Machine / ship cooperative trajectory tracking control method based on nonsingular sliding mode and RBF neural network

Abstract

The invention discloses a non-singular sliding mode and RBF neural network-based aircraft / ship cooperative trajectory tracking control method. The method comprises the steps of constructing a mathematical model of an under-actuated ship and a mathematical model of a four-rotor unmanned aerial vehicle; constructing a virtual control law of the underactuated ship according to the mathematical model of the underactuated ship based on the set expected trajectory of the underactuated ship; constructing a ship controller based on a nonsingular sliding mode and an RBF neural network; constructing an attitude control law of the unmanned aerial vehicle according to the mathematical model of the four-rotor unmanned aerial vehicle; and constructing a position control law of the unmanned aerial vehicle according to the mathematical model of the four-rotor unmanned aerial vehicle, and designing a non-singular sliding mode controller according to the position control law so as to realize vehicle / ship cooperative trajectory tracking control. The method solves the problems that in the prior art, state matching of aircraft / ship cooperation is insufficient, accurate transmission of real-time position and attitude information of a ship to the unmanned aerial vehicle cannot be effectively achieved, and meanwhile it cannot be guaranteed that the position and attitude of the unmanned aerial vehicle can rapidly respond to ship trajectory changes in the dynamic following process; and the problem of cooperative failure caused by information delay or control deviation is avoided.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of trajectory tracking and ship / aircraft cooperative control of underactuated ships, and particularly relates to a ship / aircraft cooperative trajectory tracking control method based on a non-singular sliding mode and a RBF neural network. BACKGROUND

[0002] In marine operations, a single ship or unmanned aerial vehicle often has functional limitations: the ship is limited in maneuverability and is difficult to quickly obtain information about a large range of sea areas, and under time-varying disturbances such as wind, waves and currents, the trajectory tracking accuracy is easily affected; the unmanned aerial vehicle has the advantages of flexible maneuverability and wide field of view, but is limited by endurance and payload and cannot independently complete long-time and long-distance marine operation tasks. Therefore, through ship / aircraft cooperation, the ship is used as the main operation platform and endurance support, and the unmanned aerial vehicle is used as an aerial observation, data relay or auxiliary operation unit to form a cooperative mode of "ship as base and aircraft as eye", which becomes a key technical direction for improving the efficiency and safety of marine operations.

[0003] The core requirement of the technology is to solve two core problems: one is high-precision trajectory tracking of underactuated ships, since most civilian and engineering ships are underactuated structures (without lateral propulsion devices), under the existence of model parameter uncertainty (such as inertia parameter deviation caused by ship body wear and load change) and external disturbance (time-varying effect of wind, waves and currents), the traditional control method is difficult to balance the tracking accuracy and system stability; the second is the insufficient state matching of ship / aircraft cooperation, which cannot effectively realize the accurate transmission of ship real-time position and attitude information to the unmanned aerial vehicle, and cannot guarantee that the position and attitude of the unmanned aerial vehicle can quickly respond to the changes in the ship trajectory during dynamic following, avoiding the problem of cooperative failure caused by information delay or control deviation. SUMMARY

[0004] The present application provides a ship / aircraft cooperative trajectory tracking control method based on a non-singular sliding mode and a RBF neural network to overcome the above technical problems.

[0005] In order to achieve the above purpose, the technical solution of the present application is as follows: A ship / aircraft cooperative trajectory tracking control method based on a non-singular sliding mode and a RBF neural network, specifically comprising the following steps: S1: constructing a mathematical model of an underactuated ship and a mathematical model of a quadcopter unmanned aerial vehicle; S2: based on a set expected trajectory of the underactuated ship, constructing a virtual control law of the underactuated ship according to the mathematical model of the underactuated ship; S3: constructing a ship controller based on a non-singular sliding mode and a RBF neural network according to the virtual control law of the underactuated ship; and the ship controller is used to generate a control signal for controlling the movement of the ship, and the control signal includes a longitudinal propulsion force control law and a steering torque control law. S4: constructing the attitude control law of the quadrotor based on the desired attitude of the quadrotor and the mathematical model of the quadrotor combined with the non-singular sliding mode surface; S5: constructing the position control law of the quadrotor based on the desired trajectory of the quadrotor and the mathematical model of the quadrotor combined with the non-singular sliding mode surface, and designing the non-singular sliding mode controller according to the position control law; S6: giving an expected trajectory to the underactuated ship in the preset coordinate system, making the underactuated ship realize trajectory tracking through the ship controller based on the non-singular sliding mode and the RBF neural network, obtaining the real-time position information of the underactuated ship and transmitting it to the quadrotor, setting the expected trajectory of the quadrotor by combining the given expected height, and making the quadrotor realize trajectory tracking through the non-singular sliding mode controller and the attitude control law of the quadrotor by combining the given expected attitude, thereby realizing the collaborative trajectory tracking control of the ship and the quadrotor.

[0006] Further, the mathematical model of the underactuated ship constructed in S1 has an expression of (1) In the formula, represents the position of the underactuated ship in the inertial coordinate system; represents the heading angle of the underactuated ship; respectively represent the forward speed, lateral drift speed and yaw speed of the ship in the body coordinate system; represents the inertia parameter of the underactuated ship; represents the hydrodynamic damping coefficient of the underactuated ship system; represents the model uncertainty term; respectively represent the longitudinal propulsion force and steering torque of the underactuated ship; represents the time-varying disturbance caused by the external environment; represents the first derivative of ; represents the first derivative of ; represents the first derivative of ; The mathematical model of the quadrotor constructed has an expression of (2) In the formula, represents the four expected control inputs to be designed; represents the actual position of the quadrotor; represents the actual attitude angle of the quadrotor; respectively represent the moments of inertia of the quadrotor around the preset coordinate system three axes; denotes a total mass of the quadcopter; denotes a gravitational constant; denotes a drag coefficient; denotes an external disturbance; denotes a first derivative; denotes a second derivative; denotes a first derivative; denotes a second derivative.

[0007] Further, the S2 specifically comprises the following steps: S21: based on the set underactuated ship desired trajectory, obtaining the ship position error according to the mathematical model of the underactuated ship, and the obtaining formula of the ship position error is (3) In the formula: denotes the longitudinal and transverse coordinates of the underactuated ship desired trajectory; denote the longitudinal and transverse position errors respectively; S22: obtaining the ship position error derivative by derivation of the ship position error, and the expression is (4) In the formula: denotes the first derivative of ; denotes the first derivative of ; S23: in order to make the ship position error approach to zero, the in formula (4) is regarded as a virtual control quantity, a virtual control law of the underactuated ship is designed, and the virtual control law of the underactuated ship includes the longitudinal virtual control rate and the transverse virtual control rate , and the expression is (5) In the formula: , denote design constants.

[0008] Further, the S3 specifically comprises the following steps: S31: according to the virtual control law of the underactuated ship and the mathematical model of the underactuated ship, obtaining the ship speed error, and the ship speed error includes the longitudinal speed error and the transverse speed error ; The obtaining formula of the ship speed error is (6) S32: According to the longitudinal velocity error Design a first-order nonsingular sliding mode surface ; According to the lateral velocity error Design a second-order nonsingular sliding mode surface , the expression of which is (7) In the formula: Indicates The short form of Both indicate positive odd numbers and satisfy ; Indicates a positive real number; S33: Derive the first-order nonsingular sliding mode surface And the second-order nonsingular sliding mode surface , the expression of which is (8) S34: Define the intermediate parameter quantity according to the ship position error , the expression of which is (9) In the formula: Indicates the intermediate variable and ; Then according to the intermediate parameter quantity , the relationship between the longitudinal virtual control rate And the lateral virtual control rate Is expressed as (10) In the formula: Indicates the second-order derivative of ; Indicates the first-order derivative of ; Indicates the first-order derivative of ; S35: Combine formula (1), formula (6), and formula (10) with formula (8) to obtain (11) In the formula: Indicates the first-order derivative of ; S36: Based on the RBF neural network, the model uncertain part in formula (11) Approximation processing, the expression of which is (12) In the formula: Indicates the input vector of the RBF neural network and ; Indicates transposition; a radial basis function vector representing the RBF neural network; an approximation error of the RBF neural network; and satisfying , an error bound value and ; an ideal weight vector from the hidden layer to the output layer, an element in ; S37: constructing a ship controller based on the non-singular sliding mode and the RBF neural network according to the formula (12) combined with the formula (7); and the ship controller is used to generate a control signal for controlling ship movement, the control signal including a longitudinal propulsion force control law and a steering torque control law; and the expression of the ship controller based on the non-singular sliding mode and the RBF neural network is (13) (14) In the formula: an estimated value of ; a first derivative of ; a design constant and ; respectively represent estimated values of external time-varying disturbances ; an intermediate parameter quantity and ; a design constant; a design constant and ; a design parameter and ; a design constant; respectively represent priori estimated values of .

[0009] Further, the S4 specifically includes the following steps: S41: based on setting a desired attitude of the quadcopter, acquiring an attitude error of the quadcopter according to a mathematical model of the quadcopter; The acquisition formula of the attitude error is (15) In the formula: representing a desired attitude of the quadcopter; representing a roll angle tracking error, a pitch angle tracking error, and a yaw angle tracking error; S42: Construct a sliding surface for roll angle tracking error based on attitude error. Pitch angle tracking error sliding surface and yaw angle tracking error sliding surface Its expression is (16) In the formula: All represent positive odd numbers in the design and satisfy ; All represent positive real numbers in the design; express The first derivative; S43: Sliding surface based on roll angle tracking error Pitch angle tracking error sliding surface and yaw angle tracking error sliding surface Construct the approximation rate of the sliding surface for roll angle tracking error. Pitch angle tracking error sliding surface convergence rate and the approach rate of the sliding surface of the yaw angle tracking error Its expression is (17) In the formula: All represent known positive numbers; Represents a symbolic function; S44: Approach rate of the sliding surface based on roll angle tracking error Pitch angle tracking error sliding surface convergence rate and the approach rate of the sliding surface of the yaw angle tracking error To construct the attitude control law for the drone; The attitude control law of the UAV is expressed as follows: (18).

[0010] Furthermore, S5 specifically includes the following steps: S51: By rewriting the mathematical model of a quadcopter UAV, we can obtain... (19) (20) In the formula: This indicates the virtual control input for the position of the quadcopter drone; S52: Based on the desired trajectory of a quadcopter UAV, combined with equation (2), the UAV position tracking error can be obtained, and its expression is as follows: (twenty one) In the formula: represents a desired position of the quadrotor UAV; represents a position tracking error of the quadrotor UAV; S53: constructing a non-singular sliding mode surface of the quadrotor UAV position tracking according to the quadrotor UAV position tracking error, and the expression is (22) wherein: respectively represent a lateral position tracking non-singular sliding mode surface, a longitudinal position tracking non-singular sliding mode surface and a vertical position tracking non-singular sliding mode surface of the quadrotor UAV position tracking; represents a first derivative of ; all represent designed positive odd numbers, and satisfy ; all represent known positive real numbers; S54: constructing a reaching rate of the non-singular sliding mode surface of the quadrotor UAV position tracking according to the non-singular sliding mode surface of the quadrotor UAV position tracking, and the expression is (23) wherein: all represent known normal numbers; represents a sign function; S55: constructing a position control law of the quadrotor UAV according to the combination of formula (22) and formula (23) and formula (20), and the expression is (24) and designing a non-singular sliding mode controller according to the position control law, and the expression of the non-singular sliding mode controller is (25).

[0011] Beneficial effects: the application provides a ship / UAV cooperative trajectory tracking control method based on a non-singular sliding mode and a RBF neural network, a ship / UAV control rate is designed by using a non-singular sliding mode to realize trajectory tracking control of ship / UAV cooperation, that is, a ship controller based on a non-singular sliding mode and a RBF neural network is constructed; an attitude control law of the quadrotor UAV is constructed according to a mathematical model of the quadrotor UAV and a non-singular sliding mode surface; a position control law of the quadrotor UAV is constructed according to a desired trajectory of the quadrotor UAV, a mathematical model of the quadrotor UAV and a non-singular sliding mode surface to design a non-singular sliding mode controller; the method effectively solves the problems of low trajectory tracking precision of an underactuated ship under the influence of wind, wave and flow interference and model parameter changes, and complex problems such as difficult state matching and weak anti-interference ability in the ship / UAV cooperation process, and provides a reliable theoretical basis and practical method for cooperative control of the underactuated ship and the UAV in a complex marine environment, and helps to promote efficient application of the ship / UAV cooperative system in many fields such as shipping safety protection, emergency rescue and marine operation.​ Attached Figure Description

[0012] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0013] Figure 1 The flowchart is a process for the machine / ship cooperative trajectory tracking control method based on non-singular sliding mode and RBF neural network of the present invention. Figure 2 This is a schematic diagram of the coordinate system between the underactuated ship and the quadcopter UAV in this embodiment; Figure 3 This is a graph showing the expected and actual trajectories of the vessel in this embodiment; Figure 4 This is a graph showing the missing ship position in this embodiment; Figure 5 This is a graph showing the ship's speed in this embodiment; Figure 6 This is a graph showing the ship position error in this embodiment; Figure 7 This is a graph showing the ship speed error in this embodiment; Figure 8 This is a simulation graph showing the estimated values ​​of the model's uncertainty terms in this embodiment; Figure 9 This is a simulation diagram showing the estimated upper bound of the external disturbance in this embodiment; Figure 10 This is a simulation diagram of the machine / ship cooperative motion trajectory in this embodiment; Figure 11 This is a position curve of the quadcopter UAV in the x-direction in this embodiment; Figure 12 This is a position curve of the quadcopter UAV in the y-direction in this embodiment; Figure 13 This is a position curve of the quadcopter UAV in the z-direction in this embodiment; Figure 14 This is a simulation diagram of the attitude yaw angle of the quadcopter UAV in this embodiment. Detailed Implementation

[0014] In order to make the purposes, technical solutions and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are some but not all of the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work fall within the protection scope of the present application.

[0015] The embodiment provides a ship / UAV cooperative trajectory tracking control method based on a non-singular sliding mode and a RBF neural network, as shown in Figure 1 The embodiment specifically comprises the following steps: S1: constructing a mathematical model of an underactuated ship and a mathematical model of a quadrotor unmanned aerial vehicle; Specifically, the embodiment assumes that the ship mass is uniformly distributed and left-right symmetrical, and only three degrees of freedom of forward motion, transverse drift and yaw motion are considered, and the simplified ship motion inertial coordinate system and the appendage coordinate system are as shown in Figure 2 The inertial coordinate system is shown in The starting position is shown in The north direction is shown in The east direction is shown in The appendage coordinate system is shown in The midpoint position of the ship between the bow and the stern is shown in The direction along the ship centerline to the bow is shown in The direction to the right side of the ship is shown in The dynamics model of the underactuated surface ship with model uncertainty and external disturbance, i.e. the mathematical model of the underactuated ship, is as follows: (1) In the formula, The position of the underactuated ship in the inertial coordinate system is shown in The heading angle of the underactuated ship is shown in The forward speed, transverse drift speed and yaw speed of the ship in the appendage coordinate system are shown in The inertia parameters of the underactuated ship are shown in The hydrodynamic damping coefficient of the underactuated ship system is shown in The model uncertainty term is shown in The longitudinal propulsion force and steering torque of the underactuated ship are shown in The time-varying disturbance caused by the external environment is shown in The first derivative of is shown in The first derivative of is shown in The first derivative of is shown in In addition, the unmanned aerial vehicle structure and motion principle are accurately established in this embodiment, as shown in Figure 2 An inertial coordinate system is established as shown in An attached body coordinate system is established as shown in wherein represents the lift generated by the four rotors of the unmanned aerial vehicle, assuming that the body is a rigid body, the body mass center and the geometric center of gravity are completely coincident, the body is completely symmetrical, and the body model parameters are all normal, then the dynamic model of the four-rotor unmanned aerial vehicle with external disturbance, that is, the mathematical model of the four-rotor unmanned aerial vehicle, is expressed as (2) In the formula: represents the four expected control inputs to be designed; represents the actual position of the four-rotor unmanned aerial vehicle; represents the actual attitude angle of the four-rotor unmanned aerial vehicle; respectively represent the rotational inertia of the body of the four-rotor unmanned aerial vehicle around the preset coordinate system three axes; represents the total mass of the four-rotor unmanned aerial vehicle; represents the gravitational constant; represents the drag coefficient; represents the external disturbance; represents the first derivative of ; represents the second derivative of ; represents the first derivative of ; represents the second derivative of ; The embodiment is to realize the design of the neural network-based underactuated ship trajectory tracking adaptive sliding mode control law, and the following assumptions are introduced: Assumption 1. The expected trajectory of the ship is smooth and differentiable, and the first and second derivatives exist; Assumption 2. The external disturbance acting on the underactuated ship is a time-varying disturbance, and satisfies (3) In the formula: represents the upper bound of the external environmental disturbance, and the bound value is unknown; this embodiment considers the case that the ship has model uncertainty and external disturbance, and uses neural network and non-singular sliding mode control method to complete the design of the longitudinal propulsion force and steering torque of the underactuated ship, so that it can sail along the expected trajectory; the four-rotor unmanned aerial vehicle system is divided into a position and an attitude subsystem, and the non-singular sliding mode control method is used to enable the unmanned aerial vehicle to track the sailing trajectory of the underactuated ship, so as to realize the trajectory tracking of the ship / aerial vehicle cooperation; S2: based on the set underactuated ship desired trajectory, constructing the virtual control law of the underactuated ship according to the mathematical model of the underactuated ship, specifically including the following steps: S21: based on the set underactuated ship desired trajectory, obtaining the ship position error according to the mathematical model of the underactuated ship, and the formula for obtaining the ship position error is (4) In the formula, represents the longitudinal and transverse coordinates of the underactuated ship desired trajectory; respectively represent the longitudinal and transverse position errors; S22: obtaining the derivative of the ship position error with respect to time to obtain the derivative of the ship position error, and the expression is (5) In the formula, represents the first derivative of ; represents the first derivative of ; S23: in order to make the ship position error approach zero, the in formula (5) is regarded as a virtual control variable, and the virtual control law of the underactuated ship is designed, and the virtual control law of the underactuated ship includes longitudinal virtual control rate and transverse virtual control rate , and the expression is (6) In the formula, , represents a design constant.

[0016] In addition, when , formula (6) is substituted into formula (5) to obtain (7) Construct Lyapunov function: (8) Deriving formula (8) can obtain: (9) From formula (9), it can be seen that , combined with formula (7) and formula (8), it can be seen that the position error is asymptotically convergent to zero; let , and deriving formula (6) can obtain (10) S3: Based on the virtual control law of underactuated ships, a ship controller based on non-singular sliding mode and RBF neural network is constructed; and the ship controller is used to generate control signals for controlling the motion of the ship, the control signals including longitudinal propulsion control law and steering torque control law; Specifically, the following steps are included: S31: Based on the virtual control law and mathematical model of the underactuated ship, obtain the ship speed error, wherein the ship speed error includes the longitudinal speed error. Lateral velocity error ; The formula for obtaining the ship speed error is as follows: (11) S32: To enable faster system convergence and better handle uncertainties and disturbances in the system, this embodiment uses a non-singular sliding surface instead of a traditional sliding surface. Regarding the longitudinal velocity error... Design a first-order nonsingular sliding surface Since underactuated vessels lack lateral propulsion, steering torque must be introduced during the design process. Regarding lateral velocity error Design a second-order nonsingular sliding surface Its expression is (12) In the formula: express The abbreviated form; All represent positive odd numbers and satisfy ; Represents positive real numbers; S33: For the first-order non-singular sliding surface With the second-order non-singular sliding surface The time derivative is given by: (13) S34: For ease of representation, intermediate parameter quantities are defined based on the ship's position error. Its expression is (14) In the formula: Represents intermediate variables and ; Then based on the amount of intermediate parameters It can realize the vertical virtual control rate With horizontal virtual control rate The relationship between them is represented as (15) In the formula: express The second derivative; express The first derivative; express The first derivative; S35: Combining equations (1), (11), and (15) with equation (13), we can obtain (16) In the formula: express The first derivative; S36: Due to the fact that in equation (16) Since the model is uncertain, the control law cannot be directly designed. Furthermore, the RBF neural network has strong self-learning ability and can approximate any function. Therefore, in this embodiment, the uncertain part of the model in equation (16) is based on the RBF neural network. Approximation processing, its expression is: (17) In the formula: Describes the input vector of the RBF neural network and ; Indicates transpose; Represents the radial basis function vector of an RBF neural network; Describes the approximation error of the RBF neural network; and satisfies , Indicates the error threshold and ; This represents the radial basis function vector of the neural network, and the output expression of the basis functions is: (18) In the formula: The width of the Gaussian function; For the first hidden layer of the neural network One node; The center point vector value of the Gaussian function, and the input vector They have the same dimension; Let represent the ideal weight vector from the hidden layer to the output layer, and take it as for . Make Minimum vector The value of is expressed as . (19) Due to the ideal weight in practical applications It is unattainable, so it is used in the design control rate process. The estimated value , assuming neural network weights are bounded, i.e. there exists a constant such that ; S37: constructing a ship controller based on the nonsingular sliding mode and RBF neural network according to formula (19) combined with formula (12); and the ship controller is used to generate a control signal for controlling ship movement, the control signal including a longitudinal propulsion force control law and a steering torque control law; and the expression of the ship controller based on the nonsingular sliding mode and RBF neural network is (20) The weight adaptive rate of the ship controller is designed as: (21) The parameter adaptive rate of the ship controller is designed as: (22) In the formula: represents an estimated value of ; represents a first derivative of ; represents a design constant and ; respectively represent estimated values of the upper and lower bounds of the external time-varying disturbance ; represents an intermediate parameter quantity and ; represents a design constant; represents a design constant and ; represents a design parameter and ; represents a design constant; respectively represent priori estimated values of .

[0017] The stability analysis of the ship controller based on the nonsingular sliding mode and RBF neural network is also included in the embodiment: Theorem 1. For the designed trajectory tracking system, considering the existence of model uncertainty and external environmental disturbance of the underactuated ship mathematical model (1), assuming that assumptions 1 and 2 are true, using the RBF neural network algorithm to approximate the uncertainty in the model by using the weight adaptive rate (21), and designing the parameter adaptive rate (22) to estimate the upper and lower bounds of the external environmental disturbance, under the action of the ship controller (20), by designing appropriate parameters , , , , , 、 、 、 、 、 、 、 、 、 、 , which can ensure the asymptotic stability of the ship system; The Lyapunov function (23) is constructed as follows: (23) In the formula: 、 is the parameter estimation error; 、 is the weight estimation error; Substitute equations (16), (20), (21), (22) into (23) to derive and simplify to obtain (24) According to the lemma: for any positive number , there is (25) The following inequality is considered in this embodiment: (26) (27) (28) Substitute the above equation into equation (24) to obtain (29) In the formula: (30) (31) From equation (29), it can be obtained that: (32) From equation (32), it can be known that: is uniformly bounded; from equation (29), it can be seen that , , , , , is uniformly bounded, and thus it can be known that the ship speed tracking error , is bounded, and from equations (7), (8), (9), it can be known that the ship position tracking error , is bounded, and all error signals in the ship system are uniformly bounded, i.e. the stability of the ship controller based on the non-singular sliding mode and the RBF neural network is proved.

[0018] S4: based on the desired attitude of the quadrotor UAV, according to the mathematical model of the quadrotor UAV combined with the non-singular sliding mode surface, the attitude control law of the UAV is constructed, which specifically includes the following steps: S41: based on the desired attitude of the quadrotor UAV, according to the mathematical model of the quadrotor UAV, the attitude error of the quadrotor UAV is obtained; The formula for obtaining the attitude error is (33) In the formula, represents the desired attitude of the quadrotor UAV; represents the roll angle tracking error, the pitch angle tracking error and the yaw angle tracking error; S42: in order to make the system converge quickly and better handle the uncertainty and disturbance in the system, the non-singular sliding mode surface is selected instead of the traditional sliding mode surface, i.e. according to the attitude error, the roll angle tracking error sliding mode surface , the pitch angle tracking error sliding mode surface and the yaw angle tracking error sliding mode surface are constructed, and the expression is (34) In the formula, all represent a designed positive odd number and satisfy ; all represent a designed positive real number; represents the first derivative of ; S43: according to the roll angle tracking error sliding mode surface , the pitch angle tracking error sliding mode surface and the yaw angle tracking error sliding mode surface , the approach rate of the roll angle tracking error sliding mode surface , the approach rate of the pitch angle tracking error sliding mode surface and the approach rate of the yaw angle tracking error sliding mode surface are constructed, and the expression is (35) In the formula, all represent known normal numbers; represents the sign function; S44: according to the approach rate of the roll angle tracking error sliding mode surface , the approach rate of the pitch angle tracking error sliding mode surface and the approaching rate of the yaw angle tracking error sliding mode surface , the attitude control law of the unmanned aerial vehicle is constructed. The attitude control law of the unmanned aerial vehicle is expressed as (36) For the designed attitude control law of the unmanned aerial vehicle, in order to analyze the stability of the system, the Lyapunov function is selected as: (37) Substituting formula (35) and (36) into formula (34) obtains (38) Deriving formula (37) and substituting formula (38) obtains: (39) In the formula, .

[0019] From formula (39), it can be seen that Combined with formula (34) and formula (37), it can be seen that the system is asymptotically stable.

[0020] S5: Based on the expected trajectory of the quad-rotor unmanned aerial vehicle, the position control law of the unmanned aerial vehicle is constructed according to the mathematical model of the quad-rotor unmanned aerial vehicle combined with the non-singular sliding mode surface, and the non-singular sliding mode controller is designed according to the position control law, which specifically includes the following steps: S51: The mathematical model of the quad-rotor unmanned aerial vehicle is rewritten, and the position subsystem mathematical model can be obtained, which is expressed as (40) (41) In the formula, represents the position virtual control input of the quad-rotor unmanned aerial vehicle; S52: Based on the expected trajectory of the quad-rotor unmanned aerial vehicle combined with formula (2), the unmanned aerial vehicle position tracking error can be obtained, which is expressed as (42) In the formula, represents the expected position of the quad-rotor unmanned aerial vehicle; represents the position tracking error of the quad-rotor unmanned aerial vehicle; S53: According to the unmanned aerial vehicle position tracking error, the non-singular sliding mode surface of the unmanned aerial vehicle position tracking is constructed, which is expressed as (43) In the formula, respectively represent a lateral position tracking non-singular sliding mode surface, a longitudinal position tracking non-singular sliding mode surface and a vertical position tracking non-singular sliding mode surface of the unmanned aerial vehicle position tracking; represent a first derivative of all represent a designed positive odd number, and satisfy ; all represent a known positive real number; S54: constructing a non-singular sliding mode surface approaching rate of the unmanned aerial vehicle position tracking non-singular sliding mode surface according to the unmanned aerial vehicle position tracking non-singular sliding mode surface , the expression of which is (44) wherein: all represent a known normal number; represent a sign function; S55: constructing a position control law of the unmanned aerial vehicle according to the combination of formula (41) and formula (43) with formula (44), the expression of which is (45) and designing a non-singular sliding mode controller according to the position control law, and the expression of the non-singular sliding mode controller is (46) In order to prove the stability of the system, i.e. the stability of the non-singular sliding mode controller, the Lyapunov function is selected as: (47) Deriving formula (47) and substituting formula (43), (44) and (45) into it can obtain: (48) wherein: ; It can be seen from formula (48) that Combining formula (43) with formula (47) can know that the system is asymptotically stable.

[0021] S6: giving an underactuated ship a desired trajectory in a preset coordinate system, making the underactuated ship realize trajectory tracking through a ship controller based on a non-singular sliding mode and a RBF neural network; obtaining real-time position information of the underactuated ship and transmitting it to the quad-rotor unmanned aerial vehicle, setting a desired trajectory of the quad-rotor unmanned aerial vehicle through the combination of a given desired height, and making the quad-rotor unmanned aerial vehicle realize trajectory tracking through the combination of a given desired attitude, a non-singular sliding mode controller and an attitude control law of the unmanned aerial vehicle, thereby realizing the ship / aircraft collaborative trajectory tracking control.

[0022] The embodiment also includes simulation experiments and result analysis: ​The simulation experiment in the embodiment is compared with the common linear sliding mode surface to observe the control effect of trajectory tracking, the approximation effect on the uncertain part of the ship model, and the tracking effect of the attitude and position of the unmanned aerial vehicle. The ship expected trajectory equation selected in the embodiment is as follows (49) The design parameters of the underactuated ship control law in the embodiment , , , , , , , , , , , , , , , , , , , The force and torque generated by the external disturbance are , , ; The ship motion trajectory under the non-singular sliding mode control rate in the embodiment is compared with the ship motion trajectory under the traditional sliding mode control rate, as shown in Figures 3-4 The blue dashed line in the figure is the ship motion trajectory under the traditional sliding mode control rate, the red dotted line is the ship motion trajectory under the non-singular sliding mode control rate in the embodiment, and the yellow solid line is the expected trajectory of the ship selected in the embodiment. As shown in Figures 5-7 The blue dashed line is the ship motion trajectory under the traditional sliding mode control rate, and the red solid line is the ship motion trajectory under the non-singular sliding mode control rate in the embodiment. As shown in Figure 3 It can be seen that the ship can reach the expected trajectory under the two control rates, but it can be clearly seen that the ship has a large overshoot before reaching the expected motion trajectory under the traditional sliding mode controller, and the ship motion trajectory converges faster under the control rate in the embodiment. As shown in Figures 4-5 It can be seen that the ship under the non-singular sliding mode controller converges faster than the traditional sliding mode controller in the position, forward speed, lateral drift speed, and yaw angle speed in the , directions, and has a smaller overshoot. As shown in Figure 6 It can be seen that the ship under the traditional sliding mode controller has a larger position error , in the , The average time of convergence to 0 is 130 s; the average convergence time under the non-singular sliding mode surface controller is 85 s, and the convergence speed is improved by 34.6% compared with the traditional sliding mode surface controller. From Figure 7 it can be seen that the error of the forward speed and the lateral drift speed of the ship under the traditional sliding mode surface controller , The average time of convergence to 0 is 200 s; the average convergence time under the non-singular sliding mode surface controller is 70 s, and the convergence speed is improved by 65% compared with the traditional sliding mode surface controller. The above shows that the non-singular sliding mode surface controller has certain performance improvement compared with the traditional sliding mode surface controller.

[0023] This embodiment uses the RBF neural network control algorithm to approximate the model uncertainty part, the node selection of the RBF neural network hidden layer is , the center points are uniformly distributed on , the width of the Gaussian basis function is selected as , , the initial value of the weight estimation is , ; the ship model uncertainty term is , , ; The actual model uncertainty term curve is shown by the blue solid line in Figure 8 , and the model uncertainty approximation value curve is shown by the red dashed line; The actual disturbance curve is shown by the blue solid line in Figure 9 , and the upper bound approximation value of the external disturbance is shown by the red solid line. From Figure 8 it can be seen that the RBF neural network part in the controller designed in this embodiment can estimate the uncertainty term of the ship model, and has good approximation performance; from Figure 9 it can be seen that the parameter adaptive rate designed in this embodiment can estimate the upper bound of the external environmental disturbance according to the change of the external environmental disturbance. In order to realize the ship / unmanned aerial vehicle cooperative control described in this embodiment, the real-time position of the ship in the , direction at each moment is taken as the expected position of the unmanned aerial vehicle in the , direction at each moment, the expected trajectory in the direction is selected as , and the expected attitude yaw angle is °. Under the control of the unmanned aerial vehicle position controller and the unmanned aerial vehicle attitude controller designed in this embodiment, the position and attitude image of the unmanned aerial vehicle is shown in Figures 10-14 . The expected path of the unmanned aerial vehicle is shown by the purple dashed line in Figure 10 , the actual path of the unmanned aerial vehicle is shown by the yellow solid line curve, the expected path of the ship is shown by the blue solid line, and the actual path of the ship is shown by the red dashed line. Figures 11-14The middle red solid line is the expected path of the UAV, and the blue dashed line is the actual path of the UAV. Figure 10 It can be seen that the UAV can well track the trajectory of the ship under the action of the controller of the embodiment, and the actual path of the UAV is very close to the expected path. Figures 11-14 It can be seen that the UAV can reach the expected position in the , , direction under the action of the position controller designed in the embodiment, and the yaw angle of the UAV can reach the expected attitude under the action of the attitude controller designed in the embodiment.

[0024] The method has the beneficial effects that: the control rate of the ship and the UAV is designed by using the non-singular sliding mode to realize the trajectory tracking control of the ship / UAV cooperation, the upper bound of the external disturbance is estimated by using the adaptive term in the sliding mode controller, and the uncertain term in the ship model is estimated by using the RBF neural network. The application solves the problems of low trajectory tracking precision of the underactuated ship under the influence of wind, wave and current disturbance and model parameter change, and difficult state matching and weak anti-interference ability in the ship / UAV cooperation process, and provides a reliable theoretical basis and practical method for the cooperation control of the underactuated ship and the UAV in the complex marine environment, and helps to promote the efficient application of the ship / UAV cooperation system in many fields such as shipping safety protection, emergency rescue and marine operation.

[0025] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the application, but not to limit them; although the application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that they can still modify the technical solutions recorded in the foregoing embodiments, or make equivalent replacement to part or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the application.

Claims

1. A machine / ship cooperative trajectory tracking control method based on non-singular sliding mode and RBF neural network, characterized in that, Specifically, the following steps are included: S1: Construct mathematical models of underactuated ships and quadcopter drones; S2: Based on the set desired trajectory of the underactuated vessel, construct the virtual control law of the underactuated vessel according to the mathematical model of the underactuated vessel; S3: Based on the virtual control law of underactuated ships, a ship controller based on non-singular sliding mode and RBF neural network is constructed; and the ship controller is used to generate control signals for controlling the motion of the ship, the control signals including longitudinal propulsion control law and steering torque control law; S4: Based on the desired attitude of the quadcopter UAV, the attitude control law of the UAV is constructed according to the mathematical model of the quadcopter UAV and the non-singular sliding surface. S5: Based on the desired trajectory of a quadcopter drone, the position control law of the drone is constructed by combining the mathematical model of the quadcopter drone with the non-singular sliding surface, and a non-singular sliding mode controller is designed based on the position control law. S6: Given a desired trajectory for the underactuated vessel in a preset coordinate system, the vessel is enabled to track the trajectory using a vessel controller based on non-singular sliding mode and RBF neural network; the real-time position information of the underactuated vessel is acquired and transmitted to the quadrotor UAV; the desired trajectory of the quadrotor UAV is set by combining the given desired altitude, and the quadrotor UAV is enabled to track the trajectory by combining the given desired attitude with the attitude control law of the non-singular sliding mode controller and the UAV, thereby realizing the cooperative trajectory tracking control of the machine / ship.

2. The machine / ship cooperative trajectory tracking control method based on non-singular sliding mode and RBF neural network according to claim 1, characterized in that, The mathematical model of the underactuated ship constructed in S1 is expressed as follows: （1） In the formula: This indicates the position of the underactuated vessel in the inertial coordinate system; Indicates the heading angle of an underactuated vessel; These represent the ship's forward speed, lateral drift speed, and bow roll speed in the attached coordinate system, respectively. Indicates the inertial parameters of an underactuated vessel; This represents the hydrodynamic damping coefficient of an underactuated ship system. Represents the uncertainties in the model; These represent the longitudinal propulsion force and steering moment of an underactuated vessel, respectively. This indicates time-varying interference caused by the external environment; express The first derivative; express The first derivative; express The first derivative; The mathematical model of the constructed quadcopter UAV is expressed as follows: （2） In the formula: This represents the four desired control inputs to be designed; Indicates the actual position of the quadcopter drone; This indicates the actual attitude angle of the quadcopter drone; These represent the quadcopter drone's body orbiting a preset coordinate system. Moment of inertia of the three axes; This indicates the total mass of the quadcopter drone; Represents the gravitational constant; Indicates the drag coefficient; Indicates external interference; express The first derivative; express The second derivative; express The first derivative; express The second derivative.

3. The machine / ship cooperative trajectory tracking control method based on non-singular sliding mode and RBF neural network according to claim 2, characterized in that, S2 specifically includes the following steps: S21: Based on the set desired trajectory of the underactuated vessel, the vessel position error is obtained according to the mathematical model of the underactuated vessel, and the formula for obtaining the vessel position error is as follows: （3） In the formula: The x and y coordinates represent the desired trajectory of the underactuated vessel. These represent the longitudinal position error and the lateral position error, respectively. S22: Obtain the derivative of the ship's position error by differentiating the ship's position error; its expression is as follows: （4） In the formula: express The first derivative; express The first derivative; S23: To make the ship's position error approach zero, the equation (4) is modified as follows: Treating these as virtual control variables, design virtual control laws for underactuated ships, including longitudinal virtual control laws. With horizontal virtual control rate Its expression is （5） In the formula: , This represents the design constant.

4. The machine / ship cooperative trajectory tracking control method based on non-singular sliding mode and RBF neural network according to claim 3, characterized in that, S3 specifically includes the following steps: S31: Based on the virtual control law and mathematical model of the underactuated ship, obtain the ship speed error, wherein the ship speed error includes the longitudinal speed error. With lateral velocity error ; The formula for obtaining the ship speed error is as follows: （6） S32: Based on longitudinal velocity error Design a first-order nonsingular sliding surface Based on lateral velocity error Design a second-order nonsingular sliding surface Its expression is （7） In the formula: express The abbreviated form; All represent positive odd numbers and satisfy ; Represents positive real numbers; S33: For a first-order nonsingular sliding surface With second-order nonsingular sliding surface Taking the derivative, the expression is: （8） S34: Define intermediate parameter quantities based on ship position error Its expression is （9） In the formula: Represents intermediate variables and ; Then based on the amount of intermediate parameters , will vertical virtual control rate With horizontal virtual control rate The relationship between them is represented as （10） In the formula: express The second derivative; express The first derivative; express The first derivative; S35: Combining equations (1), (6), and (10) with equation (8), we can obtain （11） In the formula: express The first derivative; S36: Based on the RBF neural network, the model uncertainty in equation (11) is addressed. Approximation processing, its expression is: （12） In the formula: Describes the input vector of the RBF neural network and ; Indicates transpose; Represents the radial basis function vector of an RBF neural network; Describes the approximation error of the RBF neural network; and satisfies , Indicates the error threshold and ; This represents the ideal weight vector from the hidden layer to the output layer. express Elements in; S37: Based on Equation (12) and Equation (7), a ship controller based on non-singular sliding mode and RBF neural network is constructed; and the ship controller is used to generate control signals for controlling the motion of the ship, the control signals including longitudinal propulsion control law and steering torque control law; Furthermore, the expression for the ship controller based on non-singular sliding mode and RBF neural network is: （13） （14） In the formula: express The estimated value; express The first derivative; Represents design constants and ; These represent external time-varying disturbances. The estimated value of the boundary; Indicates the quantity of intermediate parameters and ; Indicates design constants; Represents design constants and ; Indicate design parameters and ; Indicates design constants; They represent The prior estimate of .

5. The machine / ship cooperative trajectory tracking control method based on non-singular sliding mode and RBF neural network according to claim 4, characterized in that, S4 specifically includes the following steps: S41: Based on the desired attitude of the quadcopter drone, obtain the attitude error of the quadcopter drone according to the mathematical model of the quadcopter drone. The formula for obtaining the attitude error is as follows: （15） In the formula: This indicates the desired attitude of the quadcopter drone; This represents the tracking error of roll angle, pitch angle, and yaw angle; S42: Construct a sliding surface for roll angle tracking error based on attitude error. Pitch angle tracking error sliding surface and yaw angle tracking error sliding surface Its expression is （16） In the formula: All represent positive odd numbers in the design and satisfy ; All represent positive real numbers in the design; express The first derivative; S43: Sliding surface based on roll angle tracking error Pitch angle tracking error sliding surface and yaw angle tracking error sliding surface Construct the approximation rate of the sliding surface for roll angle tracking error. Pitch angle tracking error sliding surface convergence rate and the approach rate of the sliding surface of the yaw angle tracking error Its expression is （17） In the formula: All represent known positive numbers; Represents a symbolic function; S44: Approach rate of the sliding surface based on roll angle tracking error Pitch angle tracking error sliding surface convergence rate and the approach rate of the sliding surface of the yaw angle tracking error To construct the attitude control law for the drone; The attitude control law of the UAV is expressed as follows: （18）。 6. The machine / ship cooperative trajectory tracking control method based on non-singular sliding mode and RBF neural network according to claim 5, characterized in that, S5 specifically includes the following steps: S51: By rewriting the mathematical model of a quadcopter UAV, we can obtain... （19） （20） In the formula: This indicates the virtual control input for the position of the quadcopter drone; S52: Based on the desired trajectory of a quadcopter UAV, combined with equation (2), the UAV position tracking error can be obtained, and its expression is as follows: （21） In the formula: Indicates the desired location of the quadcopter drone; This indicates the position tracking error of a quadcopter drone; S53: Construct a non-singular sliding surface for UAV position tracking based on the UAV position tracking error; its expression is as follows: （22） In the formula: These represent the non-singular sliding surface for lateral position tracking, longitudinal position tracking, and vertical position tracking of UAVs, respectively. express The first derivative; All represent positive odd numbers in the design and satisfy the following conditions: ; All represent known positive real numbers; S54: Construct the convergence rate of the non-singular sliding surface for UAV position tracking based on the non-singular sliding surface. Its expression is （23） In the formula: All represent known positive numbers; Represents a symbolic function; S55: Based on equations (22) and (23) combined with equation (20), the position control law of the UAV is constructed, and its expression is: （24） A non-singular sliding mode controller is designed based on the position control law, and the expression of the non-singular sliding mode controller is as follows: （25）。

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