A predefined time sliding mode control method for underactuated underwater robots

Abstract

This invention discloses a predefined time sliding mode control method for underactuated underwater robots, belonging to the field of ship control technology. First, a five-degree-of-freedom mathematical model for trajectory tracking of the underactuated underwater robot is constructed. Then, a model transformation scheme is developed, and a radial basis function neural network compensation rate is designed based on minimizing the basic constant force of the learning parameters. Next, the results of the above steps are integrated over time, and an adaptive tracking controller with predefined time convergence performance is designed in conjunction with a Gaussian error function. Finally, motion control is performed through the adaptive tracking controller based on the desired trajectory and control parameters. Under the integrated predefined time controller designed in this invention, the underactuated underwater robot can effectively avoid the influence of external disturbances and unknown model parameters, achieving three-dimensional trajectory tracking control with predefined time convergence.

Description

Technical Field

[0001] This invention belongs to the field of ship control technology, specifically relating to a predefined time sliding mode control method for an underactuated underwater robot. Background Technology

[0002] The ocean is the origin and cradle of life. As an assistant to humans in exploring and developing the ocean, autonomous underwater vehicles (AUVs) are increasingly favored in ocean development due to their advantages of low cost, unmanned operation, and greater flexibility compared to other vehicles.

[0003] Tracking control of AUVs has long been a challenging problem. During AUV trajectory tracking, it is subject to external disturbances from the marine environment, perturbations of AUV model parameters, and positional factors such as asymmetric saturation of the actuators. These disturbances affect controller performance. Researchers from various countries have proposed several solutions to address these issues, such as using radial basis function neural networks (RBFNNs) to compensate for the impact of model parameter perturbations and external time-varying disturbances on AUV trajectory tracking; and adding symmetric boundary constraints to the actuators using saturation functions such as sign functions to prevent control inputs from exceeding the actuators' maximum allowable limits. Unfortunately, existing solutions neglect the impact of limited computational resources on RBFNN performance and the asymmetric saturation of the actuators caused by wear and installation errors. Therefore, to better accomplish tracking control tasks, it is necessary to comprehensively consider the computational speed of RBFNNs and the asymmetric saturation of the input when dealing with unknown disturbances during the tracking control phase.

[0004] Furthermore, it is worth noting that convergence speed, as a crucial indicator for evaluating controller performance, is currently only achieving asymptotically stable convergence in most studies. It is difficult to estimate the convergence time (CT) of the error, meaning that the AUV state may only reach the planned expected value when the CT approaches infinity. The introduction of finite-time stability and fixed-time stability alleviates this problem. Unfortunately, the convergence time of finite-time stability is affected by the initial state, and the control parameters of fixed-time stability are relatively complex, making it difficult to determine the upper bound of the convergence time. To further address this issue, it is essential to apply and develop control methods with predefined time convergence (PTS). Summary of the Invention

[0005] The purpose of this invention is to provide a predefined time sliding mode control method for underactuated underwater robots. Its feature is an adaptive predefined time sliding mode controller based on underactuated model transformation. Under the action of the integrated predefined time controller designed in this invention, the AUV can achieve three-dimensional trajectory tracking control under external disturbances and unknown model parameters.

[0006] The objective of this invention is achieved through the following technical solution:

[0007] This invention provides a predefined time sliding mode control method for an underactuated underwater robot, comprising the following steps:

[0008] Step 1: Construct a five-degree-of-freedom mathematical model for trajectory tracking of an underactuated underwater robot;

[0009] Step 2: Construct a model conversion scheme for underactuated underwater robots;

[0010] Step 3: Design the compensation rate of the radial basis function neural network based on the fundamental theorem of minimizing learning parameters;

[0011] Step 4: Integrate the results of Step 2 and Step 3 over time, and design an adaptive tracking controller with predefined time convergence performance by combining the Gaussian error function;

[0012] Step 5: Design the desired trajectory and control parameters of the underactuated underwater robot according to the task requirements; perform motion control through the adaptive tracking controller based on the desired trajectory and control parameters.

[0013] Furthermore, in step 1, a five-degree-of-freedom mathematical model for trajectory tracking of an underactuated underwater robot is constructed:

[0014]

[0015] in, Let η be the first derivative of η with respect to time; η = [x, y, z, θ, ψ] T Let v = [u, v, w, q, r] represent the position vector of the underactuated underwater robot in the fixed inertial frame of Earth. T This represents the velocity vector of the underactuated underwater robot in a fixed inertial frame on Earth. Let v be the first derivative of v with respect to time; It is a coordinate transformation matrix; M = diag(m) 11 ,m 22 .m 33 ,m 55 ,m 66 ) is the inertia matrix; The centripetal force matrix is ​​D(v) = diag(d). 11 ,d 22 ,d33 ,d 55 ,d 66 ) represents the fluid damping matrix; Let ρ be the vector of restoring force and torque generated by gravity and buoyancy, and ρ be the density of seawater. For displacement, GM L It is the vertical center of gravity height; τ = [τ u ,0,0,τ q ,τ r ] T τ is the control force matrix of the carrier. d =[τ ud ,τ vd ,τ wd ,τ qd ,τ rd ] T Unknown external disturbances to the marine environment.

[0016] Furthermore, step 2 specifically includes:

[0017] Step 2.1: By combining ψ and θ to form new state variables, all inputs [τ] are made equal to the sum of the given values. u ,τ q ,τ r ] T The new state variable appears in the error dynamic equation:

[0018]

[0019] Where l represents the design parameters; [x1, y1, z1] T These are new state variables designed through model transformation.

[0020] Step 2.2: Combine the known desired trajectory η of the underactuated underwater robot d The second derivative of the error dynamic equation for trajectory tracking is:

[0021]

[0022] in, For [e x1 e y1 e z1 ] T The second derivative, [e x1 e y1 e z1 ] T For the tracking error designed through model transformation, e x1 =x1-x d e y1 =y1-y d e z1=z1-z d F represents the unknown model parameters of the underactuated underwater robot; G represents the state-related design parameters of the underactuated underwater robot. l≠0; New control input τ1=[τ u , τ q , τ r ] T New interference τ d1 =[τ ud , τ qd , τ rd ] T .

[0023] Furthermore, step 3 specifically includes the following steps:

[0024] Step 3.1: A radial basis function neural network with n nodes is used to accurately identify model uncertainties and unknown disturbances, which can identify unknown model parameters F, G·τ1 and external disturbances G·τ. d1 It can be approximated as:

[0025] F+Gτ d1 +Gτ1=W *T H(Z)+ε(x)

[0026] in, Let Z be the weight matrix; Z = [u, v, w, q, r] T It is the input from the network; It is an error vector and has an unknown upper bound.

[0027] Step 3.2: Based on the idea of ​​minimizing the learning parameters, the maximum norm of the adaptive parameters is used in each neuron:

[0028]

[0029] Where σ=||W *T ||, σ>0; D>0.

[0030] Furthermore, step 4 specifically includes the following steps:

[0031] Step 4.1: To compensate for the asymmetric saturation of the actuator, the ERF function is designed as follows:

[0032]

[0033] in, This represents the maximum thrust and torque capability of the AUV.

[0034]

[0035] Step 4.2: Combining the error dynamics equations of the AUV from Steps 2 and 3, rewrite as follows:

[0036]

[0037] Step 4.3: Design the predefined time sliding surface s:

[0038]

[0039] In the formula, For speed error; For position tracking error; λ∈(0,1), k3 > 0; sign(·) represents the standard function symbol;

[0040] Differential with respect to time

[0041]

[0042] Where, d ν =Gτ d1 At this point, the control law τ1 and the adaptive update rate as follows:

[0043]

[0044]

[0045]

[0046] Among them, k4=k1, k3=k2, k6>0, γ1, γ2>0, T k T is the controller convergence time. k satisfy T max The upper bound of the convergence time; position tracking error and estimation error It can be done in a predefined time T k It converges inward to a sufficiently small region around the origin.

[0047] The beneficial effects of this invention are as follows:

[0048] This invention addresses the 3D trajectory tracking problem of AUVs under model uncertainty and unknown external disturbances by designing an adaptive predefined time sliding mode controller. First, a novel underactuated AUV model transformation method simplifies the controller design process. Then, a radial basis function neural network (RBN) with minimized learning parameters compensates for unknown disturbances during AUV 3D trajectory tracking. Finally, an adaptive predefined time controller is designed using Lyapunov stability proofs, enabling effective trajectory tracking of the optimal trajectory generated earlier under model uncertainty and unknown external disturbances. This forces the tracking error to converge to a small residual region within a preset time. Simulation experiments verify the effectiveness and superiority of the proposed method. Attached Figure Description

[0049] Figure 1 This is a flowchart of an adaptive predefined time control method for saturation conditions involving asymmetric actuators, provided in an embodiment of the present invention.

[0050] Figure 2 This is a mathematical model of an AUV provided in an embodiment of the present invention;

[0051] Figure 3 This is a schematic diagram of an underactuated AUV model conversion provided in an embodiment of the present invention;

[0052] Figure 4 This is a schematic diagram of asymmetric saturation function processing provided in an embodiment of the present invention;

[0053] Figure 5 These are comparison images of the 3D tracking effect of the adaptive predefined time controller provided in the embodiments of the present invention;

[0054] Figure 6 This is a schematic diagram of the adaptive rate estimation value of the adaptive predefined time controller provided in an embodiment of the present invention;

[0055] Figure 7 This is a comparison chart of the speed curves of the adaptive predefined time controller provided in the embodiments of the present invention;

[0056] Figure 8 This is a comparison diagram of the thrust and torque of the adaptive predefined time controller provided in an embodiment of the present invention;

[0057] Figure 9 This is a comparison diagram of the position and attitude of the adaptive predefined time controller provided in an embodiment of the present invention. Detailed Implementation

[0058] The embodiments of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0059] Please see Figure 1 As shown, this invention provides a predefined time sliding mode control method for an underactuated underwater robot, comprising the following steps:

[0060] Step 1: Establish a mathematical model for the underactuated underwater robot;

[0061] Specifically, in this step, the mathematical model of the underactuated underwater robot is described as follows:

[0062]

[0063] in, Let η be the first derivative of η with respect to time; η = [x, y, z, θ, ψ] T Let v represent the position vector of the underactuated underwater robot in the fixed inertial frame of Earth; v = [u, v, w, q, r] T This represents the velocity vector of the underactuated underwater robot in a fixed inertial frame on Earth. Let v be the first derivative of v with respect to time; It is a coordinate transformation matrix; M = diag(m) 11 ,m 22 ,m 33 ,m 55 ,m 66 ) is the inertia matrix; The centripetal force matrix is ​​D(v) = diag(d). 11 ,d 22 ,d 33 ,d 55 ,d 66 ) represents the fluid damping matrix; The vector of restoring force and torque generated by gravity and buoyancy. ρ is the displacement, ρ is the density of seawater, and GM is the displacement. L It is the vertical center of gravity height; τ = [τ u ,0,0,τ q ,τ r ] T τ is the control force matrix of the carrier. d =[τ ud ,τ vd ,τ wd ,τ qd ,τ rd ]T Unknown external disturbances to the marine environment are considered; the effects of high-order nonlinear hydrodynamic damping terms and roll motion on AUVs are ignored.

[0064] Please see Figure 2 As shown, step 2 is performed: design an underactuated AUV model conversion scheme to solve the trajectory tracking control difficulties caused by insufficient AUV drive;

[0065] First, to make all inputs [τ] u ,τ q ,τ r ] T In the error dynamic equation, we combine ψ and θ to form new state variables:

[0066]

[0067] Then, combined with the given AUV expected trajectory η d The error in trajectory tracking can be expressed as:

[0068]

[0069] In the formula, l represents the design parameters, [x1, y1, z1]. T , These are new state variables and tracking errors designed through model transformation. Furthermore, we can derive the second derivative in the error dynamic equation:

[0070]

[0071] Where F represents the unknown model parameters of AUV. In the formula, G is a design parameter related to the AUV condition. l≠0, new control input τ1=[τ u , τ q , τ r ] T New interference τ d1 =[τ ud , τ qd , τ rd ] T .

[0072] Regarding the second-order equation above, it is easy to see that when l≠0, the above control matrix... In other words, the matrix is ​​non-singular for any θ, ψ, and this formula always holds true.

[0073] Step 3: Construct an RBFNN compensation rate based on minimizing the learning parameters to address the impact of unknown external disturbances and AUV model parameter perturbations on controller accuracy; For the impact of external time-varying disturbances and model uncertainties in underactuated AUVs during 3D trajectory tracking, the following design is implemented:

[0074] First, an RBFNN with n nodes is used to accurately identify model uncertainties and unknown disturbances, which can identify unknown model parameters F, Gτ1, and external disturbances Gτ. d1 It can be approximated by the following expression:

[0075] F+Gτ d1 +Gτ1=W *T H(Z)+ε(x)

[0076] in, It is a weight matrix, Z = [u, v, w, q, r] T It is the input from the network. It is an error vector and has an unknown upper bound.

[0077] Then, based on the idea of ​​minimizing the learning parameters, the maximum norm of the adaptive parameters is adopted in each neuron unit. This effectively reduces the time spent on online learning, resulting in the following expression:

[0078]

[0079] Where σ=||W *T ||,σ>0;D is a positive design parameter,D>0; In this way, we only need to estimate the constant σ, and do not need to estimate the matrix W. *T This effectively reduces the computational burden and greatly improves the real-time performance of the control system.

[0080] Lemma 1 guarantees that this part holds:

[0081] Lemma 1. (Lu et al., 2020). RBF neural networks can be used in compact sets. The approximation of any given real continuous function f(Z) is accurate as follows:

[0082] f(Z) = W T S(Z)+ε,Z∈Ω Z

[0083] Where W = [w1, w2, L…, w m ] T It is the ideal constant weight value, and ε is the unknown constant that satisfies |ε|≤ε. * (where ε) *The bounded function approximation error of >0) for Z∈Ω Z S(Z)=[s1(Z),…,s l (Z)] T and s j (Z) Select s j (Z) The commonly used Gaussian function form is:

[0084]

[0085] in, h is the center vector j is the width of the Gaussian function.

[0086] Step 4: Integrate the output values ​​from Steps 2 and 3, and design an adaptive tracking controller with predefined time convergence performance using the Gaussian error function (ERF function). To address the saturation problem of the AUV's asymmetric actuator, this section employs the ERF function to improve it. The error function erf(x) (also known as the Gaussian error function) is a non-primitive function of sigmoid shape, defined as:

[0087]

[0088] In this way, the control input can be rewritten as:

[0089]

[0090] in, sign(·) represents the standard symbolic function.

[0091] The error function erf(x) is a real-valued, continuously differentiable function. It has no singularities (except at infinity) and its Taylor expansion is always convergent. Furthermore, after the above transformation, it can be changed... and Easily replace the upper and lower bounds of the asymmetric saturation function, in Figure 4 Examples were provided in the text.

[0092] To compensate for the asymmetric saturation of the actuator, the ERF function is designed as follows:

[0093]

[0094] In summary, combining steps 2 and 3, the error dynamic equation of the AUV can be rewritten as:

[0095]

[0096] In the formula, For speed error; This refers to the position tracking error; sign(·) represents the standard function symbol;

[0097] Finally, select the predefined time sliding surface as follows:

[0098]

[0099] in,

[0100] Expanding the time derivative yields:

[0101]

[0102] Where, d ν =Gτ d1 .

[0103] At this point, given the trajectory tracking control input of a predefined time controller, when the control law and adaptive update rate are selected as follows:

[0104]

[0105]

[0106]

[0107] Where k4 = k1, k5 = k2, k6 > 0;

[0108] And satisfy γ1, γ2 > 0, When the AUV is subjected to unknown external disturbances and model parameter perturbations, the controller provided by this invention can make the tracking error quickly stabilize within a predefined time.

[0109] Step 5: Based on the transformation and simplification operations described in steps 1-4, the control law and adaptive law obtained can be proven using relevant theories. The Lyapunov function V1 and the position tracking error... and estimation error It can be done in a predefined time T k The convergence time T is when the point converges to a sufficiently small region around the origin. k Satisfy the following formula Controller stability verification;

[0110] First, define the estimation error. Choose Lyapunov functions as

[0111] Differentiating with respect to time yields:

[0112]

[0113] According to step 2:

[0114]

[0115] According to step 3:

[0116]

[0117] Combining the above transformations and step 4, we can obtain:

[0118]

[0119] Rearranging and further simplifying yields:

[0120]

[0121] Where k8 > 0;

[0122] At this point, according to Lemma 2:

[0123] Lemma 2. (Xie and Chen, 2021) For the system Define a continuous function V(x) and design parameters 0 < λ < 1, T k >0 and satisfy:

[0124]

[0125] Furthermore, the trajectory of xf(t,x,d) is practically predefined time-stable (PPTS), and its region of convergence is...

[0126]

[0127] Among them, T k Indicates satisfaction The convergence time, T max The upper boundary.

[0128] According to Lemma 2, the above inequality can be transformed into:

[0129]

[0130] Substituting the control law and adaptive law from step 4, we get:

[0131]

[0132] Using Young's inequality and related mathematical theory, we can obtain:

[0133]

[0134]

[0135]

[0136]

[0137]

[0138] in,

[0139] At this point, after the above transformation, the original inequality can be transformed into:

[0140]

[0141] Summarized as follows:

[0142]

[0143] Applying Lemma 2 again, we can obtain:

[0144]

[0145] in,

[0146] Lemma 2 can be used to prove the Lyapunov function V1 and the tracking error. and estimation error Eventually converges to a sufficiently small region

[0147]

[0148] At the scheduled time T k Indicates satisfaction Q.E.D.

[0149] To demonstrate the robustness and predefined time convergence of the proposed controller under the influence of unknown external disturbances and model uncertainties, comparative experiments were conducted to verify the superiority of the proposed control algorithm, comparing it with common finite-time control methods and fixed-time control methods.

[0150] In this section, we also consider a model with 5 degrees of freedom external uncertainties, caused by the following factors: τ id =(D1sin(ω) d t)-D2)rand(·)

[0151] In the formula, D1 = 0.5 is the maximum value of the time-varying part of the disturbance, D2 = 0.5 is its constant deviation, and ω d =0.1 is its angular frequency, rand(·) is Gaussian random noise with a mean of 0 and a variance of 1, i = u,v,w,q,r.

[0152] Furthermore, to separately demonstrate the effectiveness of the designed controller within a predetermined time period, the initial condition [3, 0, 0, 0, 0] was selected. T A simulation comparison experiment was conducted with the following reference trajectory:

[0153]

[0154] To ensure fairness in the comparison process, we set the parameters as consistently as possible. The parameters in this paper are as follows:

[0155] T k =10s, λ=0.1, l=0.08, k1=k4=1.63, k2=k5=1.52, k3=k6=0.001, γ1=0.005, γ2=10, γ3=γ5=3.34, γ4=3.91, γ6=2.67,

[0156] μ j =[-1.5,1.5]×[-0.05,0.05]×[-0.15,0.15]×[-0.4,0.4]×[-0.6,0.6], h j =1.5.

[0157] Simulation results are as follows Figures 5 to 9 As shown, Figure 5 The trajectory curves show that all control methods can capture the desired trajectory to achieve the expected control objective, but the proposed method has a faster convergence speed. Figure 9 e can be observed in x1 ,e y1 ,e z1 At a predefined time T k =10s ago, the residuals had stabilized within a small range, and from Figure 7 It can be seen that the speed of the underactuated AUV also converges according to the predefined time. In addition, Figure 8 This indicates that, under the same control objectives, the compared method exhibits greater chattering. This undesirable situation leads to greater energy consumption and errors in practical applications. Figure 6 This indicates that the adaptive rate eventually stabilizes. In summary, compared to other existing methods, the proposed method exhibits smaller and smoother fluctuations in the control input, demonstrating that the entire system maintains extremely high stability against external disturbances, thus better handling various unknown disturbances encountered by underactuated AUVs during 3D trajectory tracking.

[0158] The foregoing has described one embodiment of the present invention in detail, but the content described is only a preferred embodiment of the present invention and should not be considered as limiting the scope of the present invention. All equivalent variations and improvements made within the scope of the present invention should fall within the patent coverage of the present invention.

Claims

1. A predefined time sliding mode control method for an underactuated underwater robot, characterized in that: Includes the following steps: Step 1: Construct a five-degree-of-freedom mathematical model for trajectory tracking of an underactuated underwater robot; Step 2: Construct a model conversion scheme for underactuated underwater robots; Step 3: Design the compensation rate of the radial basis function neural network based on the fundamental theorem of minimizing learning parameters; Step 4: Integrate the results of Step 2 and Step 3 over time, and design an adaptive tracking controller with predefined time convergence performance by combining the Gaussian error function; Step 5: Design the desired trajectory and control parameters of the underactuated underwater robot according to the task requirements; perform motion control through the adaptive tracking controller based on the desired trajectory and control parameters.

2. The predefined time sliding mode control method for underactuated underwater robots according to claim 1, characterized in that: In step 1, a five-degree-of-freedom mathematical model for trajectory tracking of the underactuated underwater robot is constructed: in, Let η be the first derivative of η with respect to time; η = [x, y, z, θ, ψ] T Let v represent the position vector of the underactuated underwater robot in the fixed inertial frame of Earth; v = [u, v, w, q, r] T This represents the velocity vector of the underactuated underwater robot in a fixed inertial frame on Earth. Let v be the first derivative of v with respect to time; It is a coordinate transformation matrix; M = diag(m) 11 ,m 22 .m 33 ,m 55 ,m 66 ) is the inertia matrix; The centripetal force matrix is ​​D(v) = diag(d). 11 ,d 22 ,d 33 ,d 55 ,d 66 ) represents the fluid damping matrix; Let ρ be the restoring force and torque vector generated by gravity and buoyancy, ▽ be the displacement, and GM be the displacement. L It is the vertical center of gravity height; τ = [τ u ,0,0,τ q ,τ r ] T Let τ be the control force matrix of the carrier. d =[τ ud ,τ vd ,τ wd ,τ qd ,τ rd ] T Unknown external disturbances to the marine environment.

3. The predefined time sliding mode control method for underactuated underwater robots according to claim 1, characterized in that: Step 2 specifically includes: Step 2.1: By combining y and q to form new state variables, all inputs [τ] are made equal to the sum of the given values. u ,τ q ,τ r ] T The new state variable appears in the error dynamic equation: Where l represents the design parameters; [x1, y1, z1] T The new state variables designed through model transformation; Step 2.2: Combine the known desired trajectory η of the underactuated underwater robot d The second derivative of the error dynamic equation for trajectory tracking is: in, For [e x1 e y1 e z1 ] T The second derivative, [e x1 e y1 e z1 ] T For the tracking error designed through model transformation, e x1 =x1-x d e y1 =y1-y d e z1 =z1-z d F represents the unknown model parameters of the underactuated underwater robot; G represents the state-related design parameters of the underactuated underwater robot. l≠0; New control input τ1=[τ u , τ q , τ r ] T New interference τ d1 =[τ ud , τ qd , τ rd ] T .

4. The predefined time sliding mode control method for underactuated underwater robots according to claim 1, characterized in that: Step 3 specifically includes the following steps: Step 3.1: A radial basis function neural network with n nodes is used to accurately identify model uncertainties and unknown disturbances, which can identify unknown model parameters F, G·τ1 and external disturbances G·τ. d1 It can be approximated as: F+Gτ d1 +Gτ1=W *T H(Z)+ε(x) in, Let Z be the weight matrix; Z = [u, v, w, q, r] T It is the input from the network; It is an error vector and has an unknown upper bound. Step 3.2: Based on the idea of ​​minimizing the learning parameters, the maximum norm of the adaptive parameters is used in each neuron: where σ = ||W *T ||, σ > 0; D > 0; 5. The predefined time sliding mode control method for underactuated underwater robots according to claim 1, characterized in that: Step 4 specifically includes the following steps: Step 4.1: To compensate for the asymmetric saturation of the actuator, the ERF function is designed as follows: in, This represents the maximum thrust and torque capability of the AUV. Step 4.2: The dynamic error equation of AUV, combining the results from Steps 2 and 3, is rewritten as follows: Step 4.3: Design the predefined time sliding surface s: In the formula, For speed error; For position tracking error; λ∈(0,1), k3 > 0; sign(·) represents the standard function symbol; Differential with respect to time At this point, the control law τ1 and the adaptive update rate are selected. as follows: Among them, k4=k1, k3=k2, k6>0, γ1, γ2>0, T k T is the controller convergence time. k satisfy T max The upper bound of the convergence time; position tracking error and estimation error It can be done in a predefined time T k It converges inward to a sufficiently small region around the origin.

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