Multi-robot preset performance event-triggered pre-defined time control method
By designing a fixed-time performance function and a predefined time command filter, combined with a dynamic event triggering mechanism and an RBF neural network, the problem of control performance degradation of multi-robotic arm systems under random disturbances was solved. Efficient consistency tracking and error convergence within a predefined time were achieved, simplifying controller design.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- QINGDAO UNIV OF TECH
- Filing Date
- 2026-04-03
- Publication Date
- 2026-06-09
AI Technical Summary
Existing multi-manipulator systems are prone to performance degradation under random disturbances, and consistency errors are difficult to converge within a predetermined time. Traditional methods have strict constraints on initial errors, making them unsuitable for unmeasurable or non-strict feedback systems, and they also suffer from controller structure complexity and complexity explosion problems.
A fixed-time performance function is designed, and an error transformation is performed by combining it with a barrier function. A predefined time command filter and a dynamic event triggering mechanism are introduced. An RBF neural network is used for online estimation. An adaptive backstepping method and a virtual control signal are designed to achieve consistent tracking of the multi-robotic arm system within a predefined time.
It achieves consistent tracking of multi-robotic arm systems within a preset time, ensuring that the tracking error is within the preset performance boundary, reducing the controller update frequency and communication resource consumption, simplifying controller design, and avoiding complexity explosion.
Smart Images
Figure CN122165410A_ABST
Abstract
Description
Technical Field
[0001] This application belongs to the field of robot control technology, specifically relating to a method for controlling the predefined time of preset performance events triggered by multiple robotic arms. Background Technology
[0002] Multi-arm robotic systems have attracted significant attention from the academic community in recent years due to their important applications in formation control, sensor networks, and underwater robotics. Among these applications, the cooperative control of multi-arm robotic systems has become a research hotspot, with the leader-follower consistency problem considered one of the fundamental and core issues in this area. This problem has now become an important research topic in control theory. In multi-arm robotic systems, the single-link robotic arm is typically considered as a follower. The main task is to design a suitable controller that enables the follower to accurately track the pre-specified leader's trajectory while ensuring that the tracking process remains within a preset performance boundary. However, existing methods often suffer from performance degradation in the presence of random perturbations, and the consistency error is difficult to converge to a small neighborhood within a predetermined time.
[0003] Existing technical solutions mainly combine preset performance control, command filtering backstepping, predefined time control, and event triggering mechanisms for consistent tracking control of nonlinear multi-agent systems. However, they generally suffer from the following shortcomings: Traditional preset performance methods impose strict constraints on initial errors, limiting the flexibility of application; while predefined time control can ensure that the upper bound of convergence time can be preset, it usually only provides a conservative estimate, and the controller structure is complex, making it difficult to effectively integrate with the backstepping framework; command filters lack convergence time guarantees, and the decay rate of filtering errors is unpredictable, which may affect the final tracking performance; the threshold of static event triggering mechanisms is fixed, making it difficult to achieve a balance between transient response and steady-state energy saving; in addition, most solutions rely on the measurability of all states and are mainly designed for strict feedback system structures, making it difficult to directly extend to real-world systems where states are unmeasurable or non-strict feedback. Furthermore, the backstepping design still requires analytical differentiation of the virtual control quantity, resulting in a "complexity explosion" problem. To address the aforementioned limitations, this paper proposes a fixed-time performance function to eliminate dependence on initial conditions, designs a command filter based on predefined time stability theory to ensure that filtering errors converge within a precisely set time, introduces a dynamic event triggering mechanism to further reduce communication overhead, and achieves effective control of non-strict feedback systems by using neural network output feedback and error compensation mechanisms, while avoiding complexity explosion. Summary of the Invention
[0004] This invention provides a predefined time control method for multi-robotic arm systems based on preset performance events. By designing a fixed-time performance function, constructing a predefined time command filter, and introducing a dynamic event triggering mechanism, it achieves consistent tracking of the multi-robotic arm system within a preset time, ensuring that the tracking error always meets the preset performance constraints, the filtering error converges within a precisely set time, and the controller update frequency is effectively reduced. The technical solution is as follows: A method for controlling predefined time triggering preset performance events in multi-robotic arms includes the following steps: S1. Convert the dynamic equations of the multi-robotic arm system into state equations, describe the communication topology between the multi-robotic arms using graph theory, and calculate the consistency error of the multi-robotic arm system. S2. Design a fixed-time preset performance function so that the initial error is no longer limited by the initial value constraint, and transform the error through the barrier function; S3. Utilize the approximation properties of RBF neural networks to perform online estimation of unknown functions in the dynamic equations of a multi-manipulator system; S4. Based on the adaptive backstepping method, design the required predefined time virtual control signal, adaptive rate, event triggering mechanism, and actual control feedback to the multi-robotic arm system for position state adjustment until the state of the multi-robotic arm system meets the preset performance constraints and achieves consistent tracking within the predefined time.
[0005] Preferably, in step S1, when describing the communication topology between the multi-robotic arm system, each follower robotic arm is defined as a node, and the weighted directed graph is represented as follows: ; in For a set of nodes, Let be a set of directed edges. For the leader's adjacency matrix; The formula for calculating consistency error is as follows: ; ; in For the first Each follower outputs, For the leader's output, For the leader's adjacency matrix elements; It is a weighted directed graph adjacency matrix The elements in the table are used to describe the communication topology between follower robotic arms; if the follower Able to follow Receive information, then ;otherwise Self-looping is not allowed. ; Define the adjacency matrix of the leader as If followers Being able to receive information from leaders ;otherwise, .
[0006] Preferably, step S2 removes the initial condition constraints: by constructing Makes the method applicable to arbitrary initial errors; fixed-time convergence: performance boundary within a preset time. The contraction is completed within a specified time; the performance can be preset: both transient and steady-state performance can be quantitatively controlled through design parameters.
[0007] A continuously differentiable function is defined if the following conditions are met. This is referred to as an increasing fixed-time performance scalar function. (1) It is a monotonically increasing function, which from Increase to ; (2) ,and for ; in It is a positive design constant. It is a set time independent of the initial conditions; Fixed-time performance scalar function Designed as follows: ; in It is a design constant; Based on scalar functions, the fixed-time preset performance function is constructed as follows: ; in It is a design constant. It is the initial value of the performance function.
[0008] Preferably, to ensure that the consistency error meets the preset performance, that is, for any , Construct barrier function as follows: ; ; From the fixed-time performance scalar function and the barrier function, we can obtain ,and For any initial error value They all Therefore, the pre-set performance design is independent of the initial conditions; if exist If time is bounded, then we can conclude that right Established, which indicates It can meet the preset performance requirements, with transient performance being as follows: Timely satisfaction The steady-state performance is characterized by... Timely satisfaction ; right Differentiation yields: ; ; ; These are the coefficients before the state derivative. Additional terms caused by changes in the performance function.
[0009] Preferably, in step S3, the radial basis function neural network can approximate any continuous nonlinear function. , making ; For the input vector of RBFNN, For weight vectors, This represents the weight of the m-th node; As basis functions, The number of neurons is defined in Continuous functions Ideal weights The following approximation is performed: ; To approximate the error, Given a known upper bound on the error, the ideal weight vector is defined as: ; s represents the number of hidden layer nodes in the neural network.
[0010] Preferably, the predefined time command filter in step S3 is as follows: filter error The definition is as follows: ; in For the virtual control signal to be designed, This indicates the output signal of the command filter; To ensure that synchronization errors and filtering errors converge within a predefined time, the predefined time command filter is designed as follows: ; , ,in This represents the order of the system (the state dimension of each follower). The system in this article is of order 2.
[0011] A preset time is set to ensure that the error converges within this period; coefficient of positive power term, Negative power coefficient, Convergence rate adjustment parameters For filter input; , , , This is the power term of the filtering error variable; Design constants; Preferably, in order to reduce filtering error The compensation mechanism for the impact on control performance is designed as follows: ; Setting parameters , ; , It is a compensation signal. ; is the total in-degree weight of the follower robotic arm; Auxiliary tracking error Defined as: ; Compensation signal; The converted tracking error The derivative is ; For auxiliary tracking error Taking the time derivative, we get: ; Preferably, in step S4, the event triggering conditions are as follows: The event triggering mechanism includes the following triggering conditions: ; In the formula, express Update time, It is the maximum lower bound function. express Update time, Indicates the first The measurement error of each follower Indicates the first Design parameters related to the trigger threshold of each follower. Indicates the first A controller for a follower; ; ; ; ; In the formula, yes The state of being maintained Represents a positive number parameter. yes time Update when the trigger condition is met. .
[0012] Preferably, in step S4, the predetermined time virtual control signal for the position layer of the multi-robotic arm system is: ; in ,in Represents the order of the system. It is an online estimate of the squared norm of the ideal weight vector of the position layer neural network of the nth follower multi-arm system. yes The derivative, These are design parameters related to the preset time. It is an adaptive gain parameter. It is the auxiliary tracking error. It is the basis function vector of the position-related RBF of the multi-arm system with the nth follower. These are design parameters; ; In the formula, The control gain is known. Representing the The speed layer virtual control signal of a multi-arm robotic system with a follower It is an online estimate of the norm squared of the ideal weight vector of the velocity layer neural network of the nth follower multi-arm system. yes The derivative, It is the first The basis function vectors of the velocity-dependent RBF neural network for a multi-arm robotic system with a follower.
[0013] Preferably, the position layer adaptation rate of the multi-robotic arm system is: ; ; in , , , respectively, are positive coefficients, , It is a positive function; and for any If the initial conditions are satisfied , A is a lemma function; therefore, we obtain ,in It is the squared ideal weight norm of the neural network at the i-th follower position layer. Online estimates; the velocity layer adaptive rate of the multi-manipulator system is: ; according to Similarly, we can obtain .
[0014] Compared with the prior art, the beneficial effects of this application are as follows: 1. This application utilizes a fixed-time performance function to design a preset performance boundary and combines it with a barrier function to perform error transformation, so that the consistency error of the multi-robotic arm system is no longer limited by the initial condition constraints, thus improving the applicability of the method.
[0015] 2. This application designs a predefined time command filter to ensure that the filtering error converges within a precisely set time, avoiding the problem of uncontrollable convergence speed of traditional filters and effectively improving control accuracy.
[0016] 3. This application introduces a dynamic event triggering mechanism, which adaptively adjusts the triggering threshold, significantly reducing the update frequency of the controller while ensuring control performance and reducing communication resource consumption.
[0017] 4. This application uses a predefined time command filter to replace the analytical differentiation of the virtual control signal in the traditional backstepping method, which avoids the "complexity explosion" problem and simplifies the controller design. Attached Figure Description
[0018] Figure 1 This is a flowchart of the technology of the present invention; Figure 2 Weight matrix of a multi-arm robotic system; Figure 3 Output tracking trajectory diagram of a follower in a multi-robotic arm system; Figure 4 Performance diagram of output trajectory tracking error of a follower in a multi-robotic arm system; Figure 5 Follower controller in a multi-arm robotic system ; Figure 6 This is a diagram showing the event trigger intervals for follower 1 in a multi-robotic arm system. Figure 7 The event trigger interval diagram for follower 2 in a multi-robotic arm system; Figure 8 The event trigger interval diagram for follower 3 in a multi-robotic arm system; Figure 9 Filtering error diagram of the follower in a multi-robotic arm system; Figure 10 A graph showing the adaptive parameter changes of the first step of a follower in a multi-arm robotic system; Figure 11 This is a graph showing the second-step adaptive parameter changes for the follower in a multi-arm robotic system. Detailed Implementation
[0019] 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, and not all embodiments. The present invention is described using the detection of forged value-added tax invoices as a specific implementation scenario.
[0020] This invention relates to a control method for multi-robotic arm systems, and discloses a predefined time control method for multi-robotic arm systems triggered by preset performance events. The method includes the following steps: converting the dynamic equations of the multi-robotic arm system into state equations; calculating the consistency error of the multi-robotic arm system; setting the desired convergence time; determining a fixed-time performance function; performing error transformation through a barrier function to ensure that the initial error is no longer constrained by the initial value constraints in traditional methods; using an RBF neural network to estimate the unknown functions in the system dynamic equations online; designing a predefined time command filter; setting the desired filtering convergence time; and designing a filtering compensation signal to eliminate the impact of filtering errors on control performance; detecting event triggering conditions; when the event triggering conditions are met, updating the controller input signal of the multi-robotic arm system and transmitting it to its own and neighboring robot arm controllers; and jointly using the predefined time virtual control signal, adaptive rate, and event-triggered control law to feed back to the multi-robotic arm system for position and state adjustment until the state of the multi-robotic arm system meets the preset performance constraints and achieves consistent tracking within the predefined time. The method disclosed in this invention enables multi-robotic arm systems to achieve consistent tracking within a predefined time, effectively reduces communication resource consumption, and ensures that the tracking error remains within the preset performance boundaries.
[0021] A method for controlling predefined time triggering preset performance events in multi-robotic arms includes the following steps: S1. Establish a multi-robotic arm system model, convert the dynamic equations of the multi-robotic arm system into state equations, describe the communication topology between the multi-robotic arms through graph theory, and calculate the consistency error of the multi-robotic arm system.
[0022] The dynamic equations of the multi-manipulator system are as follows: The dynamic characteristics of each follower robotic arm can be described by the following second-order differential equation: ; in, Representing the The angular position of each robotic arm Let its angular velocity be denoted as ω. For ease of controller design, define the state variables. , The system model can then be transformed into a state-space representation: ; in, Given the known control gain of the system, , These represent the position and velocity of the multi-arm robotic system, respectively. For controller input, and These represent the nonlinear terms in the system dynamics. In the robotic arm model studied in this paper... ,and The expression is , Indicates the output of a multi-robotic arm system. The length of the link. It is the acceleration due to gravity. For viscous friction damping during joint or movement processes. This indicates the mass of the robotic arm link. It is the moment of inertia of the robotic arm about its axis of rotation.
[0023] In Information transfer among followers is defined as a weighted directed graph. ,in It is a set of nodes, with each follower as a node; It is a set of directed edges formed by connecting nodes; It is a weighted adjacency matrix. If the followers Able to follow Receive information, then ;otherwise Self-looping edges are not allowed. Definition diagram The Laplace matrix is ,in When a directed graph has a spanning tree, it means that there exists a special node in the graph (called the root node), and there is a directed path from the root node to all other nodes in the graph. The adjacency matrix of the leader is defined as... If followers Being able to receive information from leaders ;otherwise, .
[0024] The communication topology of a multi-arm robotic system is represented by a weighted directed graph. It means that among them For a set of nodes, Let be a set of directed edges. The weighted adjacency matrix is used; the formula for calculating the consistency error is as follows: ; in For the first Each follower outputs, For the leader's output, The elements in the leader's adjacency matrix.
[0025] S2. Design a fixed-time preset performance function and propose a piecewise performance function so that the initial error is no longer limited by the initial value constraints in traditional methods. Error transformation is performed through a barrier function.
[0026] Definition 1: A continuously differentiable function is defined if the following conditions are met. This is referred to as an increasing fixed-time performance scalar function. (1) It is a monotonically increasing function, which from Increase to ; (2) ,and for ; in It is a positive design constant. It is a set time that is independent of the initial conditions.
[0027] According to Definition 1, a typical fixed-time performance scalar function can be designed as follows: ; in It is a design constant.
[0028] Based on the scalar function described above, the fixed-time preset performance function is constructed as follows: ; in It is a design constant. It is the initial value of the performance function.
[0029] To achieve consensus tracking control in a multi-agent system, the consensus error is defined as: ; in Indicates follower and followers The edge weights between them. If the follower Able to receive from followers The information, ;otherwise, Self-looping edges are not allowed. . Represents leaders and followers The edge weights between them. Similarly, if followers can receive information from the leader, then ;otherwise, .
[0030] To ensure that the consistency error meets the preset performance, that is, for any The barrier function is constructed as follows: ; ; From the fixed-time performance scalar function and the barrier function, we can obtain ,and This means that for any initial error value They all Therefore, the performance design in this study is presupposed to be independent of the initial conditions. Furthermore, if exist If time is bounded, then we can conclude that right Established, which indicates It can meet the preset performance requirements, with transient performance being as follows: Timely satisfaction The steady-state performance is characterized by... Timely satisfaction The boundedness problem will be solved with the help of subsequent controller design.
[0031] right Differentiation yields ; in .
[0032] S3. Utilizing the approximation characteristics of the RBF neural network, online estimation of the unknown functions in the dynamic equations of the multi-manipulator system is performed. A predefined time command filter is designed, the desired filter convergence time is set, and a filter compensation signal is designed to eliminate the impact of filter errors on control performance.
[0033] Radial basis function neural networks (RBFNNs) can approximate any continuous nonlinear function. , making , For the input vector of RBFNN, For weight vectors, As basis functions, The number of neurons is defined in Continuous functions Ideal weights The following approximation is performed: , To approximate the error, Given the upper bound of the error, the ideal weight vector Defined as: .
[0034] The predefined time command filter is defined as follows: To ensure that the synchronization error and filtering error converge within the predefined time, and to minimize the tracking error, the filtering error is defined as follows: ; in The virtual control signal to be designed.
[0035] To ensure that synchronization errors and filtering errors converge within a predefined time, the predefined time command filter is designed as follows: ; The design parameters are as follows: , ,in This represents the order of the system (the dimension of each follower's state). The system in this article is of order 2. A preset time to ensure that the error converges within this period.
[0036] To reduce filtering error The compensation mechanism for the impact on control performance is designed as follows: ; Furthermore, the auxiliary tracking error is defined as: ; The converted tracking error The derivative is: ; For auxiliary tracking error Taking the time derivative, we get: .
[0037] S4. Based on the adaptive backstepping method, design the required predefined time virtual control signal, adaptive rate, event triggering mechanism, and actual control feedback to the multi-robotic arm system for position state adjustment until the state of the multi-robotic arm system meets the preset performance constraints and achieves consistent tracking within the predefined time.
[0038] The event triggering conditions are as follows: The event triggering mechanism includes the following triggering conditions: ; In the formula, express Update time, It is the maximum lower bound function. express Update time, Indicates the first The measurement error of each follower Indicates the first Design parameters related to the trigger threshold of each follower. Indicates the first A controller for a follower; ; ; ; ; In the formula, yes The state of being maintained Represents a positive number parameter. yes time Update when the trigger condition is met. .
[0039] The predefined time first step of the multi-robotic arm system's position layer time adaptation rate is: ; Consider the following differential equation, ,in , , , respectively, are positive coefficients, , It is a positive function; and for any If the initial conditions are satisfied , Therefore, we obtain ,in It is the squared ideal weight norm of the neural network at the i-th follower position layer. The online estimate.
[0040] The virtual control signal for the position layer of the multi-robotic arm system at the predetermined time is: ; in ,in This represents the order of the system (the dimension of each follower's state). The system in this article is of order 2. It is an online estimate of the squared norm of the ideal weight vector of the position layer neural network of the nth follower multi-arm system. yes The derivative, These are design parameters related to the preset time. It is an adaptive gain parameter. It is the auxiliary tracking error. It is the basis function vector of the position-related RBF of the multi-arm system with the nth follower. These are design parameters.
[0041] The speed layer adaptive rate of the second-step multi-robotic arm system is: ; according to Similarly, we can obtain .
[0042] The virtual control signal for the speed layer of the multi-arm robotic system is: ;
[0043] In the formula, Representing the The speed layer virtual control signal of a multi-arm robotic system with a follower It is an online estimate of the norm squared of the ideal weight vector of the velocity layer neural network of the nth follower multi-arm system. yes The derivative, It is the first The basis function vectors of the velocity-dependent RBF neural network for a multi-arm robotic system with a follower.
[0044] S5. The proposed control method is analyzed and verified by stability analysis and simulation based on Lyapunov stability theory.
[0045] To verify the effectiveness of the event-triggered predefined time command filtering control method for the multi-robotic arm system based on preset performance provided in this embodiment, MATLAB simulation experiments were conducted, and detailed descriptions are provided with reference to the accompanying drawings.
[0046] The technical process of this invention is as follows: Figure 1 As shown, a multi-agent robotic arm system is considered, consisting of one leader and three followers, each of whom is a single-link robotic arm. The dynamic equations of the multi-agent robotic arm system are obtained. Graph theory is introduced to describe the communication topology between the agents, and a consensus error system is constructed. For the unknown nonlinear functions in the system's dynamic equations, a radial basis function neural network is used for online approximation. A pre-defined performance control technique is introduced, using a performance function... The convergence process of constrained consistency error is described. An error transformation method converts the constrained error system into an unconstrained system, ensuring that the synchronization error remains within a preset boundary. To avoid the "complexity explosion" problem caused by repeated differentiation of the virtual control law in traditional backstepping methods, a predefined time filter is designed. Based on the adaptive backstepping method, virtual control signals, event triggering mechanisms, actual control, and adaptive update rates are designed. Stability analysis and simulation verification of the proposed control method are performed using Lyapunov stability theory. Figure 2 The weight matrix A of the multi-arm robotic system is: ;
[0047] In the simulation experiment, the parameters of the selected single-link robotic arm model are as follows: , , , , .
[0048] Select the initial state of the system The rest of the initial states are 0.
[0049] The key signals for leaders are: .
[0050] The design parameters are: , , , , , , , , , , , , , , , , .
[0051] from Figure 3 It can be seen that the output trajectory of the follower in the multi-arm robotic system can track the reference signal of the leader quite well. Figure 4 It can be seen that the consensus error of the multi-robotic arm system is always within the preset performance line and can converge to a small neighborhood within a predefined time of 1.5s. Figure 5 The demonstration shows the controller for the follower in a multi-arm robotic system. The follower's control input is only allowed when the event triggering conditions are met. It will then be updated. Figure 6-8 The event triggering intervals of the four followers in the multi-arm robotic system were demonstrated. Figure 9 The filtering error is displayed. Figure 10 and Figure 11 The convergence of the two adaptive parameters is shown.
[0052] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention 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 or all of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.
Claims
1. A method for controlling predefined time triggering preset performance events in multi-robotic arms, characterized in that, Includes the following steps: S1. Convert the dynamic equations of the multi-robotic arm system into state equations, describe the communication topology between the multi-robotic arms using graph theory, and calculate the consistency error of the multi-robotic arm system. S2. Design a fixed-time preset performance function so that the initial error is no longer limited by the initial value constraint, and transform the error through the barrier function; S3. Utilize the approximation properties of RBF neural networks to perform online estimation of unknown functions in the dynamic equations of a multi-manipulator system; S4. Based on the adaptive backstepping method, design the required predefined time virtual control signal, adaptive rate, event triggering mechanism, and actual control feedback to the multi-robotic arm system for position state adjustment until the state of the multi-robotic arm system meets the preset performance constraints and achieves consistent tracking within the predefined time.
2. The multi-robotic arm preset performance event triggering predefined time control method according to claim 1, characterized in that, In step S1, when describing the communication topology between the multi-robotic arm system, each follower robotic arm is defined as a node, and the weighted directed graph is represented as follows: ; in For a set of nodes, Let be a set of directed edges. For the leader's adjacency matrix; The formula for calculating consistency error is as follows: ； ； in For the first Each follower outputs, For the leader's output, For the leader's adjacency matrix elements; It is a weighted directed graph adjacency matrix The elements in the table are used to describe the communication topology between follower robotic arms; if the follower Able to follow Receive information, then ;otherwise Self-looping is not allowed. ; Define the adjacency matrix of the leader as If followers Being able to receive information from leaders ; otherwise, .
3. The multi-robotic arm preset performance event triggering predefined time control method according to claim 1, characterized in that, Step S2: Remove initial condition constraints: by constructing Makes the method applicable to arbitrary initial errors; fixed-time convergence: performance boundary within a preset time. Contraction is completed within a specified timeframe; performance is preset: both transient and steady-state performance can be quantitatively controlled through design parameters. A continuously differentiable function is defined if the following conditions are met. This is referred to as an increasing fixed-time performance scalar function. (1) It is a monotonically increasing function, which from Increase to ; (2) ,and for ; in It is a positive design constant. It is a set time independent of the initial conditions; Fixed-time performance scalar function Designed as follows: ； in It is a design constant; Based on scalar functions, the fixed-time preset performance function is constructed as follows: ； in It is a design constant. It is the initial value of the performance function.
4. The multi-robotic arm preset performance event triggering predefined time control method according to claim 3, characterized in that, To ensure that the consistency error meets the preset performance, that is, for any , Construct barrier function as follows: ； ； From the fixed-time performance scalar function and the barrier function, we can obtain ,and For any initial error value They all Therefore, the pre-set performance design is independent of the initial conditions; if exist If time is bounded, then we can conclude that right Established, which indicates It can meet the preset performance requirements, with transient performance being as follows: Timely satisfaction The steady-state performance is characterized by... Timely satisfaction .
5. The multi-robotic arm preset performance event triggering predefined time control method according to claim 1, characterized in that, In step S3, the radial basis function neural network can approximate any continuous nonlinear function. , making ; For the input vector of RBFNN, For weight vectors, This represents the weight of the m-th node; As basis functions, The number of neurons is defined in Continuous functions Ideal weights The following approximation is performed: ； To approximate the error, Given a known upper bound on the error, the ideal weight vector is defined as: ； s represents the number of hidden layer nodes in the neural network.
6. The multi-robotic arm preset performance event triggering predefined time control method according to claim 5, characterized in that, Step S3 predefined time command filter is as follows: filter error The definition is as follows: ； in For the virtual control signal to be designed, This indicates the output signal of the command filter; To ensure that synchronization errors and filtering errors converge within a predefined time, the predefined time command filter is designed as follows: ； , ,in Represents the order of the system; A preset time is set to ensure that the error converges within this period; coefficient of positive power term, Negative power coefficient, Convergence rate adjustment parameters For filter input; , , , This is the power term of the filtering error variable; Design constants.
7. The multi-robotic arm preset performance event triggering predefined time control method according to claim 6, characterized in that, To reduce filtering error The compensation mechanism for the impact on control performance is designed as follows: ； Setting parameters , ; , It is a compensation signal. ; is the total in-degree weight of the follower robotic arm; Auxiliary tracking error Defined as: ； Compensation signal; The converted tracking error The derivative is ； For auxiliary tracking error Taking the time derivative, we get: ； These are the coefficients before the state derivative. Additional terms caused by changes in the performance function.
8. The multi-robotic arm preset performance event triggering predefined time control method according to claim 1, characterized in that, In step S4, the event triggering conditions are as follows: The event triggering mechanism includes the following triggering conditions: ； In the formula, express Update time, It is the maximum lower bound function. express Update time, Indicates the first The measurement error of each follower Indicates the first Design parameters related to the trigger threshold of each follower. Indicates the first A controller for a follower; ； ； ； ； In the formula, yes The state of being maintained Represents a positive number parameter. yes time Update when the trigger condition is met. .
9. The multi-robotic arm preset performance event triggering predefined time control method according to claim 8, characterized in that, In step S4, the virtual control signal for the predetermined time of the position layer of the multi-robotic arm system is: ； in ,in Represents the order of the system. It is an online estimate of the norm squared of the ideal weight vector of the position layer neural network of the nth follower multi-arm system. yes The derivative, These are design parameters related to the preset time. It is an adaptive gain parameter. It is the auxiliary tracking error. It is the basis function vector of the position-related RBF of the multi-arm system with the nth follower. These are design parameters; the virtual control signal for the predetermined time of the speed layer of the multi-robotic arm system is: ； In the formula, The control gain is known. Representing the The speed layer virtual control signal of a multi-arm robotic system with a follower It is an online estimate of the squared norm of the ideal weight vector of the velocity layer neural network of the nth follower multi-manipulator system. yes The derivative, It is the first The basis function vectors of the velocity-dependent RBF neural network for a multi-arm robotic system with a follower.
10. The multi-robotic arm preset performance event triggering predefined time control method according to claim 9, characterized in that, The position layer adaptation rate of the multi-arm robotic system is: ； ； in , , , respectively, are positive coefficients, , It is a positive function; and for any If the initial conditions are satisfied , A is a lemma function; therefore, we obtain ,in It is the squared ideal weight norm of the neural network at the i-th follower position layer. Online estimates; the velocity layer adaptive rate of the multi-manipulator system is: ； according to Similarly, we can obtain .