An underwater mechanical arm trajectory tracking control method based on RBF neural network and sliding surface

Abstract

The application provides an underwater mechanical arm trajectory tracking control method based on an RBF neural network and a sliding surface, and comprises the following steps: establishing an underwater dynamics model and an electric drive mathematical model; constructing a cascade dynamics equation group; setting an expected input signal and defining related errors; performing sliding mode function design of a fast terminal sliding surface, defining a Lyapunov equation expression and deriving the Lyapunov equation; designing a function for eliminating singular points according to a singular point problem existing in the sliding mode function, and updating the derivation function of the Lyapunov equation; designing an RBF neural network and weights to perform neural network approximation on a nonlinear term of the derivation function of the Lyapunov equation, and designing an electric drive auxiliary controller and a dynamics auxiliary controller. The application improves the fast terminal sliding control method on the basis of considering the auxiliary control of the electric drive force, not only improves the stability and convergence speed of the underwater mechanical arm trajectory tracking control method, but also can adapt to specific requirements of different application occasions.

Description

Technical Field

[0001] This invention relates to an Underwater Vehicle-Manipulator System (UVMS), specifically to a trajectory tracking and control method for an underwater manipulator based on an RBF neural network and a sliding surface. Background Technology

[0002] Underwater vehicle manipulator systems (UVMS) can control underwater manipulators to perform underwater tasks in place of humans, and are currently an effective means of developing underwater ocean energy. Typically, a UVMS is formed when a manipulator is attached to an underwater vehicle. As an important tool for underwater vehicles, UVMS is of great significance for tasks such as real-time underwater photography, underwater target reconnaissance and surveillance, marine resource development, and marine biological exploration. UVMS plays a crucial role in various underwater ocean missions and has become a focus of research for many scholars.

[0003] Due to the strong robustness of sliding mode control, various control methods have been proposed both domestically and internationally for underwater robotic arm trajectory tracking. These methods include designing controllers using sliding mode control and employing non-singular terminal sliding mode control. However, these methods lack fast convergence speeds and do not consider the impact of electric drives. Without considering the effects of electric drives, the trajectory tracking control of underwater robotic arms will suffer from low tracking accuracy. Summary of the Invention

[0004] The purpose of this invention is to propose a trajectory tracking control method for an underwater robotic arm based on RBF neural network and sliding surface, which provides more stable trajectory tracking control and faster convergence speed when considering electric drive motion.

[0005] To achieve the above objectives, the technical solution of the present invention is: a trajectory tracking and control method for an underwater robotic arm based on an RBF neural network and a sliding mode surface, comprising the following steps:

[0006] Step 1: Establish the underwater dynamics model and the electric drive mathematical model to provide a theoretical basis for the construction of the cascade dynamic equations in Step 2;

[0007] Step 2: Construct a cascaded set of dynamic equations to provide a more detailed and reliable mathematical model for the design of control methods;

[0008] Step 3: Set the desired input signal, substitute it into the cascaded dynamic equations, thereby introducing the electric drive auxiliary controller and the dynamic auxiliary controller, and defining the relevant errors;

[0009] Step 4: Design the sliding function for the fast terminal sliding surface, define the expression of the Lyapunov equation and differentiate it;

[0010] Step 5: Based on the problem of singularities in sliding mode functions, design a function to eliminate singularities and update the derivative function of the Lyapunov equation;

[0011] Step 6: Design an RBF neural network and weights to approximate the nonlinear terms of the derivative function of the Lyapunov equation in Steps 4 and 5 using a neural network.

[0012] Step 7: Design an electric drive auxiliary controller and a dynamics auxiliary controller to achieve trajectory tracking control of the underwater robotic arm.

[0013] Furthermore, in step 1, the mathematical model for the underwater robotic arm and its electric drive is established as follows:

[0014]

[0015] In the formula, q, Let represent the joint angle, the first derivative of the joint angle, and the second derivative of the joint angle, respectively. M(q) is the inertia matrix. Here is the matrix of centripetal force and Coriolis force coefficients. Let G(q) be the coefficient matrix of the water resistance term, and G(q) be the equivalent gravity vector matrix. ms The input is the dynamic control input, Δ represents the uncertain disturbance, including the influence of the underwater robot body on the underwater manipulator, I is the current, and K is the dynamic control input. me Let τ be the transformation matrix between torques. me This indicates the driving force of the motor.

[0016] Furthermore, the cascaded dynamic equations are as follows:

[0017]

[0018] Where L e Represents inductance, R e Represents resistance, K e Represents capacitance, τ e Represents a voltage source. The first derivative of electric current.

[0019] Furthermore, step 3 specifically includes:

[0020] Set the desired input signal I d And a dynamic auxiliary controller τ1 is introduced:

[0021]

[0022] Using I d The cascaded dynamic equations are updated, and an electric drive auxiliary controller τ2 is introduced:

[0023]

[0024] Define the current error η and the joint tracking error e:

[0025] η = I d -I

[0026] e = q d -q

[0027] Where M, C, D, and G are M(q), The simplification of G(q), q d For the desired current, It is the second derivative of the desired current.

[0028] Furthermore, the function of the fast terminal sliding surface is:

[0029]

[0030] Where γ1 and γ2 represent positive numbers, and γ1≥1, 0≤γ2≤1, and α1 and α2 are positive gain matrices. This is the first derivative of the joint tracking error;

[0031] The first derivative of the function of the fast terminal sliding surface is:

[0032]

[0033] Introducing auxiliary variables and μ:

[0034]

[0035] The specific expression for the Lyapunov equation is as follows:

[0036]

[0037] Where L represents the inductance coefficient matrix, the derivative of the Lyapunov equation is:

[0038]

[0039] in, Let be the first derivative of the desired input signal.

[0040] Furthermore, step 5 specifically involves: identifying singularities in the derivative function of the Lyapunov equation from step 4. To perform saturation treatment, the saturation function is designed as follows:

[0041]

[0042] in, For positive numbers, use sat(v)z ) replacement

[0043] Define the first derivative of the improved sliding surface as First derivative based on improved sliding surface By reverse engineering the new sliding surface s2, the Lyapunov function is updated as follows:

[0044]

[0045] And update the derivative function of the Lyapunov equation:

[0046]

[0047] Furthermore, step 6 specifically involves defining f1 and f2 as nonlinear terms of the derivative function of the Lyapunov equation in step 4, and approximating the nonlinear terms f1 and f2 using an RBF neural network, with the specific expression being:

[0048]

[0049] Where W1 and W2 represent the ideal weights of the neural network, h1(x) and h2(x) represent the Gaussian function output vectors, and ε1 and ε2 represent the approximation errors of the unknown nonlinear function in the network.

[0050] Replace f1 with f3, and define f3 as the nonlinear term of the derivative function of the Lyapunov equation in step 5. Approximate the nonlinear term f3 using an RBF neural network, with the specific expression as follows:

[0051]

[0052] Among them, W 1N This represents the ideal weights for the improved neural network.

[0053] Furthermore, the design of the electric drive auxiliary controller and the dynamics auxiliary controller in step 7 is specifically as follows:

[0054]

[0055] Where τ1 represents the dynamic auxiliary controller and τ2 represents the electric drive auxiliary controller.

[0056] Compared with the prior art, the present invention has the following beneficial effects:

[0057] This invention provides a cascaded sliding mode underwater manipulator trajectory tracking control method based on RBF neural network to ensure stable trajectory tracking control of the underwater manipulator in complex underwater environments. It also considers the influence of electric drive and forms a cascaded control system. Compared with the traditional non-singular fast terminal sliding mode control method, it can more stably complete the work requirements and ensure that the control system has good performance. Attached Figure Description

[0058] Figure 1 This is a spatial motion diagram of the underwater robotic arm mounted on an underwater robot according to an embodiment of the present invention;

[0059] Figure 2 This is a diagram illustrating the spatial trajectory tracking effect of the underwater robotic arm implemented in this invention.

[0060] Figure 3 This is the trajectory tracking effect of joints 1, 2, and 3 of the underwater robotic arm in an embodiment of the present invention;

[0061] Figure 4 This describes the tracking performance of the underwater robotic arm in the x, y, and z directions under an embodiment of the present invention.

[0062] Figure 5 It refers to the tracking errors of the underwater robotic arm in the x, y, and z directions in the embodiments of the present invention. Detailed Implementation

[0063] The following is in conjunction with the appendix Figure 1-5 The technical solution of the present invention will be described in detail below.

[0064] It should be noted that the following detailed descriptions are exemplary and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0065] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments according to this application. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0066] This invention proposes a trajectory tracking control method for an underwater robotic arm based on an RBF neural network and a sliding mode surface. Building upon fast terminal sliding mode control, it eliminates saturation functions at singular points, uses an RBF neural network to approximate nonlinear terms, and considers the influence of electric drive. A cascaded control trajectory tracking method is then proposed. Comparison of the tracking performance of the underwater robotic arm under ordinary sliding mode control and PD control verifies the superiority of the proposed control method.

[0067] The underwater robotic arm trajectory tracking control method based on RBF neural network and sliding surface proposed in this invention has the following specific steps:

[0068] Establish mathematical models for the underwater robotic arm and its electric drive:

[0069]

[0070] In the formula, q, Let represent the joint angle, the first derivative of the joint angle, and the second derivative of the joint angle, respectively. M(q) is the inertia matrix. Here is the matrix of centripetal force and Coriolis force coefficients. Let G(q) be the coefficient matrix of the water resistance term, and G(q) be the equivalent gravity vector matrix. ms The input is the dynamic control input, Δ represents the uncertain disturbance, including the influence of the underwater robot body on the underwater manipulator, I is the current, and K is the dynamic control input. me Let τ be the transformation matrix between torques. me This indicates the driving force of the motor.

[0071] Construct a system of cascaded dynamic equations:

[0072]

[0073] Where L e Represents inductance, R e Represents resistance, K e Represents capacitance, τ e Represents a voltage source. The first derivative of electric current.

[0074] Set the desired input signal:

[0075]

[0076] Where M, C, D, and G are M(q), The simplification of G(q), q d For the desired current, Let τ be the second derivative of the desired current, and τ1 be the auxiliary controller for the underwater robotic arm dynamics.

[0077] Using Id Update the cascade dynamic equations:

[0078]

[0079] Wherein, τ2 is the auxiliary controller of the electrical system.

[0080] The tracking error of the joint is defined as follows:

[0081] e = q d -q

[0082] The current tracking error is defined as follows:

[0083] η = I d -I

[0084] To ensure convergence quality, the fast terminal sliding surface is selected as:

[0085]

[0086] Where γ1 and γ2 represent positive integers, γ1≥1, 0≤γ2≤1, and α1 and α2 are positive gain matrices.

[0087] The first derivative of the sliding surface is further obtained as follows:

[0088]

[0089] To facilitate calculation, auxiliary variables are introduced. And μ, represented as follows:

[0090]

[0091] To ensure system stability, we can define Lyapunov functions for the error, mechanical subsystem, and electrical signal flux linkage system as follows:

[0092]

[0093] Further, the derivative function of the Lyapunov equation is obtained:

[0094]

[0095] This function contains nonlinear terms. For trajectory tracking control of an underwater robotic arm, these nonlinear terms have a certain impact on the control results; therefore, an RBF neural network is used to approximate the nonlinear terms.

[0096]

[0097] Wherein, the weights, hidden layer inputs, and approximation errors are represented by W. i h iand ε i To express.

[0098] When e→0, in f1 Singularity phenomena can occur, which can lead to controller disconnects, particularly in F1. Singularity issues may occur, so saturation is applied. The fast terminal sliding surface is modified to a non-singular fast terminal sliding surface. The saturation function is designed as follows:

[0099]

[0100] in, It is a positive number.

[0101] Use sat(v) z ) replacement Define the first derivative of the improved sliding surface as:

[0102]

[0103] First derivative based on improved sliding surface By reverse engineering the new sliding surface s2, the fast termination sliding mode controller s1 can be improved into a non-singular fast termination sliding surface s2.

[0104] To ensure system stability, its Lyapunov function is defined as follows:

[0105]

[0106] Differentiate the Lyapunov function.

[0107]

[0108] The nonlinear terms are approximated using a neural network:

[0109]

[0110] Among them, W 1N This represents the ideal weights for the improved neural network.

[0111] Therefore, auxiliary controllers τ1 and τ2 can be designed as follows:

[0112]

[0113] Where τ1 represents the dynamic auxiliary controller and τ2 represents the electric drive auxiliary controller.

[0114] Finally, a Lyapunov function is set up to verify the stability of the embodiments of the present invention:

[0115]

[0116] Furthermore, we can obtain:

[0117]

[0118] To ensure the robustness of the neural network controller, the following definition is used for weight adjustment:

[0119]

[0120] in, Indicates the adjusted weight derivative, W represents the adjusted weight derivative. 1e W represents the ideal weights of the neural network. 2e Let x represent the ideal weights of the neural network, k1 and k2 represent constants (positive), and h1(X1) and h2(X2) represent the outputs of the updated Gaussian function.

[0121] Clearly, the system error is asymptotically stable. This verifies that the non-singular fast terminal sliding mode cascade controller based on RBF neural network proposed in this invention has good stability and can ensure good stability and tracking accuracy of UVMS when performing trajectory tracking underwater.

[0122] Figure 2 This is a diagram illustrating the spatial trajectory tracking effect of the underwater robotic arm implemented according to the present invention. A comparison with RBF neural network (NN) and PD control shows that the control algorithm proposed in this invention significantly outperforms the other two control methods in terms of tracking performance.

[0123] Figure 3 This is the trajectory tracking effect of underwater robotic arm joints 1-3 under an embodiment of the present invention. A comparison with RBF neural network (NN) and PD control shows that the non-singular fast end-effector sliding surface and the neural network-based sliding mode controller proposed in this patent have higher tracking stability than PD control.

[0124] Figure 4 This describes the tracking performance of the underwater robotic arm in the x, y, and z directions under an embodiment of the present invention. A comparison with RBF neural network (NN) and PD control shows that the tracking performance in the x direction is stable in methods with neural network control. Furthermore, the RBF-based non-singular fast terminal sliding mode control method proposed in this patent can track the desired trajectory faster and more stably.

[0125] Figure 5This refers to the tracking errors of the underwater robotic arm in the x, y, and z directions under the embodiments of this invention. A comparison with RBF neural network (NN) and PD control shows that the control algorithm proposed in this invention has a smaller tracking error in the x direction compared to the other two control methods, verifying that the control algorithm proposed in this patent provides more stable tracking in the x direction.

Claims

1. A trajectory tracking control method for an underwater robotic arm based on RBF neural networks and sliding surfaces, characterized in that, Includes the following steps: Step 1: Establish the underwater dynamics model and the electric drive mathematical model to provide a theoretical basis for the construction of the cascade dynamic equations in Step 2; Step 2: Construct a cascaded set of dynamic equations to provide a more detailed and reliable mathematical model for the design of control methods; Step 3: Set the desired input signal, substitute it into the cascaded dynamic equations, thereby introducing the electric drive auxiliary controller and the dynamic auxiliary controller, and defining the relevant errors; Step 4: Design the sliding function for the fast terminal sliding surface, define the expression of the Lyapunov equation and differentiate it; Step 5: Based on the problem of singularities in sliding mode functions, design a function to eliminate singularities and update the derivative function of the Lyapunov equation; Step 6: Design an RBF neural network and weights to approximate the nonlinear terms of the derivative function of the Lyapunov equation in Steps 4 and 5 using a neural network. Step 7: Design an electric drive auxiliary controller and a dynamics auxiliary controller to achieve trajectory tracking control of the underwater robotic arm; In step 1, the mathematical models for the underwater robotic arm and its electric drive are established as follows: In the formula, q, Let represent the joint angle, the first derivative of the joint angle, and the second derivative of the joint angle, respectively. M(q) is the inertia matrix. Here is the matrix of centripetal force and Coriolis force coefficients. Let G(q) be the coefficient matrix of the water resistance term, and G(q) be the equivalent gravity vector matrix. ms The input is the dynamic control input, Δ represents the uncertain disturbance, including the influence of the underwater robot body on the underwater manipulator, I is the current, and K is the dynamic control input. me Let τ be the transformation matrix between torques. me Indicates the driving force of the motor; The cascaded dynamic equations are as follows: Where L e Represents inductance, R e Represents resistance, K e Represents capacitance, τ e Represents a voltage source. The first derivative of electric current.

2. The underwater robotic arm trajectory tracking control method based on RBF neural network and sliding surface according to claim 1, characterized in that, Step 3 specifically involves: Set the desired input signal I d And a dynamic auxiliary controller τ1 is introduced: Using I d The cascaded dynamic equations are updated, and an electric drive auxiliary controller τ2 is introduced: Define the current error η and the joint tracking error e: η=I d -I e=q d -q Where M, C, D, and G are M(q), The simplification of G(q), q d For the desired current, It is the second derivative of the desired current.

3. The underwater robotic arm trajectory tracking control method based on RBF neural network and sliding surface according to claim 2, characterized in that, The function of the fast terminal sliding surface is: Where γ1 and γ2 represent positive numbers, and γ1≥1, 0≤γ2≤1, and α1 and α2 are positive gain matrices. This is the first derivative of the joint tracking error; The first derivative of the function of the fast terminal sliding surface is: Introducing auxiliary variables and μ: The specific expression for the Lyapunov equation is as follows: Where L represents the inductance coefficient matrix, the derivative of the Lyapunov equation is: in, Let be the first derivative of the desired input signal.

4. The underwater robotic arm trajectory tracking control method based on RBF neural network and sliding surface according to claim 3, characterized in that, Step 5 specifically involves: The occurrence of singularities in the derivative function of the Lyapunov equation in step 4 To perform saturation treatment, the saturation function is designed as follows: in, For positive numbers, use sat(v) z ) alternative Define the first derivative of the improved sliding surface as First derivative based on improved sliding surface By reverse engineering the new sliding surface s2, the Lyapunov function is updated as follows: And update the derivative function of the Lyapunov equation:

5. The underwater robotic arm trajectory tracking control method based on RBF neural network and sliding surface according to claim 4, characterized in that, Step 6 specifically involves defining f1 and f2 as nonlinear terms of the derivative function of the Lyapunov equation in step 4, and approximating the nonlinear terms f1 and f2 using an RBF neural network. The specific expression is as follows: Where W1 and W2 represent the ideal weights of the neural network, h1(x) and h2(x) represent the output vector of the Gaussian function, and ε1 and ε2 represent the approximation error of the unknown nonlinear function of the network. Replace f1 with f3, and define f3 as the nonlinear term of the derivative function of the Lyapunov equation in step 5. Approximate the nonlinear term f3 using an RBF neural network, with the specific expression as follows: Among them, W 1N This represents the ideal weights for the improved neural network.

6. The underwater robotic arm trajectory tracking control method based on RBF neural network and sliding surface according to claim 5, characterized in that, The design of the electric drive auxiliary controller and the dynamics auxiliary controller in step 7 is as follows: Where τ1 represents the dynamic auxiliary controller and τ2 represents the electric drive auxiliary controller.

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