Adaptive event-triggered control method for single-link multi-manipulators with given performance

Abstract

The embodiment of this specification provides a single-link multi-manipulator adaptive event-triggered control method with given performance, which is characterized by comprising: modeling a multi-single-link manipulator system to obtain a single-link manipulator state equation with input dead zone and saturation constraints; designing the i-th single-link manipulator consistency tracking error, and designing a first virtual control law α i,1 and the adaptive law to design an adaptive event triggering mechanism; design a second virtual law α i,2 , adaptive laws, and simulation experiments were conducted using a Matlab experimental platform. To achieve the goal of convergence of the system to a given performance within a given time, the present invention combines the speed function and the performance function to obtain a speed-performance function. Through clever mixing and transformation, a transformation function and a barrier Lyapunov function are constructed, enabling the system to converge to a given performance within a given time while avoiding violations of state constraints.

Description

Technical Field

[0001] This document relates to the technical field of single-link multi-manipulator control methods, and in particular to an adaptive event-triggered control method for a single-link multi-manipulator with given performance. Background Technology

[0002] Against the backdrop of rapid scientific and technological development, multi-robotic arm collaborative working modes are widely used in various fields due to their operational flexibility, robustness, and ability to complete complex tasks. However, during various tasks, the robotic arm control system suffers from many uncertainties and unknown strong interference problems. Simultaneously, the system exhibits numerous nonlinear characteristics, posing significant challenges to the timeliness of the robotic arm system. Furthermore, existing research largely focuses on the system's convergence time, often neglecting system performance. However, rapid system convergence can degrade performance, leading to problems such as excessive overshoot and potentially even safety issues. To achieve rapid convergence, maintain performance, and compensate for nonlinearities, the control network bears a heavy communication burden, which can lead to communication delays or even packet loss when communication resources are limited. Considering these issues, a solution is urgently needed to address the problems of uncertain nonlinear multi-robotic arm systems with dead zones and saturation constraints. Summary of the Invention

[0003] This invention provides an adaptive event-triggered control method for a single-link multi-manipulator with given performance, aiming to solve the above-mentioned problems.

[0004] The adaptive event-triggered control method for a single-link multi-manipulator with given performance provided by this invention includes:

[0005] S1. Model the multi-link robotic arm system to obtain the state equation of the single-link robotic arm with input dead zone and saturation constraint;

[0006] S2. Design the consistency tracking error of the i-th single-link robotic arm, and design the first virtual control law α. i,1 And adaptive law ;

[0007] S3. Design an adaptive event triggering mechanism;

[0008] S4. Design the second virtual law α i,2 Adaptive Law and

[0009] S5. Simulation experiments were conducted based on the Matlab experimental platform.

[0010] The preset time adaptive event-triggered control method with given performance designed in this invention can effectively enable the system to converge and meet the given performance within a preset time, avoid violating state constraints, and effectively compensate for problems such as input dead zone and input saturation in multi-manipulator systems. Attached Figure Description

[0011] To more clearly illustrate the technical solutions in one or more embodiments of this specification or in the prior art, 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 recorded in this specification. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0012] Figure 1 This is a schematic diagram of an adaptive event-triggered control method for a single-link multi-manipulator with given performance according to an embodiment of the present invention;

[0013] Figure 2 This is a schematic diagram of the nonlinear input dead zone in an embodiment of the present invention;

[0014] Figure 3 This is a communication topology diagram of a virtual leader and four followers according to an embodiment of the present invention;

[0015] Figure 4 This is a schematic diagram of the reference signals and outputs of each robotic arm in an embodiment of the present invention;

[0016] Figure 5 This is a schematic diagram of the synchronization error curve in an embodiment of the present invention;

[0017] Figure 6 This is a schematic diagram of the dynamic error curve of an embodiment of the present invention;

[0018] Figure 7 This is a schematic diagram of the control signals for the robotic arm 1 according to an embodiment of the present invention;

[0019] Figure 8 This is a schematic diagram of the control signals for the robotic arm 2 according to an embodiment of the present invention;

[0020] Figure 9 This is a schematic diagram of the control signals for the robotic arm 3 according to an embodiment of the present invention;

[0021] Figure 10 This is a schematic diagram of the control signals for the robotic arm 4 according to an embodiment of the present invention;

[0022] Figure 11 This is a schematic diagram of the triggering event intervals for followers 1, 2, 3 and 4 in an embodiment of the present invention. Detailed Implementation

[0023] To enable those skilled in the art to better understand the technical solutions in one or more embodiments of this specification, the technical solutions in one or more embodiments of this specification will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this specification, and not all of the embodiments. Based on one or more embodiments of this specification, all other embodiments obtained by those skilled in the art without creative effort should fall within the protection scope of this document.

[0024] Method Implementation Examples

[0025] This invention provides an adaptive event-triggered control method for a single-link multi-manipulator with given performance. Figure 1 This is a schematic diagram of an adaptive event-triggered control method for a single-link multi-manipulator with given performance, according to an embodiment of the present invention. Figure 1 As shown, the adaptive event-triggered control method for a single-link multi-manipulator with given performance according to an embodiment of the present invention specifically includes:

[0026] S1. Model the multi-link robotic arm system to obtain the state equations of the single-link robotic arm with input dead zone and saturation constraints; Step S1 specifically includes:

[0027] A single-link robotic arm system with input dead zone and saturation constraints consists of a leader and M (M>2) followers, where the i-th (i=1,2…,M) robotic arm is modeled as follows:

[0028]

[0029] Among them, x i,1 and x i,2 J represents the angular velocity of the i-th robotic arm link. i =1 represents the moment of inertia, for f i (X)=(B i x i,2 +G i L I sin(x i,1 )), where B i =1 represents the coefficient of viscous friction, G i =9.8 represents the mass of the i-th link, l i =0.1 represents the length of the i-th link, and X = [x i,1 ,x i,2 ] T .

[0030] Figure 2Schematic diagram of the non - linear input dead zone for the embodiments of the present invention. The input dead zone model is expressed as follows:

[0031]

[0032] where, u i represents the input signal, represents the non - linear dead - zone input, S U >0 and S L <0 represent the system input saturation parameters, D r (u i ) and D l (u i ) are unknown linear functions, where r sr and r sl respectively represent the saturation breakpoints, r dr and r dl represent the dead - zone breakpoints, satisfying r sl <r dl <0<r dr <r sr .

[0033] Since has non - smooth non - linearity, thus can be approximated as:

[0034]

[0035] where, represents the approximation error, which satisfies The smoothing function κ i (u i ) can be defined as:<000022{8}>

[0036]

[0037] where, l r , l l , θ1 and θ2 are all constants greater than zero.

[0038] According to the Lagrange mean - value theorem, for the function with respect to holds, where represents the derivative of κ i (u h ), and l is an unknown constant satisfying 0 < l < 1.

[0039] Making Then Defining Then can be expressed as:

[0040]

[0041] For ease of description, we need to refer to relevant knowledge of algebraic graph theory, treating each robotic arm as a node. The information interaction in a unidirectional directed topology consisting of M (M>2) robotic arms is represented by a directed graph G=(V,E), where V={1,…,M} represents the multi-robotic arm node (1,…,M) and V… j ×V i ∈E represents an edge from robot arm node j to robot arm node i. A=[a ij ] M×M Let a represent the adjacency matrix between all robot arm nodes. If robot arm j can obtain information from robot arm i, then a ij >0, otherwise a ij =0. Define the Laplace matrix as L = DA, where the diagonal matrix D = diag[d1, d2, ..., d]. M ]∈R M ×M , For multi-arm robotic systems, an extended diagram can be used. To indicate, among which Let represent the leader 0 and multiple robotic arms (1,…,M), and

[0042] Definition 1. The synchronization error of the i-th robotic arm can be defined as:

[0043]

[0044] Where y0 represents the output signal of the virtual leader, b i Let b represent the information transfer coefficient between the leader and robotic arm i. If robotic arm i can obtain information from the leader, then b... i >0, otherwise b i =0.

[0045] For any definition in the set Ω∈R p functions on Both can be approximated and modeled using radial basis function neural networks, which are represented as follows:

[0046]

[0047] in Represents the input vector. This represents the approximation error, which has an upper bound. satisfy Denotes a basis function vector, where This can be represented by the Gaussian function as:

[0048]

[0049] Where q>1 represents the number of nodes in the Gaussian function. and π i This represents the center and width of the Gaussian function.

[0050] Optimal weight K * It can be represented as:

[0051]

[0052] Where K = [K1, K2, ... K q ] T Represents the weight vector.

[0053] To limit systematic errors within a given range, this invention introduces a class of transformation functions:

[0054] Z(t)=(Z 0 -Z ∞ )φ(t)+Z ∞ (10)

[0055] Among them, Z 0 >0 and Z ∞ >0 represents the initial and final values ​​of the function, and φ(t) represents a decay function satisfying φ(0)=1, and The systematic error z(t) satisfies |z(t)|≤Z(t).

[0056] To improve the system's convergence speed and reach the final state with given performance within a given time, a velocity function is introduced:

[0057]

[0058] Where T represents the preset time.

[0059] Combining (10) and (11), we can obtain the velocity performance function:

[0060] Z v (t)=(Z 0 -Z ∞ )φ(t)v(t)+Z ∞ (12)

[0061] This function satisfies Z. v (t)≤Z(t), where λ and Q satisfy λ≤Z 0 and λQ -1 ≤Z ∞ Construct the transformation function:

[0062]

[0063] S2. Design the consistency tracking error of the i-th single-link robotic arm, and design the first virtual control law α.i,1 And adaptive law Step S2 specifically includes:

[0064] First, using graph theory, we define the error variable for the i-th robotic arm:

[0065] z i,k =x i,k -α i,(k-1) ,k=2,…,m (14)

[0066] e i,k =g i z i,k k = 1, 2, ..., m (15)

[0067] Among them, z i,2 ,z i,3 ,…,z i,m Let α represent the error variable. i,1 ,α i,2 ,…,α i,m-1 This represents a virtual control law.

[0068] Radial basis function neural networks are used to process the unknown nonlinear components. It can be represented as

[0069]

[0070] Among them, X i,1 =[x i,1 ,x j,1 ,x j,2 ] T The upper bound of the error satisfies

[0071] Constructing the first virtual control law α i,1 And adaptive law as follows:

[0072]

[0073]

[0074] in, The design parameters satisfy γ i,1 >0 and δ i,1 >0.

[0075] S3. Design an adaptive event triggering mechanism; Step S3 specifically includes:

[0076] To reduce the pressure on the communication system and save communication resources, an adaptive event triggering mechanism was designed:

[0077]

[0078] Among them, the design parameters satisfy 0 < ζ i <1、σ i >0 and o i >0.

[0079] Define intermediate signals for:

[0080]

[0081] in, α i,2 This represents a virtual control law.

[0082] Because h i For unknown constants, use To estimate η i,2 , To represent the estimation error, then It can be represented as:

[0083]

[0084] From the above, we can conclude that:

[0085]

[0086] Among them, v i,1 and v i,2 Satisfy | v i,1 |≤1 and|v i,2 |≤1.

[0087] S4. Design the second virtual law α i,2 Adaptive Law and Step S4 specifically includes:

[0088] S041. Use radial basis function neural networks to handle the unknown parts. It can be represented as

[0089]

[0090] in, satisfy

[0091] S042. Construct the second virtual control law α i,2 Adaptive Law and

[0092]

[0093]

[0094]

[0095] in, The design parameters satisfy γ i,2 >0, δ i,2 >0 and ξ i,2 >0.

[0096] S5. Simulation experiments were conducted based on the Matlab experimental platform.

[0097] To verify the effectiveness of the proposed method, the algorithm was simulated using the Matlab experimental platform. Figure 3 A communication topology graph with one virtual leader and four followers is described. In this graph, robotic arms 1-4 and robotic arm 0 represent the four followers and the virtual leader, respectively. Assume the output signal (reference signal) of the virtual leader is y = sin(t). Based on the fundamental principles of graph theory, the adjacency matrix A and the in-degree matrix D can be obtained:

[0098]

[0099] The adaptive event-triggered cooperative controller designed in this invention is as follows:

[0100]

[0101]

[0102]

[0103]

[0104]

[0105]

[0106] The proposed control method was applied to a multi-robotic arm system, and the system parameters were set as follows: a preset time T = 1 was selected, and a given performance function Z was set. i,1 =(3.75-0.05)e 0.1t +0.05 and Z i ,2=(7.5-0.1)e 0.1t +0.1, transformation function parameter Q = 75, define the same state constraint value x. i,1 <2 and x i,2 <5, then λ i,1 =3.75 and λ i,2 =7.5, event triggering mechanism parameter set to ζ i =0.4,σ i =2.5,o i=0.3, the initial state of the system is set to x i,1 = [0.2, 0.3, -0.2, -0.3] T x i,2 = [0.1, 0.2, 0.3, 0.4] T γ i，1 =[24,28,14,20] T γ i，2 =[16,18,16,14] T δ i,j =[3,2] T ,β i,j =[0.01,0.01] T , ρ i,m =2.5.

[0107] Simulation results are as follows Figures 4-11 It can be seen that all signals are bounded. Figure 4 The reference signal and the output signals of each robotic arm are displayed, showing that the system achieves excellent consistent tracking performance within a given preset time. The synchronization error curve is shown below. Figure 5 As shown, the synchronization error enters the 5% error band after approximately 0.2 seconds, achieving the system's convergence target within the preset time of 1 second. The dynamic error curve is shown below. Figure 6 As shown, the synchronization error enters the 5% error band after approximately 0.8 seconds, achieving the system's convergence target within the preset time of 1 second. The control input signals for the four followers are as follows: Figures 7-10 As shown, the event triggering time intervals for each robotic arm are as follows: Figure 11 As shown in the figure. This control method can ensure rapid system convergence, save communication resources, and effectively compensate for nonlinear characteristics.

[0108] The following beneficial effects are achieved by employing the embodiments of the present invention:

[0109] 1. To achieve the goal of a system converging to a given performance within a given time, this invention combines a velocity function and a performance function to obtain a velocity-performance function. Through clever mixing and transformation, a transformation function and a potential barrier Lyapunov function are constructed, enabling the system to converge within a given time and satisfy the given performance, while avoiding violations of state constraints.

[0110] 2. To compensate for input dead zone and input saturation in multi-robot arms, this invention uses the Lagrange mean value theorem to rewrite the model into the form of linear input and approximate error, and constructs an adaptive event-triggered control method to achieve compensation for input dead zone and input saturation.

[0111] 3. To improve the applicability and flexibility of the proposed method, the present invention applies transformation functions and barrier Lyapunov functions at each step of the design process, so that all errors converge within a given time, making the proposed control method more applicable and flexible.

[0112] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; 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; and these 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. An adaptive event-triggered control method for a single-link multi-manipulator with given performance, characterized in that, include: S1. Model the multi-link robotic arm system to obtain the state equation of the single-link robotic arm with input dead zone and saturation constraint; S2, Design No. To ensure consistent tracking error of a single-link robotic arm, the first virtual control law was designed. And adaptive law ; S3. Design an adaptive event triggering mechanism; S4. Design the second virtual law Adaptive Law and ; S5. Simulation experiments were conducted based on the Matlab experimental platform. In step S2, the design of the first The specific tracking errors of a single-link robotic arm include: Definition of the first Error variables of a robotic arm: Formula 1: Formula 2: in, Represents the error variable. Represents a virtual control law; In step S2, the first virtual control law is designed. And adaptive law ; Radial basis function neural networks are used to process the unknown nonlinear components: Formula 3: in, The upper bound of the error satisfies , This represents the optimal weight vector of a radial basis function neural network. The radial basis function vector represents the state of the i-th robotic arm; Constructing the first virtual control law And adaptive law as follows: Formula 4: Formula 5: in, The design parameters meet and , Representing the leader and the robotic arm The information transfer coefficient between them , This represents the adjacency matrix between all robotic arm nodes. If the robotic arm Can from robotic arm If the information is obtained, then ,otherwise .

2. The method according to claim 1, characterized in that, Step S1 specifically includes: Equation 6 is used to analyze the first step of a single-link robotic arm system with input dead zone and saturation constraint. i Modeling a robotic arm; Formula 6: in, i =1,2,…, M , and Representing the The angular velocity of the robotic arm links, Represents the moment of inertia. Indicates the input signal. This indicates a nonlinear dead-zone input; ,in Represents the coefficient of viscous friction. Representing the The mass of each link. Representing the The length of each link, and .

3. The method according to claim 2, characterized in that, The dead zone input It is expressed as follows: Formula 7: in, and Indicates the system input saturation parameter. and For an unknown linear function, and These represent the saturation breakpoints, and Denotes dead zone breakpoints, where the saturation breakpoints and dead zone breakpoints satisfy... .

4. The method according to claim 3, characterized in that, The dead zone input is expressed as an approximation error and a smoothing function using Equation 8: Formula 8: in, It is represented as the approximation error, which satisfies , The smoothing function is defined as follows: Formula 9; in, , , and All are constants greater than zero; According to the Lagrange Mean Value Theorem, the function right Established, among which express The derivative of It is to satisfy The unknown constant; Make ,but ,definition ,but Represented as: Formula 10.

5. The method according to claim 1, characterized in that, The method further includes: treating each robotic arm as a node, and by... M The information interaction in a unidirectional directed topology composed of robotic arms is represented by a directed graph. This indicates, specifically, that includes: pass Represents multiple robotic arm nodes 1, 2, ... M , Indicates from the robotic arm node To the robotic arm node The edge, This represents the adjacency matrix between all robotic arm nodes. If the robotic arm Can from robotic arm If the information is obtained, then ,otherwise ; Define the Laplace matrix as Among them, the diagonal array , For multi-arm robotic systems, use extended diagrams. To indicate, among which Indicates the leader and multiple robotic arms ,and ; No. The synchronization error of a robotic arm is defined as: Formula 11; in, The output signal representing the virtual leader Representing the leader and the robotic arm The information transmission coefficient between them, if the robotic arm If information can be obtained from the leader, then ,otherwise ; For any definition in the set functions on Both can be approximated and modeled using radial basis function neural networks, which are represented as follows: Formula 12; in Represents the input vector. This represents the approximation error, which has an upper bound. satisfy , Denotes a basis function vector, where This can be represented by the Gaussian function as: Formula 13; in, This represents the number of nodes in the Gaussian function. and Indicates the center and width of the Gaussian function; Optimal weight Represented as: Formula 14; in, Represents the weight vector; By introducing a transformation function, the systematic error is limited to a given range: Formula 15; in, and This represents the initial and final values ​​of the function. Let a decay function satisfy... ,and System error satisfy ; By introducing a velocity function v ( t The number of parameters that allows the system to reach a final state with a given performance within a given time period is: Formula 16; in, Indicates the preset time; Obtain the speed performance function using formulas 17 and 18: Formula 17; The speed performance function satisfies ,in, and satisfy and ; Construct the transformation function: Official 18.

6. The method according to claim 1, characterized in that, Step S3 specifically includes: To reduce the pressure on the communication system and save communication resources, an adaptive event triggering mechanism was designed: Formula 19; Among them, the design parameters satisfy , and ; Define intermediate signals for: Formula 20; in, , Represents a virtual control law; For unknown constants, use To estimate , To represent the estimation error, then Represented as: Formula 21; From the above, we can conclude that: Official formula 22; in, and satisfy and .

7. The method according to claim 1, characterized in that, Step S4 specifically includes Radial basis function neural networks are used to handle the unknown parts. , represented as Formula 23; in, , satisfy ; Constructing the second virtual control law Adaptive Law and : Formula 24; Formula 25; Formula 26; in, , The design parameters meet , and .

8. The method according to claim 1, characterized in that, Step S5 specifically includes: According to the design of the first Consistent tracking error of a single-link robotic arm, and the first virtual control law. And adaptive law Design an adaptive event triggering mechanism and the second virtual law. Adaptive Law and The system is confirmed to converge within a finite time.

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