A multi-unmanned ship fixed time distributed formation control method based on backstepping technique

Abstract

The present application relates to a kind of multi-unmanned ship fixed time distributed formation control method based on backstepping technique, establishes the dynamic model of unmanned ship movement, and establishes the corresponding fixed time disturbance observer of each follower unmanned ship of pilot unmanned ship, for carrying out online observation and compensation;Further, the backstepping control method enabling multi-unmanned ship to form stable formation in fixed time is constructed, the controller of tracking sub-level system corresponding pilot unmanned ship is constructed by backstepping technique, for controlling pilot unmanned ship to track virtual pilot unmanned ship, and the controller of formation sub-level system corresponding each follower unmanned ship is constructed, for controlling each follower unmanned ship formation to track pilot unmanned ship, the two controllers designed have good convergence, can make the whole formation system stable and reach the expected performance requirement;And preset performance control technique is introduced in design, ensure that formation system still keeps good relative position deviation between unmanned ship under the condition of existing external interference, makes the overall formation configuration stable.

Description

TECHNICAL FIELD

[0001] The application relates to a multi-unmanned ship fixed-time distributed formation control method based on a backstepping technique and belongs to the technical field of mobile robot formation cooperative control. BACKGROUND

[0002] The ocean power strategy is a major strategy for realizing the great rejuvenation of the Chinese nation, and in order to meet the national ocean strategy demand, the application demand of unmanned ships under various new situations is increasing. With the rapid development of unmanned ship control technology in the control field, single unmanned ship control technology is increasingly perfect, and the application is also relatively wide, which can replace human beings to perform dangerous, boring and complex work in complex water conditions and harsh environments, such as coastal cruising, strategic reconnaissance, water mine clearance, sea rescue, material transportation, sea island surveying and mapping and water quality monitoring. Due to the complexity of the water surface environment and the difficulty of the operation task, a single unmanned ship is difficult to perform complex operation tasks, and multiple unmanned ships can make up for the defects of large operation error and relatively low task completion quality of a single unmanned ship, and the operation area coverage range of multiple unmanned ship formation is large, the operation space is wide, and the execution efficiency is high. In the face of complex water conditions, the control operability and operation effectiveness of a single unmanned ship are not satisfactory, and in order to make up for the performance disadvantage of a single unmanned ship, the research on the multiple unmanned ship formation control system with more operation space and higher operation efficiency is imminent and has more research value.

[0003] There are still some technical problems in the traditional multi-unmanned ship formation control method. In the formation process of the multi-unmanned ship system, various uncertain disturbances cannot be avoided, for example, the collision of waves on the ship body, the frequent change of the navigation water level, the change of the weather and climate and the like will deteriorate the formation effect, and even cause the formation system to be unable to normally operate, thereby bringing many challenges to the stability of the multi-unmanned ship formation system. In fact, these dynamic uncertain disturbances of the unmanned ship are difficult to be directly measured by the sensor. SUMMARY

[0004] The technical problem to be solved by the application is to provide a multi-unmanned ship fixed-time distributed formation control method based on a backstepping technique, which has good convergence and can make the whole formation system stable and meet the expected performance requirement.

[0005] The application adopts the following technical scheme to solve the above technical problem: the application designs a multi-unmanned ship fixed-time distributed formation control method based on a backstepping technique, based on a preset virtual leader unmanned ship expected trajectory, and the following steps are executed in real time, so as to realize the cooperative control of the distributed formation of the leader unmanned ship and each follower unmanned ship.

[0006] Step A. Establish the corresponding motion dynamics model of the virtual leader unmanned ship, the leader unmanned ship and each follower unmanned ship in the inertial coordinate system, and then enter step B;

[0007] Step B. According to the motion dynamics model of the virtual leader unmanned ship, the leader unmanned ship and each follower unmanned ship in the inertial coordinate system, establish the corresponding fixed-time disturbance observer of the leader unmanned ship and each follower unmanned ship, which is used for online observation and compensation for the poor system stability caused by the uncertain dynamic disturbance of the unmanned ship in the formation process, and then enter step C;

[0008] Step C. Based on the tracking sub-system corresponding to the virtual leader unmanned ship and the leader unmanned ship, according to the preset motion dynamics model of the virtual leader unmanned ship and the leader unmanned ship in the inertial coordinate system, construct the controller of the tracking sub-system corresponding to the leader unmanned ship, which is used for controlling the leader unmanned ship to track the virtual leader unmanned ship, and then enter step D;

[0009] Step D. Based on the formation sub-system corresponding to the leader unmanned ship and each follower unmanned ship, according to the motion dynamics model of the leader unmanned ship and each follower unmanned ship in the inertial coordinate system, construct the controller of the formation sub-system corresponding to each follower unmanned ship, which is used for controlling each follower unmanned ship to form a stable formation.

[0010] The multi-unmanned ship fixed-time distributed formation control method based on backstepping technology has the following technical effects compared with the prior art:

[0011] The multi-unmanned ship fixed-time distributed formation control method based on backstepping technology is designed, the motion dynamics model of the unmanned ship is established, and the corresponding fixed-time disturbance observer of the leader unmanned ship and each follower unmanned ship is established for online observation and compensation. Further, the backstepping control method capable of forming a stable formation of the multi-unmanned ship in a fixed time is constructed, the controller of the tracking sub-system corresponding to the leader unmanned ship is constructed by backstepping technology, which is used for controlling the leader unmanned ship to track the virtual leader unmanned ship, and the controller of the formation sub-system corresponding to each follower unmanned ship is constructed, which is used for controlling each follower unmanned ship to form a stable formation. The two controllers have good convergence, can make the whole formation system stable and meet the expected performance requirements. The preset performance control technology is introduced in the design to ensure that the unmanned ships still maintain good relative position deviation and the overall formation configuration is stable in the presence of external disturbance. BRIEF DESCRIPTION OF DRAWINGS

[0012] Figure 1is a flow chart of a multi-unmanned ship fixed time distributed formation control method based on backstepping technology designed by the application;

[0013] Figure 2 is a multi-unmanned ship distributed formation motion trajectory diagram in an inertial coordinate system;

[0014] Figure 3 is a position state output curve diagram of each unmanned ship;

[0015] Figure 4 is a velocity state output curve diagram of each unmanned ship;

[0016] Figure 5 is a forward speed and yaw angle speed disturbance estimation diagram of the leader unmanned ship;

[0017] Figure 6 is a forward speed and yaw angle speed disturbance estimation diagram of the follower unmanned ship 1;

[0018] Figure 7 is a forward speed and yaw angle speed disturbance estimation diagram of the follower unmanned ship 2;

[0019] Figure 8 is a formation tracking error diagram of the follower unmanned ship 2 in the X-axis direction;

[0020] Figure 9 is a formation tracking error diagram of the follower unmanned ship 2 in the Y-axis direction;

[0021] Figure 10 is a heading angle formation tracking error diagram of the follower unmanned ship 2. DETAILED DESCRIPTION

[0022] The specific embodiments of the application will be further described in detail below in conjunction with the accompanying drawings of the specification.

[0023] The application designs a multi-unmanned ship fixed time distributed formation control method based on backstepping technology, which is based on a preset virtual leader unmanned ship expected trajectory, as shown in Figure 1 The following steps are executed in real time to realize the cooperative control of the distributed formation containing one leader unmanned ship and each follower unmanned ship.

[0024] Step A. For the virtual leader unmanned ship, the leader unmanned ship, and each follower unmanned ship, the corresponding motion dynamics models in the inertial coordinate system are established, and then step B is entered.

[0025] In the specific design and implementation of the above step A, for the leader unmanned ship and each follower unmanned ship, the corresponding motion dynamics models are established by the following operations.

[0026] Based on the number of unmanned ships in the distributed formation n+1, and 0≤i≤n.

[0027] When 1≤i≤n corresponds to each follower unmanned ship, first for each follower unmanned ship, the kinematics model of the follower unmanned ship in the inertial coordinate system is established as follows:

[0028]

[0029] η i =[x i ,y i ,ψ i ] T ,η i represents the position vector of the ith follower unmanned ship in the inertial coordinate system, (x i , y i ) represents the position coordinates of the ith follower unmanned ship in the inertial coordinate system, ψ i ∈[0,2π] represents the angle between the ith follower unmanned ship in the inertial coordinate system and the X axis, that is, the heading angle of the ith follower unmanned ship in the inertial coordinate system; v i =[u i ,v i ,r i ] T , v i represents the velocity vector of the ith follower unmanned ship in the body coordinate system, u i , v i , r i represent the forward speed, lateral speed, and yaw rate of the ith follower unmanned ship in the body coordinate system, respectively; R(ψ i ) represents the transformation matrix between the ith follower unmanned ship in the inertial coordinate system and the body coordinate system, and R T (ψ i )R(ψ i )=I, I represents the unit matrix, ||R(ψ i )||=1, ||R(ψ i )|| represents the norm of the transformation matrix R(ψ i ), R T (ψ i )S(r i )R(ψ i )=R(ψ i )S(r i )R T (ψ i )=S(r i ), S(r i ) represents an anti-symmetric matrix,

[0030] Then, the dynamics model of each follower USV in the inertial coordinate system is established as follows:

[0031]

[0032] M i represents the inertial mass matrix of the i-th follower USV in the inertial coordinate system, m i,22 represents the element in the second row and the second column of M i C i represents the Coriolis force matrix of the i-th follower USV in the inertial coordinate system. The Coriolis force matrix is a fictitious moment introduced in the equation of motion of a body moving in a rotating reference frame to simplify the treatment of the rotating frame. D i represents the hydrodynamic damping parameter of the i-th follower USV in the inertial coordinate system, i.e., the hydrodynamic damping parameter matrix of the i-th follower USV in the inertial coordinate system, D i,11 , D i,22 , D i,23 , D i,33 , D i,32 all represent physical parameters of the USV and are greater than 0; τ i = [τ i,1 0τ i,3 ] T , τ i represents the control input matrix of the i-th follower USV, τ i,1 represents the forward control quantity of the i-th follower USV, τ i,3 represents the turning control quantity of the i-th follower USV; d i = [d i,u d i,v d i,r ] T , d i represents the lumped disturbance matrix received by the i-th follower USV during movement, which mainly comes from water waves, wind, parameter uncertainty, etc., d i,u represents the disturbance received by the i-th follower USV in the body coordinate system during forward movement, d i,v represents the disturbance received by the i-th follower USV in the body coordinate system during lateral movement, d i,r represents the disturbance received by the i-th follower USV in the body coordinate system during yaw angle movement.

[0033] ​By the dynamic control of the forward displacement, lateral displacement and yaw angle of the unmanned ship, the unmanned ship can be driven to follow the expected movement. This simplified 3-DOF unmanned ship model not only can achieve the ideal control effect, but also can simplify the control difficulty of the multi-unmanned ship system and reduce the control cost of the multi-unmanned ship formation system, and has certain practical significance.

[0034] That is, based on the follower unmanned ship kinematics model and the follower unmanned ship dynamics model in the inertial coordinate system, the corresponding motion dynamics model of the follower unmanned ship is constituted.

[0035] Then when i = 0 corresponds to the leader unmanned ship, the corresponding motion dynamics model of the leader unmanned ship in the inertial coordinate system in the tracking sub-level system is established as follows:

[0036]

[0037] η0=[x0,y0,ψ0] T , η0 represents the position vector of the leader unmanned ship in the inertial coordinate system, (x0, y0) represents the position coordinates of the leader unmanned ship in the inertial coordinate system, and ψ0∈[0, 2π] represents the angle between the leader unmanned ship in the inertial coordinate system and the X axis, that is, the heading angle of the leader unmanned ship in the inertial coordinate system; v0=[u0,v0,r0] T , v0 represents the velocity vector of the leader unmanned ship in the body coordinate system, u0, v0 and r0 respectively represent the forward velocity, lateral velocity and yaw angular velocity of the leader unmanned ship in the body coordinate system; M0 represents the inertial mass matrix of the leader unmanned ship in the inertial coordinate system; R0 represents the transformation matrix between the inertial coordinate system and the body coordinate system; τ0=[τ 0,1 0 τ 0,3 ] T , τ0 represents the control input matrix of the leader unmanned ship, τ 0,1 represents the forward control amount of the leader unmanned ship, and τ 0,3 represents the steering control amount of the leader unmanned ship; d0=[d 0,u d 0,v d 0,r ] T , d0 represents the lumped disturbance matrix of the leader unmanned ship in the motion process, d 0,u represents the disturbance of the leader unmanned ship in the body coordinate system corresponding to the forward motion process, d 0,v represents the disturbance of the leader unmanned ship in the body coordinate system corresponding to the lateral motion process, and d 0,r represents the disturbance of the leader unmanned ship in the body coordinate system corresponding to the yaw angle motion process.

[0038] Further, the preset virtual leader unmanned ship corresponding motion dynamics model is established by the following operation.

[0039] The motion dynamics model corresponding to the virtual leader unmanned ship expected trajectory (η d ,v d ) in the inertial coordinate system of the tracking sub-level system is established as follows.

[0040]

[0041] η d =[x d ,y d ,ψ d ] T ,η d represents the position vector of the virtual leader unmanned ship expected trajectory in the inertial coordinate system, (x d ,y d ) represents the position coordinates of the virtual leader unmanned ship expected trajectory in the inertial coordinate system, ψ d ∈[0,2π] represents the angle between the virtual leader unmanned ship expected trajectory in the inertial coordinate system and the X axis, that is, the heading angle of the virtual leader unmanned ship expected trajectory in the inertial coordinate system; v d =[u d ,v d ,r d ] T , v d represents the velocity vector of the virtual leader unmanned ship expected trajectory in the body coordinate system, u d , v d , r d represents the forward speed, lateral speed and yaw rate of the virtual leader unmanned ship expected trajectory in the body coordinate system; M d represents the inertial mass matrix of the virtual leader unmanned ship in the inertial coordinate system; R d represents the transformation matrix between the inertial coordinate system and the body coordinate system of the virtual leader unmanned ship; τ d =[τ d,1 0 τ d,3 ] T , τ d represents the control input matrix of the virtual leader unmanned ship, τ d,1 and τ d,3 represent the forward control quantity and steering control quantity of the leader unmanned ship, respectively.

[0042] Step B. According to the motion dynamics model of the virtual leader unmanned ship, the leader unmanned ship and each follower unmanned ship in the inertial coordinate system, a fixed time disturbance observer corresponding to the leader unmanned ship and each follower unmanned ship is established for online observation and compensation of the poor system stability caused by the uncertain dynamic disturbance suffered by the unmanned ship in the formation process, and then step C is entered.

[0043] In actual application, step B is performed according to the preset motion dynamics model of the virtual leader unmanned ship, the leader unmanned ship and each follower unmanned ship in the inertial coordinate system, in combination with the number n+1 of the unmanned ships in the distributed formation, 0≤i≤n, i=0 corresponding to the leader unmanned ship and 1≤i≤n corresponding to each follower unmanned ship, and the following operations are performed to establish the fixed time disturbance observer corresponding to the leader unmanned ship and each follower unmanned ship.

[0044] The fixed time disturbance observer corresponding to the forward velocity channel of the leader unmanned ship and each follower unmanned ship is established as follows:

[0045]

[0046]

[0047] In the formula, u i represents the forward velocity of the i-th unmanned ship in the body coordinate system, represents the estimated value of u i ; d i,u represents the disturbance suffered by the i-th unmanned ship in the body coordinate system during the forward motion process, is the estimated value of d i,u , τ i,1 represents the forward control amount of the i-th unmanned ship, is the estimated value error of u i , K i,u1 , K i,u2 , K i,u3 , α i,u1 , α i,u2 are preset adjustable parameters of the i-th unmanned ship in the inertial coordinate system corresponding to the forward motion process, and K i,u1 >0, K i,u2 >0, K i,u3 >0, α i,u1 >1, 0 i,u2 <1, and sign is a sign function.

[0048] The fixed time disturbance observer corresponding to the yaw rate channel of the leader unmanned ship and each follower unmanned ship is established as follows:

[0049]

[0050]

[0051] ψ i ∈[0,2π] represents the angle between the i-th unmanned ship and the X-axis in the inertial coordinate system, i.e., the heading angle of the i-th unmanned ship in the inertial coordinate system, is an estimated value of r i ; i,r represents the disturbance suffered by the i-th unmanned ship in the process of corresponding yaw angle movement in the body coordinate system, is an estimated value of d i,r ; τ i,3 represents the steering control amount of the i-th unmanned ship, is an estimated value error of r i ; K i,r1 , K i,r2 , K i,r3 , α i,r1 , α i,r2 are preset adjustable parameters of the i-th unmanned ship in the process of corresponding yaw angle movement in the inertial coordinate system, and K i,r1 > 0, K i,r2 > 0, K i,r3 > 0, α i,r1 > 1, 0 < α i,r2 < 1.

[0052] Step C. Based on the tracking sub-level system corresponding to the virtual leader unmanned ship and the leader unmanned ship, according to the preset motion dynamics model of the virtual leader unmanned ship and the leader unmanned ship in the inertial coordinate system, a controller of the tracking sub-level system corresponding to the leader unmanned ship is constructed for controlling the leader unmanned ship to track the virtual leader unmanned ship, and then step D is entered.

[0053] In the implementation, in step C, based on the tracking sub-level system corresponding to the virtual leader unmanned ship and the leader unmanned ship, according to the preset motion dynamics model of the virtual leader unmanned ship and the leader unmanned ship in the inertial coordinate system, first, the tracking error between the leader unmanned ship trajectory (η0, v0) and the virtual leader unmanned ship expected trajectory (η d , v d ) is defined as follows:

[0054]

[0055] η0=[x0,y0,ψ0] T, η0denotes the position vector of the leader USV in the inertial coordinate system, (x0, y0) denotes the position coordinates of the leader USV in the inertial coordinate system, ψ0∈[0, 2π] denotes the angle between the leader USV and the X axis in the inertial coordinate system, i.e. the heading angle of the leader USV in the inertial coordinate system; v0= [u0, v0, r0] T , v0denotes the velocity vector of the leader USV in the body coordinate system, u0, v0, r0denote the forward velocity, lateral velocity, yaw angular velocity of the leader USV in the body coordinate system respectively; η d = [x d , y d , ψ d ] T , η d denotes the position vector of the desired trajectory of the virtual leader USV in the inertial coordinate system, (x d , y d ) denotes the position coordinates of the desired trajectory of the virtual leader USV in the inertial coordinate system, ψ d ∈[0, 2π] denotes the angle between the desired trajectory of the virtual leader USV and the X axis in the inertial coordinate system, i.e. the heading angle of the desired trajectory of the virtual leader USV in the inertial coordinate system; v d = [u d , v d , r d ] T , v d denotes the velocity vector of the desired trajectory of the virtual leader USV in the body coordinate system, u d , v d , r d denote the forward velocity, lateral velocity, yaw angular velocity of the desired trajectory of the virtual leader USV in the body coordinate system; ξ 0,1 = [x0-x d y0-y d ψ0-ψ d ] T denotes the position error between the leader USV and the virtual leader USV in the tracking sub-system, ξ 0,2 = [u0-u d v0-v d r0-r d ] T denotes the velocity error between the leader USV and the virtual leader USV in the tracking sub-system.

[0056] Then the leader USV trajectory (η0, v0) and the desired trajectory of the virtual leader USV (η d , v dderivation of the tracking error between the tracking sub-level system leader unmanned ship trajectory (η0, v0) and the virtual leader unmanned ship desired trajectory (η d ,v d ), the error dynamic equation is as follows:

[0057]

[0058] M0 represents the inertial mass matrix of the leader unmanned ship in the inertial coordinate system; R0 represents the transformation matrix of the leader unmanned ship between the inertial coordinate system and the ship coordinate system; τ0 represents the control input matrix of the leader unmanned ship.

[0059] Then the fixed time sliding mode surface model corresponding to the tracking sub-level system is constructed as follows:

[0060]

[0061] ε0 represents the sliding mode surface, L 0,1 、L 0,2 、L 0,3 、a 0,1 、a 0,2 、b 0,1 、b 0,2 respectively represent the preset adjustable parameters greater than 0 of the fixed time sliding mode surface model corresponding to the leader unmanned ship, and satisfy a 0,1 >a 0,2 ,b 0,1 >b 0,2 , and the derivation is carried out for the fixed time sliding mode surface model to obtain the following:

[0062]

[0063] Further, the following is obtained:

[0064]

[0065] Further, the controller of the tracking sub-level system corresponding to the leader unmanned ship is as follows:

[0066]

[0067] λ 0,1 、λ 0,2 、λ 0,3 、m0、n0 respectively represent the preset positive adjustable parameters of the controller corresponding to the leader unmanned ship.

[0068] Step D. Based on the formation sub-systems corresponding to the navigator UAV and each follower UAV, and according to the motion dynamics models of the navigator UAV and each follower UAV in the inertial coordinate system, construct the controllers of the formation sub-systems corresponding to each follower UAV, which are used to control the formation of each follower UAV to track the navigator UAV.

[0069] In the specific design and implementation of step D above, based on the number of unmanned vessels (UVs) in the distributed formation (n+1) and the condition that 0 ≤ i ≤ n, when 1 ≤ i ≤ n corresponds to each follower UV, and combining the leader UV and the formation subsystems corresponding to each follower UV, the formation tracking error between each follower UV and the leader UV is first defined as follows:

[0070] ξ i,1 =η i -η0-E i

[0071] ξ i,1 =[ξ i,1x ,ξ i,1y ,ξ i,1ψ ] T ξ represents the tracking error matrix of the corresponding position loop of each follower unmanned surface vessel formation. i,1x Let ξ represent the formation tracking error of the i-th follower unmanned surface vessel in the X-axis direction corresponding to the position loop. i,1y Let ξ represent the formation tracking error of the i-th follower unmanned surface vessel at the corresponding position loop along the Y-axis. i,1ψ η represents the formation tracking error of the i-th follower unmanned surface vessel at the corresponding position loop in terms of heading angle. i Let E represent the position vector of the i-th follower UAV in the inertial coordinate system, η0 represent the position vector of the navigator UAV in the inertial coordinate system, and E represent the position vector of the navigator UAV in the inertial coordinate system. i =[x i,E y i,E ψ i,E ] T This represents the relative positional deviation matrix between each follower UAV and the leader UAV in the relative position coordinate system, x i,E This represents the relative positional deviation between each follower UAV and the leader UAV along the X-axis of the relative position coordinate system, y i,E ψ represents the relative positional deviation between each follower UAV and the leader UAV along the Y-axis of the relative position coordinate system. i,E This indicates the relative positional deviation between each follower UAV and the navigator UAV in the relative position coordinate system's heading angle.

[0072] Next, the following performance function is introduced:

[0073]

[0074] Construct a preset performance controller, wherein u i ρ represents the forward velocity of the i-th follower unmanned surface vessel in the ship's coordinate system; i Used to set the upper and lower bounds of the preset performance corresponding to the i-th follower unmanned surface vessel, when At that time, take Defined as a preset performance upper bound, when At that time, take p i =ρ i Defined as a preset performance lower bound; i This represents the preset positive design parameters corresponding to the i-th follower unmanned surface vessel, used to adjust the steady-state value of the performance boundary; t is the time variable; β i For ρ i The filtered signal, μ i For signal β i First-order differential; function in, σ i denoted as the preset adjustable parameter corresponding to the i-th follower unmanned vessel; z is the independent variable in the function f(z).

[0075] by As a constraint on position consistency tracking error, the formation tracking error ξ between the corresponding position loops of each follower UAV and the navigator UAV is considered. i,1 After conversion using preset performance control technology, the resulting error is as follows:

[0076]

[0077] ζ i Indicates the formation tracking error ξ i,1 The converted error, obtained through a preset performance control technology, is used in the design of the virtual controller for the position loop of the follower unmanned surface vessel. χ i To normalize the error, the virtual controller of the position loop of the i-th follower unmanned vessel is further obtained as follows. The introduction of preset performance control technology ensures that the formation system maintains a good relative position deviation between unmanned vessels even in the presence of external interference.

[0078]

[0079] in, κ represents the virtual control signal for the position loop of the i-th follower unmanned surface vessel. i,1 =diag(κ) i,1x ,κ i,1y ,κ i,1ψ) represents the adjustable parameter of the virtual control law corresponding to the i-th follower unmanned surface vessel, κ. i,1x κ i,1y κ i,1ψ Let represent the adjustable parameters of the virtual control law for the position loop of the i-th follower UAV in the X-axis direction, Y-axis direction, and heading angle, respectively; further, consider the tracking error of the velocity loop corresponding to the i-th follower UAV. By taking the derivative, the error dynamic equations of the corresponding velocity loop of the formation subsystem are obtained as follows:

[0080]

[0081] The velocity loop controller for the i-th follower unmanned surface vessel in the formation subsystem is then constructed as follows:

[0082]

[0083] In the formula, v i Let d represent the velocity vector of the i-th follower unmanned surface vessel in the ship's coordinate system. i M represents the lumped disturbance matrix experienced by the i-th follower unmanned surface vessel during its movement. i Let R(ψ) represent the inertial mass matrix of the i-th follower unmanned surface vessel in the inertial coordinate system. i ) represents the transformation matrix of the i-th follower unmanned surface vessel between the inertial coordinate system and the hull coordinate system; τ i Let κ represent the control input matrix of the i-th follower unmanned vessel. i,2 =diag(κ) i,2x ,κ i,2y ,κ i,2ψ ) represents the positive adjustable parameter of the speed loop controller corresponding to the i-th follower unmanned surface vessel, κ. i,2x κ i,2y κ i,2ψ ξ represents the adjustable parameters of the virtual control law for the i-th follower unmanned surface vessel in the ship's coordinate system, corresponding to its forward velocity, lateral velocity, and yaw rate. i,2 =[ξ i,2x ,ξ i,2y ,ξ i,2ψ ] T =[u i -u0 v i -v0 r i -r0] T This represents the velocity loop tracking error matrix corresponding to the i-th follower unmanned surface vessel. d i The estimate, ξ i,2x ξ represents the formation tracking error of the i-th follower unmanned surface vessel in the corresponding velocity loop along the X-axis. i,2yξ represents the formation tracking error of the i-th follower unmanned surface vessel in the Y-axis direction corresponding to the velocity loop. i,2ψ Let u represent the formation tracking error of the i-th follower unmanned surface vessel at the corresponding velocity loop in terms of heading angle. i v i r i Let u0, v0, and r0 represent the forward velocity, lateral velocity, and yaw rate of the i-th follower UAV in the ship's coordinate system, respectively, and let r0, v0, and r0 represent the forward velocity, lateral velocity, and yaw rate of the navigator UAV in the ship's coordinate system, respectively.

[0084] The fixed-time distributed formation control method for multiple unmanned vessels based on backstepping technology designed in this invention is applied in practice, considering a distributed formation system consisting of one navigator unmanned vessel and two follower unmanned vessels.

[0085] In the system, the navigator unmanned surface vessel η0 = [x0 y0 ψ0] T and follower unmanned boat η i =[x i y i ψ i ] T (i=1,2) The controller parameters are set as shown in Table 1, which shows the unmanned vessel controller parameters:

[0086] Table 1

[0087] Parameter Value Parameter Value Parameter Value [[ L 0，1 ]]> 3 Kappa 1,1x ]] 0.2 Kappa 2,1x ]] 0.1 [[ L 0，2 ]]> 2 Kappa 1,1y ]] 2 Kappa 2,1y ]] 1 [[ L 0，3 ]]> 2 Kappa 1,1ψ ]] 11 Kappa 2,1ψ ]] 10 0,1 ]]> ​ 0.3 Kappa 1,2x ]] 5 Kappa 2,2x ]] 5 0,2 ]]> ​ 1.2 Kappa 1,2y ]] 5 Kappa 2,2y ]] 5 0,3 ]]> ​ 0.8 Kappa 1,2ψ ]] 5 Kappa 2,2ψ ]] 5 a 0,1 ]]> 7 x E,1 ]]> -1 x E,2 ]]> 1 a 0,2 ]]> 5 [ y E,1 ]] 1 x E,2 ]]> 1 b 0,1 ]]> 5 E,1 ]]> ​ 0 x E,2 ]]> 0 b 0,2 ]]> 9 1,x ]]> ​ 0.5 2,x ]]> ​ 1 m0 5 1,y ]]> ​ 0.5 2,y ]]> ​ 0.5 [n0] 9 1,ψ ]]> ​ 0.5 2,ψ ]]> ​ 0.5

[0088] The parameter settings for the fixed-time interference observers of each unmanned vessel in the system are shown in Table 2:

[0089] Table 2

[0090]

[0091]

[0092] The interference experienced by the navigator unmanned surface vessel in a multi-unmanned surface vessel formation is d0 = [d 0,1 d 0,2 d 0,3 ] T Table 3 shows the true values ​​of interference, i.e. Figure 5 The shown values ​​are the true values ​​of the forward velocity and yaw rate of the Navigator unmanned surface vessel, and... Figure 5 The estimated value of the interference is obtained from the corresponding fixed-time interference observer.

[0093] Table 3

[0094] Parameter Value d 0,1 ]]> 5 sin(0.2t) d 0,2 ]]> 3 sin(2t) d 0,3 ]]> 6 sin(0.2t)

[0095] The interference experienced by follower unmanned vessel 1 in a multi-unmanned vessel formation is d1 = [d1,1 d 1,2 d 1,3 ] T The true values ​​of interference are shown in Table 4, i.e. Figure 6 The forward velocity and yaw rate of the follower unmanned surface vessel 1 shown are disturbed from the true values, and Figure 6 The estimated value of the interference is obtained from the corresponding fixed-time interference observer.

[0096] Table 4

[0097] Parameter Value d 1,1 ]]> 6 sin(0.2t) d 1,2 ]]> 3 sin(2t) d 1,3 ]]> 7 sin(0.2t)

[0098] The interference experienced by follower unmanned vessel 2 in a multi-unmanned vessel formation is d2 = [d 2,1 d 2,2 d 2,3 ] T The true values ​​of interference are shown in Table 5, i.e. Figure 7 The forward velocity and yaw rate of the follower unmanned surface vessel 2 shown are disturbed from the true values, and Figure 7 The estimated value of the interference is obtained from the corresponding fixed-time interference observer.

[0099] Table 5

[0100] Parameter Value d 2,1 ]]> 7 sin(0.2t) d 2,2 ]]> 3 sin(2t) d 2,3 ]]> 6 sin(0.2t)

[0101] Unmanned vessel inertial mass matrix M i And the surface disturbance interference coefficient matrix D of unmanned vessels i as follows:

[0102]

[0103] The initial position of each unmanned vessel is η d (0) =

[000] T η0(0)=[100.1] T η1(0)=[020.2] T η2(0)=[0-10.3] T The virtual navigator unmanned surface vessel control input is set to: τ d =[32sin(0.2πt)] 2 022cos(0.1πt)] T This allows the virtual navigator to navigate along a sinusoidal trajectory, guiding the other unmanned surface vessels (USVs) in the formation to follow its path. One USV and two follower USVs form a triangular formation, following the virtual navigator's trajectory in formation navigation. Figure 2 The following describes the distributed formation motion trajectories of multiple unmanned surface vessels in the inertial coordinate system shown, and where... Figure 3 The position and status output curves of each unmanned vessel are shown below. Figure 4The speed state output curves of each unmanned surface vessel shown are as follows: Figure 8 , Figure 9 , Figure 10 The diagrams sequentially illustrate the formation tracking error of the follower unmanned surface vessel 2 in the X-axis direction, the Y-axis direction, and the heading angle direction.

[0104] The aforementioned technical solution employs a fixed-time distributed formation control method for multiple unmanned surface vessels (USVs) based on backstepping technology. This method establishes a dynamics model of the USVs and creates fixed-time disturbance observers for the leader USV and each follower USV for online observation and compensation. Furthermore, it constructs a backstepping control method that enables multiple USVs to form a stable formation at a fixed time. This involves building a controller for the leader USV in the tracking sub-system using backstepping technology, controlling the leader USV to track the virtual leader USV, and a controller for each follower USV in the formation sub-system, controlling the follower USVs to track the leader USV in formation. The designed controllers exhibit good convergence, ensuring the stability of the entire formation system and achieving the expected performance requirements. Additionally, the design incorporates pre-set performance control technology to ensure that the USVs maintain a good relative positional deviation even under external disturbances, thus maintaining overall formation stability.

[0105] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A fixed-time distributed formation control method for multiple unmanned vessels based on backstepping technology, characterized in that: Based on the preset virtual navigator unmanned surface vessel's expected trajectory, the following steps are executed in real time to achieve collaborative control of a distributed formation consisting of one navigator unmanned surface vessel and each follower unmanned surface vessel. Step A. Establish motion dynamics models in the inertial coordinate system for the virtual navigator UAV, the navigator UAV, and each follower UAV, and then proceed to Step B; Step B. Based on the motion dynamics models of the virtual navigator UAV, the navigator UAV, and each follower UAV in the inertial coordinate system, establish fixed-time disturbance observers for the navigator UAV and each follower UAV. These observers are used to conduct online observation and compensation for the poor system stability caused by uncertain dynamic disturbances experienced by the UAVs during formation. Then proceed to Step C. In step B above, based on the preset motion dynamics models of the virtual navigator UAV, the leader UAV, and each follower UAV in the inertial coordinate system, combined with the number of UAVs in the distributed formation... , ,as well as Corresponding to the Navigator unmanned surface vessel, For each follower UAV, establish a fixed-time interference observer for the navigator UAV and each follower UAV as follows; The following fixed-time interference observers are established for the leader unmanned surface vessel (USV) and each follower USV, corresponding to their respective forward velocity channels: ； In the formula, Indicates the first The forward velocity of the unmanned surface vessel in the hull coordinate system express The estimated value; Indicates the first The disturbances experienced by an unmanned surface vessel during its forward motion in the hull coordinate system. for The estimated value, Indicates the first The forward control quantity of an unmanned vessel for The estimation error, , , , , All are the first The preset adjustable parameters for the forward motion of each unmanned surface vessel in the inertial coordinate system, and , , , , , It is a symbolic function; The following fixed-time interference observers were established for the navigator unmanned surface vessel (USV) and each follower USV, corresponding to their respective yaw rate channels: ； , Indicates the first The angle between the unmanned surface vessel and the X-axis in the inertial coordinate system, i.e., the angle between the first and second unmanned surface vessels, is... The heading angle of an unmanned surface vessel in an inertial coordinate system. for The estimated value; Indicates the first The disturbances experienced by an unmanned surface vessel during its yaw angle motion in the ship's coordinate system. for The estimated value, Indicates the first The steering control quantity of an unmanned vessel for The estimation error, , , , , All are the first The preset adjustable parameters for the yaw angle motion process of an unmanned surface vessel in an inertial coordinate system, and , , >0, , ; Step C. Based on the virtual navigator UAV and the tracking subsystem corresponding to the navigator UAV, according to the preset motion dynamics models of the virtual navigator UAV and the navigator UAV in the inertial coordinate system, construct the controller of the tracking subsystem corresponding to the navigator UAV, which is used to control the navigator UAV to track the virtual navigator UAV, and then proceed to step D. Step D. Based on the formation subsystem corresponding to the navigator UAV and each follower UAV, according to the motion dynamics model of the navigator UAV and each follower UAV in the inertial coordinate system, construct the controller of each follower UAV corresponding to the formation subsystem, which is used to control each follower UAV to form a formation and track the navigator UAV. In step D above, the number of unmanned ships in the distributed formation is used as a basis. ,as well as ,when When dealing with each follower UAV, considering the leader UAV and the formation subsystems corresponding to each follower UAV, the formation tracking error between each follower UAV and the leader UAV is first defined as follows: ； This represents the tracking error matrix for the corresponding position loop of each follower unmanned surface vessel (USV) formation. Indicates the first The formation tracking error of the follower unmanned surface vessel at the corresponding position loop in the X-axis direction. Indicates the first The formation tracking error of the follower unmanned surface vessel at the corresponding position loop in the Y-axis direction. Indicates the first The formation tracking error of a follower unmanned surface vessel at the corresponding position loop in terms of heading angle. Indicates the first The position vector of the follower unmanned surface vessel in the inertial coordinate system. This represents the position vector of the navigator unmanned surface vessel in the inertial coordinate system. This represents the relative positional deviation matrix between each follower UAV and the leader UAV in the relative positional coordinate system. This represents the relative positional deviation between each follower UAV and the leader UAV along the X-axis of the relative position coordinate system. This represents the relative positional deviation between each follower UAV and the leader UAV along the Y-axis of the relative position coordinate system. This indicates the relative positional deviation between each follower UAV and the navigator UAV in the relative position coordinate system heading angle; Next, the following performance function is introduced: ； Constructing a preset performance controller, wherein, Indicates the first The forward velocity of the follower unmanned surface vessel in the hull coordinate system; Used to set the The upper and lower bounds of the preset performance corresponding to each follower unmanned surface vessel, when At that time, take Defined as a preset performance upper bound, when At that time, take Defined as a preset performance lower bound; Indicates the first The preset positive design parameters corresponding to each follower unmanned surface vessel are used to adjust the steady-state value of the performance boundary; t is a time variable; for The filtered signal, For signal First-order differential; function ,in, , , Indicates the first Preset adjustable parameters corresponding to each follower unmanned surface vessel; It is a function The independent variable in; by To satisfy the position consistency tracking error constraint, the formation tracking error is addressed for the corresponding position loops between each follower UAV and the navigator UAV. After conversion using preset performance control technology, the resulting error is as follows: ； Indicates formation tracking error The converted error, obtained through a preset performance control technology, is used in the design of the virtual controller for the position loop of the follower unmanned surface vessel. , To normalize the error, further obtain the first... The virtual controller for the position loop of the follower unmanned surface vessel is as follows: ； in, Indicates the first The virtual control signal for the position loop of the follower unmanned surface vessel. For the first Each follower unmanned surface vessel corresponds to a virtual control law with adjustable parameters. They represent the first The virtual control law for the position loop of the follower unmanned surface vessel in the X-axis direction, Y-axis direction, and heading angle has positive adjustable parameters; further targeting the first Tracking error of the velocity loop of each follower unmanned surface vessel By taking the derivative, the error dynamic equations of the corresponding velocity loop of the formation subsystem are obtained as follows: ； Then construct the first sub-system of the formation The speed loop controller for the follower unmanned surface vessel is as follows: ； In the formula, Indicates the first The velocity vector of the follower unmanned surface vessel in the ship's coordinate system Indicates the first The lumped disturbance matrix experienced by a follower unmanned surface vessel during its movement. Indicates the first The inertial mass matrix of the follower unmanned surface vessel in the inertial coordinate system. Indicates the first The transformation matrix of a follower unmanned surface vessel between the inertial coordinate system and the hull coordinate system; Indicates the first The control input matrix of a follower unmanned surface vessel For the first The adjustable parameters of the speed loop controller for each follower unmanned surface vessel are positive. They represent the first The virtual control law for each follower unmanned surface vessel in the ship's coordinate system, corresponding to forward velocity, lateral velocity, and yaw rate, has positive adjustable parameters. Indicates the first The velocity loop tracking error matrix for each follower unmanned surface vessel. express The estimate, Indicates the first The formation tracking error of the follower unmanned surface vessel in the corresponding velocity loop along the X-axis direction. Indicates the first The formation tracking error of the follower unmanned surface vessel in the Y-axis direction corresponding to the velocity loop. Indicates the first The formation tracking error of a follower unmanned surface vessel in terms of the corresponding velocity loop at the heading angle. , , They represent the first The forward velocity, lateral velocity, and yaw rate of the follower unmanned surface vessel in the ship's coordinate system. , , These represent the forward velocity, lateral velocity, and yaw rate of the Navigator unmanned surface vessel in the ship's coordinate system, respectively.

2. The multi-unmanned vessel fixed-time distributed formation control method based on backstepping technology according to claim 1, characterized in that: In step A, for the navigator unmanned surface vessel and each follower unmanned surface vessel, the following operations are performed to establish corresponding kinematic dynamic models; Based on the number of unmanned ships in the distributed formation ,as well as ; when For each follower UAV, the kinematic model of the follower UAV in the inertial coordinate system is established as follows: ； , Indicates the first The position vector of the follower unmanned surface vessel in the inertial coordinate system. Indicates the first The position coordinates of the follower unmanned surface vessel in the inertial coordinate system. Indicates the first The angle between the follower unmanned surface vessel and the X-axis in the inertial coordinate system, i.e., the angle between the first and second unmanned surface vessels. The heading angle of the follower unmanned surface vessel in the inertial coordinate system; , Indicates the first The velocity vector of the follower unmanned surface vessel in the ship's coordinate system , , They represent the first The forward velocity, lateral velocity, and yaw rate of the follower unmanned surface vessel in the ship's coordinate system; , Indicates the first The transformation matrix of the follower unmanned surface vessel between the inertial coordinate system and the hull coordinate system, and , Represents the identity matrix. , Represents the transformation matrix norm, , , , Describe an antisymmetric matrix. ; Next, for each follower unmanned surface vessel (USV), the following dynamic model of the USV in the inertial coordinate system is established: ； , Indicates the first The inertial mass matrix of the follower unmanned surface vessel in the inertial coordinate system. express The element in the 2nd row and 2nd column; , , Indicates the first The Coriolis force matrix of a follower unmanned surface vessel in an inertial coordinate system; , Indicates the first The hydrodynamic damping parameter of the follower unmanned surface vessel in the inertial coordinate system, i.e., the first... The hydrodynamic damping parameter matrix of a follower unmanned surface vessel in the inertial coordinate system. , , , , All of these represent the physical parameters of the unmanned vessel, and all are greater than 0; , Indicates the first The control input matrix of a follower unmanned surface vessel Indicates the first The forward control quantity of a follower unmanned vessel Indicates the first The steering control quantity of a follower unmanned vessel; , Indicates the first The lumped disturbance matrix experienced by a follower unmanned surface vessel during its movement. Indicates the first The disturbances experienced by a follower unmanned surface vessel during its forward motion in the ship's coordinate system. Indicates the first The disturbances experienced by a follower unmanned surface vessel during its lateral motion in the ship's coordinate system. Indicates the first The disturbances experienced by a follower unmanned surface vessel during its yaw angle motion in the ship's coordinate system; That is, based on the kinematic model and dynamic model of the follower unmanned surface vessel in the inertial coordinate system, the corresponding motion dynamic model of the follower unmanned surface vessel is constructed. Then when The motion dynamics model of the navigator UAV in the inertial coordinate system of the tracking subsystem is established as follows: ； , This represents the position vector of the navigator unmanned surface vessel in the inertial coordinate system. This represents the position coordinates of the navigator unmanned surface vessel in the inertial coordinate system. This represents the angle between the Navigator UAV and the X-axis in the inertial coordinate system, i.e., the heading angle of the Navigator UAV in the inertial coordinate system. , This represents the velocity vector of the navigator unmanned surface vessel in the ship's coordinate system. , , These represent the forward velocity, lateral velocity, and yaw rate of the Navigator unmanned surface vessel in the ship's coordinate system, respectively. This represents the inertial mass matrix of the navigator unmanned surface vessel in the inertial coordinate system. This represents the transformation matrix of the navigator unmanned surface vessel between the inertial coordinate system and the hull coordinate system; , This represents the control input matrix of the navigator unmanned surface vessel. This indicates the forward control parameters of the navigator unmanned surface vessel. This indicates the steering control parameters of the navigator unmanned surface vessel; , This represents the lumped interference matrix experienced by the navigator unmanned surface vessel during its movement. This represents the disturbances experienced by the navigator unmanned surface vessel during its forward motion in the ship's coordinate system. This represents the disturbances experienced by the navigator unmanned surface vessel during its lateral motion in the ship's coordinate system. This indicates the disturbances experienced by the navigator unmanned surface vessel during its yaw angle motion in the ship's coordinate system.

3. The multi-unmanned vessel fixed-time distributed formation control method based on backstepping technology according to claim 1, characterized in that: In step A, the motion dynamics model corresponding to the preset virtual navigator unmanned vessel is established by performing the following operations: Establish the desired trajectory of the virtual navigator unmanned surface vessel in the inertial coordinate system within the tracking subsystem. The corresponding kinematics model is as follows: ； , This represents the position vector of the virtual navigator unmanned surface vessel in the inertial coordinate system, indicating its desired trajectory. This represents the position coordinates of the virtual navigator unmanned surface vessel in the inertial coordinate system, indicating its desired trajectory. This represents the angle between the desired trajectory of the virtual navigator unmanned surface vessel (USV) in the inertial coordinate system and the X-axis, i.e., the heading angle of the desired trajectory of the USV in the inertial coordinate system. , This represents the velocity vector of the virtual navigator unmanned surface vessel in the ship's coordinate system, indicating its desired trajectory. , , This represents the forward velocity, lateral velocity, and yaw rate of the virtual navigator unmanned surface vessel in the ship's coordinate system, representing the expected trajectory of the virtual navigator unmanned surface vessel. The inertial mass matrix represents the virtual navigator unmanned surface vessel in the inertial coordinate system; This represents the transformation matrix of the virtual navigator unmanned surface vessel between the inertial coordinate system and the hull coordinate system; , This represents the control input matrix of the virtual navigator unmanned surface vessel. and These represent the forward control and steering control quantities of the navigator unmanned surface vessel, respectively.

4. The multi-unmanned vessel fixed-time distributed formation control method based on backstepping technology according to claim 1, characterized in that: In step C, based on the virtual navigator UAV and its corresponding tracking subsystem, and according to the preset motion dynamics models of the virtual navigator UAV and the navigator UAV in the inertial coordinate system, the trajectory of the navigator UAV is first defined. Expected trajectory of the virtual navigator unmanned vessel The tracking errors between them are as follows: ； , This represents the position vector of the navigator unmanned surface vessel in the inertial coordinate system. This represents the position coordinates of the navigator unmanned surface vessel in the inertial coordinate system. This represents the angle between the Navigator UAV and the X-axis in the inertial coordinate system, i.e., the heading angle of the Navigator UAV in the inertial coordinate system. , This represents the velocity vector of the navigator unmanned surface vessel in the ship's coordinate system. , , These represent the forward velocity, lateral velocity, and yaw rate of the Navigator unmanned surface vessel in the ship's coordinate system, respectively. , This represents the position vector of the virtual navigator unmanned surface vessel in the inertial coordinate system, indicating its desired trajectory. This represents the position coordinates of the virtual navigator unmanned surface vessel in the inertial coordinate system, indicating its desired trajectory. This represents the angle between the desired trajectory of the virtual navigator unmanned surface vessel (USV) in the inertial coordinate system and the X-axis, i.e., the heading angle of the desired trajectory of the USV in the inertial coordinate system. , This represents the velocity vector of the virtual navigator unmanned surface vessel in the ship's coordinate system, indicating its desired trajectory. , , This represents the forward velocity, lateral velocity, and yaw rate of the virtual navigator unmanned surface vessel in the ship's coordinate system, representing the expected trajectory of the virtual navigator unmanned surface vessel. This represents the positional error between the navigator UAV and the virtual navigator UAV in the tracking subsystem. This represents the speed error between the navigator UAV and the virtual navigator UAV in the tracking subsystem. Next, the trajectory of the Navigator unmanned surface vessel. Expected trajectory of the virtual navigator unmanned vessel The trajectory of the navigator unmanned surface vessel in the tracking subsystem is obtained by differentiating the tracking error between the two. Expected trajectory of the virtual navigator unmanned vessel The dynamic equation for the error between them is as follows: ； This represents the inertial mass matrix of the navigator unmanned surface vessel in the inertial coordinate system. This represents the transformation matrix of the navigator unmanned surface vessel between the inertial coordinate system and the hull coordinate system; This represents the control input matrix of the navigator unmanned surface vessel; Then, the fixed-time sliding mode surface model corresponding to the tracking subsystem is constructed as follows: ； Indicates the sliding surface. , , , , , , These represent preset adjustable parameters (greater than 0) of the fixed-time sliding mode surface model corresponding to the Navigator unmanned surface vessel, and satisfy the following conditions: , And by differentiating the sliding mode surface model with respect to a fixed time, we obtain the following: ； Further results were obtained: ； The controller for the corresponding Navigator unmanned surface vessel in the tracking subsystem is thus obtained as follows: ； , , , , These represent the preset positive adjustable parameters of the controller corresponding to the Navigator unmanned surface vessel.

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