Radial basis neural network adaptive based trajectory tracking control method for robot arm

Abstract

The application discloses a kind of based on radial basis neural network self-adapting mechanical arm trajectory tracking control method, comprising: using radial basis neural network to the nonlinear modeling information in mechanical arm system equation is approximated;Adaptive law is set;Wherein, adaptive law is obtained by constructing the relationship between control law, sliding surface, mechanical arm joint angular velocity tracking error and the weight matrix of radial basis neural network;The parameter in mechanical arm system equation is updated in real time by adaptive law, realizes mechanical arm trajectory tracking.For the dynamics modeling of system and external disturbance part, radial basis neural network RBF is used to replace adaptive parameter, so that the whole system has higher robustness, and the influence of system modeling uncertainty and external disturbance is minimized.

Description

Technical Field

[0001] This invention relates to the field of robotic arm control technology, and more specifically to a robotic arm trajectory tracking control method based on radial basis function neural network adaptation. Background Technology

[0002] A robotic arm is a complex multi-input multi-output (MIMO) system, exhibiting strong nonlinearity, coupling, and time-varying characteristics. The robotic arm system is inherently fraught with uncertainties, primarily encompassing two aspects: 1) structural uncertainties, generally including unknown dynamic modeling information, dynamic and static friction, and system parameter perturbations; 2) non-structural uncertainties, typically caused by external environmental disturbances, measurement errors, sampling delays, and actuator saturation. Considering these factors, obtaining a high-precision dynamic model of the robotic arm is virtually impossible. Modeling errors further affect the accuracy of the robotic arm's trajectory tracking; therefore, designing a high-precision, robust, and adaptive controller is of paramount importance.

[0003] Operational underwater robots can effectively assist people in underwater salvage, rescue, and other tasks. Most of these robots are equipped with robotic arms for operation. Regarding the trajectory tracking problem of these robotic arms, scholars both domestically and internationally have proposed various trajectory tracking control algorithms, including PID control, discrete control, adaptive backpropagation control, robust control, fuzzy control, and variable structure control, as well as many different control methods derived from combining these approaches. Among them, sliding mode control (SMC) has advantages such as simple structure and high control accuracy, and is widely used in the field of robotic arm control.

[0004] Traditional linear sliding mode control requires a large control gain for significant external disturbances, which increases system uncertainty. To address this issue, a Radial Basis Function Neural Network (RBFNN) is introduced on top of the linear sliding mode controller to dynamically estimate the uncertain control behavior of the system. Compared to traditional methods, RBFNN-SMC can effectively reduce potential uncertainties. However, current linear sliding mode controllers incorporating RBFNNs do not achieve high tracking accuracy for any given continuous or discontinuous trajectory under external disturbances and uncertainties. Summary of the Invention

[0005] In view of this, embodiments of the present invention provide a robotic arm trajectory tracking control method based on radial basis function neural network (RBF) to solve the technical problem that linear sliding mode controllers that introduce RBF neural networks in the prior art have low tracking accuracy for a given arbitrary continuous or discontinuous trajectory under external disturbances and uncertainties.

[0006] This invention provides a robotic arm trajectory tracking control method based on radial basis function neural network adaptation, comprising:

[0007] Radial basis neural networks are used to approximate the nonlinear modeling information in the system equations of the robotic arm;

[0008] An adaptive law is set; the adaptive law is obtained by constructing the relationship between the control law, the sliding surface, the tracking error of the robot arm joint angular velocity, and the weight matrix of the radial basis neural network.

[0009] By using an adaptive law to update the parameters in the system equations of the robotic arm in real time, the trajectory tracking of the robotic arm can be achieved.

[0010] Optionally, it also includes:

[0011] The equations of the robotic arm system are constructed based on the positive definite diagonal matrix, the system control input of the robotic arm system, the control torque vectors of each joint of the robotic arm system, the Coriolis force matrix of the robotic arm system, the gravity vector of the robotic arm system, the friction vector of the robotic arm system, and the bounded differentiable external disturbances.

[0012] Optionally, it also includes:

[0013] The Gaussian function is used as the radial basis function of the radial basis neural network.

[0014] Optionally, the weight matrix error of the radial basis neural network can be obtained based on the difference between the actual radial basis neural network output and the ideal radial basis neural network output, the radial basis neural network approximation error vector, and the sliding surface.

[0015] The output of the ideal radial basis function neural network includes the output vector of the Gaussian function, the weight matrix of the ideal radial basis function neural network, and the approximation error vector of the radial basis function neural network; the weight matrix of the ideal radial basis function neural network includes a first weight matrix about the nonlinear part and a second weight matrix about the system control input; the approximation error vector of the radial basis function neural network includes the error of the nonlinear part and the error of the system control input.

[0016] Optionally, it also includes: analyzing system stability using a Lyapunov function; wherein the Lyapunov function is constructed based on the error of the weight matrix of the radial basis function neural network, the sliding surface, external disturbances, and switching control inputs.

[0017] Optionally, the stability conditions of the robotic arm control system can be obtained based on Lyapunov function stability analysis:

[0018] The positive definite diagonal matrix in the switching control item is greater than the sum of the product of the maximum value of the nonlinear part error and the system control input, the maximum value of the system control input error, and the maximum value of the external disturbance.

[0019] Optionally, the switching control input can be constructed based on the control coefficient, the maximum value of the external disturbance, the sliding surface, and a sign function about the sliding surface.

[0020] Optionally, the sliding surface can be defined via non-singular terminal control.

[0021] Beneficial effects of the embodiments of the present invention:

[0022] This invention provides a robotic arm trajectory tracking control method based on radial basis function neural network (RBF). The RBF neural network is used to adaptively replace parameters in the dynamic modeling and external disturbance parts of the system, which makes the system more robust and minimizes the uncertainty of system modeling and the impact of external disturbances. Attached Figure Description

[0023] The features and advantages of the invention will be more clearly understood by referring to the accompanying drawings, which are schematic and should not be construed as limiting the invention in any way. In the drawings:

[0024] Figure 1 A flowchart of a robotic arm trajectory tracking control method based on radial basis function neural network adaptive method is shown in an embodiment of the present invention;

[0025] Figure 2 A radial basis function neural network structure diagram is shown in an embodiment of the present invention. Detailed Implementation

[0026] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0027] A rigid robotic arm with n-degree-of-freedom rotary joints is a nonlinear control object, inherently possessing numerous uncertainties. These uncertainties are nonlinear and difficult to measure, affecting the accuracy of system modeling and further impacting the dynamic performance of the control system. Therefore, this embodiment employs an RBF radial basis function neural network to approximate the nonlinear modeling parameters of the robotic arm system, achieving more precise and robust control.

[0028] like Figure 1 and Figure 2 As shown, this embodiment of the invention provides a robotic arm trajectory tracking control method based on radial basis function neural network adaptation, including:

[0029] Step S10: Use a radial basis neural network to approximate the nonlinear modeling information in the robotic arm system equations.

[0030] In this embodiment, the equations of the robotic arm system are constructed based on the positive definite diagonal matrix, the system control input of the robotic arm system, the control torque vectors of each joint of the robotic arm system, the Coriolis force matrix of the robotic arm system, the gravity vector of the robotic arm system, the friction vector of the robotic arm system, and the bounded differentiable external disturbances.

[0031] Considering external disturbances and uncertainties, the dynamic equations of a rotationally rigid n-degree-of-freedom manipulator are as follows:

[0032]

[0033] In the formula: These are the angles, angular velocities, and angular acceleration vectors of each joint of the robotic arm; Let be the positive definite symmetric inertia matrix of the robotic arm system; Here are the centripetal force and Coriolis force matrices for the robotic arm system; Let be the gravity vector of the robotic arm system; This is the frictional force vector of the robotic arm system, which represents the uncertainty of the system. This represents the control torque vector of each joint in the robotic arm system; This refers to the external time-varying disturbance of the robotic arm system.

[0034] by As the system output, in the formula, x1, x2 ∈ R n The system control input is τ∈R n Considering the uncertainties and friction in the robotic arm system, the robotic arm system can be constructed as follows:

[0035]

[0036] In the formula, M diag It is a positive definite diagonal matrix; d(t,x)∈R n It is a bounded and differentiable external disturbance.

[0037] The nonlinear modeling information in equation (2) is approximated using an RBF neural network, allowing...

[0038]

[0039] In the formula, f(x)∈R n×n , representing the nonlinear part, g(x)∈R n , representing the system control input. The system equations are rewritten as:

[0040]

[0041] RBF neural networks consist of an input layer, hidden layers, and an output layer, and their network structure is as follows: Figure 2 As shown. The Gaussian function is used as the radial basis function of the radial basis neural network:

[0042]

[0043] In the formula, c j b is the center vector of the Gaussian function of node j; j This represents the width of the Gaussian nucleus of the neuron. The ideal network output is:

[0044]

[0045] In the formula, N is the number of output neurons; h j Let h(x) be the Gausky function of the j-th neuron, where h(x) ∈ R. N P is the output vector of the Gaussian function; x is the network input; P * ∈R n×N Q * ∈R n×N Let f(x) and g(x) be the ideal network weight matrices that approximate f(x) and g(x), respectively; ε1∈R n ,ε2∈R n Let ε1 be the network approximation error vector, satisfying |ε1|≤ε max1 ,|ε2|≤ε max2 .

[0046] The error of the weight matrix of the radial basis function neural network is obtained by using the difference between the actual output of the radial basis function neural network and the output of the ideal radial basis function neural network, the radial basis function neural network approximation error vector and the sliding surface.

[0047] The output of the ideal radial basis function neural network includes the output vector of the Gaussian function, the weight matrix of the ideal radial basis function neural network, and the approximation error vector of the radial basis function neural network; the weight matrix of the ideal radial basis function neural network includes a first weight matrix about the nonlinear part and a second weight matrix about the system control input; the approximation error vector of the radial basis function neural network includes the error of the nonlinear part and the error of the system control input.

[0048] The actual output of the RBF neural network is:

[0049]

[0050] in, This is the weight matrix of the RBF neural network.

[0051] As an optional implementation, the sliding surface s is defined by non-singular terminal control.

[0052] The system's total control input is:

[0053]

[0054] make

[0055]

[0056] Substituting equation (8) into equation (9), we get:

[0057]

[0058] In the formula,

[0059] Let p and q be the error values, where p and q are both positive odd numbers.

[0060] Step S20: Set the adaptive law; wherein, the adaptive law is obtained by constructing the relationship between the control law, the sliding surface, the tracking error of the robot arm joint angular velocity and the weight matrix of the radial basis neural network.

[0061] The adaptive law is designed as follows:

[0062]

[0063] Step S30: The parameters in the robot arm system equation are updated in real time using an adaptive law to achieve robot arm trajectory tracking.

[0064] As an optional implementation, system stability is analyzed using Lyapunov functions; wherein the Lyapunov functions are constructed based on the weight matrix error of the radial basis function neural network, the sliding surface, external disturbances, and switching control inputs.

[0065] As an optional implementation, the stability conditions of the robotic arm control system are obtained based on Lyapunov function stability analysis:

[0066] The positive definite diagonal matrix in the switching control item is greater than the sum of the product of the maximum value of the nonlinear part error and the system control input, the maximum value of the system control input error, and the maximum value of the external disturbance.

[0067] The positive definite diagonal matrix in the switching control term is made greater than the sum of the product of the maximum value of the nonlinear error and the system control input, the maximum value of the system control input error, and the maximum value of the external disturbance. Alternatively, the switching control input can be constructed based on the control coefficients, the maximum value of the external disturbance, the sliding surface, and a sign function about the sliding surface.

[0068] The Lyapunov function is chosen as follows:

[0069]

[0070] In the formula, γ1,γ2>0,γ1,γ2∈R n Differentiating the above equation, we get:

[0071]

[0072] In the formula, let μ > ε max1 τ+ε max2 +γ max We can obtain:

[0073]

[0074] According to Lyapunov's stability theorem, under the action of the adaptive law, the system can converge to a stable state from any initial state in a finite time.

[0075] As described above, this invention provides a robotic arm trajectory tracking control method based on radial basis function neural network (RBN), comprising: approximating the nonlinear modeling information in the robotic arm system equation using RBN; setting an adaptive law; wherein the adaptive law is obtained by constructing the relationship between the control law, sliding surface, robotic arm joint angular velocity tracking error and the weight matrix of the RBN; and updating the parameters in the robotic arm system equation in real time through the adaptive law to achieve robotic arm trajectory tracking.

[0076] In this invention, a radial basis function (RBF) neural network is used for adaptive parameter replacement in the dynamic modeling and external disturbance part of the system, which makes the system as a whole more robust and minimizes the uncertainty of system modeling and the impact of external disturbance.

[0077] Although embodiments of the invention have been described in conjunction with the accompanying drawings, those skilled in the art can make various modifications and variations without departing from the spirit and scope of the invention, and such modifications and variations all fall within the scope defined by the appended claims.

Claims

1. A robotic arm trajectory tracking control method based on radial basis function neural network adaptation, characterized in that, include: We use a radial basis function neural network to approximate the nonlinear modeling information in the robotic arm system equations, let: In the formula, with As system output, , It is a positive definite diagonal matrix. , indicating the nonlinear part, This indicates the system control input section. These are the angle and angular velocity vectors of each joint of the robotic arm; Let be the positive definite symmetric inertia matrix of the robotic arm system; The centripetal force and Coriolis force matrix of the robotic arm system; Let be the gravity vector of the robotic arm system; This is the frictional force vector of the robotic arm system, which represents the uncertainty of the system. The system equations are rewritten as follows: In the formula, For a bounded and differentiable external disturbance, This represents the control torque vector of each joint in the robotic arm system; The ideal network output is: In the formula, This represents the number of output neurons; For the first N Gaussian function of 1 neuron This is the output vector of the Gaussian function; For network input; Approximation and The ideal network weight matrix; Let be the network approximation error vector, satisfying ; The actual output of the RBF neural network is: in, , Here is the weight matrix of the RBF neural network; An adaptive law is set; wherein, the adaptive law is obtained by constructing the relationship between the control law, the sliding surface, the tracking error of the robot arm joint angular velocity, and the weight matrix of the radial basis neural network; The parameters in the system equations of the robotic arm are updated in real time by the adaptive law to achieve robotic arm trajectory tracking. The equations of the robotic arm system are constructed based on the positive definite diagonal matrix, the system control input of the robotic arm system, the control torque vectors of each joint of the robotic arm system, the Coriolis force matrix of the robotic arm system, the gravity vector of the robotic arm system, the friction vector of the robotic arm system, and the bounded differentiable external disturbances. The error of the weight matrix of the radial basis neural network is obtained based on the difference between the actual output of the radial basis neural network and the output of the ideal radial basis neural network, the approximation error vector of the radial basis neural network, and the sliding surface. The output of the ideal radial basis function neural network includes the output vector of the Gaussian function, the weight matrix of the ideal radial basis function neural network, and the approximation error vector of the radial basis function neural network; the weight matrix of the ideal radial basis function neural network includes a first weight matrix about the nonlinear part and a second weight matrix about the system control input; the approximation error vector of the radial basis function neural network includes the nonlinear part error and the system control input error. The stability of the system is analyzed using a Lyapunov function; wherein the Lyapunov function is constructed based on the error of the weight matrix of the radial basis function neural network, the sliding surface, external disturbances, and switching control inputs; and the stability conditions of the robotic arm control system are obtained based on the stability analysis of the Lyapunov function. The positive definite diagonal matrix in the switching control item is greater than the sum of the product of the maximum value of the nonlinear part error and the system control input, the maximum value of the system control input error, and the maximum value of the external disturbance; The switching control input is constructed based on the control coefficient, the maximum value of the external disturbance, the sliding surface, and a sign function about the sliding surface.

2. The robotic arm trajectory tracking control method based on radial basis function neural network adaptive control according to claim 1, characterized in that, Also includes: The Gaussian function is used as the radial basis function of the radial basis neural network.

3. The robotic arm trajectory tracking control method based on radial basis function neural network adaptive control according to claim 1, characterized in that, The sliding surface is defined by non-singular terminal control.

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