A novel non-singular fast terminal sliding mode control method and system for underwater robots

Abstract

To address the problem of low posture control accuracy in underwater robot systems due to strong nonlinearity, high coupling, and internal parameter uncertainty, this paper proposes a novel non-singular fast terminal sliding mode control method with finite-time convergence. First, the existing dynamic equations of the underwater robot are given and transformed into trajectory tracking error equations to facilitate the application of the sliding mode control method. Then, a non-singular fast terminal sliding mode surface and its corresponding controller are proposed. The controller is analyzed and verified to effectively avoid singularity problems. Simulations are performed on a six-degree-of-freedom underwater robot as the control object, and its posture control is compared with a typical non-singular fast terminal sliding mode control method. The simulation results show that this method can converge the posture to the target posture within a finite time, with higher relative convergence accuracy, and effectively suppresses the effects of internal parameter uncertainty and external interference in the underwater robot system.

Description

Technical Field

[0001] The present invention belongs to the technical field of trajectory tracking control methods for underwater vehicles, and in particular to a novel non-singular fast terminal sliding mode control method and system for underwater robots. Background Art

[0002] Autonomous underwater vehicles (AUVs) are now widely used to perform various underwater missions, such as underwater ore mining, underwater rescue, and oceanographic research. Accurate pose tracking is crucial for successful underwater operations. However, the uncertain underwater environment exacerbates the strong nonlinearity, high coupling, and internal parameter uncertainty of AUV dynamic models. This severely impacts controller performance and poses significant challenges to controller design. Therefore, designing an effective controller is a key task in AUV design.

[0003] Traditional underwater robot controller design often uses linear control methods to achieve the goal of tracking the underwater robot's posture and converge the error to zero. Although this control method is effective, the asymptotic convergence achieved can easily lead to excessively long convergence times, resulting in poor real-time robot control. To address this shortcoming, many advanced nonlinear control methods have been proposed in recent years, such as backstepping control, model predictive control, adaptive control, and sliding mode control. Among these control methods, sliding mode control has attracted the attention of researchers due to its strong robustness, simplicity, and insensitivity to parameter changes, and has been rapidly applied to underwater robot controller design. Wang et al. proposed terminal sliding mode control (TSMC). This method accelerates convergence near the equilibrium point by adding a nonlinear term with a power series less than 1 to the sliding surface, achieving finite-time convergence. Yu et al. proposed fast terminal sliding mode control (FTSMC), which converges faster than terminal sliding mode control when the initial position is far from or close to the equilibrium point. While each of these control methods has its advantages, terminal sliding mode control and fast terminal sliding mode control involve negative power terms when designing controllers, which can lead to singularities when the system converges to an equilibrium point. Elmokadem et al. proposed nonsingular fast terminal sliding mode control to address this singularity by modifying the terminal sliding surface.

[0004] Application No. 2016107072404 discloses a method for terminal sliding mode control of an underwater vehicle based on time delay estimation. This method uses time delay estimation techniques to estimate the lumped uncertainty of the underwater vehicle's closed-loop control system, making the entire control algorithm independent of the system model. Based on this, a fast nonsingular terminal sliding mode hyperplane and a fast terminal sliding mode reaching law are combined to derive a continuous fast nonsingular terminal sliding mode trajectory tracking control method for an underwater vehicle based on time delay estimation. This method uses time delay estimation techniques to estimate the lumped uncertainty of the underwater vehicle's closed-loop control system, but it still cannot accurately estimate the uncertainty, and control accuracy needs to be further improved. Summary of the Invention

[0005] The purpose of the present invention is to provide a novel non-singular fast terminal sliding mode control method and system for an underwater robot, which transforms the dynamic equation of the underwater robot into a trajectory tracking error equation for the application of the sliding mode control method, and then proposes a non-singular fast terminal sliding mode surface and its corresponding controller, which can make the posture converge to the target posture within a finite time with higher relative convergence accuracy, and effectively suppress the influence of internal parameter uncertainty and external interference of the underwater robot system.

[0006] The technical solutions for achieving the purpose of the present invention are:

[0007] A novel non-singular fast terminal sliding mode control method for an underwater robot comprises the following steps:

[0008] S01: Construct the dynamic equations of the underwater robot;

[0009] S02: transform the dynamic equation into the trajectory tracking error equation;

[0010] S03: Design the sliding surface of non-singular fast terminal sliding mode control as:

[0011]

[0012] Where, e is the tracking error, is the first-order derivative of e, α and β are control parameters, α=diag([α1,…,α6]), β=diag([β1,…,β6]), and α i ,β i >0,i=1,…,6,0 i <p i <2q i Represents an internal parameter and is an odd number, i=1,…6;

[0013] S04: Apply non-singular fast terminal sliding mode control to the underwater robot. The control input is designed as:

[0014] ​

[0015] where σ = [σ1,…,σ6] T represents the internal parameter, σ i >0,|D i | <L i ,i=1,2,…,6,L i represents the upper bound of the i-th dimension D, B, f, and D are the parameters of the trajectory tracking error equation, and Lsign(s) is the sign function of the upper and lower bounds of interference.

[0016] In the preferred technical solution, the construction of the dynamic equation of the underwater robot in step S01 includes:

[0017] The dynamic model of underwater robot is generally described as follows:

[0018]

[0019] Among them, η∈R 6×1 =[η1,η2] T =[x,y,z,φ,θ,ψ] T Represents the position and angle of the underwater robot in the world coordinate system, ν∈R 6×1 =[ν1,ν2] T =[u,v,w,p,q,r] T Represents the speed and angular velocity of the underwater robot in the motion coordinate system, τ∈R 6×1 =[F e ,M e ] T =[X,Y,Z,K,M,N] T Represents the external force and torque applied to the underwater robot; M∈R 6×6 is the inertia matrix; C(ν)∈R 6×6 is the rigid body Coriolis term and centripetal term matrix; D(ν) is the fluid damping matrix; G(η) is the gravity and buoyancy matrix, d(η,t) is the external disturbance to the underwater robot, and t is time; are the speed and angular velocity of the underwater robot in the world coordinate system, is the speed and angular velocity of the underwater robot in the motion coordinate system;

[0020] The coordinate transformation matrix between the motion coordinate system and the world coordinate system is expressed by J(η):

[0021]

[0022] R(η),T(η) are:

[0023]

[0024]

[0025] Where, s(·), c(·), t(·) represent sin(·), cos(·), tan(·), respectively; θ is the pitch angle, ψ is the pitch angle, and φ is the roll angle;

[0026] The dynamic model transformation in the world coordinate system is:

[0027]

[0028] in: A is the equation parameter, is the second-order derivative of η, which represents the acceleration and angular acceleration of the underwater robot in the world coordinate system.

[0029] In the preferred technical solution, the error between the dynamic parameters calculated by the underwater robot and the actual dynamic parameters is dis represents external interference, △ represents the amount of change, |d n (η,t)|≤L gn ,n=1,2,3,4,5,6,L gn is the upper bound of the nth dimension.

[0030] In the preferred technical solution, the sign(·) in the controller is used to offset model uncertainty and external interference;

[0031] When the tracking error approaches the equilibrium point, the It tends to zero, thereby weakening the influence of (L|s|-sD) and suppressing the disturbance, where L represents the upper bound of D and |s| represents the absolute value of the sliding surface;

[0032] If the system state variables reach the equilibrium point, k = 0, the uncertainty of the model parameters and external interference will not have any impact on the system, and sliding mode control without chattering will be performed.

[0033] The present invention also discloses a novel non-singular fast terminal sliding mode control system for an underwater robot, comprising:

[0034] Dynamic equation building module, which builds the dynamic equations of underwater robots;

[0035] Trajectory tracking error transformation module, transforming the dynamic equation into trajectory tracking error equation;

[0036] The sliding surface design module designs the sliding surface of non-singular fast terminal sliding mode control as follows:

[0037]

[0038] Where, e is the tracking error, is the first-order derivative of e, α and β are control parameters, α=diag([α1,…,α6]), β=diag([β1,…,β6]), and α i ,β i >0,i=1,…,6,0 i <p i <2q i Represents an internal parameter and is an odd number, i=1,…6;

[0039] The non-singular fast terminal sliding mode control module applies non-singular fast terminal sliding mode control to the underwater robot. The control input is designed as:

[0040]

[0041] where σ = [σ1,…,σ6] T represents the internal parameter, σ i >0,|D i | <L i ,i=1,2,…,6,L i represents the upper bound of the i-th dimension D, B, f, and D are the parameters of the trajectory tracking error equation, and Lsign(s) is the sign function of the upper and lower bounds of interference.

[0042] In the preferred technical solution, the step of constructing the dynamic equation of the underwater robot in the dynamic equation construction module includes:

[0043] The dynamic model of underwater robot is generally described as follows:

[0044]

[0045] Among them, η∈R 6×1 =[η1,η2] T =[x,y,z,φ,θ,ψ] T Represents the position and angle of the underwater robot in the world coordinate system, ν∈R 6×1 =[ν1,ν2] T =[u,v,w,p,q,r] T Represents the speed and angular velocity of the underwater robot in the motion coordinate system, τ∈R 6×1 =[F e ,M e ] T =[X,Y,Z,K,M,N] T Represents the external force and torque applied to the underwater robot; M∈R 6×6 is the inertia matrix; C(ν)∈R 6×6 ​is the rigid body Coriolis term and centripetal term matrix; D(ν) is the fluid damping matrix; G(η) is the gravity and buoyancy matrix, and d(η,t) is the external disturbance to the underwater robot; are the speed and angular velocity of the underwater robot in the world coordinate system, is the speed and angular velocity of the underwater robot in the motion coordinate system;

[0046] The coordinate transformation matrix between the motion coordinate system and the world coordinate system is expressed by J(η):

[0047]

[0048] R(η),T(η) are:

[0049]

[0050]

[0051] Where, s(·), c(·), t(·) represent sin(·), cos(·), tan(·), respectively; θ is the pitch angle, ψ is the pitch angle, and φ is the roll angle;

[0052] The dynamic model transformation in the world coordinate system is:

[0053]

[0054] in: A is the equation parameter, is the second-order derivative of η, which represents the acceleration and angular acceleration of the underwater robot in the world coordinate system.

[0055] In the preferred technical solution, the error between the dynamic parameters calculated by the underwater robot and the actual dynamic parameters is dis represents external interference, △ represents the amount of change, |d n (η,t)|≤L gn ,n=1,2,3,4,5,6,L gn is the upper bound of the nth dimension.

[0056] In the preferred technical solution, the sign(·) in the controller is used to offset model uncertainty and external interference;

[0057] When the tracking error approaches the equilibrium point, the It tends to zero, thereby weakening the influence of (L|s|-sD) and suppressing the disturbance, where L represents the upper bound of D and |s| represents the absolute value of the sliding surface;

[0058] If the system state variables reach the equilibrium point, k = 0, the uncertainty of the model parameters and external interference will not have any impact on the system, and sliding mode control without chattering will be performed.

[0059] The present invention further discloses a computer storage medium on which a computer program is stored. When the computer program is executed, the novel non-singular fast terminal sliding mode control method of the underwater robot is realized.

[0060] Compared with the prior art, the present invention has the following significant advantages:

[0061] 1. This paper proposes a new finite-time nonsingular fast terminal sliding mode control method. This method provides a new sliding surface and a corresponding controller design, minimizing chattering effects and effectively suppressing the effects of internal parameter uncertainty and external interference during the underwater robot's pose convergence. Using the Lyapunov function, this method demonstrates that the underwater robot system can converge within a finite time. Furthermore, the controller is verified to be nonsingular, preventing sudden large control inputs from damaging the underwater robot system during execution, further facilitating practical system application and expansion. This method enables the pose to converge to the target pose within a finite time, with higher relative convergence accuracy, and effectively suppresses the effects of internal parameter uncertainty and external interference in the underwater robot system.

[0062] 2. Due to model uncertainty and time-varying external disturbances, the controller uses sign(·) to counteract these uncertainties. When the tracking error approaches equilibrium, k, which is related to the tracking error, approaches zero, weakening the influence of (L|s| - sD), thereby suppressing disturbances. If the system can achieve an ideal sliding mode, when the state variables reach equilibrium, k = 0. Model parameter uncertainty and external disturbances have no impact on the system, resulting in chatter-free sliding mode control. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 Flowchart of a novel non-singular fast terminal sliding mode control method for underwater robots according to a preferred embodiment;

[0064] Figure 2 This is a principle block diagram of a novel non-singular fast terminal sliding mode control system for an underwater robot according to a preferred embodiment;

[0065] Figure 3 is the convergence curve of η[4](roll);

[0066] Figure 4 Control input for singular problems;

[0067] Figure 5 The control input required for Euler angle convergence;

[0068] Figure 6 is the X channel convergence curve;

[0069] Figure 7 is the Y channel convergence curve;

[0070] Figure 8 is the Z channel convergence curve;

[0071] Figure 9 is the roll angle channel convergence curve;

[0072] Figure 10 is the convergence curve of the pitch angle channel;

[0073] Figure 11 is the yaw angle channel convergence curve. DETAILED DESCRIPTION

[0074] The principle of this invention is as follows: This method provides a new sliding mode surface and a corresponding controller design, which minimizes the impact of chattering and effectively suppresses the effects of internal parameter uncertainty and external interference during the underwater robot's posture convergence. Using the Lyapunov function, this method proves that the underwater robot system can converge within a finite time and verifies that the controller is non-singular. The overall structure of this paper is: first, an AUV model is given and transformed into a dynamic model in the world coordinate system. Then, the designed non-singular fast terminal sliding mode control method and corresponding proof are presented. Finally, the proposed method is applied to the AUV model simulation and analyzed and explained.

[0075] Example 1:

[0076] like Figure 1 As shown in FIG, a novel non-singular fast terminal sliding mode control method for an underwater robot includes the following steps:

[0077] S01: Construct the dynamic equations of the underwater robot;

[0078] S02: transform the dynamic equation into the trajectory tracking error equation;

[0079] S03: Design the sliding surface of non-singular fast terminal sliding mode control as:

[0080]

[0081] Where, e is the tracking error, is the first-order derivative of e, α and β are control parameters, α=diag([α1,…,α6]), β=diag([β1,…,β6]), and α i ,β i >0,i=1,…,6,0 i <p i <2q​i Represents an internal parameter and is an odd number, i=1,…6;

[0082] S04: Apply non-singular fast terminal sliding mode control to the underwater robot. The control input is designed as:

[0083]

[0084] where σ = [σ1,…,σ6] T represents the internal parameter, σ i >0,|D i | <L i ,i=1,2,…,6,L i represents the upper bound of the i-th dimension D, B, f, and D are the parameters of the trajectory tracking error equation, and Lsign(s) is the sign function of the upper and lower bounds of interference.

[0085] In one embodiment, constructing the dynamic equation of the underwater robot in step S01 includes:

[0086] The dynamic model of underwater robot is generally described as follows:

[0087]

[0088] Among them, η∈R 6×1 =[η1,η2] T =[x,y,z,φ,θ,ψ] T Represents the position and angle of the underwater robot in the world coordinate system, ν∈R 6×1 =[ν1,ν2] T =[u,v,w,p,q,r] T Represents the speed and angular velocity of the underwater robot in the motion coordinate system, τ∈R 6×1 =[F e ,M e ] T =[X,Y,Z,K,M,N] T Represents the external force and torque applied to the underwater robot; M∈R 6×6 is the inertia matrix; C(ν)∈R 6×6 is the rigid body Coriolis term and centripetal term matrix; D(ν) is the fluid damping matrix; G(η) is the gravity and buoyancy matrix, and d(η,t) is the external disturbance to the underwater robot; Represents the speed and angular velocity of the underwater robot in the world coordinate system, Represents the speed and angular velocity of the underwater robot in the motion coordinate system;

[0089] The coordinate transformation matrix between the motion coordinate system and the world coordinate system is expressed by J(η):

[0090]

[0091] R(η),T(η) are described as:

[0092]

[0093]

[0094] Where, s(·), c(·), t(·) represent sin(·), cos(·), tan(·), respectively; θ is the pitch angle, ψ is the pitch angle, and φ is the roll angle;

[0095] The dynamic model transformation in the world coordinate system is:

[0096]

[0097] in: A is the equation parameter, is the second-order derivative of η, which represents the acceleration and angular acceleration of the underwater robot in the world coordinate system.

[0098] In one embodiment, the error between the dynamic parameters measured by the underwater robot and the actual dynamic parameters is dis represents external interference, △ represents the amount of change, |d n (η,t)|≤L gn ,n=1,2,3,4,5,6,L gn is the upper bound of the nth dimension.

[0099] In one embodiment, the sign(·) in the controller is used to offset model uncertainty and external disturbances;

[0100] When the tracking error approaches the equilibrium point, the It tends to zero, thereby weakening the influence of (L|s|-sD) and suppressing the disturbance, where L represents the upper bound of D and |s| represents the absolute value of the sliding surface;

[0101] If the system state variables reach the equilibrium point, k = 0, the uncertainty of the model parameters and external interference will not have any impact on the system, and sliding mode control without chattering will be performed.

[0102] In another embodiment, a computer storage medium stores a computer program, which, when executed, implements the novel non-singular fast terminal sliding mode control method for an underwater robot.

[0103] In another embodiment, Figure 2 As shown in FIG, a novel non-singular fast terminal sliding mode control system for underwater robots includes:

[0104] A dynamic equation construction module 10 is used to construct the dynamic equation of the underwater robot;

[0105] The trajectory tracking error conversion module 20 converts the dynamic equation into a trajectory tracking error equation;

[0106] The sliding surface design module 30 designs the sliding surface of the non-singular fast terminal sliding mode control as follows:

[0107]

[0108] Where, e is the tracking error, is the first-order derivative of e, α and β are control parameters, α=diag([α1,…,α6]), β=diag([β1,…,β6]), and α i ,β i >0,i=1,…,6,0 i <p i <2q i Represents an internal parameter and is an odd number, i=1,…6;

[0109] The non-singular fast terminal sliding mode control module 40 applies non-singular fast terminal sliding mode control to the underwater robot, and the control input is designed as:

[0110]

[0111] where σ = [σ1,…,σ6] T represents the internal parameter, σ i >0,|D i | <L i ,i=1,2,…,6,L i represents the upper bound of the i-th dimension D, B, f, and D are the parameters of the trajectory tracking error equation, and Lsign(s) is the sign function of the upper and lower bounds of interference.

[0112] Specifically, the following describes the workflow of the novel underwater robot non-singular fast terminal sliding mode control system using a preferred embodiment as an example:

[0113] AUV model and problem description

[0114] 2.1 AUV model description

[0115] The dynamic model of underwater robot is generally described as follows:

[0116]

[0117] Where: η∈R 6×1 =[η1,η2] T ​=[x,y,z,φ,θ,ψ] T Represents the position and angle of the underwater robot in the world coordinate system; ν∈R 6×1 =[ν1,ν2] T =[u,v,w,p,q,r] T Represents the speed and angular velocity of the underwater robot in the motion coordinate system; τ∈R 6×1 =[F e ,M e ] T =[X,Y,Z,K,M,N] T Represents the external force and torque applied to the underwater robot. 6×6 is the inertia matrix; C(ν)∈R 6×6 is the rigid body Coriolis term and centripetal term matrix; D(ν) is the fluid damping matrix; G(η) is the gravity and buoyancy matrix, d(η,t) is the external disturbance to the underwater robot, and t is time.

[0118] The position and posture of the underwater robot are usually described in the world coordinate system. The coordinate transformation matrix between the motion coordinate system and the world coordinate system is represented by J(η), which is described as:

[0119]

[0120] R(η),T(η) are described as:

[0121]

[0122]

[0123] Among them: s(·), c(·), t(·) represent sin(·), cos(·), tan(·).

[0124] The pitch angle θ is bounded, Avoid singularity problems in J(η) caused by rotation.

[0125] In actual systems, the hydrodynamic coefficients, external disturbances, errors in system modeling, etc. may cause inaccurate system parameters. dis represents external interference. In actual systems, velocity, acceleration, and uncertain parameters are all bounded, so |d n (η,t)|≤L gn ,n=1,2,3,4,5,6,L gn is the upper bound of the nth dimension.

[0126] τ represents the thrust and torque decomposed into six degrees of freedom, which satisfies the matrix relationship with the direction of the thruster force of the actual AUV model.

[0127] The first equation in model (1) is denoted as 1.a, and the second equation is denoted as 1.b.

[0128] In order to obtain the dynamic model in the world coordinate system, substitute 1.a into 1.b in formula (1) and perform the transformation to obtain:

[0129]

[0130] in: D=-(MJ1) -1 d(η,t),D represents the perturbation parameter matrix of the trajectory tracking error equation.

[0131] The relevant derivations are as follows:

[0132] Eliminate ν in the motion coordinate system in equation 1.b with η in the world coordinate system. Derivative the following equation, we can get

[0133]

[0134] Right now:

[0135]

[0136]

[0137] Since the rotation matrix R(η) is an orthogonal matrix, R -1 (η)=R T (η), which can be expressed as a compact matrix:

[0138]

[0139] Use J1 and J2 to replace the corresponding position matrices:

[0140]

[0141] Substitute (4) into (1.b) and have to:

[0142]

[0143] Deformed:

[0144]

[0145] Among them: J -1 (η)=J1,

[0146] Let B = (MJ1) -1 ,A=(MJ1) -1 (MJ2J -1(η)-C(v)J -1 (η)-D(v)J -1 (η)),D=-(MJ1) -1 d(η,t)

[0147] At this point, Equation 1 is transformed into a dynamic model in the world coordinate system, preparing for subsequent sliding mode control applications.

[0148] symbol[·] g =|·| g sign(·), where g is a positive number, sign() is the sign function, |·| g =diag(|·|1,…,|·| n ]), is the Euclidean norm.

[0149] 2.2.1 Basic Definitions and Lemmas

[0150] Definition 1: Given a sliding surface s, movement on the sliding surface is called an ideal sliding mode, which can be described as:

[0151]

[0152] Definition 2: Given a sliding surface s, movement within a range of δ near the sliding surface is called an actual sliding mode, which can be described as:

[0153]

[0154] Lemma 1: Suppose there is a continuous, smooth, unbounded positive definite function V(x) such that:

[0155]

[0156] Where: c>0, b>0, 0<α<1, at this time V(x) reaches V(x)=0 in a finite time, and the stabilization time T x (x0) satisfies:

[0157]

[0158] 3 Non-singular fast terminal sliding mode control

[0159] In the case of uncertain internal parameters and uncertain external disturbances in underwater robots, this section proposes a novel non-singular fast terminal sliding mode control method. Based on this method, an underwater robot controller is designed. This controller can effectively control the target posture, that is, the actual posture η converges to the target posture η within a finite time. d , let e = η - η d , represents the tracking error. The target poses in this paper are all constant, so All are 0. Transform equation (3) into a function of the tracking error:

[0160]

[0161] Where:

[0162] 3.1 Traditional Fast Terminal Sliding Mode Control

[0163] The sliding surface is described as:

[0164]

[0165] Where:

[0166] m = diag([m1,…,m6]), n = diag([n1,…,n6])

[0167] , m i , n i > 0, i = 1,…,6, p < q represent internal parameters and are positive odd numbers. When the system state variables reach the sliding surface (i.e., s = 0), the values of m, n, p, and q can be adjusted to meet different requirements of the system for the rapidity of convergence. When e is far from the equilibrium point, e provides a large convergence speed; when e is close to the equilibrium point, converges faster than e. From any initial state e(0) ≠ 0, the time for the system state to converge to the equilibrium point is:

[0168]

[0169] For the underwater robot system (7), by adopting the design method of equation (8), the control input τ can be obtained to satisfy:

[0170]

[0171] Where: γ1, γ2, m, n, represent parameter diagonal matrices, and the diagonal parameters are all positive. When e = 0, the in the control input τ will become extremely large, resulting in a non - bounded τ and a singularity problem. In ideal sliding mode control, see Definition 1, as long as the system enters the sliding mode (s = 0) that is and q < p < 2p, at this time no singularity problem will occur.

[0172] 3.2 Nonsingular Fast Terminal Sliding Mode Control

[0173] In actual systems, due to calculation errors and uncertainties, the sliding mode will not maintain the theoretical value (s = 0), but will move within the range of δ near s = 0 (see Definition 2). When the system state is close to the equilibrium point, the frequency of singular problems increases. In order to effectively solve the singular problem of fast terminal sliding mode control, the following sliding mode surface is proposed:

[0174]

[0175] Among them, α=diag([α1,…,α6]), β=diag([β1,…,β6]), and α i ,β i >0,i=1,…,6.0 i <p i <2q i Represents an internal parameter and is an odd number, i=1,…6.

[0176] Theorem 1: Apply non-singular fast terminal sliding mode control (11) to the underwater robot system (7), and the control input is designed as:

[0177]

[0178] Where, the parameters α, β, p, q are defined as shown in Equation (11), and σ = [σ1,…,σ6] T represents the internal parameter, σ i >0,|D i | <L i ,i=1,2,…,6,L i Represents the upper bound of the i-th dimension D. The designed control input is non-singular and its corresponding non-singular fast terminal sliding surface converges in finite time, and the state variables of the system also converge and stabilize in finite time.

[0179] The proof process is as follows:

[0180] Because p,q are positive odd numbers, and q i <p i <2q i ,so 2q-p is an odd number, which can make It is not a negative power, because When it approaches zero, τ becomes infinite, that is, there is no singular problem. Similarly, the other terms in τ do not contain negative powers, and there is no singular problem.

[0181] The finite time convergence is proved below. For the sliding surface (11), along the system dynamics derivative for:

[0182] ​

[0183] Where: Represents the second derivative of the error e.

[0184] Let We get:

[0185]

[0186] Select the Lyapunov function:

[0187]

[0188] Differentiate Equation (14) and substitute Equation (13) into it, we get

[0189] <s

[0190] Since 0 < p < 2q and p is odd, so p - q and p + q are even, we get That is, k > 0, Finally Satisfies Lyapunov stability.

[0191] Equation (16) can be further written as the following inequality:

[0192]

[0193] Where: α, kσ > 0, Similar to Lemma 1, it can be obtained that the nonsingular fast terminal sliding mode surface can converge in finite time. After the system enters the sliding mode quickly, there is The time for the tracking error of the underwater robot to converge to the equilibrium point is the same as Equation (6), so the tracking error converges in finite time, and thus Theorem 1 is proved.

[0194] An important prerequisite for the nonsingular fast terminal sliding mode to remain nonsingular is 0 < q < p < 2q, and p and q are odd.

[0195] To further reduce the chattering of the sliding mode control and the designed control input needs to be continuous, the saturation function sat(s / φ) can be used.

[0196] Due to model uncertainty and time-varying external disturbances, the sign(·) in the controller is used to counteract these uncertainties. When the tracking error approaches equilibrium, k, which is related to the tracking error, approaches zero, weakening the influence of (L|s| - sD) and thus suppressing disturbances. If the system can operate in an ideal sliding mode, as defined in Definition 1, when the system state variables reach equilibrium, k = 0. Model parameter uncertainty and external disturbances have no impact on the system, resulting in chatter-free sliding mode control.

[0197] To verify the effectiveness of the proposed non-singular fast terminal sliding mode control, a simulation and analysis of the pose tracking of an underwater robot system is performed. To illustrate the effectiveness of the proposed method (NFTSMC-1), the classic non-singular fast terminal sliding mode control (NFTSMC-2) and non-singular terminal sliding mode control (NTSMC) are introduced for comparison and explanation.

[0198] The parameters of the three sliding mode controls are shown in the table below:

[0199]

[0200] The relevant parameters in the table are expressed using parameters from the original literature. α, β, γ1, and γ2 affect convergence speed and accuracy. The choice of each parameter depends on project requirements and hardware availability. This paper uses the coefficients in the figure to facilitate algorithm comparison. p, q, K1, and K2 are all 6-row, 1-column column vectors, with all elements being identical. Therefore, Table 1 uses constants to better represent matrices with identical parameters.

[0201] The following is a simulation result of the convergence of the roll degree of freedom in the six degrees of freedom when there is no uncertain external interference and the model parameters are determined, as shown in the figure: Figure 3 shown.

[0202] Figure 3 This curve shows the roll angle convergence from an initial value of 5 to a target value of 0 under ideal conditions with no uncertain external disturbances and precise model parameters. This demonstrates that the proposed nonsingular fast terminal sliding mode control method converges within a finite time. The variances of the roll angle convergence errors calculated for the three methods from convergence to the equilibrium point to the end of the simulation are 1.8139e-05, 5.9779e-04, and 0.0172, respectively. This demonstrates that the proposed nonsingular fast terminal sliding mode control method converges with high accuracy and is effective.

[0203] Next, the simulation results of the underwater robot in six degrees of freedom are given under the actual situation of uncertain external interference, where the initial value η=[-5,5,1,5,5,90], Target value η d =[0,0,0,0,0,0]T .

[0204] Figure 4 A sliding mode control method with a singular problem is applied to the underwater robot, that is, the six-degree-of-freedom control input obtained by Equation 8. As can be seen from the figure, when the tracking error of the six degrees of freedom converges to near 0, due to the existence of the singular problem, the tracking error suddenly becomes large in a short period of time, causing the underwater robot to be unstable. Figure 5 The curves of the control input required for the roll angle, pitch angle, and yaw angle errors to converge to 0 are u(4), u(5), and u(6) respectively. It can be seen from the figure that the non-singular fast terminal sliding mode control proposed in this paper will not have extremely large control input values ​​when the tracking error converges to near 0, and there is no singular phenomenon.

[0205] Figure 6-11 Convergence curves for the six degrees of freedom (DOFs) are plotted from their initial values ​​to their target values. The figures demonstrate that the proposed nonsingular fast terminal sliding mode control converges within a finite time. A zoomed-in view of the convergence plot shows that the proposed sliding mode control exhibits smaller chatter amplitude than the other two compared sliding mode controls, reducing chatter and improving the system's tracking accuracy. The proposed sliding mode control method exhibits smoother curves, faster convergence, and superior control performance, validating the correctness and effectiveness of the designed finite-time nonsingular fast terminal sliding mode control.

[0206] in conclusion

[0207] This paper proposes a novel nonsingular fast terminal sliding mode control method. The controller designed by this method is nonsingular. Using the Lyapunov function, the method demonstrates finite-time convergence and can mitigate the effects of internal parameter inaccuracies and external disturbances on the system. Finally, the method is applied to an underwater robot system to achieve relatively precise position and posture control of the system. Experimental results demonstrate that all six degrees of freedom converge rapidly to near zero, and the method achieves more accurate convergence than representative nonsingular fast terminal sliding mode control methods.

[0208] The above embodiments are preferred implementation modes of the present invention, but the implementation modes of the present invention are not limited to the above embodiments. Any other changes, modifications, substitutions, combinations, and simplifications that do not deviate from the spirit and principles of the present invention should be considered as equivalent replacement methods and are included in the scope of protection of the present invention.

Claims

1. A novel non-singular fast terminal sliding mode control method for underwater robots, characterized by: The following steps are involved: S01: Construct the dynamic equations of the underwater robot; S02: transform the dynamic equation into the trajectory tracking error equation; S03: Design the sliding surface of non-singular fast terminal sliding mode control as: Where, e is the tracking error, is the first-order derivative of e, α and β are control parameters, α=diag([α1,…,α6]), β=diag([β1,…,β6]), and α i ,β i >0,i=1,…,6,0 i <p i <2q i Represents an internal parameter and is an odd number, ​ S04: Apply non-singular fast terminal sliding mode control to the underwater robot. The control input is designed as: where σ = [σ1,…,σ6] T represents the internal parameter, σ i >0,|D i | <L i ,i=1,2,…,6,L i represents the upper bound of the i-th dimension D, B, f, and D are the parameters of the trajectory tracking error equation, and Lsign(s) is the sign function of the upper and lower bounds of interference; The sign function sign(·) in the controller is used to offset model uncertainty and external disturbances. When the tracking error approaches the equilibrium point, k, which is related to the tracking error, tends to zero, thereby weakening the influence of (L|s|-sD), thereby suppressing the disturbance, where L represents the upper bound of D and |s| represents the absolute value of the sliding surface.

2. The novel non-singular fast terminal sliding mode control method for underwater robots according to claim 1 is characterized in that: The dynamic equations of the underwater robot constructed in step S01 include: The dynamic model of underwater robot is generally described as follows: Among them, η∈R 6×1 =[η1,η2] T =[x,y,z,φ,θ,ψ] T Represents the position and angle of the underwater robot in the world coordinate system, ν∈R 6×1 =[ν1,ν2] T =[u,v,w,p,q,r] T Represents the speed and angular velocity of the underwater robot in the motion coordinate system, τ∈R 6×1 =[F e ,M e ] T =[X,Y,Z,K,M,N] T Represents the external force and torque applied to the underwater robot; M∈R 6×6 is the inertia matrix; C(ν)∈R 6×6 is the rigid body Coriolis term and centripetal term matrix; D(ν) is the fluid damping matrix; G(η) is the gravity and buoyancy matrix, d(η,t) is the external disturbance to the underwater robot, and t is time; are the speed and angular velocity of the underwater robot in the world coordinate system, is the speed and angular velocity of the underwater robot in the motion coordinate system; The coordinate transformation matrix between the motion coordinate system and the world coordinate system is expressed by J(η): R(η),T(η) are: Where, s(·), c(·), t(·) represent sin(·), cos(·), tan(·), respectively; θ is the pitch angle, ψ is the pitch angle, and φ is the roll angle; The dynamic model transformation in the world coordinate system is: in: A is the equation parameter, is the second-order derivative of η, which represents the acceleration and angular acceleration of the underwater robot in the world coordinate system.

3. The novel non-singular fast terminal sliding mode control method for underwater robots according to claim 2 is characterized in that: The error between the dynamic parameters calculated by the underwater robot and the actual dynamic parameters dis represents external interference, △ represents the amount of change, |d n (η,t)|≤L gn ,n=1,2,3,4,5,6,L gn is the upper bound of the nth dimension.

4. The novel non-singular fast terminal sliding mode control method for underwater robots according to claim 1 is characterized in that: Related to tracking error If the system state variables reach the equilibrium point, k = 0, the uncertainty of the model parameters and external disturbances will not affect the system, and sliding mode control without chattering is performed.

5. A novel non-singular fast terminal sliding mode control system for underwater robots, characterized by: include: Dynamic equation building module, which builds the dynamic equations of underwater robots; Trajectory tracking error transformation module, transforming the dynamic equation into trajectory tracking error equation; The sliding surface design module designs the sliding surface of non-singular fast terminal sliding mode control as follows: Where, e is the tracking error, is the first-order derivative of e, α and β are control parameters, α=diag([α1,…,α6]), β=diag([β1,…,β6]), and α i ,β i >0,i=1,…,6,0 i <p i <2q i Represents an internal parameter and is an odd number, The non-singular fast terminal sliding mode control module applies non-singular fast terminal sliding mode control to the underwater robot. The control input is designed as:​ where σ = [σ1,…,σ6] T represents the internal parameter, σ i >0,|D i | <L i ,i=1,2,…,6,L i represents the upper bound of the i-th dimension D, B, f, and D are the parameters of the trajectory tracking error equation, and Lsign(s) is the sign function of the upper and lower bounds of interference; The sign function sign(·) in the controller is used to offset model uncertainty and external disturbances. When the tracking error approaches the equilibrium point, k, which is related to the tracking error, tends to zero, thereby weakening the influence of (L|s|-sD), thereby suppressing the disturbance, where L represents the upper bound of D and |s| represents the absolute value of the sliding surface.

6. The novel non-singular fast terminal sliding mode control system for underwater robots according to claim 5 is characterized in that: The dynamic equations of the underwater robot constructed in the dynamic equation construction module include: The dynamic model of underwater robot is generally described as follows: Among them, η∈R 6×1 =[η1,η2] T =[x,y,z,φ,θ,ψ] T Represents the position and angle of the underwater robot in the world coordinate system, ν∈R 6×1 =[ν1,ν2] T =[u,v,w,p,q,r] T Represents the speed and angular velocity of the underwater robot in the motion coordinate system, τ∈R 6×1 =[F e ,M e ] T =X,Y,Z,K,M,N] T Represents the external force and torque applied to the underwater robot; M∈R 6×6 is the inertia matrix; C(ν)∈R 6×6 is the rigid body Coriolis term and centripetal term matrix; D(ν) is the fluid damping matrix; G(η) is the gravity and buoyancy matrix, and d(η,t) is the external disturbance to the underwater robot; are the speed and angular velocity of the underwater robot in the world coordinate system, is the speed and angular velocity of the underwater robot in the motion coordinate system; The coordinate transformation matrix between the motion coordinate system and the world coordinate system is expressed by J(η): R(η),T(η) are: Where, s(·), c(·), t(·) represent sin(·), cos(·), tan(·), respectively; θ is the pitch angle, ψ is the pitch angle, and φ is the roll angle; The dynamic model transformation in the world coordinate system is: in: A is the equation parameter, is the second-order derivative of η, which represents the acceleration and angular acceleration of the underwater robot in the world coordinate system.

7. The novel non-singular fast terminal sliding mode control system for underwater robots according to claim 6 is characterized in that: The error between the dynamic parameters calculated by the underwater robot and the actual dynamic parameters dis represents external interference, △ represents the amount of change, |d n (η,t)|≤L gn ,n=1,2,3,4,5,6,L gn is the upper bound of the nth dimension.

8. The novel non-singular fast terminal sliding mode control system for underwater robots according to claim 5 is characterized in that: Related to tracking error If the system state variables reach the equilibrium point, k = 0, the uncertainty of the model parameters and external interference will not have any impact on the system, and sliding mode control without chattering will be performed.

9. A computer storage medium having a computer program stored thereon, characterized in that: When the computer program is executed, the novel non-singular fast terminal sliding mode control method for underwater robots according to any one of claims 1 to 4 is implemented.

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