Autonomous underwater vehicle system hyper-torsion sliding mode control trajectory tracking method based on improved adaptive law

Abstract

The application discloses a kind of based on the trajectory tracking method of super torsion sliding mode control of autonomous underwater robot system of improved self-adaptive law, comprising the following steps: establishing the dynamics model of 6 degrees of freedom autonomous underwater robot;6 degrees of freedom autonomous underwater robot dynamics model is converted, constructs the dynamics model of 6 degrees of freedom autonomous underwater robot based on lumped disturbance;Design synchronous time estimator to estimate lumped disturbance in the dynamics model of 6 degrees of freedom autonomous underwater robot based on lumped disturbance;Based on sliding surface, synchronous time estimator and hyperbolic tangent function, design the super torsion sliding mode controller based on improved self-adaptive law, realize trajectory tracking to autonomous underwater robot.The application can effectively inhibit the chattering of system while guaranteeing the rapid convergence of system.

Description

TECHNICAL FIELD

[0001] The application relates to a trajectory tracking method for an autonomous underwater vehicle system based on improved adaptive law super-twisting sliding mode control, and belongs to the technical field of trajectory tracking of autonomous underwater vehicle systems. BACKGROUND

[0002] In recent years, autonomous underwater vehicles are widely used in various marine research tasks such as ocean data collection, marine search and rescue, and marine resource development due to their convenience and irreplaceability. However, autonomous underwater vehicles are a typical high nonlinear system, and the inherent strong coupling dynamics, model uncertainty, external disturbances (wind, waves, strong currents, etc.) and physical constraints of actuators may affect the control performance of the system. In order to achieve better tracking effect, various advanced motion control schemes are proposed to solve the trajectory tracking control problem of autonomous underwater vehicles, including adaptive control, sliding mode control, backstepping control, model predictive control and fuzzy logic control methods.

[0003] Among the above methods, sliding mode control is widely used due to its strong robustness. However, the traditional sliding mode control method has the problems of chattering and uncertain convergence time. In order to reduce the convergence time of the system, a terminal sliding mode control scheme is proposed, and Chinese patent application CN113110532A proposes an adaptive terminal sliding mode control method, which effectively improves the convergence time of the system. However, due to the existence of switching terms in the control law, the chattering of the control becomes a problem that is difficult to avoid in the design process; Chinese patent application CN115903859A discloses an adaptive integral sliding mode control scheme, which increases the integral element to weaken the chattering phenomenon of the controller as much as possible.

[0004] Although the above-mentioned methods can improve the performance of the system, some unknown system dynamics are still inevitable. In order to solve this problem, a variety of observers and adaptive disturbance rejection methods have been proposed, such as nonlinear disturbance observer and. The nonlinear disturbance observer designed in Chinese patent application CN114895569A can obtain better tracking results for the centralized uncertainty of the system. Another adaptive algorithm is designed in Chinese patent application CN112859891A to ensure that the system has strong robustness and good performance. However, all these methods have the problems of slow convergence speed or overestimation of adaptive gain. SUMMARY

[0005] The application provides a trajectory tracking method for an autonomous underwater robot system based on improved adaptive law super torsion sliding mode control, a super torsion sliding mode controller based on improved adaptive law is designed based on a sliding mode surface, a synchronous time estimator and a hyperbolic tangent function, trajectory tracking of the autonomous underwater robot is realized, and chattering of the system can be effectively inhibited while ensuring rapid convergence of the system.

[0006] The technical scheme of the application is:

[0007] A trajectory tracking method for an autonomous underwater robot system based on improved adaptive law super torsion sliding mode control, comprising the following steps:

[0008] S1, a 6-DOF (degree of freedom) autonomous underwater robot dynamics model is established; the 6-DOF autonomous underwater robot dynamics model is converted to construct a 6-DOF autonomous underwater robot dynamics model based on lumped disturbance;

[0009] S2, a synchronous time estimator is designed to estimate the lumped disturbance in the 6-DOF autonomous underwater robot dynamics model based on lumped disturbance;

[0010] S3, a super torsion sliding mode controller based on improved adaptive law is designed based on a sliding mode surface, a synchronous time estimator and a hyperbolic tangent function, and trajectory tracking of the autonomous underwater robot is realized.

[0011] Further, the 6-DOF autonomous underwater robot dynamics model based on lumped disturbance has the following expression:

[0012]

[0013] wherein, is defined as the lumped disturbance, represents the derivative of the linear velocity and angular velocity vector υ in the moving coordinate system; respectively represent the nominal parts of M, C(υ), D(υ) and g(η); ΔM, ΔC(υ), ΔD(υ) and Δg(η) represent the uncertain part of M, C(υ), D(υ) and g(η) respectively; M is an inertia matrix, C(υ) is a Coriolis centripetal force matrix, D(υ) is a dynamic resistance and buoyancy matrix due to water flow, g(η) is a restoring force vector caused by the combined action of buoyancy and gravity, d is an external disturbance caused by the environment, and τ is a control input torque.

[0014] Further, the S2 comprises:

[0015] The 6-DOF autonomous underwater robot dynamics model based on lumped disturbance is converted to obtain a converted 6-DOF autonomous underwater robot dynamics model based on lumped disturbance;

[0016] According to the intermediate variable, an auxiliary variable is designed;

[0017] According to the auxiliary variable, an error variable is defined; and an output variable is defined according to the error variable;

[0018] According to the error variable and the output variable, a synchronous time estimator is designed to obtain an estimated value of the lumped disturbance.

[0019] Further, the S3 comprises:

[0020] S3.1, a hyperbolic tangent function is designed:

[0021]

[0022] wherein: i = 1, 2,..., 6; u iM is a normal number; τ represents τ(u) = τ ε u and τ(u) ∈ R 6 , R 6 represents a 6-order vector, τ i (u i ) represents the i-th element of τ(u), u i is the i-th element of u, and u represents a super-torsional sliding mode controller based on an improved adaptive law; τ ε = diag(τ 1ε , τ 2ε ,..., τ 6ε ), u iε = εu i + (1-ε)u0, u0 = 0, and ε is an unknown constant satisfying 0 < ε < 1;

[0023] S3.2, a sliding mode surface is designed:

[0024] The system tracking error e of the autonomous underwater vehicle is defined as:

[0025]

[0026] wherein η d represents an expected value of η; η = [x, y, z, φ, θ, ψ] T , x, y and z are respectively the positions of surge, sway and heave, and φ, θ and ψ are respectively the attitudes of roll, pitch and yaw;

[0027] A recursive terminal sliding mode surface is given, and the sliding mode surface is used to drive the autonomous underwater vehicle to track the expected trajectory:

[0028]

[0029] Wherein, s represents a sliding mode variable;Alpha, beta, c1>0, gamma, c2>0;A, b are positive integers satisfying a < b; Denotes the derivative of the system tracking error e of the autonomous underwater robot; Is an intermediate variable;Sig γ (E) = |e| γ Sign (e);

[0030] S3.3, based on the sliding mode surface, the synchronization time estimator and the hyperbolic tangent function, an improved adaptive law-based super-twisting sliding mode controller u is designed, and the expression is as follows:

[0031]

[0032] Wherein, Denotes the nominal part of M, and M is an inertia matrix;J denotes a Jacobian matrix; I=1,...,6, e i Denotes the i-th element of the system tracking error e of the autonomous underwater robot; Is an estimated value of the lumped disturbance obtained according to the synchronization time estimator; Denotes a super-twisting sliding mode adaptive law;F, delta1 are intermediate variables;

[0033]

[0034] The beneficial effects of the present application are as follows: for a 6-DOF autonomous underwater robot system, an improved adaptive law-based adaptive super-twisting sliding mode control trajectory tracking method for the autonomous underwater robot system is designed, the method estimates the lumped disturbance by introducing an auxiliary variable and an output variable to design a synchronization time estimator, so that the system convergence error can converge to zero within the synchronization time;The further constructed improved adaptive law-based super-twisting sliding mode controller can accelerate the system convergence while suppressing system chattering;In summary, the adaptive super-twisting sliding mode controller based on the synchronization time estimator can effectively overcome the influence of the lumped disturbance on the control gain, so that a smoother control input can be obtained, the effect of reducing chattering is achieved, and the convergence time of the system can be reduced, so that the system can maintain significant tracking performance when subjected to complex uncertainty. BRIEF DESCRIPTION OF DRAWINGS

[0035] Figure 1 Is a control flow diagram of the present application;

[0036] Figure 2 Is a position tracking effect diagram of the 6-DOF autonomous underwater robot under different expected trajectories;

[0037] Figure 3An effect schematic diagram of position tracking error e of 6 degrees of freedom of an autonomous underwater robot under different expected trajectories according to the present application;

[0038] Figure 4 An effect schematic diagram of control input u of each degree of freedom of an autonomous underwater robot according to the present application;

[0039] Figure 5 An effect schematic diagram of estimation of a synchronization time estimator according to the present application;

[0040] Figure 6 An error μ of estimation of a synchronization time estimator according to the present application e An effect schematic diagram. DETAILED DESCRIPTION

[0041] In order to make the objects, 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. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative work under the premise that the embodiments are not deviated from the scope of the present application. It should be noted that, in the case of no conflict, the embodiments in the present application and the features in the embodiments can be combined with each other at will.

[0042] Embodiment 1: As shown in the figure, a trajectory tracking method of an autonomous underwater robot system super-twisting sliding mode control based on improved adaptive law, comprising the following steps: Figures 1-6

[0043] S1, a 6-degree-of-freedom autonomous underwater robot dynamics model is established; the 6-degree-of-freedom autonomous underwater robot dynamics model is converted to construct a 6-degree-of-freedom autonomous underwater robot dynamics model based on lumped disturbance; the initial position information of the autonomous underwater robot, the adaptive super-twisting sliding mode controller parameters are initialized;

[0044] S2, a synchronization time estimator is designed to estimate the lumped disturbance in the 6-degree-of-freedom autonomous underwater robot dynamics model based on lumped disturbance;

[0045] S3, a super-twisting sliding mode controller based on improved adaptive law is designed based on sliding surface, synchronization time estimator and hyperbolic tangent function to realize trajectory tracking of the autonomous underwater robot.

[0046] Further, the S1 comprises:

[0047] ​S1.1, in order to study the motion law of autonomous underwater vehicle AUV, to determine its position and attitude and speed, the two rectangular coordinate system is adopted, respectively, ground coordinate system and motion coordinate system, the kinematics equation of AUV needs to establish the conversion relationship between the two coordinate systems, namely the linear velocity and angular velocity conversion matrix of AUV. The kinematics equation of AUV can be expressed as:

[0048]

[0049] Wherein: the Jacobian matrix J(η) represents the derivative of position in ground coordinate system And the relationship between the linear velocity and angular velocity vector υ in the motion coordinate system; η = [x, y, z, φ, θ, ψ] T , x, y, z are the longitudinal, lateral and heave positions, φ, θ, ψ represent the roll, pitch and yaw attitudes, υ = [u*, v, w, p, q, r] T , u*, v, w, p, q, r are the longitudinal, lateral, heave, roll, pitch and yaw velocities.

[0050] The dynamic model of 6 degrees of freedom autonomous underwater vehicle is established:

[0051]

[0052] Wherein: M is the inertia matrix, C(υ) is the Coriolis centripetal force matrix, D(υ) is the dynamic resistance and buoyancy matrix caused by water flow, g(η) is the restoring force vector caused by the combined action of buoyancy and gravity, d is the external disturbance caused by the environment, τ is the control input torque.

[0053] S1.2, in practical application, the accurate AUV dynamics is usually unavailable, therefore, the parameter matrix and vector in AUV dynamics model can be divided into nominal term part and uncertain term part, M, C(υ), D(υ), g(η) can be expressed as:

[0054]

[0055] Wherein: Respectively, M, C(υ), D(υ), g(η) are the nominal part; ΔM, ΔC(υ), ΔD(υ), Δg(η) are the uncertain term part of M, C(υ), D(υ), g(η). According to the above description, the original AUV dynamics model can be converted into the 6 degrees of freedom autonomous underwater vehicle dynamics model based on the lumped disturbance, the expression is as follows:

[0056]

[0057] Wherein: Defined as the lumped disturbance, Let represent the derivative of υ.

[0058] Furthermore, S2 specifically includes:

[0059] The dynamic model of the 6-DOF autonomous underwater robot based on lumped perturbation, i.e., Equation (4), is transformed as follows:

[0060]

[0061] Where: σ1 is a positive constant;

[0062] Let intermediate variables Based on the intermediate variable, design an auxiliary variable υ. a The expression is:

[0063]

[0064] Based on the auxiliary variables, define an error variable υ. e Based on the error variable υ e Define an output variable y1, with the following expression:

[0065]

[0066] Where σ2 is a positive constant.

[0067] According to the error variable υ a Given the output variable y1, design a synchronization time estimator:

[0068]

[0069] in: This is an estimate of the lumped disturbance. It is the error variable υ e The estimated value, for The derivative of ; σ3 and σ4 are two positive constants; β1 and β2 are two positive odd numbers, and satisfy . This represents the error of the synchronization time estimator.

[0070] Furthermore, a stability proof of the synchronization time estimator proposed in this invention is provided:

[0071] Define the estimation error of the synchronization time estimator as μ e According to equation (7), we can obtain:

[0072]

[0073] Combining equation (8), we can obtain the estimation error expression for the synchronization time estimator:

[0074]

[0075] So we have:

[0076]

[0077] The derivative of the estimation error of the synchronization time estimator can be obtained accordingly

[0078]

[0079] The Lyapunov function is selected as:

[0080]

[0081] Where T represents transposition;

[0082] Taking the differential of V with respect to time t: g

[0083]

[0084] Based on (14), it can be obtained that for any t≥T g , the Lyapunov function and the error of the synchronization time estimator satisfy the following:

[0085]

[0086] Therefore, the synchronization time T g is:

[0087]

[0088] According to it can be obtained that for any t≥T g , there is: μ e ≡0, which indicates that the estimation error of the synchronization time estimator can converge to zero.

[0089] Further, the S3 comprises:

[0090] S3.1 Designing a hyperbolic tangent function

[0091] The control input torque τ is defined as an unknown input saturated actuator thrust, which can be specifically expressed as:

[0092]

[0093] Where: i=1, 2,..., 6; u iM is a normal number; τ represents τ(u)=τ ε u and τ(u)∈R 6 , R 6 represents a 6-order vector, τ​i (u i ) represents the i-th element of τ(u), u i is the i-th element of u, u represents the super-twisting sliding mode controller based on the improved adaptive law; τ ε = diag(τ 1ε , τ 2ε , …, τ 6ε ), u iε = εu i + (1 - ε)u0, u0 = 0, ε is an unknown constant satisfying 0 < ε < 1.

[0094] S3.2, design sliding surface

[0095] The system tracking error e of the autonomous underwater vehicle is defined as:

[0096]

[0097] where η d represents the desired value of η;

[0098] Given a recursive terminal sliding surface, the autonomous underwater vehicle is driven to track the desired trajectory using this sliding surface:

[0099]

[0100] where s represents the sliding variable; a, b, c1 > 0, g, c2 > 0; a, b are positive integers satisfying a < b; represents the derivative of the system tracking error e of the autonomous underwater vehicle.

[0101] From equation (19), it can be seen that when the initial condition of the integral term is set to the sliding variable s = 0. This means that the system can start directly from the sliding surface without going through the reaching phase. This feature speeds up the response of the system and enhances the global robustness of the system.

[0102] S3.3 Design super-twisting sliding mode controller

[0103] Since τ = τ eq + τ sw ; according to the equivalent control principle, the equivalent control law τ eq is designed as:

[0104]

[0105] where: represents the nominal part of the inertia matrix M; J represents the Jacobian matrix; Λ = diag(|e iγ-1 ), i = 1,..., 6, e i denotes the i-th element of the system error e of the autonomous underwater vehicle; υ is the linear and angular velocity vector in the moving coordinate system, C is the Coriolis centripetal force matrix, D is the dynamic drag and buoyancy matrix due to water flow, and g is the restoring force vector caused by the combined action of buoyancy and gravity;

[0106] The switching term is used to compensate for the difference between the expected dynamics and the actual dynamics due to model uncertainty and external disturbances. In order to reduce the chattering of the system, an improved super-torsional sliding mode adaptive law is used as in equations (23) and (24). Combined with the improved super-torsional sliding mode adaptive law, the controller robust term τ sw can be designed as

[0107]

[0108] wherein, is the estimated value of the lumped disturbance;

[0109] Based on the fact that the gain of the ordinary adaptive law is prone to overestimation, and in order to effectively reduce the chattering of the system, the above robust control term uses the following super-torsional sliding mode adaptive law:

[0110]

[0111] wherein: i = 1,..., 6, ζ1, ζ2, γ1, k1 and k2 are all normal numbers; is a very small number; s i denotes the i-th element of s;

[0112] Based on the sliding surface, the synchronization time estimator and the hyperbolic tangent function, the super-torsional sliding mode controller u based on the improved adaptive law is designed, expressed as:

[0113]

[0114] Further, the Lyapunov function is selected to prove the stability of the robot:

[0115] In the case of s = 0, we can get This means that and have the same convergence time. Therefore, the second term of equation (19) is equivalent to:

[0116]

[0117] Solving equation (26) this Bernoulli equation gives:

[0118]

[0119] Therefore, the time for the robot to reach the inner layer sliding surface s from the outer layer sliding surface satisfies:

[0120]

[0121] When , the Lyapunov function of the system is defined as V = 0.5e 2 , the derivative of V with respect to time can be obtained:

[0122]

[0123] where: Λ2 = α|e| + β|e| γ According to the finite time stability theory, the time for the tracking error e to converge to zero satisfies:

[0124]

[0125] In order to prove the finite time convergence of the approximation stage, a new state variable

[0126]

[0127] The derivative with respect to time is:

[0128]

[0129] where: is bounded and can be expressed as is a constant, and has δ is a small constant (0.2 in the embodiment of the present application): I represents the unit matrix.

[0130] The following Lyapunov function is selected:

[0131]

[0132] where: γ1, γ2 are normal numbers; is the upper limit of the gain; P is a positive definite matrix.

[0133]

[0134] The derivative of V0 with respect to time is:

[0135]

[0136] wherein:

[0137]

[0138] wherein κ, μ represent normal numbers;

[0139] According to formula (24), we can obtain:

[0140]

[0141] If If formula (38) is satisfied, the matrix Q is positive definite.

[0142]

[0143] It is easy to obtain Ω = QΩ1= Ω1Q, Ω1is a positive definite symmetric matrix, when Q is a symmetric positive definite matrix, Ω is also a positive definite symmetric matrix, therefore, when formula (38) is satisfied, we can obtain:

[0144]

[0145] wherein λ min (Q) represents the minimum eigenvalue of the matrix Q.

[0146] Since We can obtain:

[0147]

[0148] wherein λ max (P) and λ min (P) are the maximum and minimum eigenvalues of P, and

[0149]

[0150] wherein r2 = min (r1, k1, k2), is the upper limit of the gain; the adaptive gain K 1i and K 2i are bounded, that is Therefore:

[0151]

[0152] The derivative of V is:

[0153]

[0154] The derivative of can be divided into three cases for discussion:

[0155] Case 1: |s i|>ζ1 and Adaptive law as well as By setting It can be obtained Right now:

[0156]

[0157] The value will change as By increasing until equations (45) and (38) are satisfied, the sliding surface will reach a very small neighborhood in a finite amount of time.

[0158] Case 2: |s j If |<ζ1, j=1 or j=2, the adaptive law takes:

[0159]

[0160] Lyapunov function V b satisfy:

[0161]

[0162] In the formula, This represents the upper limit of the gain.

[0163] when It will increase linearly with velocity k2 until... if Easy to obtain According to equation (43), the sign of the Lyapunov function may change, after which |s i | will be greater than ζ1, once |s i |>ζ1, similar to case 1, the sliding surface will converge to a small neighborhood in a finite time |s i |≤ζ1.

[0164] Case 3: |s n |<ζ1, n=1,2, the Lyapunov function satisfies:

[0165]

[0166] Similar to case 2, the sliding mode variable will converge to a small neighborhood in a finite amount of time.

[0167] In summary, the tracking error can converge to zero within a finite amount of time.

[0168] Example 2: To verify the feasibility of the method described in Example 1, this example presents a control simulation experiment of the above control method on a 6-DOF autonomous underwater robot. The specific parameter settings are as follows:

[0169] The matrix M, C(v), D(v) and g(i) include physical constants m ii d ii ,g1 and g2 (i = 1,..., 6). It should be noted that in the design of the control input, these physical constants are assumed to be unknown. That is, when designing the control input, there is no physical constant information of the 6-DOF AUV.

[0170] M = diag(m 11 ,m 22 ,m 33 ,m 44 m 55 ,m 66 )

[0171] D(v) = diag(d 11 ,d 22 ,d 33 ,d 44 ,d 55 ,d 66 )

[0172]

[0173] During the simulation, a 20% model uncertainty is added to all nominal hydrodynamic values. The simulation is carried out in discrete form with a sampling time T s = 0.001 s. These settings are common to all simulations. The desired reference trajectory is a three-dimensional helical diving curve, denoted as: η d = [2cos(0.3πt), -2sin(0.3πt), πt / 15, 0, 0, π / 6] T

[0174] The initial position information of the AUV is The external disturbance d is set as: d = [0.2sin3t, 0.1cos4t, 0.1sin3t, 0.1cos4t, 0.15sin4t, 0.15sin6t] T .

[0175] The parameters of the adaptive hyper-torsional sliding mode controller of the application are as follows:

[0176] Table 1 Adaptive hyper-torsional sliding mode controller parameters

[0177]

[0178] Substituting the above parameters into the controller u and the 6-DOF underwater robot dynamics model formula (4) of the application, the simulation results are: the position tracking response curve and the position tracking error response curve of the 6-DOF autonomous underwater robot are as follows:Figure 2 and Figure 3 The control input response curve of 6 degrees of freedom is shown in Figure 4 The estimation result and estimation error of the synchronization time estimator are shown in Figure 5 6

[0179] It can be seen from Figure 2 and Figure 3 that the system has a short rising time, a fast response speed and a small tracking error, and has a good control performance.

[0180] It can be seen from Figure 4 that the system has a good tracking performance, and the adaptive super-twisting sliding mode controller makes the controller have a small chattering.

[0181] It can be seen from Figure 5 and Figure 6 that the designed synchronization time estimator has a good estimation performance, can effectively observe the disturbance, and the estimation error of the lumped disturbance in the 6 degrees of freedom can converge to zero at the same time in about 1s.

[0182] The specific embodiments of the present application are described in detail above with reference to the drawings, but the present application is not limited to the above-described embodiments, and various changes can be made within the knowledge of those skilled in the art without departing from the purpose of the present application.​​

Claims

1. A super-twisting sliding mode control trajectory tracking method for autonomous underwater vehicle system based on improved adaptive law, characterized in that, The method comprises the following steps: S1, establishing a 6-DOF autonomous underwater robot dynamics model; converting the 6-DOF autonomous underwater robot dynamics model to construct a 6-DOF autonomous underwater robot dynamics model based on lumped disturbance; S2, designing a synchronization time estimator to estimate the lumped disturbance in the 6-DOF autonomous underwater robot dynamics model based on lumped disturbance; S3, designing an improved adaptive law-based hyper-torsional sliding mode controller based on a sliding mode surface, a synchronization time estimator and a hyperbolic tangent function to realize trajectory tracking of the autonomous underwater robot; The 6-DOF dynamics model of the autonomous underwater vehicle based on lumped perturbation is expressed as follows: ; The S2 specifically comprises: The 6-DOF autonomous underwater robot dynamics model based on lumped disturbance is converted into the following: ； wherein: is a constant; is defined as the lumped disturbance, denotes the derivative of the linear and angular velocity vectors in the moving coordinate system ; , , , denote the nominal parts of , , , ; denote the uncertainty parts of , , , ; is the inertia matrix, is the Coriolis centripetal force matrix, is the dynamic resistance and buoyancy matrix for the water flow, is the restoring force vector caused by the combined action of the buoyancy and gravity, is the external disturbance caused by the environment, is the control input torque; Let an intermediate variable ; depending on the intermediate variable, design a helper variable , the expression is: ； An error variable is defined in terms of the auxiliary variable An output variable is defined in terms of the error variable An output variable is defined in terms of the error variable with the following specific expression ； wherein: is a constant; According to the error variable and the output variable , a synchronization time estimator is designed: ； wherein: is an estimate of the aggregate disturbance, is an error variable is an estimate of the aggregate disturbance, is a derivative of ; are two positive odd integers, and satisfy ; ; ; ; is an error of the synchronization time estimator; The S3 comprises: S3.1, designing a hyperbolic tangent function: ； wherein: ; is a constant; denotes and , denotes a 6th order vector, denotes the i-th element of , the i-th element of , denotes a super-twisting sliding mode controller based on an improved adaptive law; , , , , is an unknown constant satisfying . S3.2, designing a sliding mode surface: System tracking error for autonomous underwater robots is defined as: ； wherein denotes the expected value of , are the positions of surge, sway, heave, respectively, denote the attitudes of roll, pitch, yaw, respectively; Given a recursive terminal sliding mode surface, the sliding mode surface is used to drive the autonomous underwater robot to track the desired trajectory: ； ； wherein, represents a sliding mode variable; , ; to meet a positive integer; represents a system tracking error of the autonomous underwater vehicle derivative of; , is an intermediate variable; ; S3.

3. Designing hyper-torsion sliding mode controller based on improved adaptive law based on sliding surface, synchronization time estimator and hyperbolic tangent function , expression: ； wherein represents the nominal part of , is the inertia matrix; represents the Jacobian matrix; , , represents the system tracking error of the autonomous underwater vehicle the first element of ; is the estimate of the lumped disturbance obtained from the synchronized time estimator; represents the super-twisting sliding mode adaptive law; , is an intermediate variable; , .

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