization step size, I1=χ and I i+1 =I i +χ; therefore, the quantization error of the uniform quantizer is bounded and is expressed as S22, define the ideal trajectory as Where p0(t) represents a continuously differentiable parameter trajectory; the ideal parameter trajectory p0(t) is differentiable, and its first-order derivative and second-order derivative are bounded; this indicates that the ideal parameter trajectory is smooth and stable over time, that is, there is a constant p m satisfy S23. Use a graph of Λ = {T, Φ} to represent the relationship between N USVs and the virtual leader; where Φ = (i, j)∈Γ×Γ represents the set of edges, T = n0,n1,...,n N represents a set of points; in this case, n i ,n j represents the communication flow from node j to node i; the adjacency matrix is ​​expressed as If there is an edge (n j ,n i )∈Φ, then a ij =1; otherwise, a ij =0; S24. The main control objective is to enable each USV in the ship formation system to track the time-varying ideal trajectory in the presence of communication bandwidth constraints and obstacles, that is: ||p i (t)-p j (t)||≥ R ,||p i (t)-p k (t)||≥ R 0 Among them, P i (t) = [x i ,y i ] T represents the actual position of each USV in the formation, P 0id (t) represents the position deviation of each USV relative to the parameter trajectory of the virtual leader, and ζ represents a constant greater than zero.

4. The method for collision and obstacle avoidance control of a multi-unmanned vessel formation with event triggering and signal quantization according to claim 1 is characterized in that: Step S3 specifically includes: S31. According to step S11 and step S12, the mathematical model of the i-th USV is rewritten as follows: in, ν i =[u i ,v i ,r i ] T , is a rotation matrix that satisfies: S32. Design ESO as follows: Among them, σ i Represents a constant greater than 0, and is the observer state, and Represent the quantized state variables and Q(ν i )=[Q(u i ),Q(v i ),Q(r i )] T Observed values ​​of Represents the system uncertainty term The observed value of Represents a rotation matrix based on state quantization, and satisfies: S33. Definition And satisfy: Then we get: S34. Due to That is, there is a constant greater than 0 and Make S35. Define the dynamic error of ESO as follows: in, S36. For the designed ESO, there exists a positive definite matrix Satisfy given conditions so that the ESO error subsystem is input-state stable; S37, Order in, And The derivative is: in, S38, according to There is a constant greater than 0 satisfy S39. In order to analyze the stability of the ESO error system, two inequalities are given as follows: Among them, ∈ i >0, represents the yaw angular velocity r i The upper bound of and satisfies S310. Design the Lyapunov function as follows: in, Represents a positive definite matrix, and the derivative of the above formula is: S311, when satisfied Then we get: Among them, 0<o i <1, and satisfies in and The matrices The maximum and minimum eigenvalues ​​of ; S312, due to Then we get: Therefore, the error subsystem of the ESO is input-state stable.

5. The method for collision and obstacle avoidance control of a multi-unmanned vessel formation with event triggering and signal quantization according to claim 1 is characterized in that: Step S4 specifically includes: S41. In the kinematic subsystem, the guidance law is designed based on the actuator to achieve tracking of the ideal trajectory of the unmanned ship formation; S42. In the dynamic subsystem, a linear analytical model is used to describe the quantization process. At the same time, event-triggered drive and actuator fault-tolerant strategies are introduced into the system. Based on the sliding mode control strategy, the system control law and adaptive law are designed.

6. The method for collision and obstacle avoidance control of a multi-unmanned vessel formation with event triggering and signal quantization according to claim 5 is characterized in that: Step S41 specifically includes: S411. Define the distributed formation tracking error as follows: in, represents the estimated value of the actual position of multiple USVs in the formation, P0 represents the actual position of the virtual leader, represents a rotation matrix that satisfies: S412. Derivative the defined distributed formation tracking error to obtain: in, S413. In order to eliminate the influence of under-actuation on the USV formation control system, the error transformation is set as follows: Wherein, δ0∈R is a constant greater than 0, and combined with step S412, we can obtain: Among them, h i =diag{l i ,δ0}, S414, Definition From step S3, we know that the designed ESO is stable and the observation error can converge to a smaller residual set. Therefore, the kinematic guidance law based on ESO is designed as follows: in, S415. According to step S414, the kinematic error is expressed as 7. The method for collision and obstacle avoidance control of a multi-unmanned vessel formation with event triggering and signal quantization according to claim 5 is characterized in that: Step S42 specifically includes: S421, according to the nonlinear dynamic mathematical model of the under-actuated unmanned vessel established in step S12, obtain: And simplify the above formula to: in, S422. Let Q(τ iu ) = q 11iu (t)τ iu + q 12iu (t), Q(τ ir ) = q[[ID=1,11]] 11ir (t)τ ir [[ID= / *14* / 14]]+ q 12ir (t), and: Among them, q 11iu (t) and q 11ir (t) is an unknown parameter; since the sign remains unchanged during the entire quantization process, it can be seen from the above formula that q 11iu (t)>0,q 11ir (t)>0; In addition, if |τ iu (t)|<b and|τ ir (t)|<b, considering Q(τ iu (t)) and Q(τ ir (t)) is bounded, then q 12iu (t) and q 12ir (t) is also bounded and satisfies S423. Considering the fault tolerance of the actuator, design the controller as follows: where, q1(t) = ρ0q 11 (t), q2(t) = ρ0q 12 (t), 0 < ρ0 < 1; S424. Set the control target of the dynamics subsystem as follows: Wherein, a1 and a2 are both small positive integers; S425. Define the integral sliding film surface as follows: Among them, b iu >0 and b ir >0, take the derivative of the above formula and we get: S426. According to step S425, the following is obtained: Among them, ρ iu ,μ iu ,ρ ir ,μ ir Both represent constants greater than 0, μ iu ≥μ ud ,μ ir ≥μ rd And μ ud >0,μ rd >0; S427, due to q 1iu (t) and q 1ir (t) is unknown and time-varying, so an adaptive method is used to estimate its boundary. In order to prevent the singular problem when the estimated value tends to zero, q 1iu (t) and q 1ir (t) is estimated; the time-varying gain η is defined iu =1 / q 1iu (t) min and η ir =1 / q 1ir (t) min , where q 1iu (t) min and q 1ir (t) min q 1iu (t) and q 1ir (t), therefore, the USV formation tracking control law is designed as follows: Among them, γ1,γ2,c iu ,c ir ,ω iu ,ω ir ,ε iu ,ε ir and All represent constants greater than 0; S428. Based on the quantization of the control system, an event trigger mechanism with a time-varying threshold is introduced, which is defined as follows: Where m represents the time constant, Indicates angle tracking, satisfying S429, there exists a continuous time-varying coefficient θ i (t) satisfies, θ i (t k+1 )=±1,θ i (t k )=0, and |θ i (t)|≤1, we can get Right now The analysis is as follows: When θ i When (t) = 0, We get t=t k , When |θ i (t)|<1, Get t∈[t k ,t k+1 ); When |θ i When (t)|=1, The corresponding time at this time is t=t k+1 ,get For t∈[t k+1 ,t k+2 ] is the same as above, so at any moment it satisfies Among them, θ i (t) satisfies θ i (t k )=0,θ(t k+1 )=±1, and |θ i (t)|≤1; S4210, based on s iu , s ir , The kinematic and dynamic error systems are expressed as:

8. The method for collision and obstacle avoidance control of a multi-unmanned vessel formation with event triggering and signal quantization according to claim 1 is characterized in that: Step S5 specifically includes: S51, consider the USV formation tracking and collision avoidance control system with quantization, actuator fault tolerance and event triggering mechanism, combined with the designed ESO, kinematic guidance rate, dynamic control rate and adaptive law, with state s iu ,s ir , and input Ω iu ,Ω ir The formation tracking control system is stable in terms of input state, the tracking error can converge to a small residual set, and all signals in the designed control system are uniformly and ultimately bounded; S52. Define the Lyapunov function as follows: And take the derivative of the defined Lyapunov function: S53, combining step S426 and step S52 to obtain: S54. Combining the designed control law and adaptive law, we get: Among them, q 1i (t)>0 and S55. Due to but make So we get: S56, taking into account Then we get: S57, due to and Then we get: in, S58, due to Then we get: S59, Order Then we get: in, is a positive constant and satisfies S510, Definitions u =[s 1u ,s 2u ,...,s Nu ] T , s r =[s 1r ,s 2r ,...,s Nr ] T , S511, outside the collision avoidance zone therefore Then we get: S512, note Then we get: in, Therefore, the designed error subsystem with event triggering, actuator fault tolerance, and input and state quantization is input-to-state stable and satisfies: Among them, R c =diag{1,1 / 2γ1η 1u ,...,1 / 2c1h Nu ,1 / 2c2h 1r ,...,1 / 2c2h Nr }; S513: In the collision avoidance area, the following is obtained: S514、Notice Then we get: in, Therefore, the designed error subsystem with event triggering, actuator fault tolerance, and input and state quantization is input-to-state stable and satisfies: S515, due to s iu , s ir is bounded, so is bounded; according to step S414, the kinematic guidance signal u ic and r ic are bounded, so we can prove that u i and r i is bounded; from step S3, we can see that ESO is stable in the input state. According to the cascade system stability theory, combined with step S512, ||Y(t)|| is bounded and satisfies:

Images