Mechanical arm fixed time trajectory tracking control method based on non-singular terminal sliding mode

Abstract

The application is suitable for the technical field of trajectory tracking, and provides a mechanical arm fixed-time trajectory tracking control method based on a non-singular terminal sliding mode, which comprises the following steps: based on data of a pre-designed expected trajectory of a joint of a mechanical arm, a dynamics model of the mechanical arm with uncertainty is established; based on the dynamics model, real-time trajectory data of the joint of the mechanical arm is obtained; based on the data of the expected trajectory and the real-time trajectory data, a sliding mode signal is determined; based on the dynamics model, a radial basis neural network is constructed, and a weight updating law of a weight vector of the radial basis neural network is determined by using a projection method; the sliding mode signal and the adaptive updating law of the weight vector of the radial basis neural network are applied to a controller of the mechanical arm, so that the joint of the mechanical arm can track the expected trajectory in a fixed time. The method can improve the accuracy of estimating the model parameter uncertainty of the mechanical arm and external disturbance in various complex scenes.

Description

Technical Field

[0001] This application belongs to the field of trajectory tracking technology, and in particular relates to a fixed-time trajectory tracking control method for a robotic arm based on non-singular terminal sliding mode. Background Technology

[0002] Robotic arms can assist or replace humans in performing tasks accurately and efficiently in extreme situations, playing a crucial role, especially in semiconductor equipment. Because semiconductor equipment manufacturing involves complex steps and demands high precision, ensuring the robotic arm moves accurately along pre-set joint trajectories is key to achieving these complex tasks. However, the physical parameters of the robotic arm vary with load, external disturbances, and motion characteristics, introducing uncertainties. Therefore, accurately estimating the model parameter uncertainties and external disturbances of the robotic arm through advanced control methods, enabling it to quickly and precisely track the desired trajectory, is of significant practical importance.

[0003] In recent years, many control strategies for nonlinear systems have been proposed. Most of these methods can use a robotic arm as the controlled object, including PID control, robust control, fuzzy control, feedback linearization, neural network control, and sliding mode control. However, when dealing with various complex scenarios, the above control strategies are still not accurate enough in estimating the uncertainty of the model parameters of the robotic arm and external disturbances. Summary of the Invention

[0004] To overcome the problems existing in related technologies, this application provides a fixed-time trajectory tracking control method for a robotic arm based on non-singular terminal sliding mode, which can improve the accuracy of estimating the uncertainty of the model parameters of the robotic arm and external disturbances in various complex scenarios.

[0005] This application is achieved through the following technical solution:

[0006] In a first aspect, embodiments of this application provide a fixed-time trajectory tracking and control method for a robotic arm based on non-singular terminal sliding mode, including:

[0007] Based on the data of the preset expected trajectory of the joints of the robotic arm, a dynamic model of the robotic arm with uncertainty is established.

[0008] Based on the dynamic model, real-time trajectory data of the joints of the robotic arm is obtained;

[0009] Based on the desired trajectory data and real-time trajectory data, the sliding mode signal is determined;

[0010] Based on the dynamic model, a radial basis function neural network is constructed, and the weight update law of the weight vector of the radial basis function neural network is determined by the projection method.

[0011] The adaptive update law of the sliding mode signal and the weight vector of the radial basis neural network is applied to the controller of the robotic arm to obtain the tracking trajectory of the robotic arm's joints at a fixed time.

[0012] In one possible implementation of the first aspect, the robotic arm dynamics model is represented as:

[0013]

[0014] Where q is the acceleration signal of the joint of the robotic arm. τ is the second derivative of q; M is the inertia matrix of the robotic arm; τ is the control torque; The model uncertainty is composed of the parameter uncertainty of the robotic arm and external disturbances, where, M(q)=M0(q)+M Δ (q)∈R n×n g(q)=g0(q)+g Δ (q)∈R n It is the gravity vector; It is the centrifugal Coriolis force matrix; d∈R n It is a bounded external disturbance torque.

[0015] In one possible implementation of the first aspect, the sliding mode signal is determined based on the desired trajectory data and the real-time trajectory data, including:

[0016] The error signal is obtained by comparing the real-time trajectory data of the robotic arm's joints with the data of the pre-designed desired trajectory;

[0017] Based on the error signal, the dynamic model of the uncertain robotic arm is reconstructed into a second-order system;

[0018] Based on the design of a second-order system, a non-singular fast terminal sliding surface is constructed to generate a sliding signal.

[0019] In one possible implementation of the first aspect, the error signal is represented as:

[0020]

[0021] Where e1 and e2 are error signals, and x1(t) = q(t), x d (t)=q d (t), For x d The derivative of , where q(t) is the position signal of the joint of the robotic arm. Let q(t) be the derivative of q. d (t) represents the position signal of the joint of the robotic arm considering external disturbance torque.

[0022] In one possible implementation of the first aspect, the sliding mode signal is represented as:

[0023]

[0024] in, The coefficients of the sliding mode variable are time-varying coefficients related to the error e1, represented by a diagonal matrix. express, Where i = 1, 2, ..., n, α, β, p, g, k are constants, and α > 0, β > 0, k > 1, v1 > 1, and p, g satisfy gk > 1, 1 / v1 <pk<1。

[0025] In one possible implementation of the first aspect, a radial basis function neural network is constructed based on a dynamic model, and an adaptive update law for the weight vector of the radial basis function neural network is designed according to the projection method, including:

[0026] Based on the approximation principle, radial basis functions are constructed, i.e., radial basis neural networks;

[0027] Based on the radial basis functions, determine the optimal approximation equation of the radial basis functions for the nonlinear function;

[0028] The Gaussian function is chosen as the activation function for the radial basis function neural network;

[0029] The output vector of the radial basis function is determined by approximating the uncertainty of the dynamic model based on the optimal approximation equation and the activation function of the radial basis neural network.

[0030] The approximation error of the radial basis neural network is defined based on the output vector after approximating the uncertainty of the dynamic model using radial basis functions.

[0031] The output vector of the dynamic model after approximation by the radial basis function with uncertainty and the approximation error of the radial basis neural network are based on the projection method to determine the weight update law of the weight vector of the radial basis neural network.

[0032] In one possible implementation of the first aspect, the output vector of the radial basis function after approximating the uncertainty of the dynamic model is represented as:

[0033]

[0034] Among them, w i =[w i1 ,w i2 ,…,w im ] T It is a weight vector, z i =[e 1i ,e 2i ] TIt is the input to the neural network; σ i =[σ i1 ,σ i2 ,…,σ im ] T ∈R m Indicates the activation function;

[0035] The approximation error of a radial basis function neural network is expressed as:

[0036]

[0037] in, This represents the optimal weight vector.

[0038] In one possible implementation of the first aspect, the weight update law of the weight vector of the radial basis neural network is determined using a projection method, including:

[0039] Design the update law for the weight vector of a radial basis neural network;

[0040] The update law for the weight vector of a modified radial basis function neural network;

[0041] By using the projection method to ensure the boundedness of the weight vector, the weight update law of the final radial basis function neural network is determined; the weight update law of the final radial basis function neural network weight vector is expressed as:

[0042]

[0043] in, ρ i and ξ i It is bounded; δ is the learning rate of the neural network, δ>0; a constant vector. satisfy

[0044] In one possible implementation of the first aspect, the control torque of the robotic arm's controller is expressed as:

[0045]

[0046] Where M0 is the inertia matrix of the robotic arm; v1, v2, v3, and v4 are constants; The coefficients of the sliding mode variable, They are two diagonal matrices; K s For controller parameters; For x d The second derivative of .

[0047] In a second aspect, this application provides a terminal device, including a memory and a processor, wherein the memory stores a computer program that can run on the processor, characterized in that, when the processor executes the computer program, it implements the fixed-time trajectory tracking control method for a robotic arm based on non-singular terminal sliding mode as described in any of the first aspects.

[0048] The beneficial effects of the embodiments in this application compared with the prior art are:

[0049] In this embodiment, by combining the sliding mode signal with the weight update law of the radial basis neural network weight vector determined by the projection method, it is possible to enable the joint motion to quickly track the desired trajectory when the dynamic model of the robotic arm is uncertain. This ensures the boundedness of the weight vector, addresses various complex scenarios of robotic arm control, and improves the accuracy of estimating the uncertainty of the robotic arm model parameters and external disturbances.

[0050] It is understandable that the beneficial effects of the second aspect mentioned above can be found in the relevant descriptions in the first aspect mentioned above, and will not be repeated here.

[0051] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and are not intended to limit this specification. Attached Figure Description

[0052] To more clearly illustrate the technical solutions in the embodiments of this application, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0053] Figure 1 This is a flowchart illustrating a fixed-time trajectory tracking control method for a robotic arm based on non-singular terminal sliding mode, provided in an embodiment of this application.

[0054] Figure 2 This is a schematic diagram of the operation process of the fixed-time trajectory tracking control method for a robotic arm based on non-singular terminal sliding mode provided in an embodiment of this application in the system;

[0055] Figure 3 This is a schematic diagram of the position and speed tracking of joint 1 of the robotic arm of a robot provided in an embodiment of this application;

[0056] Figure 4 This is a schematic diagram of the position and speed tracking of joint 2 of the robotic arm of a robot provided in an embodiment of this application;

[0057] Figure 5This refers to the tracking error of joint 1 of the robotic arm of a robot provided in one embodiment of this application;

[0058] Figure 6 This refers to the tracking error of joint 2 of the robotic arm of a robot provided in one embodiment of this application;

[0059] Figure 7 This is a schematic diagram of the controller input torque provided in an embodiment of this application;

[0060] Figure 8 It is the norm of the weight vector of the radial basis neural network provided in one embodiment of this application;

[0061] Figure 9 This is a schematic diagram of the structure of a terminal device provided in an embodiment of this application. Detailed Implementation

[0062] In the following description, specific details such as particular system architectures and techniques are set forth for illustrative purposes and not for limitation, in order to provide a thorough understanding of the embodiments of this application. However, those skilled in the art will understand that this application may also be implemented in other embodiments without these specific details. In other instances, detailed descriptions of well-known systems, apparatuses, circuits, and methods have been omitted so as not to obscure the description of this application with unnecessary detail.

[0063] It should be understood that, when used in this application specification and the appended claims, the term "comprising" indicates the presence of the described features, integrals, steps, operations, elements and / or components, but does not exclude the presence or addition of one or more other features, integrals, steps, operations, elements, components and / or a collection thereof.

[0064] It should also be understood that the term “and / or” as used in this application specification and the appended claims means any combination of one or more of the associated listed items and all possible combinations, and includes such combinations.

[0065] As used in this application specification and the appended claims, the term "if" may be interpreted, depending on the context, as "when," "once," "in response to determination," or "in response to detection." Similarly, the phrase "if determined" or "if detected [the described condition or event]" may be interpreted, depending on the context, as meaning "once determined," "in response to determination," "once detected [the described condition or event]," or "in response to detection [the described condition or event]."

[0066] Furthermore, in the description of this application and the appended claims, the terms "first," "second," "third," etc., are used only to distinguish descriptions and should not be construed as indicating or implying relative importance.

[0067] The phrases "in one embodiment," "in some embodiments," "in other embodiments," "in still other embodiments," etc., appearing in different parts of this application specification do not necessarily refer to the same embodiment, but rather mean "one or more, but not all, embodiments," unless otherwise specifically emphasized. The terms "comprising," "including," "having," and variations thereof mean "including but not limited to," unless otherwise specifically emphasized.

[0068] To enable those skilled in the art to better understand the present invention, the technical solutions in the embodiments of this application will be clearly and completely described below in conjunction with the accompanying drawings and specific implementation methods.

[0069] Figure 1 This is a flowchart illustrating a fixed-time trajectory tracking control method for a robotic arm based on non-singular terminal sliding mode, according to an embodiment of this application. (Refer to...) Figure 1 and Figure 2 The detailed process of the fixed-time trajectory tracking and control method for the robotic arm based on non-singular terminal sliding mode is as follows:

[0070] In step 101, a dynamic model of the robotic arm with uncertainty is established based on the data of the preset expected trajectory of the joints of the robotic arm.

[0071] Specifically, based on the preset desired trajectory data of the robotic arm's joints, a dynamic model of the robotic arm is established using the Euler-Lagrange method. The dynamic model of the robotic arm is expressed as follows:

[0072]

[0073] in, The acceleration signal of the joints of the robotic arm; τ(t)∈R n To control the torque; d(t)∈R n It is a bounded external disturbance torque; M(q)=M0(q)+M Δ (q)∈R n×n Here is the inertia matrix of the robotic arm; M0(q) is the matrix of the centrifugal Coriolis force. g0(q) is known, M Δ (q), g Δ (q) is unknown but bounded; g(q) = g0(q) + g Δ (q)∈R n This is the gravity vector.

[0074] Based on the nonsingularity of the inertia matrix M0(q), it can be known that M0(q) is invertible. Therefore, the dynamic model of the robotic arm can be expressed as:

[0075]

[0076] in, The model uncertainty is composed of the parameter uncertainty of the robotic arm and external disturbances.

[0077] The uncertainty term κ in the above robotic arm dynamics model generally satisfies the following conditions: all terms of κ are bounded, and for all κ... i (i = 1, 2, ..., n) all satisfy Among them, κ id (i = 1, 2, ..., n) are unknown positive constants.

[0078] In step 102, real-time trajectory data of the joints of the robotic arm are obtained based on the dynamic model.

[0079] For example, the preset desired trajectory of the joints of the robotic arm under the dynamic model is input into the controller of the robotic arm to obtain real-time trajectory data when the joints of the robotic arm move according to the preset desired trajectory.

[0080] In step 103, the sliding mode signal is determined based on the data of the desired trajectory and the real-time trajectory data.

[0081] For example, the modeling uncertainty κ(t) is a state-dependent function. Specifically, in this application, the composite disturbance κ(t) depends on the position q and velocity of the robot's manipulator joints. And the torque input τ. Since the input power of the robotic arm is limited by electrical energy, and the robot's motion space is finite and continuous, it is assumed that the modeling uncertainty κ(t) is bounded.

[0082] Specifically, step 103 may include: obtaining an error signal by comparing the real-time trajectory data of the joints of the robotic arm with the data of the pre-designed desired trajectory; reconstructing the dynamic model of the uncertain robotic arm into a second-order system based on the error signal; and designing a non-singular fast end-effector sliding surface based on the second-order system to construct a sliding signal.

[0083] For example, the error signal is obtained by subtracting the real-time state trajectory of the robotic arm joint from the pre-designed desired trajectory.

[0084] Define x1(t) = q(t), x d (t)=q d (t), reconstructing the robotic arm's dynamics model into a second-order system:

[0085]

[0086] Further define the error signal:

[0087]

[0088] Substituting equation (3) into equation (4) yields a second-order system with the error signal as the state variable:

[0089]

[0090] A non-singular fast terminal sliding surface is designed based on the error signal, and the sliding signal is represented as follows:

[0091]

[0092] in, It is a diagonal matrix; Represented as:

[0093]

[0094] Where α>0, β>0, k>1, v1>1, and p, g satisfy gk>1, 1 / v1 <pk<1。

[0095] coefficients of sliding mode variables It is a time-varying coefficient related to the error e1. Compared with the traditional non-singular terminal sliding mode, this sliding mode variable can quickly pull the system state error back to the sliding surface s=0 when the system state error is large, thus enhancing its convergence performance. From equation (6), the derivative of the sliding mode signal s is:

[0096]

[0097] in, They are two diagonal matrices. Represented as:

[0098]

[0099] The principle of sliding mode variable structure control in this application mainly refers to forcibly changing the trajectory of the system state during the dynamic response process, causing the system state to move on the sliding surface until it reaches an equilibrium point. Its advantages include insensitivity to parameter uncertainties and external disturbances, strong robustness, and suitability for nonlinear robotic arm systems with high tracking accuracy requirements. However, its disadvantage is that the system state is prone to bouncing back and forth on the sliding surface, generating chattering, which can easily damage hardware. Therefore, sliding mode variable structure control is usually combined with other techniques to suppress chattering, such as neural networks and adaptive control.

[0100] The aforementioned sliding mode signal is a computationally simple sliding mode variable. Compared to traditional non-singular terminal sliding mode, this sliding mode variable can quickly pull the system back to the sliding surface s=0 when the system state error is large, thus enhancing its convergence performance. Furthermore, this sliding mode variable satisfies fixed-time convergence. Its fixed-time stability can be easily proven using a Lyapunov function.

[0101] For example, when hour,

[0102]

[0103] The expression for the Lyapunov function is defined as follows:

[0104]

[0105] The derivative of the system trajectory represented by equation (10) is expressed as:

[0106]

[0107] Based on the relevant lemmas of the fixed-time stability theory relied upon by the controller design, see equations (27)-(29). The sliding mode variable satisfies fixed-time stability, and the error signal contained in the sliding mode variable will converge to the origin within a fixed time T, where the expression for the fixed time T is:

[0108]

[0109] In step 104, a radial basis function neural network is constructed based on the dynamic model, and the weight update law of the weight vector of the radial basis function neural network is determined by the projection method.

[0110] Specifically, step 104 may include:

[0111] Step 1041: Construct radial basis functions, i.e., radial basis neural networks, based on the approximation principle.

[0112] For example, in step 1041, if f(x): R n →R is a continuous function defined on a compact set Ω, f(w,x):R m ×R n →R is a continuous approximation function that depends on w and x. Therefore, the key to approximation is determining the optimal parameters w. * , such that the distance function d * satisfy:

[0113] d * (f(w * ,x),f(x))≤∈ (14)

[0114] Here, ∈ is an acceptable small value.

[0115] The approximation process of a nonlinear function by radial basis functions can be expressed by the following formula:

[0116]

[0117] Among them, z i =[z i1 ,z i2 ,…,z iq ] T ∈R q Let w represent the input vector. i =[w i1 ,w i2 ,…,w im ] T ∈R m Let m represent the weight vector, m≥2 represent the number of hidden nodes, and σ represent the weight vector. i =[σ i1 ,σ i2 ,…,σ im ] T ∈R m f represents the activation function. i ∈R.

[0118] Step 1042: Determine the optimal approximation equation of the radial basis functions for the nonlinear function based on the radial basis functions.

[0119] For example, by adjusting the size of m, It can accurately approximate nonlinear continuous functions f i The optimal approximation equation can be expressed as:

[0120]

[0121] Where, ε i It is the error of the approximation function, satisfying It is a very small positive number. It is the optimal weight vector.

[0122] Step 1043: Select the Gaussian function as the activation function for the radial basis function neural network.

[0123] For example, the radial basis function neural network is a classic neural network activation function, and its kernel function is generally chosen as a Gaussian function, which is expressed as:

[0124]

[0125] Where k = 1, 2, ..., m.μ ik =[μik1 ,μ ik2 ,…,μ ikq ] T It is the center of the receptive field, η ik It is the width of the Gaussian kernel function.

[0126] Step 1044: Determine the output vector of the radial basis function after approximating the uncertainty of the dynamic model based on the optimal approximation equation and the activation function of the radial basis neural network.

[0127] For example, the output vector of the radial basis function after approximating the model uncertainty is represented as:

[0128]

[0129] Among them, w i =[w i1 ,w i2 ,…,w im ] T It is a weight vector, z i =[e 1i ,e 2i ] T It is the input to the neural network.

[0130] Step 1045: Define the approximation error of the radial basis neural network based on the output vector after approximating the uncertainty of the dynamic model using radial basis functions.

[0131] Define the neural network approximation error as:

[0132]

[0133] in, This represents the optimal weight vector.

[0134] Step 1046: Based on the output vector of the dynamic model after approximation of the uncertainty by the radial basis function and the approximation error of the radial basis neural network, and the weight update law of the weight vector of the radial basis neural network determined by the projection method.

[0135] Specifically, in step 1046, the weight update law of the weight vector of the radial basis function neural network is determined by the projection method, including: designing the update law of the weight vector of the radial basis function neural network; correcting the update law of the weight vector of the radial basis function neural network; and using the projection method to ensure the boundedness of the weight vector and determine the final weight update law of the weight vector of the radial basis function neural network.

[0136] The final weight update law for the radial basis function neural network is expressed as:

[0137]

[0138] in, ρ i and ξ i It is bounded; δ is the learning rate of the neural network, δ>0; a constant vector. satisfy

[0139] definition The weight update law of the neural network is designed as follows:

[0140]

[0141] Where δ>0 is the learning rate of the neural network, and since μ is an unknown variable, the update law is modified as follows:

[0142]

[0143] An adaptive parametric projection method is used to ensure the boundedness of the weight vector, defining a constant vector. satisfy The update law for the neural network weight vector is designed based on projection as follows:

[0144]

[0145] in, ρ i and ξ i It is bounded.

[0146] Radial basis function (RBF) neural networks can map the relationship between known input and output data, exhibiting excellent function approximation capabilities. Theoretically, RBF-based methods can accurately approximate nonlinear functions by adjusting the number of hidden layer nodes. For uncertain systems where the specific mathematical model is unknown, RBF-based methods can effectively provide data for constructing controllers; the adaptive adjustment of weights is typically obtained using the Lyapunov stability criterion.

[0147] The neural network weight vector employs an adaptive update law designed using the projection method, ensuring that the weight vector remains within a bounded range during the learning process.

[0148] To prove the weight vector Given the boundedness of Lyapunov functions, we choose the Lyapunov function:

[0149]

[0150] According to the projection adaptive update law (23), there are two cases that need to be considered.

[0151] Case 1: If or We can obtain:

[0152]

[0153] Scenario 2: If We can obtain:

[0154]

[0155] As can be seen from the above proof, as long as the following conditions are met... The projection update law (23) always guarantees

[0156] In step 105, the adaptive update law of the sliding mode signal and the weight vector of the radial basis neural network is applied to the controller of the robotic arm to obtain the desired tracking trajectory of the robotic arm's joints at a fixed time.

[0157] The controller design relies on the following lemmas related to fixed-time stability theory:

[0158] Lemma 1: For nonlinear systems f(0)=0, x(t∈R n The time T for the system to reach equilibrium r It is uniformly bounded and independent of the initial state value, that is:

[0159]

[0160] The system is stable over a fixed period of time.

[0161] Lemma 2: For nonlinear systems f(0)=0, x(t∈R n Suppose there exists a Lyapunov function V(x(t)) that satisfies:

[0162]

[0163] Where α, β, p, g, k are positive constants satisfying pk < 1, gk > 1. If is a bounded positive constant, then the state of the system can be determined in a fixed time T. r Converging inward to the region Ω, where

[0164]

[0165]

[0166] Based on the above-mentioned fixed-time stability criteria, a controller is designed to ensure that the robotic arm can quickly and accurately track the desired trajectory within a fixed time. The controller design is as follows:

[0167]

[0168] In one embodiment of the present invention, the effectiveness of the proposed controller is verified using a dual-link rigid robotic arm. Define x1 = [x 11 ,x 12 ] T As the joint angles of the robotic arm, the relevant matrices in the mathematical model of the two-link robot are then given:

[0169]

[0170] in, p3=m2l1l c2 p4 = m1l c2 +m2l1,p5=m2l c2 m i and l i Let m1 = 2.00 (kg), m2 = 0.85 (kg), l1 = 0.35 (m), and l2 = 0.31 (m), respectively, be the mass and length of connecting rod i. i It is the moment of inertia of link i. l ci It is the center of mass of the i-th link; g = 9.8 (m / s²) 2 ).

[0171] The robot's initial position and velocity are:

[0172] x 11 (0)=x 12 (0) = 1.5 (rad), x 21 (0)=x 22 (0)=0(rad / s) (32)

[0173] The desired trajectory is set as follows:

[0174] x d =[0.1sin(0.5t)+cos(0.5t), 0.1cos(t)+cos(t)] T (33)

[0175] Where t∈[0,t m ], t m =10(s).

[0176] The disturbance torque is:

[0177] d(t)=[0.1sin(0.5t)+0.25cos(0.5t),0.25sin(0.5t)+0.1sin(0.5t)] T (34)

[0178] The controller parameters are set as follows: α = 2.5, β = 2.5, p = 0.5. v1 = 2, v4 = 2, K s =10, δ a =0.001.

[0179] Simulation results are as follows Figure 2-7 As shown, where, Figure 3 This shows that the angular position and angular velocity of joint 1 of the two-link rigid robotic arm can track the desired joint angular trajectory with high precision, speed and stability. Figure 4 This indicates that the angular position and angular velocity of joint 2 of the two-link rigid robotic arm can track the desired joint angular trajectory with high precision, speed and stability. Figure 5 This indicates that the tracking errors of the angular position and angular velocity of joint 1 can converge to zero quickly; Figure 6 This indicates that the tracking errors of the angular position and angular velocity of joint 2 can quickly converge to zero. Figure 7 This indicates that the control signal of the proposed controller is continuous and bounded; Figure 8 This indicates that the weight vector of the radial basis function neural network is always bounded during the learning process. Simulation results show that the proposed control method is feasible and effective.

[0180] This invention proposes a fixed-time trajectory tracking control method for a robotic arm based on non-singular terminal sliding mode. Considering the model uncertainty of the robotic arm, a computationally simple non-singular terminal sliding surface is designed, which enables the trajectory tracking error to converge rapidly within a fixed time. The convergence time is independent of the initial state, and the upper bound of the error convergence time is independent of the initial state of the system, being determined only by the controller parameters. The model uncertainty of the robotic arm is approximated using a radial basis function neural network to estimate and compensate for the model uncertainty. Considering the boundedness of the weights, an adaptive update law for the weight vector is designed based on the projection method to ensure the boundedness of the weight vector. This method can cope with various complex scenarios in robotic arm control and improves the accuracy of estimating the model parameter uncertainty and external disturbances of the robotic arm.

[0181] It should be understood that the sequence number of each step does not imply the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiments of this application.

[0182] Corresponding to the fixed-time trajectory tracking control method for robotic arms based on non-singular terminal sliding mode in the above embodiments, this application also provides a terminal device, see [link to relevant documentation]. Figure 9The terminal device 300 may include at least one processor 310 and a memory 320, wherein the memory 320 stores a computer program 321 that can run on the at least one processor 310, and the processor 310 executes the computer program to implement the steps in any of the above method embodiments, for example... Figure 1 Steps 101 to 105 in the illustrated embodiment.

[0183] For example, a computer program may be divided into one or more modules / units, one or more of which are stored in memory 320 and executed by processor 310 to complete this application. The one or more modules / units may be a series of computer program segments capable of performing specific functions, which describe the execution process of the computer program in terminal device 300.

[0184] Those skilled in the art will understand that Figure 9 This is merely an example of a terminal device and does not constitute a limitation on the terminal device. It may include more or fewer components than shown, or combine certain components, or different components, such as input / output devices, network access devices, buses, etc.

[0185] The processor 310 can be a central processing unit (CPU), or other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor can be a microprocessor or any conventional processor.

[0186] The memory 320 can be an internal storage unit of the terminal device or an external storage device, such as a plug-in hard drive, a smart media card (SMC), a secure digital card (SD), or a flash card. The memory 320 is used to store the computer program and other programs and data required by the terminal device. The memory 320 can also be used to temporarily store data that has been output or will be output.

[0187] The bus can be an Industry Standard Architecture (ISA) bus, a Peripheral Component Interconnect (PCI) bus, or an Extended Industry Standard Architecture (EISA) bus, etc. Buses can be categorized as address buses, data buses, control buses, etc. For ease of illustration, the buses shown in the accompanying drawings are not limited to a single bus or a single type of bus.

[0188] The fixed-time trajectory tracking control method for robotic arms based on non-singular terminal sliding mode provided in this application embodiment can be applied to terminal devices such as computers, tablets, laptops, netbooks, and personal digital assistants (PDAs). This application embodiment does not impose any restrictions on the specific type of terminal device.

[0189] This application also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps in the various embodiments of the above-described robotic arm fixed-time trajectory tracking control method based on non-singular terminal sliding mode.

[0190] This application provides a computer program product that, when run on a mobile terminal, enables the mobile terminal to execute the steps described in the various embodiments of the robotic arm fixed-time trajectory tracking control method based on non-singular terminal sliding mode.

[0191] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, all or part of the processes in the methods of the above embodiments of this application can be implemented by a computer program instructing related hardware. The computer program can be stored in a computer-readable storage medium, and when executed by a processor, it can implement the steps of the various method embodiments described above. The computer program includes computer program code, which can be in the form of source code, object code, executable files, or certain intermediate forms. The computer-readable medium can include at least: any entity or device capable of carrying computer program code to a photographing device / terminal device, a recording medium, a computer memory, a read-only memory (ROM), a random access memory (RAM), an electrical carrier signal, a telecommunication signal, and a software distribution medium. Examples include USB flash drives, portable hard drives, magnetic disks, or optical disks. In some jurisdictions, according to legislation and patent practice, computer-readable media cannot be electrical carrier signals or telecommunication signals.

[0192] In the above embodiments, the descriptions of each embodiment have different focuses. For parts that are not described in detail or recorded in a certain embodiment, please refer to the relevant descriptions of other embodiments.

[0193] Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of this application.

[0194] In the embodiments provided in this application, it should be understood that the disclosed apparatus / network devices and methods can be implemented in other ways. For example, the apparatus / network device embodiments described above are merely illustrative. For instance, the division of modules or units is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed may be through some interfaces; the indirect coupling or communication connection between devices or units may be electrical, mechanical, or other forms.

[0195] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment according to actual needs.

[0196] The above-described embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application, and should all be included within the protection scope of this application.

Claims

1. A method for tracking and controlling a robotic arm's fixed-time trajectory based on non-singular terminal sliding mode, characterized in that, include: Based on the data of the preset expected trajectory of the joints of the robotic arm, a dynamic model of the robotic arm with uncertainty is established. Based on the dynamic model, real-time trajectory data of the joints of the robotic arm are obtained; Based on the data of the desired trajectory and the real-time trajectory data, the sliding mode signal is determined; Based on the aforementioned dynamic model, a radial basis function neural network is constructed, and the weight update law of the weight vector of the radial basis function neural network is determined using the projection method. The sliding mode signal and the weight update law of the radial basis neural network are applied to the controller of the robotic arm to obtain the expected tracking trajectory of the joints of the robotic arm at a fixed time. The dynamic model of the robotic arm is represented as follows: in, The acceleration signal of the joints of the robotic arm; Here is the inertia matrix of the robotic arm; To control the torque; The model uncertainty is composed of the parameter uncertainty of the robotic arm and external disturbances, wherein, , , This represents the position signal of the robotic arm's joints; It is the gravity vector; It is the centrifugal Coriolis force matrix. for The derivative; It is a bounded external disturbance torque; The determination of the sliding mode signal based on the desired trajectory data and the real-time trajectory data includes: The error signal is obtained by comparing the real-time trajectory data of the joints of the robotic arm with the data of the pre-designed desired trajectory. Based on the error signal, the dynamic model of the uncertain robotic arm is reconstructed into a second-order system; Based on the second-order system, a non-singular fast terminal sliding surface is designed, and the sliding signal is constructed. The error signal is represented as: in, and The error signal is... , , , for The derivative, This refers to the position signals of the joints of the robotic arm. for The derivative, The position signal of the joints of the robotic arm takes into account external disturbance torque; The sliding mode signal is represented as: in, The coefficient of the sliding mode variable is related to the error. The relevant time-varying coefficients are represented by a diagonal matrix. express, ,in, , It is a constant, and , , , ,and , satisfy , ; The control torque of the controller of the robotic arm is expressed as: in, M 0. The inertia matrix of the robotic arm; , , , It is a constant; The coefficients of the sliding mode variable, They are two diagonal matrices; K s For controller parameters; for The second derivative, This is the output vector of the radial basis function after approximating the uncertainty of the dynamic model.

2. The fixed-time trajectory tracking control method for a robotic arm based on non-singular terminal sliding mode as described in claim 1, characterized in that, The process of constructing a radial basis function neural network based on the dynamic model and designing the weight update law of the weight vector of the radial basis function neural network according to the projection method includes: Based on the approximation principle, radial basis functions are constructed, i.e., radial basis neural networks; Based on the radial basis functions, determine the optimal approximation equation of the radial basis functions for the nonlinear function; A Gaussian function is selected as the activation function for the radial basis function neural network; Based on the optimal approximation equation and the activation function of the radial basis neural network, the output vector of the radial basis function after approximating the uncertainty of the dynamic model is determined; The approximation error of the radial basis neural network is defined based on the output vector after approximating the uncertainty of the dynamic model using the radial basis functions. The output vector of the dynamic model after the radial basis function approximates the uncertainty and the approximation error of the radial basis neural network are based on the radial basis function, and the weight update law of the weight vector of the radial basis neural network is determined by the projection method.

3. The fixed-time trajectory tracking and control method for a robotic arm based on non-singular terminal sliding mode as described in claim 1 or 2, characterized in that, The output vector of the radial basis function after approximating the uncertainty of the dynamic model is expressed as: in, It is a weight vector. It is the input to the neural network; Indicates the activation function; The approximation error of the radial basis neural network is expressed as: in, , Represents the optimal weight vector. This represents the error in the weight vector.

4. The fixed-time trajectory tracking control method for a robotic arm based on non-singular terminal sliding mode as described in claim 3, characterized in that, The weight update law for the weight vector of the radial basis function neural network is determined using the projection method, including: Design the update law for the weight vector of a radial basis neural network; The update law for the weight vector of the radial basis neural network is modified; The boundedness of the weight vector is ensured using the projection method, and the weight update law of the final radial basis function neural network weight vector is determined; the weight update law of the final radial basis function neural network weight vector is expressed as: in, , and It is bounded; It is the learning rate of the neural network. constant vector ,satisfy .

5. A terminal device, comprising a memory and a processor, wherein the memory stores a computer program executable on the processor, characterized in that, When the processor executes the computer program, it implements the fixed-time trajectory tracking control method for a robotic arm based on non-singular terminal sliding mode as described in any one of claims 1 to 4.

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