Longitudinal formation control method for multiple unmanned ships based on nonlinear extended state observer

Abstract

The application discloses a multi-unmanned ship longitudinal formation control method based on a nonlinear extended state observer, which comprises the following steps: establishing a kinematics model, a dynamics model and a wind disturbance model, and analyzing the boundedness of the wind disturbance; designing a time corresponding distance increment rule, generating a reference signal, constructing a longitudinal scheduling mixer and a double closed-loop control framework; establishing a kinematics error system, and deducing an error dynamics equation; designing a backstepping controller with a geometric compensation term, and outputting an expected speed; establishing an extended state system, and introducing equivalent external disturbances as extended states; designing a nonlinear extended state observer, outputting disturbance estimation values as disturbance compensation signals, and deducing a nonlinear error system; designing an integral sliding mode surface with a continuous nonlinear function; establishing a speed error system, and designing an integral sliding mode controller with a saturation function and disturbance feedforward compensation. The application designs a robust control and collaborative compensation mechanism for the working condition of narrow channel and superimposed wind disturbance, so as to guarantee the safe and reliable operation of the multi-ship longitudinal formation.

Description

Technical Field

[0001] This invention relates to the field of autonomous driving technology for unmanned vessels, specifically a longitudinal formation control method for multiple unmanned vessels based on a nonlinear extended state observer. Background Technology

[0002] In collaborative operations involving multiple unmanned vessels in narrow waterways, longitudinal formation can significantly improve navigation efficiency and waterway utilization for tasks such as marine exploration and environmental monitoring, and is widely used in maritime operations. However, under actual working conditions, existing formation control technologies still have many problems.

[0003] I. Control Challenges Arising from Narrow Waterways

[0004] On the one hand, formation control is prone to problems such as error propagation and occlusion: once intra-formation errors propagate and are amplified along the formation direction, they may lead to decreased formation stability or even formation disintegration; at the same time, occlusion effects weaken relative measurement or communication quality, making tracking errors more likely to accumulate. In addition, curvature changes and path geometric constraints in narrow waterways further exacerbate the above-mentioned error accumulation, making it more difficult for the formation to maintain stability and consistency in critical conditions such as turning and bottleneck passage.

[0005] On the other hand, fixed-formation formations often face problems of "occupancy expansion" and insufficient environmental adaptability in narrow waterways: the effective passage width of the formation increases, making it prone to conflicts with waterway boundaries or obstacle constraints, reducing passage feasibility and coordinated maneuverability. Under directional connection or restricted communication conditions, rigid formations with "expansion footprints" may also weaken the robustness of attitude and heading tracking, thereby affecting formation consistency and coordinated maneuver effectiveness.

[0006] To adapt to narrow waterways, and even waterways with self-intersecting structures, existing technologies have proposed control strategies for single-path passage and longitudinal formation maintenance, including self-organized longitudinal formation control, unguided longitudinal formation control, and reconfigurable adaptive longitudinal formation methods, in order to maintain the target formation and suppress error propagation.

[0007] II. Control Challenges Arising from Gust Disturbance

[0008] In real sea conditions, gusts of wind can introduce unstable lateral loads and additional moments, easily causing lateral positional shifts, heading disturbances, and speed fluctuations, thereby reducing the formation's motion consistency and ability to maintain formation. Especially in longitudinal formation scenarios, gust disturbances may induce a continuous accumulation of relative errors within the formation, further weakening heading stability and trajectory tracking accuracy in narrow waterways.

[0009] Under strong winds such as gusts, unmanned surface vessels (USVs) are typically subjected to relatively smooth, low-frequency external force / torque disturbances. For these disturbances, existing technologies utilize extended state observers (ESOs) to perform online estimation of the combined disturbances and provide compensation for the controller. Furthermore, existing technologies have proposed observer structures combining neural networks or deep learning to assess model uncertainties and enhance the estimation capability for time-varying wind disturbances, thereby improving heading stability and trajectory tracking performance. Some technical solutions combine fixed-time observers and sliding mode control to suppress sideslip and lateral drift induced by strong winds, thus improving system robustness and convergence speed.

[0010] In general, existing technologies mainly focus on longitudinal formation geometric constraints, distributed coordination, and disturbance compensation, and have developed certain solutions for single operating conditions. However, under the conditions of narrow waterways and superimposed gust disturbances, there are still problems such as insufficient formation error suppression capabilities and the need to enhance the robustness of formation maintenance. There is an urgent need to design robust control and coordinated compensation mechanisms for such operating conditions to ensure the safe and reliable operation of multi-ship longitudinal formations. Summary of the Invention

[0011] To address the technical challenges of insufficient formation error suppression and enhanced formation robustness under conditions of overlapping narrow channels and gust disturbances, this invention proposes a longitudinal formation control method for multiple unmanned vessels based on a nonlinear extended state observer, comprising the following steps:

[0012] S1. Establish kinematic model, dynamic model, and wind disturbance model, and conduct boundedness analysis of wind disturbance;

[0013] S2. Design time-corresponding interval increment rules, generate unmanned vessel reference signals, construct a longitudinal scheduling mixer, build a dual closed-loop control framework, and use the output of the longitudinal scheduling mixer as the reference input of the dual closed-loop control framework.

[0014] S3. Establish the kinematic error system and derive the error dynamics equation; design a backstepping controller that introduces a geometric compensation term, and output the desired velocity vector as the reference input of the velocity tracking controller;

[0015] S4. Establish an extended state system and introduce an equivalent external disturbance as an extended state; design a nonlinear extended state observer and output the disturbance estimate as a disturbance compensation signal; define the observation error and the disturbance estimation error, and derive the nonlinear error system.

[0016] S5. Design an integral sliding surface and introduce a continuous nonlinear function; establish a velocity error system and substitute it into the dynamic model; design an integral sliding controller with saturation function and disturbance feedforward compensation.

[0017] Further, step S1 includes the following steps:

[0018] S1.1 Establish the inertial coordinate system and the body coordinate system, define the pose of the unmanned vessel in the inertial coordinate system and its velocity state in the body coordinate system; construct a rotation matrix to map the vessel velocity to the inertial coordinate system to obtain the kinematic equations; the kinematic model is used to construct the pose error system and serve as the design basis for the backstepping controller.

[0019] S1.2 Select the longitudinal channel and yaw channel as the control objects, and the control inputs are the longitudinal propulsion force and the yaw control torque; establish the nonlinear dynamic equations with additional mass and hydrodynamic damping, and obtain the controlled dynamic expressions of the longitudinal velocity and the turning velocity respectively; unify the modeling uncertainty and external disturbances into the equivalent external disturbance term to provide the observation objects for the design of the nonlinear extended state observer;

[0020] S1.3 Establish a wind disturbance model. Based on aerodynamic modeling, construct wind force and moment terms and convert them into external disturbances in the longitudinal and yaw channels. Use a first-order Gauss-Markov process to describe the time-varying characteristics of gusts and introduce time constant and steady-state variance parameters to characterize the randomness of gusts.

[0021] S1.4 Perform a boundedness analysis of wind disturbance. Assuming that the wind speed component has an upper bound, we obtain that the wind-induced equivalent external disturbance has a constant bound. We give the boundedness of disturbance as a prerequisite for the subsequent bounded convergence of the nonlinear extended state observer error and the robustness analysis of the integral sliding mode control method.

[0022] Further, step S1.1 specifically involves: the body coordinate system and the inertial coordinate system being represented as follows: and For the first Unmanned boat, longitudinal speed Along the longitudinal axis lateral speed Along the horizontal axis Steering speed Represents rotational motion along the vertical axis, coordinate system This indicates the position and heading angle of the unmanned surface vessel in the inertial coordinate system. Indicates the yaw angle for counterclockwise rotation;

[0023] Let the first The pose of an unmanned surface vessel in an inertial coordinate system Unmanned surface vessel velocity state in body coordinate system The kinematic model behavior of the unmanned vessel is as follows:

[0024]

[0025]

[0026] in, This represents the rotation matrix from the body coordinate system to the inertial coordinate system;

[0027] Step S1.2 specifically involves the following: The dynamic model behavior of the unmanned vessel is as follows:

[0028]

[0029]

[0030]

[0031]

[0032] in, Indicates control input, For longitudinal propulsion; For yaw control torque; This represents the effective translational inertia in the longitudinal direction, including the added mass. This represents the effective translational inertia in the lateral direction, including the added mass. This represents the effective yaw moment of inertia, including additional inertia. , , , , , , For the longitudinal nonlinear damping-related hydrodynamic derivative, This represents the second-order transverse damping term. This indicates the yaw lateral damping coupling term. The damping coupling term represents the lateral yaw. This represents the second-order yaw damping term. Indicates the second lateral moment. This indicates the yaw lateral moment coupling term. This indicates the lateral yaw moment coupling term. Indicates the second yaw moment; Indicates the first The equivalent external disturbance caused by gusts of wind to an unmanned vessel;

[0033] Step S1.3 specifically refers to: The equivalent external disturbance caused by gusts of wind to the unmanned vessel is , This represents the equivalent disturbance of longitudinal gusts. This indicates the equivalent disturbance of lateral gusts. The equivalent disturbance of yaw gust is expressed as follows:

[0034]

[0035]

[0036]

[0037] in, air density, and The longitudinal and lateral windward projected areas. and These are the drag coefficients for the longitudinal and lateral directions. This represents the longitudinal wind speed component. This represents the lateral wind speed component. For wind power equivalent effector;

[0038] Described by a first-order Gauss-Markov model as:

[0039]

[0040]

[0041] in, and For longitudinal and lateral gust disturbances, the time constants are given. and The steady-state standard deviations are given by longitudinal and lateral gust disturbances. and Zero-mean, unit-intensity Gaussian white noise;

[0042] Step S1.4 specifically involves: Since wind speed is limited by airflow and has an upper limit, therefore... Make:

[0043] .

[0044] Furthermore, step S2 is detailed as follows:

[0045] S2.1 Design a formation mixer to generate fixed-interval longitudinal formation aiming points; a reference trajectory for virtual navigation. Parameterized by arc length, the increment along the trajectory is... The status and reference trajectory of virtual navigation Synchronize the current index at the current time. Reference trajectory Recorded as The virtual navigation reference trajectory is input into the formation mixer, and a time delay is added based on the current unmanned vessel number. , Given the expected lag distance interval between two adjacent ships along the reference trajectory, we obtain the first... A reference trajectory following the unmanned vessel ,Right now The reference trajectory is then distributed to the corresponding unmanned vessels.

[0046] S2.2 Establish a dual-closed-loop control framework for the unmanned surface vessel. This framework includes a backstepping controller, an integral sliding mode controller, and a nonlinear extended state observer, wherein:

[0047] Reference trajectory vector From pose reference and speed reference composition, For vertical position reference, For horizontal position reference, For heading angle reference, For longitudinal velocity reference, For lateral velocity reference, For yaw speed reference;

[0048] Attitude error vector From pose reference With the actual pose state vector Obtained by subtraction. For longitudinal error, For lateral error, This refers to the heading angle error;

[0049] The backstep controller receives the attitude error vector. And generate the desired velocity vector. , For the desired longitudinal velocity, For the desired lateral velocity, The desired yaw speed;

[0050] velocity error vector Depend on With the actual velocity state vector We obtain the result by taking the difference, where For longitudinal velocity error, This refers to the yaw speed error;

[0051] Integral sliding mode controller receives velocity error vector and generate control input vectors. , For vertical input, For steering input;

[0052] The estimated vector of gust disturbance Input the integral sliding mode controller for gust disturbance compensation. To estimate longitudinal disturbances, To estimate yaw disturbance;

[0053] Nonlinear extended state observer for estimating gust disturbances .

[0054] Furthermore, step S3 is as follows:

[0055] S3.1 For the pose tracking problem in an inertial coordinate system, design a backstepping controller; for the first... For an unmanned surface vessel, the kinematic model is reconstructed, and the expression is:

[0056]

[0057] To track the current pose reference of the unmanned surface vessel The error in the inertial coordinate system is defined as... , , In the body reference frame, attitude error The expression is as follows:

[0058]

[0059] Based on the errors in the kinematic model and the inertial coordinate system, the kinematic error system can be expressed in the following form:

[0060]

[0061]

[0062]

[0063] S3.2 The backstepping controller is designed as follows:

[0064]

[0065] in, This is an error coupling term; and Positive control gain, The position error modulus; and Geometric compensation for heading deviation , and This is the error proportion term. For reference angular velocity;

[0066] Consider the first If the kinematic error system of an unmanned surface vessel under the action of a backstepping controller exists... , , and Then the machine coordinate system error , and Will follow It converges asymptotically to zero.

[0067] Furthermore, step S4 specifically includes:

[0068] S4.1 Based on the dynamic model, it is rewritten in the following form:

[0069]

[0070]

[0071] in, Indicates longitudinal velocity. Indicates yaw speed; and Indicates longitudinal control input and steering control input; This indicates a longitudinal disturbance. Indicates yaw disturbance;

[0072] Based on the longitudinal dynamics model and the yaw dynamics model, the expanding system is rewritten in the following form:

[0073]

[0074]

[0075] in, This indicates the longitudinal velocity estimate. This indicates the yaw speed estimate; This represents the longitudinal perturbation estimate. This indicates the yaw disturbance estimate; and These represent the effective longitudinal inertia and effective turning inertia of the unmanned vessel, respectively. ; express The derivative of express The derivative;

[0076] S4.2. Based on the extended system, the nonlinear extended state observer is expressed as:

[0077]

[0078]

[0079] The longitudinal and steering speed estimation errors are expressed as follows:

[0080]

[0081]

[0082] The longitudinal and steering disturbance estimation errors are expressed as follows:

[0083]

[0084]

[0085] in, and These represent the speed estimates for longitudinal direction and steering, respectively. and These represent the disturbance estimates for the longitudinal direction and steering, respectively. and These represent the longitudinal and steering speed gains, respectively. and These represent the disturbance gains for longitudinal direction and steering, respectively. and These represent longitudinal speed and steering speed, respectively. ;

[0086] S4.3. Based on the extended system and the nonlinear extended state observer, the nonlinear error system is established as follows:

[0087]

[0088]

[0089] Consider the first A nonlinear error system and a nonlinear extended state observer for an unmanned vessel, if exist and And longitudinal velocity gain and longitudinal perturbation gain satisfy:

[0090]

[0091]

[0092] Then the longitudinal velocity estimation error and longitudinal disturbance estimation error Converging to a bounded region;

[0093] If it exists and And steering speed gain and steering disturbance gain satisfy:

[0094]

[0095]

[0096] Then the steering speed estimation error and steering disturbance estimation error It converges to the bounded region.

[0097] Furthermore, step S5 specifically includes:

[0098] S5.1, To track the desired longitudinal velocity and desired yaw speed Design an integral sliding mode controller;

[0099] The expression for the unmanned surface vessel's airframe velocity error is as follows:

[0100]

[0101]

[0102] in, For longitudinal velocity error, This refers to the yaw speed error; For longitudinal velocity, The turning speed, i.e. ; For the desired longitudinal velocity, The desired yaw speed, i.e. ;

[0103] set up For integral sliding surfaces, The expression for the integral sliding surface, representing the observer gain, is:

[0104]

[0105]

[0106] in, , ; and It is a nonlinear function, and its expression is:

[0107]

[0108]

[0109] in, and It is a tuning parameter that depends on the order of the nonlinear convergence rate near the origin; and It is a smoothing parameter used to suppress jitter;

[0110] S5.2, Based on velocity error and integral sliding surface, there exists and and longitudinal steering speed error system , Based on the longitudinal dynamics model and the yaw dynamics model, the derivative of the integral sliding surface is rewritten in the following form:

[0111]

[0112]

[0113] S5.3, Let , These are the fal function parameters for longitudinal direction and steering, respectively;

[0114] The saturation function is defined as follows:

[0115]

[0116]

[0117] in, This is the boundary layer width, used to ensure smooth behavior in the region near the sliding surface;

[0118] set up For the controller gain, the integral sliding mode controller is designed as follows:

[0119]

[0120]

[0121] Considering the nonlinear error system and the integral sliding mode controller, for the gain , , and Longitudinal velocity error and yaw speed error They converge to the bounded region respectively.

[0122] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0123] 1. Introduce a time delay term into the formation mixer so that each following ship can automatically generate a longitudinal pre-aiming reference point that meets the desired interval along the reference trajectory, thereby achieving stable maintenance of longitudinal formation spacing and reducing the risk of error propagation and formation stretching under conditions of turning in narrow channels and speed changes.

[0124] 2. Introduce a pair of trigonometric function geometric compensation terms in the backstep controller to perform bounded and continuous nonlinear mapping and correction of heading deviation, thereby improving the attitude convergence speed and tracking consistency in scenarios with large deviations / curvature changes, while avoiding control mismatch and overcorrection caused by linear approximation.

[0125] 3. By introducing saturation and decay functions into the sliding mode controller, the control chattering and high-frequency switching are effectively suppressed while ensuring robustness, improving the continuity and smoothness of the control input, reducing the impact of the actuator and energy consumption fluctuations, thereby improving the overall stability and engineering feasibility of the system. Attached Figure Description

[0126] Figure 1 This is a design diagram for the mapping and propulsion of the body coordinate system to the inertial coordinate system of this invention.

[0127] Figure 2 This is a schematic diagram of the longitudinal formation pre-aiming point mixer of the present invention.

[0128] Figure 3 This is a schematic diagram of the dual closed-loop control framework of the present invention.

[0129] Figure 4 This is a comparison diagram of the actual trajectory of the experimental vessel and the guiding reference trajectory in an embodiment of the present invention.

[0130] Figure 5 This is a comparison diagram of the experimental hull position error and heading error in an embodiment of the present invention.

[0131] Figure 6 This is a comparison diagram of the inter-hull spacing and the expected spacing in the embodiments of the present invention.

[0132] Figure 7 This is a schematic diagram of the experimental hull disturbance estimation in an embodiment of the present invention.

[0133] Figure 8 This is a schematic diagram of the experimental hull control input in an embodiment of the present invention.

[0134] Figure 9 This is a flowchart of the multi-unmanned vessel longitudinal formation control method of the present invention. Detailed Implementation

[0135] To better understand the above-mentioned objectives, features, and advantages of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments and features described in these embodiments can be combined with each other.

[0136] A multi-unmanned vessel longitudinal formation control method based on a nonlinear extended state observer, such as Figure 9 As shown, it includes the following steps:

[0137] S1. Establish kinematic model, dynamic model, and wind disturbance model, and conduct boundedness analysis of wind disturbance;

[0138] S2. Design time-corresponding interval increment rules, generate unmanned vessel reference signals, construct a longitudinal scheduling mixer, build a dual closed-loop control framework, and use the output of the longitudinal scheduling mixer as the reference input of the dual closed-loop control framework.

[0139] S3. Establish the kinematic error system and derive the error dynamics equation; design a backstepping controller that introduces a geometric compensation term, and output the desired velocity vector as the reference input of the velocity tracking controller;

[0140] S4. Establish an extended state system and introduce an equivalent external disturbance as an extended state; design a nonlinear extended state observer and output the disturbance estimate as a disturbance compensation signal; define the observation error and the disturbance estimation error, and derive the nonlinear error system.

[0141] S5. Design an integral sliding surface and introduce a continuous nonlinear function; establish a velocity error system and substitute it into the dynamic model; design an integral sliding controller with saturation function and disturbance feedforward compensation.

[0142] The following is a detailed explanation of each step.

[0143] Step S1 is as follows:

[0144] S1.1 Establish the inertial coordinate system and the body coordinate system, define the pose of the unmanned vessel in the inertial coordinate system and its velocity state in the body coordinate system; construct a rotation matrix to map the vessel velocity to the inertial coordinate system to obtain the kinematic equations; the kinematic model is used to construct the pose error system and serves as the design basis for the backstepping controller.

[0145] Specifically, such as Figure 1 As shown, Indicates the length above the waterline. This indicates the maximum depth at the waterline. and Let represent the body coordinate system and the inertial coordinate system, respectively. For the ... Unmanned boat, longitudinal speed Along the longitudinal axis lateral speed Along the horizontal axis Steering speed Represents rotational motion along the vertical axis, coordinate system This indicates the position and heading angle of the unmanned surface vessel in the inertial coordinate system. It indicates the yaw angle of counterclockwise rotation.

[0146] Let the first The pose of an unmanned surface vessel in an inertial coordinate system Unmanned surface vessel velocity state in body coordinate system The kinematic model behavior of the unmanned vessel is described as follows:

[0147]

[0148]

[0149] in, This represents the rotation matrix from the body coordinate system to the inertial coordinate system.

[0150] S1.2 Select the longitudinal channel and yaw channel as the control objects, and the control inputs are the longitudinal propulsion force and the yaw control torque; establish nonlinear dynamic equations with additional mass and hydrodynamic damping, and obtain the controlled dynamic expressions of longitudinal velocity and steering velocity respectively; integrate the modeling uncertainty and external disturbance into the equivalent external disturbance term to form a standard structure of "known term + control input + equivalent external disturbance", which provides an object for the design of nonlinear extended state observer.

[0151] Specifically, the dynamic model behavior of the unmanned vessel is described as follows:

[0152]

[0153]

[0154]

[0155]

[0156] in, Indicates control input, For longitudinal propulsion; For yaw control torque; This represents the effective translational inertia in the longitudinal direction, including the added mass. This represents the effective translational inertia in the lateral direction, including the added mass. This represents the effective yaw moment of inertia, including additional inertia. , , , , , , For the longitudinal nonlinear damping-related hydrodynamic derivative, This represents the second-order transverse damping term. This indicates the yaw lateral damping coupling term. The damping coupling term represents the lateral yaw. This represents the second-order yaw damping term. Indicates the second lateral moment. This indicates the yaw lateral moment coupling term. This indicates the lateral yaw moment coupling term. Indicates the second yaw moment; Indicates the first The equivalent external disturbance caused by gusts of wind to the unmanned vessel.

[0157] S1.3. Establish a wind disturbance model. Based on aerodynamic modeling, construct wind force and moment terms (obtained from air density, windward area, drag coefficient, and relative wind speed, etc.) and represent them as external disturbances in the longitudinal and yaw channels. Use a first-order Gauss-Markov process to describe the time-varying characteristics of gusts, and introduce time constants and steady-state variance parameters to characterize the randomness of gusts.

[0158] No. Equivalent external disturbance caused by gusts of wind to an unmanned vessel , This represents the equivalent disturbance of longitudinal gusts. This indicates the equivalent disturbance of lateral gusts. The equivalent disturbance of yaw gust is shown below:

[0159]

[0160]

[0161]

[0162] in, air density, and The longitudinal and lateral windward projected areas. and These are the drag coefficients for the longitudinal and lateral directions. This represents the longitudinal wind speed component. This represents the lateral wind speed component. For wind power equivalent effector;

[0163] Described by a first-order Gauss-Markov model as:

[0164]

[0165]

[0166] in, and For longitudinal and lateral gust disturbances, the time constants are given. and The steady-state standard deviations are given by longitudinal and lateral gust disturbances. and Zero-mean, unit-intensity Gaussian white noise;

[0167] S1.4 Perform a boundedness analysis of wind disturbance. Assuming that the wind speed component has an upper bound, we obtain that the wind-induced equivalent external disturbance has a constant bound. We give the boundedness of disturbance as a prerequisite for the subsequent bounded convergence of the nonlinear extended state observer error and the robustness analysis of the integral sliding mode control method.

[0168] Because wind speed is limited by airflow and has an upper limit, therefore... Make:

[0169] .

[0170] Step S2 is as follows:

[0171] S2.1 Design a formation mixer to generate, for example, Figure 2 The longitudinal formation aiming points with fixed intervals are shown; in Figure 2 In the middle, the reference trajectory of virtual navigation Parameterized by arc length, the increment along the trajectory is... The status and reference trajectory of virtual navigation Synchronize the current index at the current time. Reference trajectory Recorded as The virtual navigation reference trajectory is input into the formation mixer, and a time delay is added based on the current unmanned vessel number. , Given the expected lag distance interval between two adjacent ships along the reference trajectory, we obtain the first... A reference trajectory following the unmanned vessel ,Right now The reference trajectory is then distributed to the corresponding unmanned vessels.

[0172] S2.2 Establish a dual closed-loop control framework for unmanned vessels, such as... Figure 3 As shown, the dual-closed-loop control framework consists of a backstepping controller, an integral sliding mode controller, and a nonlinear extended state observer, wherein:

[0173] Reference trajectory vector From pose reference and speed reference composition, For vertical position reference, For horizontal position reference, For heading angle reference, For longitudinal velocity reference, For lateral velocity reference, For yaw speed reference;

[0174] Attitude error vector From pose reference With the actual pose state vector Obtained by subtraction. For longitudinal error, For lateral error, This refers to the heading angle error;

[0175] The backstep controller receives the attitude error vector. And generate the desired velocity vector. , For the desired longitudinal velocity, For the desired lateral velocity, The desired yaw speed;

[0176] velocity error vector Depend on With the actual velocity state vector We obtain the result by taking the difference, where For longitudinal velocity error, This refers to the yaw speed error;

[0177] Integral sliding mode controller receives velocity error vector and generate control input vectors. , For vertical input, For steering input;

[0178] The estimated vector of gust disturbance Input the integral sliding mode controller for gust disturbance compensation. For longitudinal disturbance estimation, For steering disturbance estimation;

[0179] Nonlinear extended state observer for estimating gust disturbances .

[0180] Step S3 is as follows:

[0181] S3.1 For the pose tracking problem in an inertial coordinate system, design a backstepping controller; for the first... For an unmanned surface vessel, the kinematic model is reconstructed, and the expression is:

[0182]

[0183] To track the current pose reference of the unmanned surface vessel The error in the inertial coordinate system is defined as... , , In the body reference frame, attitude error The expression is as follows:

[0184]

[0185] Based on the errors in the kinematic model and the inertial coordinate system, the kinematic error system can be expressed in the following form:

[0186]

[0187]

[0188]

[0189] S3.2 The backstepping controller is designed as follows:

[0190]

[0191] in, This is an error coupling term; and Positive control gain, The position error modulus; and Geometric compensation for heading deviation , and This is the error proportion term. This is the reference angular velocity term.

[0192] Consider the first If the kinematic error system of an unmanned surface vessel under the action of a backstepping controller exists... , , and Then the machine coordinate system error , and Will follow It converges asymptotically to zero.

[0193] Proof: Choose the following Lyapunov functions:

[0194]

[0195] in, .right Differentiation yields:

[0196]

[0197] Substituting the kinematic error system into the equation and applying the backstepping controller, we obtain the following expression:

[0198]

[0199] Therefore, the kinematic error system is asymptotically stable, that is... , and All converge asymptotically to zero.

[0200] Step S4 is as follows:

[0201] S4.1 Based on the dynamic model, it is rewritten in the following form:

[0202]

[0203]

[0204] in, Indicates longitudinal velocity. Indicates yaw speed; and Indicates longitudinal control input and steering control input; This indicates a longitudinal disturbance. This indicates a yaw disturbance.

[0205] Based on the longitudinal dynamics model and the yaw dynamics model, the expanding system is rewritten in the following form:

[0206]

[0207]

[0208] in, This indicates the longitudinal velocity estimate. This indicates the yaw speed estimate; This represents the longitudinal perturbation estimate. This indicates the yaw disturbance estimate; and These represent the effective longitudinal inertia and effective turning inertia of the unmanned vessel, respectively. ; express The derivative of express The derivative of .

[0209] S4.2. Based on the extended system, the nonlinear extended state observer is expressed as:

[0210]

[0211]

[0212] The longitudinal and steering speed estimation errors are expressed as follows:

[0213]

[0214]

[0215] The longitudinal and steering disturbance estimation errors are expressed as follows:

[0216]

[0217]

[0218] in, and These represent the speed estimates for longitudinal direction and steering, respectively. and These represent the disturbance estimates for the longitudinal direction and steering, respectively. and These represent the longitudinal and steering speed gains, respectively. and These represent the disturbance gains for longitudinal direction and steering, respectively. and These represent longitudinal speed and steering speed, respectively. .

[0219] S4.3 Establish a nonlinear error system.

[0220] Based on the extended system and the nonlinear extended state observer, the nonlinear error system is established as follows:

[0221]

[0222]

[0223] Consider the first A nonlinear error system and a nonlinear extended state observer for an unmanned vessel, if exist and And longitudinal velocity gain and longitudinal perturbation gain satisfy:

[0224]

[0225]

[0226] Then the longitudinal velocity estimation error and longitudinal disturbance estimation error Converging to a bounded region;

[0227] If it exists and And steering speed gain and steering disturbance gain satisfy:

[0228]

[0229]

[0230] Then the steering speed estimation error and steering disturbance estimation error Converging to a bounded region;

[0231] Proof: Taking the longitudinal channel as an example, for a nonlinear error system, the Lyapunov function is chosen as:

[0232]

[0233] It is rewritten as:

[0234]

[0235] right After differentiation, its expression is obtained as follows:

[0236]

[0237] Based on the nonlinear error system, we obtain:

[0238]

[0239] make ,in:

[0240]

[0241] equation It is on the error plane by and The parabola formed;

[0242]

[0243] It should be noted that and These represent the effective longitudinal inertia and effective steering inertia of the unmanned vessel, respectively. Because... , , can be obtained According to longitudinal velocity gain and longitudinal perturbation gain According to the inequality conditions, and This means It is a downward-opening parabola with no real zero solutions, therefore It is negative. In and The area below the intersection, This leads to speed estimation error. and disturbance estimation error Convergence. In the region above the intersection point, It may be a positive value.

[0244] But because the boundary is ,Right now If this is determined, then the error amplitude will always remain bounded. Therefore, It is bounded. Similarly, It is also bounded.

[0245] The proof for the yaw channel is the same as the proof for the longitudinal channel, and will not be elaborated further here.

[0246] Therefore, the nonlinear extended state observer is effective for extended state systems.

[0247] Step S5 is as follows:

[0248] S5.1, To track the desired longitudinal velocity and desired yaw speed Design an integral sliding mode controller.

[0249] The expression for the unmanned surface vessel's airframe velocity error is as follows:

[0250]

[0251]

[0252] in, For longitudinal velocity error, This refers to the yaw speed error; For longitudinal velocity, The turning speed, i.e. ; For the desired longitudinal velocity, The desired yaw speed, i.e. ;

[0253] set up For integral sliding surfaces, The expression for the integral sliding surface, representing the observer gain, is:

[0254]

[0255]

[0256] in, , ; and It is a nonlinear function, and its expression is:

[0257]

[0258]

[0259] in, and It is a tuning parameter that depends on the order of the nonlinear convergence rate near the origin; and It is a smoothing parameter used to suppress jitter.

[0260] S5.2, Based on velocity error and integral sliding surface, there exists and and longitudinal steering speed error system , Based on the longitudinal dynamics model and the yaw dynamics model, the derivative of the integral sliding surface is rewritten in the following form:

[0261]

[0262]

[0263] S5.3, Let , These are the fal function parameters for longitudinal direction and steering, respectively.

[0264] The saturation function is defined as follows:

[0265]

[0266]

[0267] in, This is the boundary layer width, used to ensure smooth behavior in the region near the sliding surface.

[0268] set up For the controller gain, the integral sliding mode controller is designed as follows:

[0269]

[0270]

[0271] Considering the nonlinear error system and the integral sliding mode controller, for the gain , , and Longitudinal velocity error and yaw speed error They converge to the bounded region respectively.

[0272] Proof: Using longitudinal velocity error Taking the proof as an example, based on the velocity error system and the integral sliding surface, a Lyapunov function is selected:

[0273]

[0274] Integral sliding mode controller input The derivative is:

[0275]

[0276] According to consideration the The case of a nonlinear error system and a nonlinear extended state observer for an unmanned vessel, exists. Makes all All We can conclude that:

[0277]

[0278] There are two possible scenarios as follows: If Then we can obtain:

[0279]

[0280] exist To ensure .like Then it exists:

[0281]

[0282] It should be noted that when hour, Strictly negative definite. For integral sliding mode controllers... , , and In this situation, After transient decay, it converges to a bounded region. Therefore, the velocity error... Ultimately, it is confined to a smaller, bounded area.

[0283] For yaw speed error The proof is consistent with the above longitudinal velocity error. The proof is similar and will not be elaborated further here.

[0284] Example:

[0285] To verify the effectiveness of the longitudinal formation control method of the present invention, field experiments were conducted.

[0286] The following parameters are set for the three isomorphic unmanned surface vessels:

[0287] Waterline length of unmanned vessels Maximum draft Longitudinal inertia, lateral inertia, and steering inertia are respectively , and The secondary longitudinal damping coefficient and the tertiary longitudinal damping coefficient are respectively and The secondary lateral damping coefficient, lateral cross-coupling damping term, steering damping term, and secondary steering damping coefficient are respectively , , and The torques related to lateral velocity and steering velocity are respectively , and The three steering damping moments are .

[0288] The trajectory tracking curves of the three unmanned boats are as follows: Figure 4 As shown.

[0289] No. Longitudinal position error of an unmanned vessel Lateral position error and heading angle error like Figure 5 As shown, the longitudinal error of the three unmanned ships Fast convergence and maintain Within the range, lateral error Reduce to And maintain within this range, heading angle error Stay at about Within the specified range, the dual-closed-loop control framework exhibits reliable convergence under gust disturbances, with a small steady-state error range.

[0290] The spacing between the three unmanned boats is as follows: Figure 6 As shown. After a brief transition period, the spacing between ships gradually converges to the target spacing. During steady-state operation, the deviation between the inner vessel spacing and the target spacing is basically controlled within 0 to... Within the specified range, the proposed formation mixer effectively reduces the deviation of ship-to-ship spacing from the target value, with the actual ship-to-ship spacing gradually approaching the target curve spacing along the reference path.

[0291] The estimation disturbance of the nonlinear extended state observer is as follows Figure 7 As shown, the three unmanned vessels exhibited significant transient responses in the initial phase. The nonlinear extended state observer continuously estimated the slowly changing perturbation components throughout the experiment.

[0292] The control input of the integral sliding mode controller is as follows Figure 8 As shown. According to the integral sliding mode controller, the longitudinal control input (longitudinal thrust) is... And steering control input (yaw moment) The ships remained within a defined range. Experimental results show that, under gusty conditions in narrow waterways, the longitudinal formation control strategy effectively maintains a small tracking error and a fixed ship spacing.

[0293] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for longitudinal formation control of multiple unmanned vessels based on a nonlinear extended state observer, characterized in that, Includes the following steps: S1. Establish kinematic model, dynamic model, and wind disturbance model, and conduct boundedness analysis of wind disturbance; S2. Design time-corresponding interval increment rules, generate unmanned vessel reference signals, construct a longitudinal scheduling mixer, build a dual closed-loop control framework, and use the output of the longitudinal scheduling mixer as the reference input of the dual closed-loop control framework. S3. Establish the kinematic error system and derive the error dynamics equation; Design a backstepping controller that incorporates a geometric compensation term, and output the desired velocity vector as the reference input for the velocity tracking controller; S4. Establish an extended state system and introduce the equivalent external disturbance as an extended state. Design a nonlinear extended state observer that outputs a disturbance estimate as a disturbance compensation signal; define the observation error and the disturbance estimation error, and derive the nonlinear error system; S5. Design an integral sliding surface and introduce a continuous nonlinear function; establish a velocity error system and substitute it into the dynamic model; design an integral sliding controller with saturation function and disturbance feedforward compensation.

2. The multi-unmanned vessel longitudinal formation control method based on a nonlinear extended state observer according to claim 1, characterized in that, Step S1 includes the following steps: S1.1 Establish the inertial coordinate system and the body coordinate system, define the pose of the unmanned vessel in the inertial coordinate system and its velocity state in the body coordinate system; construct a rotation matrix to map the vessel velocity to the inertial coordinate system to obtain the kinematic equations; the kinematic model is used to construct the pose error system and serve as the design basis for the backstepping controller. S1.2 Select the longitudinal channel and yaw channel as the control objects, and the control inputs are the longitudinal propulsion force and the yaw control torque; establish the nonlinear dynamic equations with additional mass and hydrodynamic damping, and obtain the controlled dynamic expressions of the longitudinal velocity and the turning velocity respectively; unify the modeling uncertainty and external disturbances into the equivalent external disturbance term to provide the observation objects for the design of the nonlinear extended state observer; S1.3 Establish a wind disturbance model. Based on aerodynamic modeling, construct wind force and moment terms and convert them into external disturbances in the longitudinal and yaw channels. Use a first-order Gauss-Markov process to describe the time-varying characteristics of gusts and introduce time constant and steady-state variance parameters to characterize the randomness of gusts. S1.4 Perform a boundedness analysis of wind disturbance. Assuming that the wind speed component has an upper bound, we obtain that the wind-induced equivalent external disturbance has a constant bound. We give the boundedness of disturbance as a prerequisite for the subsequent bounded convergence of the nonlinear extended state observer error and the robustness analysis of the integral sliding mode control method.

3. The multi-unmanned vessel longitudinal formation control method based on a nonlinear extended state observer according to claim 2, characterized in that, Step S1.1 specifically involves: the body coordinate system and the inertial coordinate system being represented as follows: and For the first Unmanned boat, longitudinal speed Along the longitudinal axis lateral speed Along the horizontal axis Steering speed Represents rotational motion along the vertical axis, coordinate system This indicates the position and heading angle of the unmanned surface vessel in the inertial coordinate system. Indicates the yaw angle for counterclockwise rotation; Let the first The pose of an unmanned surface vessel in an inertial coordinate system Unmanned surface vessel velocity state in body coordinate system The kinematic model behavior of the unmanned vessel is as follows: in, This represents the rotation matrix from the body coordinate system to the inertial coordinate system; Step S1.2 specifically involves the following: The dynamic model behavior of the unmanned vessel is as follows: in, Indicates control input, For longitudinal propulsion; For yaw control torque; This represents the effective translational inertia in the longitudinal direction, including the added mass. This represents the effective translational inertia in the lateral direction, including the added mass. This represents the effective yaw moment of inertia, including additional inertia. , , , , , , For the longitudinal nonlinear damping-related hydrodynamic derivative, This represents the second-order transverse damping term. This indicates the yaw lateral damping coupling term. The damping coupling term represents the lateral yaw. This represents the second-order yaw damping term. Indicates the second lateral moment. This indicates the yaw lateral moment coupling term. This indicates the lateral yaw moment coupling term. Indicates the second yaw moment; Indicates the first The equivalent external disturbance caused by gusts of wind to an unmanned vessel; Step S1.3 specifically refers to: The equivalent external disturbance caused by gusts of wind to the unmanned vessel is , This represents the equivalent disturbance of longitudinal gusts. This indicates the equivalent disturbance of lateral gusts. The equivalent disturbance of yaw gust is expressed as follows: in, air density, and The longitudinal and lateral windward projected areas. and These are the drag coefficients for the longitudinal and lateral directions. This represents the longitudinal wind speed component. This represents the lateral wind speed component. For wind power equivalent effector; Described by a first-order Gauss-Markov model as: in, and For longitudinal and lateral gust disturbances, the time constants are given. and The steady-state standard deviations are given by longitudinal and lateral gust disturbances. and Zero-mean, unit-intensity Gaussian white noise; Step S1.4 specifically involves: Since wind speed is limited by airflow and has an upper limit, therefore... Make: 。 4. The multi-unmanned vessel longitudinal formation control method based on a nonlinear extended state observer according to claim 3, characterized in that, Step S2 is as follows: S2.1 Design a formation mixer to generate fixed-interval longitudinal formation aiming points; a reference trajectory for virtual navigation. Parameterized by arc length, the increment along the trajectory is... The status and reference trajectory of virtual navigation Synchronize the current index at the current time. Reference trajectory Recorded as The virtual navigation reference trajectory is input into the formation mixer, and a time delay is added based on the current unmanned vessel number. , Given the expected lag distance interval between two adjacent ships along the reference trajectory, we obtain the first... A reference trajectory following the unmanned vessel ,Right now The reference trajectory is then distributed to the corresponding unmanned vessels. S2.2 Establish a dual-closed-loop control framework for the unmanned surface vessel. This framework includes a backstepping controller, an integral sliding mode controller, and a nonlinear extended state observer, wherein: Reference trajectory vector From pose reference and speed reference composition, For vertical position reference, For horizontal position reference, For heading angle reference, For longitudinal velocity reference, For lateral velocity reference, For yaw speed reference; Attitude error vector From pose reference With the actual pose state vector Obtained by subtraction. For longitudinal error, For lateral error, This refers to the heading angle error; The backstep controller receives the attitude error vector. And generate the desired velocity vector. , For the desired longitudinal velocity, For the desired lateral velocity, The desired yaw speed; velocity error vector Depend on With the actual velocity state vector We obtain the result by taking the difference, where For longitudinal velocity error, This refers to the yaw speed error; Integral sliding mode controller receives velocity error vector and generate control input vectors. , For vertical input, For steering input; The estimated vector of gust disturbance Input the integral sliding mode controller for gust disturbance compensation. To estimate longitudinal disturbances, To estimate yaw disturbance; Nonlinear extended state observer for estimating gust disturbances .

5. The multi-unmanned vessel longitudinal formation control method based on a nonlinear extended state observer according to claim 4, characterized in that, Step S3 is as follows: S3.1 For the pose tracking problem in an inertial coordinate system, design a backstepping controller; for the first... For an unmanned surface vessel, the kinematic model is reconstructed, and the expression is: To track the current pose reference of the unmanned surface vessel The error in the inertial coordinate system is defined as... , , In the body reference frame, attitude error The expression is as follows: Based on the errors in the kinematic model and the inertial coordinate system, the kinematic error system can be expressed in the following form: S3.2 The backstepping controller is designed as follows: in, This is an error coupling term; and Positive control gain, The position error modulus; and Geometric compensation for heading deviation , and This is the error proportion term. For reference angular velocity; Consider the first If the kinematic error system of an unmanned surface vessel under the action of a backstepping controller exists... , , and Then the machine coordinate system error , and Will follow It converges asymptotically to zero.

6. The multi-unmanned vessel longitudinal formation control method based on a nonlinear extended state observer according to claim 4, characterized in that, Step S4 is as follows: S4.1 Based on the dynamic model, it is rewritten in the following form: in, Indicates longitudinal velocity. Indicates yaw speed; and Indicates longitudinal control input and steering control input; This indicates a longitudinal disturbance. Indicates yaw disturbance; Based on the longitudinal dynamics model and the yaw dynamics model, the expanding system is rewritten in the following form: in, This indicates the longitudinal velocity estimate. This indicates the yaw speed estimate; This represents the longitudinal perturbation estimate. This indicates the yaw disturbance estimate; and These represent the effective longitudinal inertia and effective turning inertia of the unmanned vessel, respectively. ; express The derivative of express The derivative; S4.

2. Based on the extended system, the nonlinear extended state observer is expressed as: The longitudinal and steering speed estimation errors are expressed as follows: The longitudinal and steering disturbance estimation errors are expressed as follows: in, and These represent the speed estimates for longitudinal direction and steering, respectively. and These represent the disturbance estimates for the longitudinal direction and steering, respectively. and These represent the longitudinal and steering speed gains, respectively. and These represent the disturbance gains for longitudinal direction and steering, respectively. and These represent longitudinal speed and steering speed, respectively. ; S4.

3. Based on the extended system and the nonlinear extended state observer, the nonlinear error system is established as follows: Consider the first A nonlinear error system and a nonlinear extended state observer for an unmanned vessel, if exist and And longitudinal velocity gain and longitudinal perturbation gain satisfy: Then the longitudinal velocity estimation error and longitudinal disturbance estimation error Converging to a bounded region; If it exists and And steering speed gain and steering disturbance gain satisfy: Then the steering speed estimation error and steering disturbance estimation error It converges to the bounded region.

7. The multi-unmanned vessel longitudinal formation control method based on a nonlinear extended state observer according to claim 4, characterized in that, Step S5 is as follows: S5.1, To track the desired longitudinal velocity and desired yaw speed Design an integral sliding mode controller; The expression for the unmanned surface vessel's airframe velocity error is as follows: in, For longitudinal velocity error, This refers to the yaw speed error; For longitudinal velocity, The turning speed, i.e. ; For the desired longitudinal velocity, The desired yaw speed, i.e. ; set up For integral sliding surfaces, The expression for the integral sliding surface, representing the observer gain, is: in, , ; and It is a nonlinear function, and its expression is: in, and It is a tuning parameter that depends on the order of the nonlinear convergence rate near the origin; and It is a smoothing parameter used to suppress jitter; S5.2, Based on velocity error and integral sliding surface, there exists and and longitudinal steering speed error system , Based on the longitudinal dynamics model and the yaw dynamics model, the derivative of the integral sliding surface is rewritten in the following form: S5.3, Let , These are the fal function parameters for longitudinal direction and steering, respectively; The saturation function is defined as follows: in, This is the boundary layer width, used to ensure smooth behavior in the region near the sliding surface; set up For the controller gain, the integral sliding mode controller is designed as follows: Considering the nonlinear error system and the integral sliding mode controller, for the gain , , and Longitudinal velocity error and yaw speed error They converge to the bounded region respectively.

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