Fixed-time formation control method for underactuated underwater vehicles based on event-triggered mechanism

Abstract

This paper proposes a fixed-time formation control method for underactuated underwater vehicles based on an event-triggered mechanism. A virtual AUV is designed to track a leader AUV that only provides position information. An adaptive speed regulation law is also designed for the virtual AUV, enabling the follower AUV to track a reference position via the virtual AUV without requiring the leader's speed information. Furthermore, an event-triggered mechanism is introduced to intelligently trigger the follower AUV's control input. This method designs an adaptive speed regulator for the virtual AUV, transforming the formation control problem into a trajectory generation and tracking problem. An event-triggered mechanism is added to the controller to reduce the number of executions and minimize actuator wear and resource waste. Theoretical analysis demonstrates that all tracking errors in the closed-loop system converge to the neighborhood of the origin within a fixed time. This method has the advantages of low information transmission, low computational complexity, and low energy consumption, making it valuable for implementation in marine engineering.

Description

Technical Field

[0001] The present invention relates to the technical field of underwater vehicles, and in particular to a fixed-time formation control method for under-actuated underwater vehicles based on an event triggering mechanism. Background Art

[0002] With the continuous advancement of marine technology worldwide, autonomous underwater vehicles (AUVs) have been applied in a variety of fields, including underwater target detection, deep-sea resource surveys, and seafloor mapping. AUVs offer advantages such as maneuverability and stealth. Due to the limited capabilities of individual AUVs, utilizing multiple AUVs in formation to perform missions is a recently developed technology that can significantly enhance the detection performance of multiple AUVs.

[0003] Several methods have been proposed to achieve the desired AUV formation, including behavior-based methods, topology-based methods, virtual structures, and leader-follower methods. The leader-follower method is widely cited due to its simplicity and reliability. Compared to fully actuated AUVs, underactuated AUVs have fewer actuators than degrees of freedom, posing a significant challenge to the design of formation controllers.

[0004] In the leader-follower formation method, the desired position of the follower AUV at a given moment is determined by the leader AUV's actual position at that moment. Because the follower AUV's actual position deviates from the desired position, the follower AUV must communicate with the leader AUV in real time and obtain its transmitted position and velocity information for corresponding calculations. However, due to the unique underwater environment, the transmission capacity and stability of underwater acoustic communication are poor, making it impossible to transmit excessive amounts of information in real time and stably. If formation coordination could be achieved solely through the transmission of position information, the amount of information exchange between AUVs could be effectively reduced, improving communication stability.

[0005] Several studies have investigated the formation control problem without velocity information, most of which focus on the field of UAV formations. Methods such as the consistency method, the virtual rigid body method, and the azimuth stiffness method aim to achieve UAV formation tasks with minimal communication. However, due to the under-actuated nature of AUVs, it is difficult to perform system linearization similar to UAVs, making methods from the UAV formation field difficult to simply transfer to the AUV field. Regarding the formation problem of under-actuated ocean vehicles (such as unmanned ships and underwater vehicles) without velocity information, some researchers have proposed methods such as using an integral state observer to observe the unknown velocity of the leader and using the virtual trajectory method to calculate the corresponding velocity information based on position information to achieve velocity-information-free formation control. However, these studies often focus on formation problems within a two-dimensional plane, and the settling time of their controllers is often uncertain, which severely limits the control performance of AUVs and introduces instability to AUV formation operations. Furthermore, most of the controllers proposed in these studies require continuous signal transmission. This control method, which requires continuous sampling period information transmission, is called time-triggered control. However, when the sampling period is short, the information transmitted in adjacent sampling periods is often very close, which will cause the control output calculated by the AUV to be calculated with only slight changes, which is not conducive to the stable control of the under-actuated AUV. Summary of the Invention

[0006] Due to the limited underwater acoustic communication resources of AUV clusters under weak connectivity, and to address the problems existing in existing formation control methods without speed information, this paper proposes a fixed-time formation control method for underactuated underwater vehicles based on an event-triggered mechanism. A virtual AUV is designed to track a leader AUV that only provides position information, and an adaptive speed regulation law is designed for the virtual AUV, so that the follower AUV does not need the speed information of the leader AUV when tracking the reference position through the virtual AUV. In addition, an event-triggered mechanism is introduced to realize intelligent triggering of the follower AUV control input.

[0007] The technical solution of the present invention is:

[0008] The present invention proposes a fixed-time formation control method for underactuated underwater vehicles based on an event-triggered mechanism. The formation is a leader AUV-follower AUV formation, and the AUVs are underactuated AUVs. The leader AUV only provides its own position information to the follower AUV. The formation control method includes the following steps:

[0009] Step 1: The follower AUV obtains the position vector η of the leader AUV l =[x l ,y l , z l ,θ l ,ψ l ] T, using the fixed time adaptive speed regulation law, the position and attitude vector η of the virtual AUV is calculated v and velocity vector υ v ;

[0010] Step 2: The position and attitude vector η of the virtual AUV obtained from step 1 v Take the three-dimensional position information η 1v =[x v ,y v ,z v ] T and the velocity vector υ of the virtual AUV obtained from step 1 v Take the speed information υ 1v =[u v ,q v ,r v ] T According to η 1v and υ 1v , the expected speed υ of the follower AUV is calculated using the expected speed guidance law d ;

[0011] Step 3: υ obtained from step 2 d =[u d ,q d ,r d ] T , using the dynamic controller of the underactuated follower AUV:

[0012]

[0013] The control input provided by the underactuated AUV thruster is τ = [τ u ,τ q ,τ r ]; where m 11 、m 22 、m 33 、m 55 、m 66 is a parameter in the simplified AUV generalized mass matrix; d 11 d 22 d 33 d 55 d 66 is a parameter in the damping parameter matrix; α u ,β u ,μ u ,α q ,β q ,μ q ,α r ,β r ,μ r ,δ is a set constant;

[0014] Step 4: Construct the event triggering mechanism of the dynamics controller:

[0015]

[0016] And control output according to the event trigger mechanism; where μ j ,μ j,1 ,μ j,2 is a positive constant, j=u,q,r,μ j >μ j,1 +μ j,2 ; The event triggering error representing the event triggering mechanism; It represents the output after the event triggering mechanism, which is the control output of the follower AUV dynamic controller; tanh() is a hyperbolic function; μ j,1 ,μ j,2 Both represent the setting parameters in the event trigger mechanism; t j,k Indicates a triggering moment; t j,k+1 Indicates t j,k The next trigger moment.

[0017] Furthermore, in step 1, the fixed time adaptive speed regulation law is

[0018]

[0019] where η v =[x v ,y v ,z v ,θ v ,ψ v ] T is the position and attitude vector of the virtual AUV. During the initial calculation, the initial value of the position and attitude vector of the virtual AUV is set; J v Coordinate transformation matrix; sig(η e ) a =sig(η e )|η e | a , sig(η e ) b =sig(η e )|η e | b , superscripts a and b are set constants; K1 and K2 are set positive definite matrices; η e is the error vector between the position and attitude vector of the virtual AUV and the expected position and attitude vector of the follower AUV; σ is a set constant; υ U is the unknown constant that constrains the velocity vector of the navigator, It is Uestimated value of;

[0020] By calculating the velocity vector υ of the virtual AUV v The position and attitude vector of the virtual AUV in the next cycle is obtained by integration, and fixed-time adaptive speed regulation is achieved through iterative calculation.

[0021] Furthermore, according to the formula

[0022]

[0023] get where σ η is a set constant, k3 and k4 are set constants, Γ1 is a set constant, and the superscript n is a set constant. Furthermore, the expected position and attitude vector η of the follower AUV is r for

[0024] η r =η l + J(η l )l

[0025] in θ l , ψ l are the pitch and yaw angles of the leader AUV.

[0026] Furthermore, in step 2, the desired speed guidance law is

[0027]

[0028] in ε is a set constant; K3 and K4 are set positive definite matrices; χ is the trajectory tracking error expressed in the body coordinate system, Intermediate variable η re =η 1f -η 1v , J 1f is the coordinate transformation matrix; m and n are set constants; ρ=[ρ,0,0] T , set the parameter ρ>0; η g is the virtual guidance trajectory.

[0029] Furthermore, the virtual guidance trajectory η g =η 1v -J 1f ρ.

[0030] Furthermore, in step 4, when When satisfied, the actual control output of the follower AUV actuator Update to the control output τ required for the current calculation j (t j,k ),t∈[t j,k,t j,k+1 ), otherwise the actual control output Keep the original value.

[0031] Beneficial effects

[0032] The present invention proposes a fixed-time formation control method for underactuated underwater vehicles based on an event-triggered mechanism, which realizes formation control of a master-slave structure of multiple underactuated AUVs. This method designs an adaptive speed regulator for the virtual AUVs, transforming the formation control problem into a trajectory generation and tracking problem; and adds an event-triggered mechanism to the controller to reduce the number of executions and minimize actuator wear and resource waste. Theoretical analysis proves that all tracking errors in the closed-loop system can converge to the neighborhood of the origin within a fixed time. The algorithm has the advantages of small information transmission volume, low computational complexity, and low energy consumption. This makes the proposed method valuable for implementation in marine engineering.

[0033] Additional aspects and advantages of the present invention will be set forth in part in the description which follows and, in part, will be obvious from the description which follows, or may be learned by practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the following description of the embodiments with reference to the accompanying drawings, in which:

[0035] Figure 1 : AUV coordinate system definition;

[0036] Figure 2 : Leader-follower formation configuration;

[0037] Figure 3 : Formation controller structure diagram;

[0038] Figure 4 : The formation trajectory of the AUV formation in three-dimensional space;

[0039] Figure 5 : virtual AUV tracking error;

[0040] Figure 6 : Virtual AUV adaptive parameters;

[0041] Figure 7 : follower AUV position error;

[0042] Figure 8 : follower AUV speed error;

[0043] Figure 9 : τ of follower AUV1 u , τ v , τ rThe event trigger interval;

[0044] Figure 10 : τ of follower AUV2 u , τ v , τ r The event trigger interval;

[0045] Figure 11 : Compare with the triggering times of continuous time triggering;

[0046] Figure 12 : Comparison of AUV motion trajectories;

[0047] Figure 13 : Position error comparison;

[0048] Figure 14 : Speed ​​error comparison. DETAILED DESCRIPTION

[0049] The following describes in detail embodiments of the present invention. The embodiments are exemplary and intended to explain the present invention, but are not to be construed as limiting the present invention.

[0050] Since the communication resources and computing power of AUVs are very limited, in order to ensure the operational stability of AUVs as much as possible, it is necessary to fully utilize their limited communication resources and computing resources. To this end, this embodiment proposes an event-triggered fixed-time leader-follower formation control method for multiple underactuated AUVs. Specifically, for the three-dimensional formation control problem of underactuated AUVs without speed information transmission, a virtual reference AUV with a fixed-time adaptive speed regulation law is designed, so that the follower AUV does not need the speed information of the leader AUV when tracking the reference position through the virtual AUV within a fixed time; and based on the virtual AUV, a fixed-time desired speed guidance law is designed, which calculates the speed information of the follower AUV reaching the desired position within a fixed time based on the position information provided by the leader AUV and the speed information provided by the virtual AUV; finally, based on the desired speed guidance law, a fixed-time event-triggered dynamic controller is designed to enable the follower AUV to reach the desired speed within a fixed time. At the same time, an event triggering mechanism is added to the designed dynamic controller, thereby reducing the system's computational workload and the frequency of data transmission between actuators, thereby fully utilizing communication resources.

[0051] (1) AUV model

[0052] The coordinate system and AUV model in this embodiment are as follows Figure 1 As shown. Figure 1 In the equation, O-xyz is the global coordinate system, and its origin is fixed at a fixed point on the water surface. B -x B y B zB It is the body coordinate system, whose coordinate origin coincides with the center of mass of the AUV.

[0053] In the global coordinate system, use the vector η1 = [x, y, z] T represents the position of the AUV, η2 = [φ, θ, ψ] T Represents the attitude angle of the AUV in the global coordinate system. Where x, y, z are the positions of the AUV, φ, θ, ψ are the azimuth angles of the AUV in the global coordinate system; vector υ1 = [u, v, w] T and υ2=[p,q,r] T Respectively represent the linear velocity and angular velocity of the AUV in the body coordinate system, where u, v, w, p, q, r represent the forward, roll, heave, roll, pitch, and yaw velocities in the body coordinate system. The state of the AUV is represented by (1) and (2):

[0054]

[0055] Where: J1 and J2 are the reversible coordinate transformation matrices of υ1 and υ2 from the body coordinate system to the global coordinate system.

[0056] The dynamic equation of AUV is as follows:

[0057]

[0058] Where: M is the generalized mass matrix of the AUV; C(υ) is the Coriolis centripetal force matrix; D(υ) is the damping parameter matrix; g(η) is the restoring force matrix. η = [η1; η2] is the position state and attitude state of the AUV; υ = [υ1; υ2] is the velocity state vector of the AUV. τ = [τ u ,τ q ,τ r ] is the control input provided by the underactuated AUV thruster, τ u ,τ q ,τ r Represent the control inputs in the three speed directions of u, q and r respectively.

[0059] In order to facilitate the design of the controller, the AUV kinematic model is simplified and the following assumptions are defined:

[0060] 1) AUV about O B -x B y B and O B -y B z B The plane is symmetrical

[0061] 2) The origin of the AUV body coordinate system coincides with the center of gravity of the AUV;

[0062] 3) This paper ignores the rolling motion and high-order hydrodynamic damping of the AUV;

[0063] The kinematic and dynamic equations of the AUV can be described as follows:

[0064]

[0065] Where m 11 、m 22 、m 33 、m 55 、m 66 is a parameter in the simplified AUV generalized mass matrix; d 11 d 22 d 33 d 55 d 66 is a parameter in the damping parameter matrix. 11 =X u +X u|u| |u|,d 22 =Y v +Y v|v| |v|,d 33 =Z w +Z w|w| |w|,d 55 =M q +M q|q| |q|,d 66 =N r +N r|r| |r|;X u 、Y v 、Z w 、M q 、N r are the linear hydrodynamic damping coefficients of u, v, w, q, and r, respectively. Xu|u|, Yv|v|, Zw|w|, Mq|q||, Nr|r| are the nonlinear hydrodynamic damping coefficients of u, v, w, q, and r, respectively; GZ = ρgΔGM z sinθ is the recovery force term, ρ is the density of the liquid, g is the acceleration due to gravity, Δ is the volume of liquid discharged by the AUV, GM z is the distance between the AUU's center of buoyancy and center of gravity.

[0066] (2) Formation Controller Design

[0067] For the leader AUV, the desired trajectory is a time-varying trajectory that contains position and velocity information corresponding to time. For the follower AUV, its desired position is determined by the actual position of the leader AUV at the current moment. When the actual position of the follower AUV deviates from the desired position, it is necessary to communicate with the leader AUV and pass the position and velocity information to the follower AUV. However, the transmission capability and stability of underwater acoustic communication are weak. Only sending position information for formation collaboration can effectively reduce information exchange and improve communication stability. In order to realize the formation of the leader-follower formation, the controller is divided into a leader AUV-virtual AUV structure and a virtual AUV-follower AUV structure.

[0068] like Figure 3 As shown in the figure, the formation controller consists of three parts: a fixed-time adaptive speed regulation law module, a fixed-time desired speed guidance law module, and a fixed-time event-triggered dynamics controller.

[0069] The adaptive speed regulation law module calculates the virtual AUV based on the position information of the leader AUV, which provides reference speed information for the follower AUV. This eliminates the need for the leader AUV to directly provide speed information to the follower AUV, thus saving communication resources. The fixed-time desired speed guidance law module calculates the speed information required for the follower AUV to reach the desired position within a fixed time based on the position information provided by the leader AUV and the speed information provided by the virtual AUV. The fixed-time event-triggered dynamic controller provides the follower AUV's actuator with the power required to reach the desired speed within a fixed time, and uses an event-triggered mechanism to reduce information transmission.

[0070] (3) Formation structure and fixed-time adaptive speed regulation law

[0071] like Figure 2 As shown, in this embodiment, a master-slave formation method is adopted. The formation includes three AUVs, divided into one leader AUV and two follower AUVs. The leader AUV only needs to navigate normally, while the follower AUVs need to maintain a certain distance from the leader AUV to form the required formation.

[0072] Definition η l =[x l ,y l , z l ,θ l ,ψ l ] T Vector is the position and attitude vector of the leader AUV, which is a known quantity obtained through communication; η r =[x r ,y r , z r ,θ r ,ψ r] T is the desired position and attitude vector of the follower AUV;

[0073] is the desired relative position vector of the leader AUV and the follower AUV, ρ d ,λ d are the expected distance and the expected angle respectively, which are also known quantities that can be obtained through communication.

[0074] η r =η l + J(η l )l

[0075] in θ l , ψ l are the pitch and yaw angles of the leader AUV, so that the desired position of the follower AUV can be calculated.

[0076] In order to make the virtual AUV quickly track the desired position, an adaptive speed control law is proposed to adjust the speed of the virtual AUV.

[0077] Define the error vector η between the virtual AUV and the desired position and posture vector e =[e1,e2,e3,e4,e5] T , e1, e2, e3, e4, e5 are the errors between the virtual AUV and the expected position in x, y, z, θ, ψ variables respectively.

[0078] η e =η r -η v (6)

[0079] where η v =[x v ,y v ,z v ,θ v ,ψ v ] T is the position and attitude vector of the virtual AUV.

[0080] In order to ensure that the virtual AUV can track the desired position within a fixed time, the adaptive speed control law is designed as follows:

[0081]

[0082] Where K1 and K2 are set positive definite matrices. e ) a =sig(η e )|η e | a , sig(η e )b =sig(η e )|η e | b , superscripts a and b are set constants; It is U The estimated value of the navigator's velocity vector υ l Affected by the unknown constant υ U Constraints; define the error vector υ v is the velocity vector of the virtual AUV; σ is a set constant; J v is the coordinate transformation matrix;

[0083] for Design the following adaptive function:

[0084]

[0085] Among them, σ η is a set constant, k3 and k4 are set constants, Γ1 is a set constant, and the superscript n is a set constant.

[0086] In the fixed time adaptive speed regulation law, based on the initial setting value of the virtual AUV position attitude vector, the error between the virtual AUV and the expected position is calculated, and then the unknown constant v is calculated according to formula (8): U The estimated value of the virtual AUV is obtained by using the adaptive speed regulation formula (7). The virtual AUV position and attitude vector of the next cycle is obtained by integrating the virtual AUV speed vector. The fixed-time adaptive speed regulation is realized by iterative calculation.

[0087] For the above adaptive speed regulation law, the stability proof is given below. The selected Lyapunov function is as follows:

[0088]

[0089] Taking the time derivative of V1:

[0090] Expressed as:

[0091]

[0092]

[0093] Expressed as:

[0094]

[0095] According to (10) and (11), we can get:

[0096]

[0097] as well as

[0098]

[0099] V1 is rewritten as

[0100]

[0101] Rewrite as

[0102]

[0103] Therefore, it is possible to obtain

[0104]

[0105] In the above formula It can be seen that the designed adaptive speed regulation law is actually fixed-time stable.

[0106] (4) Fixed-time desired speed guidance law

[0107] In the fixed-time adaptive speed regulation law, the speed vector υ of the virtual AUV is v The design of includes the coordinate system transformation matrix, so in addition to x v ,y v ,z v We also need to use the posture variable θ v ,ψ v , so η v And the subsequent variables are five-dimensional variables. However, since the position and velocity information of the virtual AUV have been obtained, based on the properties of AUV under-actuation and backstepping, only the position information η of the virtual AUV is needed when designing the desired velocity guidance law. 1v =[x v ,y v ,z v ] T and speed information υ 1v =[u v ,q v ,r v ] T , so the variables in the expected velocity guidance law are three-dimensional.

[0108] Virtual guidance trajectory η g The definition is as follows:

[0109] η g =η 1v -J 1f ρ (17)

[0110] Where ρ = [ρ, 0, 0] T , where the parameter ρ>0, J 1f is the coordinate transformation matrix:

[0111]

[0112] where θ f , ψ f are the pitch and yaw angles of the follower AUV.

[0113] The error between the follower AUV and the virtual AUV is defined as follows:

[0114] η ge =η 1f -η g (18)

[0115] where η 1f =[x f ,y f ,z f ] T is the actual position of the follower AUV; the error η ge Expressed in the body coordinate system, it is convenient to derive the controller:

[0116]

[0117] definition

[0118] χ=e g -ρ (20)

[0119] Therefore, it satisfies Intermediate variable η re =η 1f -η 1v ; χ is the trajectory tracking error expressed in the body coordinate system. If χ = 0, then η re = 0. Therefore, the problem can be transformed into how to stabilize χ.

[0120] The time derivative of χ is

[0121]

[0122] in ε is a set constant; r f and q f are the actual yaw and pitch speeds of the follower AUV, and the expected speed of the follower AUV is υ d =[u d ,q d ,r d ] T , the velocity error is defined as υ e =υf -υ d Formula (21) can be rewritten as

[0123]

[0124] In order to achieve a fixed convergence time, the desired velocity guidance law is designed as follows:

[0125]

[0126] Where K3 and K4 are assumed positive definite matrices, υ 1v =[u v ,q v ,r v ] T ; m and n are set constants; υ e and χ can converge to a small bounded field in constant time.

[0127] For the above-mentioned desired velocity guidance law, the stability proof is given below. The selected Lyapunov function is as follows:

[0128] V2=χ T χ (24)

[0129] Take the time derivative of V2:

[0130]

[0131] consider The desired velocity guidance law is rewritten as follows:

[0132]

[0133] Then we get the following inequality:

[0134]

[0135] Applying (27) and (28) to equation (26) yields:

[0136]

[0137] (5) Fixed-time event-triggered dynamics controller

[0138] According to the fixed time desired speed guidance law, if υ e =[u e ,q e ,r e ] TIf it converges to 0 within a fixed time, then χ is stable within a fixed time. In order to ensure the convergence of the velocity error, the forward dynamics controller τ of the underactuated follower AUV is set based on the dynamic model and the fixed time convergence theory. u Designed for

[0139]

[0140] Based on the dynamic model and fixed time convergence theory, the pitch dynamic controller τ of the underactuated follower AUV is q Designed for

[0141]

[0142] Based on the dynamic model and fixed-time convergence theory, the yaw dynamics controller τ of the underactuated follower AUV is r Designed for

[0143]

[0144] In the above formula, α u ,β u ,μ u ,α q ,β q ,μ q ,α r ,β r ,μ r ,δ is a set constant.

[0145] And construct the event triggering mechanism of the dynamics controller as

[0146]

[0147] where μ j ,μ j,1 ,μ j,2 (j=u,q,r) is a positive constant, μ j >μ j,1 +μ j,2 ; The event triggering error representing the event triggering mechanism; It represents the output after the event triggering mechanism, which is the control output of the follower AUV dynamic controller; tanh() is a hyperbolic function; μ j,1 ,μ j,2 Both represent the setting parameters in the event trigger mechanism; t j,k Indicates a triggering moment; t j,k+1 Indicates t j,k The next trigger moment; when When satisfied, the actual control output of the follower AUV actuator Update to the control output τ required for the current calculation j (t j,k ),t∈[t j,k ,t j,k+1 ), otherwise the actual control output Keep the original value.

[0148] Considering the event trigger mechanism, there is the following formula

[0149]

[0150] can be rewritten as

[0151]

[0152] In order to prove that the velocity error of the AUV can converge to the minimum neighborhood of zero under the above control input, the following Lyapunov equation can be designed:

[0153]

[0154] Taking the derivative of equation (36) we have

[0155]

[0156] In the above formula, α=min{α u ,α q ,α r},β=min{β u ,β q ,β r},Δ2=3ιδ。

[0157] (6) Overall stability analysis

[0158] In order to prove that all signals in the closed-loop system satisfy practical fixed-time stability, define the overall Lyapunov function V s =V1+V2+V3, take V s The derivative of is:

[0159]

[0160] make

[0161]

[0162] Δ3=Δ1+Δ2

[0163] It can be seen that the error signal in a closed-loop system such as is actually fixed-time stable, and all error signals converge to the residual set

[0164]

[0165] To avoid the Zeno behavior representing an infinite number of triggering moments in a finite time, we prove that there exists satisfy The minimum time interval between events. We can obtain

[0166]

[0167] because is the interval t∈[t j,k ,t j,k+1 ), equation (39) can be rewritten as

[0168]

[0169] This means It is bounded. It can be expressed as

[0170]

[0171] ζ j (j=u,q,r) is a positive constant. Then we can get

[0172]

[0173] Right now

[0174]

[0175] This proves that Zeno behavior is impossible.

[0176] (VII) Simulation Verification

[0177] To verify the effectiveness of the designed underactuated AUV formation control strategy, we conducted a numerical simulation of the trajectory tracking control of underactuated underwater vehicles in a three-dimensional environment. The AUV model parameters are shown in Table 1.

[0178] Table 1 AUV model parameters

[0179]

[0180] To illustrate the stability of the closed-loop system, consider a leader AUV and two follower AUVs to build the required formation. The leader-follower formation configuration parameters are chosen as: d1 =ρ d2 =3m,λ d1 =λ d2 =π / 2.

[0181] The initial conditions of the pilot AUV are set as

[0182]

[0183] When t≤30s, the forward speed of the leader AUV is set to u l =0.5m / s, the yaw angular velocity of the leader AUV is set to r l =0rad / s. When t>30s, the forward speed of the leader AUV is set to u l =0.5m / s, the yaw angular velocity of the leader AUV is set to r l =π / 160rad / s.

[0184] The initial conditions of the virtual AUV are set as

[0185]

[0186] The initial condition of the follower AUV is set to

[0187]

[0188] The main parameters of the proposed controller are shown in Table 2. Simulation duration: 300 seconds; simulation step: 0.001 seconds. The control input signal is saturated in the range [-100, 100] N (Nm).

[0189] Table 2 Design parameters of controller

[0190]

[0191] The simulation results are as follows Figure 4-Figure 8 As shown. Figure 4 As shown in Figure 2, the AUV formation tracking results show that under the algorithm proposed in this invention, two follower AUVs starting from random positions can track the reference path and build the required formation. Figure 5 In , we can easily verify that the error between the virtual AUV and the reference path can converge to a small neighborhood of the origin. The adaptive parameters of the virtual AUV speed are as follows Figure 6 As shown, Figure 7 The position error of the follower AUV under different initial conditions is shown. It can be seen that the error state of the follower AUV can converge to the origin of a small neighborhood within a fixed time. The above simulation experimental results show that the fixed-time formation controller designed by this invention can achieve a formation that meets the performance requirements without the leader AUV's speed information.

[0192] The transient and steady-state responses of the proposed controller both achieve satisfactory closed-loop performance. Figures 9-11 The trigger interval and frequency of the control signal are displayed, which avoids the continuous execution of the actuator and reduces energy consumption.u ,τ q ,τ r The trigger frequency is less than the number of consecutive triggers.

[0193] In this embodiment, the designed tracking controller is compared with the robust fixed-time tracking controller designed in the reference [S.An, L.Wang, and Y.He, “Robust fixed-time tracking control for underactuated AUVs based on fixed-time disturbance observer,” Ocean Eng., vol. 266, p. 112567, Dec. 2022, doi: 10.1016 / j.oceaneng.2022.112567.]. The initial position condition of the AUV is set to The initial velocity condition of the AUV is set to The AUV model parameters are shown in Table 1. The controller parameters used in this paper are listed in Table 2. The parameters of the robust fixed-time tracking controller were selected based on references. The desired trajectory was chosen to have a simulation duration of 150 seconds and a simulation step of 0.001 seconds. The control input signal was saturated within the range [-100, 100] N (N·m).

[0194] Figure 12-14 The comparison results between the controller proposed in this paper and the controller designed in the reference are shown in Figure 2. Figure 12 As shown. Figure 13 As shown in Figure 2, under both controllers, the AUV can track the desired time-varying trajectory. However, the controller designed by the present invention has a faster convergence speed. Figure 14 It can be observed that the overall velocity error of the underactuated AUV using the controller designed by the present invention decreases to the minimum neighborhood of zero more quickly.

[0195] Although the embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention without departing from the principles and purpose of the present invention.

Claims

1. A fixed-time formation control method for underactuated underwater vehicles based on an event-triggered mechanism, characterized by: The formation is a leader AUV-follower AUV formation, and the AUVs are underactuated AUVs, and the leader AUV only provides its own position information to the follower AUV; the formation control method includes the following steps: Step 1: The follower AUV obtains the position vector η of the leader AUV l =[x l ,y l , z l ,θ l ,ψ l ] T , using the fixed time adaptive speed regulation law, the position and attitude vector η of the virtual AUV is calculated v and velocity vector υ v ; Step 2: The position and attitude vector η of the virtual AUV obtained from step 1 v Take the three-dimensional position information η 1v =[x v ,y v ,z v ] T and the velocity vector υ of the virtual AUV obtained from step 1 v Take the speed information υ 1v =[u v ,q v ,r v ] T According to η 1v and υ 1v , the expected speed υ of the follower AUV is calculated using the expected speed guidance law d ; Step 3: υ obtained from step 2 d =[u d ,q d ,r d ] T , using the dynamic controller of the underactuated follower AUV: The control input provided by the underactuated AUV thruster is τ = [τ u ,τ q ,τ r ]; where m 11 、m 22 、m 33 、m 55 、m 66 is a parameter in the simplified AUV generalized mass matrix; d 11 d 22 d 33 d 55 d 66 is a parameter in the damping parameter matrix; α u ,β u ,μ u ,α q ,β q ,μ q ,α r ,β r ,μ r ,δ is a set constant; Step 4: Construct the event triggering mechanism of the dynamics controller: And control output according to the event trigger mechanism; where μ j ,μ j,1 ,μ j,2 is a positive constant, j=u,q,r,μ j >μ j,1 +μ j,2 ; The event triggering error representing the event triggering mechanism; It represents the output after the event triggering mechanism, which is the control output of the follower AUV dynamic controller; tanh() is a hyperbolic function; μ j,1 ,μ j,2 Both represent the setting parameters in the event trigger mechanism; t j,k Indicates a triggering moment; t j,k+1 Indicates t j,k The next trigger moment.

2. The method for controlling a fixed-time formation of underactuated underwater vehicles based on an event-triggered mechanism according to claim 1, characterized in that: In step 1, the fixed time adaptive speed regulation law is where η v =[x v ,y v ,z v ,θ v ,ψ v ] T is the position and attitude vector of the virtual AUV. During the initial calculation, the initial value of the position and attitude vector of the virtual AUV is set; J v Coordinate transformation matrix; sig(η e ) a =sig(η e )|η e | a , sig(η e ) b =sig(η e )|η e | b , superscripts a and b are set constants; K1 and K2 are set positive definite matrices; η e is the error vector between the position and attitude vector of the virtual AUV and the expected position and attitude vector of the follower AUV; σ is a set constant; υ U is the unknown constant that constrains the velocity vector of the navigator, It is U estimated value of; By calculating the velocity vector υ of the virtual AUV v The position and attitude vector of the virtual AUV in the next cycle is obtained by integration, and fixed-time adaptive speed regulation is achieved through iterative calculation.

3. The method for controlling a fixed-time formation of underactuated underwater vehicles based on an event-triggered mechanism according to claim 2, characterized in that: According to the formula get where σ η is a set constant, k3 and k4 are set constants, Γ1 is a set constant, and the superscript n is a set constant.

4. The method for controlling a fixed-time formation of underactuated underwater vehicles based on an event-triggered mechanism according to claim 2, characterized in that: The expected position and attitude vector η of the follower AUV r for or r =the l +J(h l )l in θ l , ψ l are the pitch and yaw angles of the leader AUV.

5. The method for controlling a fixed-time formation of underactuated underwater vehicles based on an event-triggered mechanism according to claim 1, characterized in that: In step 2, the desired speed guidance law is in ε is a set constant; K3 and K4 are set positive definite matrices; χ is the trajectory tracking error expressed in the body coordinate system, Intermediate variable η re =η 1f -η 1v , J 1f is the coordinate transformation matrix; m and n are set constants; ρ=[ρ,0,0] T , set the parameter ρ>0; η g is the virtual guidance trajectory.

6. The method for controlling a fixed-time formation of underactuated underwater vehicles based on an event-triggered mechanism according to claim 5, characterized in that: Virtual guidance trajectory η g =η 1v -J 1f ρ.

7. The method for controlling a fixed-time formation of underactuated underwater vehicles based on an event-triggered mechanism according to claim 1, characterized in that: In step 4, when When satisfied, the actual control output of the follower AUV actuator Update to the control output τ required for the current calculation j (t j,k ),t∈[t j,k ,t j,k+1 ), otherwise the actual control output Keep the original value.

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