Virtual ship guided under-actuated ship trajectory tracking input saturation constraint control method

Abstract

The invention discloses an under-actuated ship trajectory tracking input saturation constraint control method guided by a virtual ship. The method comprises the following steps: acquiring a simplified three-degree-of-freedom under-actuated ship kinematics and dynamics coupling mathematical model; designing a robust linear disturbance observer of the under-actuated ship under the complex sea condition according to the converted external disturbance; an auxiliary state variable used for trajectory tracking is introduced, and a robust linear disturbance observer is combined to construct a time domain nonlinear disturbance observer; constructing a virtual ship according to the coupling mathematical model, and obtaining a tracking error variable according to the virtual ship and the coupling mathematical model; based on a time domain nonlinear disturbance observer, a longitudinal propulsive force sliding mode control law and a steering torque sliding mode control law are obtained according to the tracking error variable, so that under-actuated ship trajectory tracking input saturation constraint control is achieved. The technical problems that a traditional ship trajectory tracking control method is low in trajectory fitting precision, conservative in disturbance compensation, large in control input impact and the like are solved.

Description

Technical Field

[0001] The present invention relates to the field of underactuated ship motion control, and in particular to a virtual ship-guided input saturation constraint control method for underactuated ship trajectory tracking. Background Art

[0002] The shipping industry plays an irreplaceable role in global trade. With the continuous expansion of the scale of international trade and the growing demand for maritime transportation, higher requirements are put forward for ship navigation control technology. Due to the structural limitations of the propulsion system, underactuated ships face many control problems in complex marine environments: on the one hand, the increasing congestion of waterways has increased the requirements for navigation accuracy; on the other hand, environmental disturbances such as wind, waves, and currents will significantly affect the ship's maneuverability. Therefore, it is of great significance to develop more advanced trajectory tracking technology. There are three significant limitations in existing research: firstly, virtual ship guidance strategies mostly focus on the kinematic level and lack deep coupling with dynamic control; secondly, disturbance compensation relies on conservative boundary value estimation and fails to fully utilize the time-domain characteristics of disturbances; thirdly, the initial transient impact of control inputs is ignored, resulting in a shortened service life of actuators. Summary of the Invention

[0003] The present invention provides a virtual ship-guided input saturation constraint control method for underactuated ship trajectory tracking to overcome the above technical problems.

[0004] To achieve the above object, the technical solution of the present invention is as follows:

[0005] A virtual ship-guided input saturation constraint control method for underactuated ship trajectory tracking specifically includes the following steps:

[0006] S1: Obtain a coupled mathematical model of the kinematics and dynamics of a simplified three-degree-of-freedom underactuated ship;

[0007] And the coupled mathematical model includes saturated input constraints and time-domain disturbances of unknown external environments;

[0008] S2: Transform the time-domain disturbances of the unknown external environment to obtain transformed external disturbances;

[0009] Design a robust linear disturbance observer for the underactuated ship under complex sea conditions according to the transformed external disturbances;

[0010] Construct a time-domain nonlinear disturbance observer by introducing auxiliary state variables for trajectory tracking and combining with the robust linear disturbance observer;

[0011] S3: Based on set conditional assumptions, construct a virtual ship for obtaining the desired ship trajectory according to the coupled mathematical model, and obtain a tracking error variable according to the virtual ship and the coupled mathematical model;

[0012] Based on the time-domain nonlinear disturbance observer, the derivative of the tracking error variable is obtained according to the tracking error variable;

[0013] S4: Based on the nonlinear control theory, the first sliding surface and the second sliding surface of the improved global fast terminal sliding mode surface are obtained according to the tracking error variable;

[0014] The first equivalent control law is obtained according to the first sliding surface combined with the derivative of the tracking error variable, and the longitudinal thrust sliding mode control law is constructed according to the first equivalent control law;

[0015] The second equivalent control law is obtained according to the second sliding surface combined with the derivative of the tracking error variable, and the steering torque sliding mode control law is constructed according to the second equivalent control law;

[0016] S5: According to the longitudinal thrust sliding mode control law and the steering torque sliding mode control law, the control of the input saturation constraint of the underactuated ship trajectory tracking is realized.

[0017] Furthermore, the coupled mathematical model of the three-degree-of-freedom underactuated ship described in S1 has the following expression

[0018]

[0019] where: x, y, ψ represent the forward displacement, cross drift displacement and heading angle of the ship in the inertial coordinate system; represents the first derivative of x, y, ψ; u, v, r represent the velocity vector of the ship in the attached coordinate system, that is, the surge angular velocity, sway angular velocity and yaw angular velocity of the ship; represents the first derivative of u, v, r; m 11 , m 22 , m 33 represent the mass components of the ship in the surge direction, sway direction and yaw direction; d 11 , d 22 , d 33 represent the damping coefficients of the ship in the surge direction, sway direction and yaw direction; τ wu , τ wv , τ wr represent the unknown external environmental time-domain disturbances of the ship in the navigation coordinate system; τ us , M rs represent the longitudinal thrust moment and the steering moment of the ship's saturated input, and

[0020] τ u , M r represent the control inputs without saturated input constraints; τ um represents the constraint threshold of the longitudinal thrust moment; M rm represents the constraint threshold of the steering moment.

[0021] Further, S2 specifically includes the following steps:

[0022] S21: Convert the time-domain disturbance of the unknown external environment to obtain the converted external disturbance;

[0023] And the expression of the converted external disturbance is

[0024]

[0025] S22: Design a robust linear disturbance observer for the underactuated ship under complex sea conditions according to the converted external disturbance; and the expression of the robust linear disturbance observer is

[0026]

[0027] In the formula: represents the estimated value of τ wu , τ wr ; K1, K2 represent positive design parameters; represents the first derivative of;

[0028] S23: Construct an auxiliary state variable for trajectory tracking, and its expression is

[0029]

[0030] In the formula: μ1, μ2 represent the introduced auxiliary state variables for trajectory tracking; represents the first derivative of μ1, μ2;

[0031] S24: Construct a time-domain nonlinear disturbance observer according to the auxiliary state variable combined with the robust linear disturbance observer, and the expression of the time-domain nonlinear disturbance observer is

[0032]

[0033] Further, S3 specifically includes the following steps:

[0034] S31: Set conditional assumptions, which include:

[0035] Assumption 1: The virtual underactuated ship model is an ideal model without any external disturbances and with its model parameters determined;

[0036] Assumption 2: The center of gravity of the virtual underactuated ship model coincides with the origin of the appendage coordinate system, and the hull is symmetric left and right;

[0037] Assumption 3: The unknown external environmental disturbances and their rates of change acting on the virtual underactuated ship are bounded, that is, A i ≤ ||τ wi || ≤ B i , i = u, r; A i , B i , C i represent bounded parameters;

[0038] S32: Based on the set conditional assumptions, construct a virtual ship for obtaining the desired trajectory of the ship according to the coupled mathematical model, and the expression of the virtual ship is

[0039]

[0040] where: x d , y d , ψ d represent the desired position and heading of the virtual underactuated ship respectively; u d , v d , r d represent the desired longitudinal velocity, lateral velocity and yaw angular velocity of the virtual underactuated ship respectively; τ ud , M rd represent the desired reference input, and represents the first derivative of x d , y d , ψ d ; represents the first derivative of u d , v d , r d ;

[0041] S33: Obtain the tracking error variable according to the virtual ship and the coupled mathematical model;

[0042] And the expression of the tracking error variable is

[0043]

[0044] where: u e , v e , r e represent the longitudinal velocity error, lateral velocity error and yaw angular velocity error respectively;

[0045] Based on the time-domain nonlinear disturbance observer, obtain the derivative of the tracking error variable according to the tracking error variable, and its expression is

[0046]

[0047] where: represents the first derivative of u e , v e , r e ;

[0048] Furthermore, the method for constructing the longitudinal propulsion force sliding mode control law in the S4 includes the following steps:

[0049] S41: Based on the nonlinear control theory, obtain the first sliding mode surface s1 of the improved global fast terminal sliding mode surface according to the tracking error variable;

[0050] And the expression of the first sliding mode surface s1 is

[0051]

[0052] In the formula: α1, β1 represent positive design parameters; p1, q1 represent positive odd numbers and p1 > q1; τ represents the integral variable with respect to the time parameter t; u e represents the abbreviated form of u e (τ);

[0053] S42: Take the derivative of the first sliding mode surface s1, and combine with the derivative of the tracking error variable to obtain

[0054]

[0055] In the formula: represents the first derivative of s1;

[0056] S43: Construct the first equivalent control law τ ueq , and the expression of the first equivalent control law τ ueq is

[0057]

[0058] S44: Let Then construct the longitudinal thrust sliding mode control law according to the first equivalent control law, and the expression of the longitudinal thrust sliding mode control law is

[0059]

[0060] In the formula: γ1 represents a positive design parameter; p2, q2 represent positive odd numbers; τ us represents the longitudinal thrust sliding mode control law.

[0061] Furthermore, the method for constructing the steering torque sliding mode control law in S4 includes the following steps:

[0062] S100: Based on the nonlinear control theory, obtain the second sliding mode surface s2 of the improved global fast terminal sliding mode surface according to the tracking error variable;

[0063] And the expression of the second sliding mode surface s2 is

[0064]

[0065] Where: α2 and β2 represent positive design parameters; p3 and q3 represent positive odd numbers.

[0066] S101: By taking the derivative of the second sliding surface s2, we can obtain

[0067]

[0068] Where: represents the second derivative of v e ; represents the first derivative of s2;

[0069] S102: Define the intermediate parameter W and Substitute it into formula (15) to obtain

[0070]

[0071] S103: Define the intermediate parameters P and Q and

[0072]

[0073] Based on the intermediate parameters P and Q, obtain the second equivalent control law M according to formula (16) req , and the expression of the second equivalent control law M req is

[0074]

[0075] S104: Construct the steering torque sliding mode control law M' according to the second equivalent control law req , and the expression of the steering torque sliding mode control law M' req is

[0076]

[0077] Where: γ2 represents a positive design parameter; p4 and q4 represent positive odd numbers.

[0078] Beneficial effects: The present invention provides an underactuated ship trajectory tracking input saturation constraint control method guided by a virtual ship. By constructing a coupled mathematical model of a three-degree-of-freedom underactuated ship, designing a new type of global fast terminal sliding surface without imaginary parts, and combining a time-domain nonlinear disturbance observer and a piecewise saturation input constraint strategy, a longitudinal propulsive force sliding mode control law and a steering torque sliding mode control law are constructed to achieve the control of underactuated ship trajectory tracking input saturation constraint, effectively solving the technical problems such as low trajectory fitting accuracy, conservative disturbance compensation, and large control input impact existing in traditional methods, and greatly improving the control accuracy and control efficiency of underactuated ship trajectory tracking input saturation constraint control. Description of the Drawings

[0079] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0080] Figure 1 It is a flowchart of the underactuated ship trajectory tracking input saturation constraint control method for virtual ship guidance of the present invention;

[0081] Figure 2 It is a schematic diagram of the nautical coordinate system and the appendage coordinate system of the ship motion in this embodiment;

[0082] Figure 3 It is a curve graph of the desired trajectory and the actual trajectory of the ship in this embodiment;

[0083] Figure 4 It is a curve graph of the desired position and the actual position of the ship in the x-axis direction in this embodiment;

[0084] Figure 5 It is a curve graph of the desired position and the actual position of the ship in the y-axis direction in this embodiment;

[0085] Figure 6 It is a curve graph of the longitudinal speed of the ship in this embodiment;

[0086] Figure 7 It is a curve graph of the lateral speed of the ship in this embodiment;

[0087] Figure 8 It is a curve graph of the yaw angular velocity of the ship in this embodiment;

[0088] Figure 9 It is a curve graph of the time history of the longitudinal speed tracking error in this embodiment;

[0089] Figure 10 It is a curve graph of the time history of the lateral speed tracking error in this embodiment;

[0090] Figure 11 It is a curve graph of the time history of the yaw angular velocity tracking error in this embodiment;

[0091] Figure 12 It is a simulation comparison graph of the longitudinal propulsion torque in this embodiment;

[0092] Figure 13 It is a simulation comparison graph of the steering torque in this embodiment;

[0093] Figure 14 It is the external disturbance estimation in this embodiment curve graph;

[0094] Figure 15 For the external disturbance estimation in this embodiment of the curve graph. Detailed implementation manner

[0095] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0096] This embodiment provides a virtual ship-guided underactuated ship trajectory tracking input saturation constraint control method, as Figure 1 shown, which specifically includes the following steps:

[0097] S1: Obtain a coupled mathematical model of the kinematics and dynamics of a simplified three-degree-of-freedom underactuated ship;

[0098] And the coupled mathematical model includes saturation input constraints and time-domain disturbances of the unknown external environment;

[0099] Specifically, the expression of the coupled mathematical model of the three-degree-of-freedom underactuated ship is

[0100]

[0101] where: x, y, ψ represent the forward displacement, transverse drift displacement, and heading angle of the ship in the inertial coordinate system; represents the first derivative of x, y, ψ; u, v, r represent the velocity vector of the ship in the attached body coordinate system, that is, the surge angular velocity, sway angular velocity, and yaw angular velocity of the ship; represents the first derivative of u, v, r; m 11 , m 22 , m 33 represent the mass components of the ship in the surge direction, sway direction, and yaw direction; d 11 , d 22 , d 33 represent the damping coefficients of the ship in the surge direction, sway direction, and yaw direction; τ wu , τ wv , τ wr represent the unknown external environment time-domain disturbances of the ship in the navigation coordinate system; τ us , M rs represent the longitudinal propulsion torque and steering torque of the ship's saturation input, and

[0102]

[0103] Among them, τ u , M r represents the control input without saturated input constraint; τ um represents the constraint threshold of the longitudinal propulsion moment; M rm represents the constraint threshold of the steering moment;

[0104] In this embodiment, the underactuated ship model only considers the simplified Fossen ship motion model with three degrees of freedom: forward motion, side drift motion, and yaw motion. This model is based on hydrodynamics theory, considering the ship's mass, buoyancy, and resistance, etc., and can accurately describe the ship's motion in water. The simplified ship motion inertial coordinate system and the appended body coordinate system are as Figure 2 shown; among them, X0O0Y0 is the navigation coordinate system or the inertial coordinate system; O0 is the starting point; O0X0 is the due north direction; O0Y0 is the due east direction; x0O b y0 is the appended body coordinate system; O b x0 points to the bow direction; O b y0 points to the starboard side;

[0105] S2: Convert the time-domain disturbance of the unknown external environment to obtain the converted external disturbance;

[0106] Design a robust linear disturbance observer for the underactuated ship in complex sea conditions according to the converted external disturbance; by introducing an auxiliary state variable for trajectory tracking and combining it with the robust linear disturbance observer, construct a time-domain nonlinear disturbance observer;

[0107] Specifically, it includes the following steps:

[0108] S21: Convert the time-domain disturbance of the unknown external environment to obtain the converted external disturbance;

[0109] And the expression of the converted external disturbance is

[0110]

[0111] S22: Design a robust linear disturbance observer for the underactuated ship in complex sea conditions according to the converted external disturbance; in this embodiment, the disturbance observer method can estimate the external environmental disturbance and compensate it into the ship controller in real time to improve the robustness of the ship control system in complex sea conditions;

[0112] And the expression of the robust linear disturbance observer is

[0113]

[0114] In the formula: represents τ wu, τ wr Estimated value; K1, K2 represent positive design parameters; represents The first derivative of

[0115] S23: In reality, only each trajectory tracking state in the ship trajectory tracking system can be measured, but its derivative is difficult to measure. Therefore, auxiliary state variables for trajectory tracking are constructed to design a time-domain nonlinear disturbance observer;

[0116] And the constructed auxiliary state variables for trajectory tracking have the expression

[0117]

[0118] Where: μ1, μ2 represent the introduced auxiliary state variables for trajectory tracking; represents the first derivative of μ1, μ2;

[0119] S24: According to the auxiliary state variables and combined with a robust linear disturbance observer, a time-domain nonlinear disturbance observer is constructed, and the expression of the time-domain nonlinear disturbance observer is

[0120]

[0121] In order to facilitate stability analysis in this embodiment, it also includes defining the observation error of the disturbance observer:

[0122]

[0123] For τ in Equation (8) wu and τ wr Taking the derivative gives:

[0124]

[0125] Taking the derivative of both sides of Equation (8) and substituting Equation (9) gives:

[0126]

[0127] S3: Based on the set conditional assumptions, a virtual ship for obtaining the desired ship trajectory is constructed according to the coupled mathematical model, and the tracking error variable is obtained according to the virtual ship and the coupled mathematical model;

[0128] Based on the time-domain nonlinear disturbance observer, the derivative of the tracking error variable is obtained according to the tracking error variable;

[0129] Specifically, it includes the following steps:

[0130] S31: Set conditional assumptions, which include:

[0131] Hypothesis 1: The virtual underactuated ship model is an ideal model without any external interference and with its model parameters determined;

[0132] Hypothesis 2: The center of gravity of the virtual underactuated ship model coincides with the origin of the appendage coordinate system, and the hull is symmetric left and right;

[0133] Hypothesis 3: The unknown external environmental interference and its rate of change acting on the virtual underactuated ship are bounded, i.e., A i ≤||τ wi ||≤B i , i = u,r; A i ,B i ,C i represent bounded parameters;

[0134] S32: Based on the set conditional hypotheses, a virtual ship for obtaining the desired trajectory of the ship is constructed according to the coupled mathematical model, and the expression of the virtual ship is

[0135]

[0136] where: x d ,y d ,ψ d represent the desired position and heading of the virtual underactuated ship respectively; u d ,v d ,r d represent the desired longitudinal velocity, lateral velocity and yaw angular velocity of the virtual underactuated ship respectively; τ ud ,M rd represents the desired reference input. In theory, in this embodiment, only by reasonably assigning values to u d and r d of the virtual underactuated ship, the virtual underactuated ship can be made to generate the desired trajectory, and v d can be solved from u d and r d , and then τ ud and τ rd can be solved, and represents the first derivative of x d ,y d ,ψ d ; represents the first derivative of u d ,v d ,r d ;

[0137] S33: Obtain the tracking error variable according to the virtual ship and the coupled mathematical model;

[0138] And the expression of the tracking error variable is

[0139]

[0140] where: u e , v e , r e represent the longitudinal velocity error, the lateral velocity error, and the yaw angular velocity error;

[0141] Based on the time-domain nonlinear disturbance observer, the derivative of the tracking error variable is obtained according to the tracking error variable, and its expression is

[0142]

[0143] where: represents the first derivative of u e , v e , r e ;

[0144] S4: Based on the nonlinear control theory, the first sliding surface and the second sliding surface of the improved global fast terminal sliding mode surface are obtained according to the derivative of the tracking error variable; in this embodiment, in order to avoid generating an imaginary part during the operation process, an improved integral-type global fast terminal sliding plane is introduced to design the control law;

[0145] The first equivalent control law is obtained according to the first sliding surface in combination with the derivative of the tracking error variable, and the longitudinal thrust sliding mode control law is constructed according to the first equivalent control law;

[0146] In a specific embodiment, the method for constructing the longitudinal thrust sliding mode control law includes the following steps:

[0147] S41: Based on the nonlinear control theory, the first sliding surface s1 of the improved global fast terminal sliding mode surface is obtained according to the tracking error variable;

[0148] And the expression of the first sliding surface s1 is

[0149]

[0150] where: α1, β1 represent positive design parameters; p1, q1 represent positive odd numbers and p1 > q1; τ represents the integral variable with respect to the time parameter t; u e represents the abbreviated form of u e (τ);

[0151] S42: The first sliding surface s1 is differentiated, and combined with the derivative of the tracking error variable, it can be obtained that

[0152]

[0153] where: represents the first derivative of s1;

[0154] S43: Construct the first equivalent control law τ according to formula (15). ueq Moreover, the expression of the first equivalent control law τ ueq is

[0155]

[0156] S44: Let Then, construct the longitudinal thrust sliding mode control law according to the first equivalent control law, and the expression of the longitudinal thrust sliding mode control law is

[0157]

[0158] In the formula: γ1 represents a positive design parameter; p2, q2 represent positive odd numbers; τ us represents the longitudinal thrust sliding mode control law;

[0159] Obtain the second equivalent control law based on the second sliding surface and the derivative of the tracking error variable, and construct the steering torque sliding mode control law according to the second equivalent control law;

[0160] In a specific embodiment, the method for constructing the steering torque sliding mode control law includes the following steps:

[0161] S100: Based on the nonlinear control theory, obtain the second sliding surface s2 of the improved global fast terminal sliding mode surface according to the tracking error variable;

[0162] Moreover, the expression of the second sliding surface s2 is

[0163]

[0164] In the formula: α2, β2 represent positive design parameters; p3, q3 represent positive odd numbers;

[0165] S101: Take the derivative of the second sliding surface s2, and we can get

[0166]

[0167] In the formula: represents the second derivative of v e ; represents the first derivative of s2;

[0168] S102: Define the intermediate parameter W and Substitute it into formula (19), and we can get

[0169]

[0170] S103: Define the intermediate parameters P and Q and

[0171]

[0172] and based on the intermediate parameters P and Q, obtain the second equivalent control law M according to formula (20) req , and the second equivalent control law M req has an expression of

[0173]

[0174] S104: Construct the steering torque sliding mode control law M′ according to the second equivalent control law req , and the steering torque sliding mode control law M′ req has an expression of

[0175]

[0176] wherein: γ2 represents a positive design parameter; p4 and q4 represent positive odd numbers;

[0177] S5: According to the longitudinal thrust sliding mode control law and the steering torque sliding mode control law, realize the control of the underactuated ship trajectory tracking input saturation constraint;

[0178] In this embodiment, by constructing a coupled mathematical model of a three-degree-of-freedom underactuated ship, designing a new global fast terminal sliding mode surface without imaginary part, and combining a time-domain nonlinear disturbance observer and a piecewise saturation input constraint strategy, a longitudinal thrust sliding mode control law and a steering torque sliding mode control law are constructed to realize the control of the underactuated ship trajectory tracking input saturation constraint, effectively solving the technical problems existing in the traditional method, such as low trajectory fitting accuracy, conservative disturbance compensation, and large control input impact, and greatly improving the control accuracy and control efficiency of the underactuated ship trajectory tracking input saturation constraint control.

[0179] The stability analysis of the method in this embodiment includes:

[0180] Theorem 1: For the designed trajectory tracking system, the ship mathematical model is as shown in formula (1), the saturated input is limited as shown in formula (2), the nonlinear disturbance observer is designed as shown in formula (6), the virtual control law of the virtual underactuated ship is designed as shown in formula (10), the sliding mode surface of the longitudinal thrust moment control law is designed as shown in formula (14), and the sliding mode surface of the steering thrust moment is designed as shown in formula (18). Under the conditions of assumptions 1 to 3, there exist appropriate parameters such that the closed-loop system is asymptotically stable;

[0181] Construct the following Lyapunov function:

[0182]

[0183] Taking the derivative of both sides of formula (24) gives:

[0184]

[0185] Convert Equation (25) into the following form:

[0186]

[0187] where V1 and V2 are:

[0188]

[0189] In this embodiment, to make it only needs to satisfy that either V1 or V2 is less than 0, and the other is less than or equal to 0. It can be known from Equation (27) that V2 < 0; so it only needs to prove that V1 ≤ 0;

[0190] Considering Hypothesis 3, according to Young's inequality, we can get:

[0191]

[0192] where a1 and a2 are positive constants, then

[0193]

[0194] To ensure V1 ≤ 0, it is necessary to satisfy:

[0195]

[0196] To satisfy Equation (30), it only needs to select appropriate K1 and K2 to achieve:

[0197]

[0198] To sum up, it only needs to satisfy K1 > a1, K2 > a2, and at the same time, the values of K1 and K2 need to be able to offset and the positive contributions of, that is, it can make V1 ≤ 0, thus proving

[0199] This embodiment also includes simulation experiments and discussions:

[0200] To verify the effectiveness of the controller designed by the method of this embodiment, a remote patrol ship named BAY CLASS of the Singapore Customs is used as the simulation object for simulation verification. The specific ship parameters are as follows: m 11 = 1.2×10 5 kg, m 22 = 1.779×10 5 kg, m 33 = 6.36×10 7 kg, d 11 = 2.15×10 4 kg / s, d22 = 1.47×10 5 kg / s, d 33 = 8.02×10 6 kg / s;

[0201] This simulation experiment is compared with the controller using the second-order sliding surface. The simulation curve under the control law in this embodiment is set as A, and the simulation curve of the comparative test second-order sliding mode control system is set as B; The parameter selection in this embodiment is through continuous simulation experiments to optimize the parameters and find more suitable parameters as much as possible until the best control effect is achieved; The design parameters of the controller in this embodiment are as follows: K1 = 10, K2 = 10, α1 = 2.15, α2 = 6.5, β1 = 1, β2 = 0.01, p1 = 9, q1 = 5, p2 = 7, q2 = 5, b1 = 10, b2 = 0.1, p3 = 3, q3 = 1, b3 = 10, b4 = 0.1, p4 = 3, q4 = 1; The saturation input constraint of the controller is taken as τ um = 1.4×10 5 , M rm = 7.5×10 6 ;

[0202] The external disturbance is taken as τ wu = 10 4 (sin(0.2t) + cos(0.5t)), τ wr = 10 5 (sin(0.5t) + cos(0.3t));

[0203] The designed virtual ship is as follows:

[0204]

[0205] Although the simulation diagram can intuitively show which method has a better control effect, in order to further compare the performance of the controller in this embodiment with that of the comparative experiment controller, the root mean square error (RMSE) is introduced. RMSE can measure the difference between the observed value and the true value. The larger the RMSE value, the greater the degree of deviation of the data from the true value, that is, the greater the fluctuation; Therefore, RMSE can be used to compare the convergence effects of the controller in this embodiment and the comparative experiment. The RMSE solution formula is as follows;

[0206]

[0207] In the formula: l i represents the expected value; represents the approximation value;

[0208] Such as Figures 3 to 5The figure shows the position tracking of a ship over time. The initial position and yaw angle of the ship are selected as (x0; y0; ψ0) = (0; 0; 0); from Figures 3 to 5 It can be seen that both controllers can finally accurately track the desired trajectory, but the controller in this embodiment is significantly better than the controller in the comparative experiment in terms of fitting the desired course; the RMSE is used to further compare the fitting effects of the controller in this embodiment and the controller in the comparative experiment. The RMSE of the tracking trajectory position is calculated from the simulation experiment, as shown in Table 1;

[0209] Table 1. RMSE of the tracking trajectory

[0210] Table 1 RMSE of the tracking trajectory

[0211]

[0212] In this embodiment, the lower the RMSE, the better the fitting effect of the tracking trajectory to the desired trajectory. It can be calculated that the fitting effect of the controller in this embodiment on the tracking trajectory and the desired trajectory is improved by about 34% compared with the fitting effect of the comparative experiment;

[0213] As Figures 6 to 8 shown is the comparison between the ship's directional speed and its desired speed. It can be seen from the figure that the longitudinal speed, lateral speed, and yaw angular velocity can finally track the desired speed; as Figures 9 to 11 shown is the curve of the error of the ship's state changing with time. From Figure 9 it can be seen that for both the controller designed in this embodiment and the controller used as a comparative experiment, the tracking error of the longitudinal speed can be maintained within a certain range, but it is obvious that the convergence effect of the controller in this embodiment is better than that of the comparative experiment controller. From Figures 10 to 11 it can be seen that for both the controller in this embodiment and the comparative experiment controller, the tracking errors of the lateral speed and yaw angular velocity can converge to 0; the RMSE is used to compare the convergence effects of the longitudinal speed errors of the controller in this embodiment and the comparative experiment controller. The RMSE of the longitudinal speed is calculated from the simulation experiment, as shown in Table 2;

[0214] Table 2. RMSE of the longitudinal velocityTable 2RMSE of the longitudinal velocity

[0215]

[0216] It can be obtained from Table 2 that the convergence effect of the controller u e in this embodiment is improved by about 48% compared with the convergence effect of the comparative experiment.

[0217] As Figures 12 to 13 shown is the curve of ship control input varying with time. It can be seen from Figure 12 that although the controller in this embodiment constrains the longitudinal propulsive force, it does not exceed the constraint conditions. Instead, it causes a certain overshoot in the negative value of the control input, which is also the place that needs to be further improved in this embodiment. It can be seen from Figure 13 that without the saturation input constraint, its steering moment far exceeds the constraint conditions. After the saturation input constraint is imposed on it, it does not affect the subsequent operation of the controller. As Figures 14 to 15 shown is the estimation of the unknown external disturbance of the ship by the designed nonlinear disturbance observer. It can be seen from the figure that the designed disturbance observer can accurately estimate the external disturbance of the ship.

[0218] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some 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 present invention.

Claims

1. A control method for trajectory tracking of an underactuated ship with input saturation constraints guided by a virtual ship, characterized in that Specifically, it includes the following steps: S1: Obtain a coupled mathematical model of the kinematics and dynamics of a three-degree-of-freedom underactuated ship; And the coupled mathematical model includes saturation input constraints and time-domain disturbances of the unknown external environment; S2: Transform the time-domain disturbance of the unknown external environment to obtain a transformed external disturbance; Design a robust linear disturbance observer for the underactuated ship under complex sea conditions according to the transformed external disturbance; Construct a time-domain nonlinear disturbance observer by introducing an auxiliary state variable for trajectory tracking and combining it with the robust linear disturbance observer; S3: Based on the set conditional assumptions, construct a virtual ship for obtaining the desired trajectory of the ship according to the coupled mathematical model, and obtain the tracking error variable according to the virtual ship and the coupled mathematical model; Based on the time-domain nonlinear disturbance observer, obtain the derivative of the tracking error variable according to the tracking error variable; S4: Based on the nonlinear control theory, obtain the first sliding mode surface and the second sliding mode surface of the improved global fast terminal sliding mode surface according to the tracking error variable; Obtain the first equivalent control law according to the first sliding mode surface combined with the derivative of the tracking error variable, and construct the longitudinal propulsion force sliding mode control law according to the first equivalent control law; Obtain the second equivalent control law according to the second sliding mode surface combined with the derivative of the tracking error variable, and construct the steering moment sliding mode control law according to the second equivalent control law; S5: Realize the control of the input saturation constraint of the underactuated ship trajectory tracking according to the longitudinal propulsion force sliding mode control law and the steering moment sliding mode control law.

2. A virtual ship-guided underactuated ship trajectory tracking input saturation constraint control method according to claim 1, characterized in that, The expression of the coupled mathematical model of the three-degree-of-freedom underactuated ship described in S1 is Where: x, y, ψ represent the forward displacement, transverse drift displacement, and course angle of the ship in the inertial coordinate system; denote the first derivatives of x, y, ψ; u, v, r represent the velocity vector of the ship in the body-fixed coordinate system, i.e., the surge angular velocity, sway angular velocity, and yaw angular velocity of the ship; denote the first derivatives of u, v, r; m 11 , m 22 , m 33 represent the mass components of the ship in the surge direction, sway direction, and yaw direction; d 11 , d 22 , d 33 represent the damping coefficients of the ship in the surge direction, sway direction, and yaw direction; τ wu , τ wv , τ wr represent the unknown external environmental time-domain disturbances of the ship in the navigation coordinate system; τ us , M rs represent the longitudinal propulsion torque and steering torque of the saturated input of the ship, and τ u ,M r represents the control input without saturated input constraint; τ um represents the constraint threshold of longitudinal propulsion torque; M rm represents the constraint threshold of steering torque.

3. A virtual ship-guided underactuated ship trajectory tracking input saturation constraint control method according to claim 2, characterized in that The specific steps of S2 include the following: S21: Transform the time-domain disturbance of the unknown external environment to obtain a transformed external disturbance; And the expression of the transformed external disturbance is S22: Design a robust linear disturbance observer for the underactuated ship under complex sea conditions according to the transformed external disturbance; and the expression of the robust linear disturbance observer is In the formula: represents τ wu , τ wr is the estimated value; K1 and K2 represent positive design parameters; represents the first derivative of S23: Construct an auxiliary state variable for trajectory tracking, and its expression is where: μ1, μ2 represent the introduced auxiliary state variables for trajectory tracking; represents the first derivative of μ1, μ2; S24: Construct a time-domain nonlinear disturbance observer according to the auxiliary state variable combined with the robust linear disturbance observer, and the expression of the time-domain nonlinear disturbance observer is 4. A virtual ship-guided underactuated ship trajectory tracking input saturation constraint control method according to claim 3, characterized in that The specific steps of S3 include the following: S31: Set conditional assumptions, which include: Assumption 1: The virtual underactuated ship model is an ideal model without any external disturbances and with its model parameters determined; Assumption 2: The center of gravity of the virtual underactuated ship model coincides with the origin of the appendage coordinate system, and the hull is symmetric left and right; Hypothesis 3: The unknown external environmental disturbances and their rates of change acting on the virtual underactuated ship are bounded, i.e., A i ≤||τ wi ||≤B i , A i , B i , C i represent bounded parameters; S32: Based on the set conditional assumptions, construct a virtual ship for obtaining the desired trajectory of the ship according to the coupled mathematical model, and the expression of the virtual ship is where: x d , y d , ψ d respectively represent the desired position and heading of the virtual underactuated ship; u d , v d , r d respectively represent the desired longitudinal speed, lateral speed and yaw angular velocity of the virtual underactuated ship; τ ud , M rd represent the desired reference input, and represents the first derivative of x d , y d , ψ d ; represents the first derivative of u d , v d , r d ; S33: Obtain the tracking error variable according to the virtual ship and the coupled mathematical model; And the expression of the tracking error variable is where: u e , v e , r e represent the longitudinal velocity error, the lateral velocity error, and the yaw angular velocity error; Based on the time-domain nonlinear disturbance observer, obtain the derivative of the tracking error variable according to the tracking error variable, and its expression is In the formula: represents the first derivative of u e , v e , r e .

5. A virtual ship-guided underactuated ship trajectory tracking input saturation constraint control method according to claim 4, characterized in that The method of constructing the longitudinal propulsion force sliding mode control law in S4 specifically includes the following steps: S41: Based on the nonlinear control theory, obtain the first sliding mode surface s1 of the improved global fast terminal sliding mode surface according to the tracking error variable; And the expression of the first sliding mode surface s1 is where: α1, β1 represent positive design parameters; p1, q1 represent positive odd numbers and p1 > q1; τ represents an integration variable with respect to the time parameter t; u e denotes the abbreviation of u e (τ); S42: Differentiate the first sliding surface s1, and combined with the derivative of the tracking error variable, it can be obtained that In the formula: represents the first derivative of s1; S43: Construct the first equivalent control law τ according to formula (11) ueq , and the expression of the first equivalent control law τ ueq is S44: Let Then, a longitudinal thrust sliding mode control law is constructed according to the first equivalent control law, and the expression of the longitudinal thrust sliding mode control law is In the formula: represents a positive design parameter; p2 and q2 represent positive odd numbers; τ us represents the longitudinal thrust sliding mode control law.

6. A virtual ship-guided underactuated ship trajectory tracking input saturation constraint control method according to claim 4, characterized in that The method for constructing the steering torque sliding mode control law in S4 specifically includes the following steps: S100: Based on the nonlinear control theory, obtain the second sliding surface s2 of the improved global fast terminal sliding mode according to the tracking error variable; And the expression of the second sliding surface s2 is In the formula: α2, β2 represent positive design parameters; p3, q3 represent positive odd numbers; S101: Differentiate the second sliding surface s2, and it can be obtained that In the formula: represents the second derivative of v e ; represents the first derivative of s2 S102: Define an intermediate parameter W and Substitute it into formula (15) to obtain S103: Define the intermediate parameters P and Q and And based on the intermediate parameters P and Q, obtain the second equivalent control law M according to formula (16). req , and the second equivalent control law M req has the following expression S104: Construct the steering torque sliding mode control law M' according to the second equivalent control law req , and the expression of the steering torque sliding mode control law M' req is as follows In the formula: represents a positive design parameter; p4 and q4 represent positive odd numbers.

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