Under-actuated underwater robot predefined time sliding mode control method

Abstract

The invention discloses an under-actuated underwater robot predefined time sliding mode control method, and belongs to the technical field of ship control. Firstly, a five-degree-of-freedom mathematical model for trajectory tracking of the under-actuated underwater robot is constructed; then constructing a model conversion scheme, and designing a radial basis function neural network compensation rate according to the basic constant force of the minimized learning parameter; thirdly, integrating the results of the above steps on time, and designing an adaptive tracking controller with predefined time convergence performance in combination with a Gaussian error function; and finally, according to the expected trajectory and the control parameters, motion control is performed through the adaptive tracking controller. Under the action of the integrated predefined time controller designed by the invention, the under-actuated underwater robot can effectively avoid the influence of external interference and unknown model parameters on the under-actuated underwater robot, and three-dimensional trajectory tracking control with predefined time convergence is realized.

Description

Technical Field

[0001] The invention belongs to the technical field of ship control, and in particular relates to a predefined time sliding mode control method for an underactuated underwater robot. Background Art

[0002] The ocean is the origin and cradle of life. As an assistant to human exploration and development of the ocean, autonomous underwater vehicles (AUVs) are becoming increasingly popular in ocean development due to their low cost, unmanned operation and greater flexibility than other vehicles.

[0003] Tracking control of AUV is a long-standing and challenging problem. When AUV performs trajectory tracking tasks, it will be disturbed by external unknown disturbances from the marine environment, perturbations of AUV model parameters, and asymmetric saturation of actuators, etc. These disturbances affect the performance of the controller. In response to these problems, researchers from various countries have proposed a variety of solutions, such as applying radial basis function neural network (RBFNN) to compensate for the impact of model parameter perturbations and external time-varying disturbances on AUV trajectory tracking; and adding symmetrical boundary constraints to actuators through saturation functions such as sign functions to avoid control inputs exceeding the maximum limit allowed by the actuator. Unfortunately, in the existing solutions, the impact of limited computing resources on the performance of RBF NN and the occurrence of asymmetric saturation of actuators caused by actuator wear and installation errors are ignored. Therefore, in order to better complete the tracking control task, it is necessary to comprehensively consider the operation speed of RBFNN and the asymmetric situation of input saturation when there are unknown disturbances in the tracking control stage.

[0004] In addition, it is worth noting that the convergence speed is an important indicator that cannot be ignored in evaluating the performance of the controller. At present, most studies have only achieved asymptotic stable convergence, and it is difficult to estimate the convergence time (CT) of the error, which means that the AUV state may only reach the expected value of the plan when CT approaches infinity. The introduction of finite-time stability and fixed-time stability alleviates this problem. Unfortunately, the convergence time of finite-time stability will be affected by the initial state, the control parameters of fixed time are relatively complex, and it is difficult to determine the supremum of the convergence time. In order to further solve this problem, it is necessary to apply and develop a control method with predefined time convergence (PTS). Summary of the invention

[0005] The purpose of the present invention is to provide a predefined time sliding mode control method for an underactuated underwater robot, which is characterized by an adaptive predefined time sliding mode controller based on underactuated model conversion. Under the action of the integrated predefined time controller designed by the present invention, the three-dimensional trajectory tracking control of the AUV is realized under the condition of external interference and unknown model parameters.

[0006] The purpose of the present invention is achieved through the following technical solutions:

[0007] The present 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 an underactuated underwater robot model conversion scheme;

[0010] Step 3: Design the radial basis neural network compensation rate according to the basic theorem of minimizing learning parameters;

[0011] Step 4: Integrate the results of step 2 and step 3 with respect to time, and design an adaptive tracking controller with a predefined time convergence performance in combination with the Gaussian error function;

[0012] Step 5: According to the task requirements, the desired trajectory and control parameters of the underactuated underwater robot are designed; and according to the desired trajectory and control parameters, motion control is performed through the adaptive tracking controller.

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

[0016] Furthermore, the step 2 specifically includes:

[0017] Step 2.1: By combining ψ and θ to form a new state variable, all inputs [τ u ,τ q ,τ r ] T Appearing in the error dynamics equation, the new state variable:

[0018]

[0019] Where l is the design parameter; [x1, y1, z1] T New state variables designed through model transformation.

[0020] Step 2.2: Combine the known expected trajectory η of the underactuated underwater robot d , the second-order derivative of the error dynamic equation of trajectory tracking is:

[0021]

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

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

[0024] Step 3.1: Use a radial basis function neural network with n nodes to accurately identify model uncertainty and unknown disturbances. The unknown model parameters F, G·τ1 and external disturbance G·τ d1 It can be approximated as:

[0025] F+Gτ d1 +Gτ1=W *T H(Z)+ε(x)

[0026] in, is the weight matrix; Z = [u, v, w, q, r] T is the input of the network; is the error vector with an unknown upper bound

[0027] Step 3.2: According to the idea of ​​minimizing the learning parameters, the maximum norm of the adaptive parameters is adopted in each neural unit:

[0028]

[0029] where σ=||W *T ||, σ>0; D>0.

[0030] Furthermore, the 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, is the maximum capability of the AUV thrust and torque;

[0034]

[0035] Step 4.2: Combining Step 2 and Step 3, the error dynamics of AUV is rewritten as:

[0036]

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

[0038]

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

[0040] Differentiate with respect to time

[0041]

[0042] Among them, d ν =Gτ d1 , at this time the control rate 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 is the controller convergence time, T k satisfy T max is the upper bound of convergence time; position tracking error and estimation error Can be in the predefined time T k The inner convergence is to a sufficiently small region around the origin.

[0047] The beneficial effects of the present invention are:

[0048] The present invention designs an adaptive predefined time sliding mode controller for the three-dimensional trajectory tracking problem of AUV under model uncertainty and unknown external disturbances. First, the design process of the controller is simplified based on a novel underactuated AUV model conversion method. Then, the radial basis function neural network based on minimizing learning parameters compensates for the unknown interference in the three-dimensional trajectory tracking process of AUV. Finally, an adaptive predefined time controller is designed through the Lyapunov stability proof, so that it can effectively track the optimal trajectory generated in the above text under the condition of model uncertainty and external unknown disturbances, and then force the tracking error to converge to a smaller residual region within a preset time. The effectiveness and superiority of this method are verified by simulation experiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 is a flow chart of an adaptive predefined time control method under saturation conditions of an asymmetric actuator provided by an embodiment of the present invention;

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

[0051] Figure 3 1 is a schematic diagram of an underactuated AUV model conversion provided by an embodiment of the present invention;

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

[0053] Figure 5 This is a comparison diagram of the three-dimensional tracking effect of the adaptive predefined time controller provided by an embodiment of the present invention;

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

[0055] Figure 7 is a speed curve comparison diagram of an adaptive predefined time controller provided by an embodiment of the present invention;

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

[0057] Fig. 9 It is a comparison diagram of the position and posture of the adaptive predefined time controller provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0058] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0059] See also Figure 1 As shown, the present invention is a predefined time sliding mode control method for an underactuated underwater robot, comprising the following steps:

[0060] Step 1, establishing a mathematical model of an underactuated underwater robot;

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

[0062]

[0063] in, is the first-order derivative of η with respect to time; η=[x,y,z,θ,ψ] T represents the position vector of the underactuated underwater robot in the earth-fixed inertial system; v = [u, v, w, q, r] T represents the velocity vector of the underactuated underwater vehicle in the Earth-fixed inertial system, is the first-order derivative of v with respect to time; is the coordinate transformation matrix; M = diag (m 11 ,m 22 ,m 33 ,m 55 ,m 66 ) is the inertia matrix; is the Coriolis centripetal force matrix; D(v)=diag(d 11 ,d 22 ,d 33 ,d 55 ,d 66 ) is the fluid damping matrix; are the restoring force and torque vectors due to gravity and buoyancy, is the displacement, ρ is the seawater density, GM L is the vertical metacentric height; τ = [τ u ,0,0,τ q ,τ r ] T is the control force matrix of the carrier, τ d =[τ ud ,τ vd ,τ wd ,τ qd ,τ rd ]T The unknown external disturbance brought by the marine environment; the influence of high-order nonlinear hydrodynamic damping and rolling motion on the AUV is ignored.

[0064] See also Figure 2 As shown, proceed to step 2: design an under-actuated AUV model conversion scheme to solve the trajectory tracking control difficulties caused by insufficient AUV drive;

[0065] First, in order to make all inputs [τ u ,τ q ,τ r ] T In the error dynamics equation, we form a new state variable by combining ψ and θ:

[0066]

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

[0068]

[0069] Where l is the design parameter, [x1, y1, z1] T , It is the new state variable and tracking error designed by model transformation. Then we can derive the second-order derivative in the error dynamic equation:

[0070]

[0071] Among them, F is the unknown model parameter of AUV, In the formula, G is the design parameter related to the AUV state, l≠0, the new control input τ1=[τ u , τ q , τ r ] T , the new disturbance τ d1 =[τ ud , τ qd , τ rd ] T .

[0072] For the second-order equation in the above formula, it can be easily seen that when l≠0, the above control matrix That is to say, the matrix is ​​non-singular for any θ, ψ, and this formula always holds.

[0073] Step 3: Construct an RBFNN compensation rate based on minimizing learning parameters to solve the impact of unknown external disturbances and AUV model parameter perturbations on controller accuracy. In view of the impact of external time-varying disturbances and model uncertainty in the three-dimensional trajectory tracking of underactuated AUV, the following design is performed:

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

[0075] F+Gτ d1 +Gτ1=W *T H(Z)+ε(x)

[0076] in, is the weight matrix, Z = [u,v,w,q,r] T is the input of the network, is the error vector with an unknown upper bound

[0077] Then, according to the idea of ​​minimizing learning parameters, the maximum norm of adaptive parameters is adopted in each neuron unit. In this way, the time consumption of online learning can be effectively reduced, and the following expression can be obtained:

[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, but not the matrix W *T . This effectively reduces the computational burden and greatly improves the real-time performance of the control system.

[0080] Lemma 1 ensures that this part holds:

[0081] Lemma 1. (Lu et al, 2020). RBF neural network can be used in a compact set Approximate any given real continuous function f(Z) with the accuracy of:

[0082] f(Z)=W T S(Z)+ε,Z∈Ω Z

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

[0084]

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

[0086] Step 4: Integrate the output values ​​of steps 2 and 3, and design an adaptive tracking controller with predefined time convergence performance in combination with the Gaussian error function (ERF function); Aiming at the saturation problem of the asymmetric actuator of AUV, the ERF function is used in this section to improve it. The error function erf(x) (also called Gaussian error function) is a non-elementary function of Sigmoid shape, defined as:

[0087]

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

[0089]

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

[0091] The error function erf(x) is a real-valued continuously differentiable function that has no singularities (except at infinity) and whose Taylor expansion always converges. And after the above transformation, it can be obtained by changing and Easily replace the upper and lower limits of the asymmetric saturation function, in Figure 4 Examples are given in the text.

[0092] In order 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 AUV can be rewritten as:

[0095]

[0096] In the formula, is the speed error; is the position tracking error; sign(·) represents the standard function sign;

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

[0098]

[0099] in,

[0100] Expanding the time derivative gives:

[0101]

[0102] Among them, d ν =Gτ d1 .

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

[0104]

[0105]

[0106]

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

[0108] And satisfy γ1,γ2>0, When it is established, for the AUV in the presence of unknown external disturbances and model parameter perturbations, the controller provided by the present invention can achieve rapid stabilization of the tracking error within a predefined time.

[0109] Step 5: Based on the control rate and adaptive rate obtained by the transformation and simplification operations described in steps 1 to 4, it can be proved by relevant theories that the Lyapunov function V1 and the position tracking error and estimation error Can be in the predefined time T k Converges to a sufficiently small area around the origin, and the convergence time is T k Satisfy the following formula Proof of controller stability;

[0110] First, define the estimation error Select the Lyapunov function as

[0111] Differentiating with respect to time gives:

[0112]

[0113] According to step 2:

[0114]

[0115] According to step 3:

[0116]

[0117] Combining the above transformation with step 4, we can get:

[0118]

[0119] Rearranging and further simplifying gives:

[0120]

[0121] Among them, k8>0;

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

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

[0124]

[0125] And the trajectory of xf(t,x,d) is practical predefined time stable (PPTS), and its convergence region is

[0126]

[0127] Among them, T k Express satisfaction The convergence time, T max For the upper bound.

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

[0129]

[0130] Substituting the control rate and adaptive rate in step 4 yields:

[0131]

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

[0133]

[0134]

[0135]

[0136]

[0137]

[0138] in,

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

[0140]

[0141] Arranged:

[0142]

[0143] Using Lemma 2 again we can get:

[0144]

[0145] in,

[0146] Lemma 2 shows that the Lyapunov function V1 and the tracking error and estimation error Eventually converge to a sufficiently small area

[0147]

[0148] At scheduled time T k Express satisfaction The proof is complete.

[0149] In order to demonstrate the robustness and predefined time convergence of the proposed controller under the influence of unknown external disturbances and model uncertainties, comparative experiments are carried out to verify the superiority of the proposed control algorithm compared with the common finite time control method and fixed time control method.

[0150] In this section, we also consider a model with 5-DOF external uncertain disturbances. The external disturbances are generated by the following factors: τ id =(D1sin(ω d t)-D2)rand(·)

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

[0152] In addition, in order to demonstrate the effectiveness of the designed controller within the predetermined time, the initial conditions [3, 0, 0, 0, 0] are selected. T The simulation comparison experiment is carried out with the following reference trajectory:

[0153]

[0154] To ensure fairness in the comparison process, we set the parameters as consistent as possible. The parameters in this article 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] The simulation results are as follows Figures 5 to 9 As shown, Figure 5 is the trajectory curve, indicating that all control methods can capture the expected trajectory to achieve the expected control target, but the proposed method has a faster convergence speed. Fig. 9 It can be observed that e x1 ,e y1 ,e z1 At a predefined time T k = 10s ago, they have stabilized in a small residual range and Figure 7 It can be seen that the speed of the underactuated AUV also converges according to the predefined time. In addition, Figure 8 It shows that under the condition of achieving the same control goal, the compared methods have greater jitter. This unpleasant situation will lead to greater energy consumption and errors in practical applications. Figure 6 It shows that the adaptation rate finally tends to be stable. In summary, compared with other existing methods, the control input fluctuation of the method proposed in this paper is smaller and smoother, which shows that the whole system maintains a very high stability to external interference, that is, it can better handle various unknown interferences in the three-dimensional trajectory tracking of under-actuated AUV.

[0158] The above is a detailed description of an embodiment of the present invention, but the content is only a preferred embodiment of the present invention and cannot be considered to limit the scope of implementation of the present invention. All equivalent changes and improvements made within the scope of the present invention should fall within the scope of the patent coverage of the present invention.

Claims

1. A predefined time sliding mode control method for an underactuated underwater robot, characterized in that: The following steps are involved: Step 1: Construct a five-degree-of-freedom mathematical model for trajectory tracking of an underactuated underwater robot; Step 2: Construct an underactuated underwater robot model conversion scheme; Step 3: Design the radial basis neural network compensation rate according to the basic theorem of minimizing learning parameters; Step 4: Integrate the results of step 2 and step 3 with respect to time, and design an adaptive tracking controller with a predefined time convergence performance in combination with the Gaussian error function; Step 5: According to the task requirements, the desired trajectory and control parameters of the underactuated underwater robot are designed; and according to the desired trajectory and control parameters, motion control is performed through the adaptive tracking controller.

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

3. The predefined time sliding mode control method for an underactuated underwater robot according to claim 1, characterized in that: The step 2 specifically includes: Step 2.1: By combining y and q to form a new state variable, all inputs [τ u ,τ q ,τ r ] T Appearing in the error dynamics equation, the new state variable: Where l is the design parameter; [x1, y1, z1] T New state variables designed through model transformation. Step 2.2: Combine the known expected trajectory η of the underactuated underwater robot d , the second-order derivative of the error dynamic equation of trajectory tracking is: in, For x1 、e y1 、e z1 ] T The second derivative of [e x1 、e y1 、e z1 ] T is the tracking error designed by model transformation, e x1 =x1-x d 、e y1 =y1-y d 、e z1 =z1-z d ; F is the unknown model parameter of the underactuated underwater robot; G is the design parameter related to the state of the underactuated underwater robot, l≠0; new control input τ1=[τ u , τ q , τ r ] T , the new disturbance τ d1 =[τ ud , τ qd , τ rd ] T .

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

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