Event-triggered multi-robot system random pre-defined time control method

Abstract

The application discloses an event-triggered multi-robot arm system random pre-defined time control method, belongs to the technical field of control, and is used for time control of a multi-robot arm system. The method comprises a dynamic equation of the multi-robot arm system, introduces a trajectory error tracking system, and obtains consensus error of the multi-robot arm system. Unknown functions in the dynamic equation of the multi-robot arm system are estimated, a virtual control signal, an adaptive rate and an event-triggered mechanism are designed according to an adaptive backstepping method, and stability analysis is performed in combination with Lyapunov stability theory. The application simultaneously adopts the event-triggered mechanism, reduces the update frequency of the controller, and in combination with experiments, the consensus error of the multi-robot arm system can converge into a small neighborhood within a pre-defined time of 1s.

Description

Technical Field

[0001] This invention discloses a random predefined time control method for a multi-robotic arm system based on event triggering, belonging to the field of control technology. Background Technology

[0002] Multi-arm robotic systems have attracted widespread attention from researchers due to their broad applications in formation control, sensor networks, and underwater robotics. Cooperative control of multi-arm robotic systems has become a key research focus, with the leader-follower consensus problem widely recognized as a fundamental and crucial issue in this field. This problem has become a research focus in control theory, treating the single-link robotic arm in a multi-arm system as a follower, with the main goal of designing a controller that enables the follower to accurately track the trajectory of the designated leader. Current technologies for multi-arm robotic systems suffer from performance degradation when considering random disturbances, and the consensus error failing to converge to a small neighborhood within a predetermined time. Summary of the Invention

[0003] The purpose of this invention is to provide a random predefined time control method for multi-robotic arm systems based on event triggering, so as to solve the problem in the prior art that the consensus error of multi-robotic arm systems cannot converge to a small neighborhood within a predetermined time.

[0004] Event-triggered multi-robotic arm systems employ random predefined time control methods, including:

[0005] S1, the dynamic equations of the multi-manipulator system;

[0006] S2. Introduce a trajectory error tracking system to obtain the consensus error of the multi-robotic arm system;

[0007] S3. Estimate the unknown functions in the dynamic equations of the multi-manipulator system;

[0008] S4. Based on the adaptive backstepping method, design the virtual control signal, adaptive rate, and event triggering mechanism;

[0009] S5. Perform stability analysis using Lyapunov stability theory.

[0010] S1 includes:

[0011] ;

[0012] ;

[0013] ;

[0014] ;

[0015] ;

[0016] In the formula, The differential symbol, For the first The position of a follower multi-arm robotic system For the first The speed of a multi-arm robotic system with a follower For time, yes The diffusion term function, For a standard one-dimensional Wiener process, For the first A follower's controller input, For the first The output of a multi-arm robotic system with a follower For the first The drift term function of each follower for The diffusion term function, For the first The state of a multi-arm robotic system with a follower. For the first The rotational inertia of the servo motor of the follower Representing the The mass of the link of the follower. Represents gravitational acceleration. Representing the The length of the link in the follower's link Indicates the first Damping coefficient of a follower.

[0017] S2 includes:

[0018] ;

[0019] In the formula, , It is the first Consensus error of a multi-arm robotic system with multiple followers , Corresponding to , consensus error, yes The maximum value, Representing the Virtual control signals for the position of a multi-arm robotic system with a follower Reference signals representing leaders, Indicates the first One follower This represents the parameters of a multi-arm robotic system, including position and velocity. For the first The position of a multi-arm robotic system with a follower;

[0020] if Receive from Information, variables ,otherwise ;

[0021] if Receive information from the leader, variables ,otherwise .

[0022] S3 includes using the approximation properties of fuzzy logic systems to estimate unknown functions:

[0023] ;

[0024] ;

[0025] ;

[0026] ;

[0027] ;

[0028] ;

[0029] ;

[0030] In the formula, It is an unknown function. It is the independent variable of the unknown function. It is a fuzzy logic system. It is an estimation error. It is a positive number. It is a weighted vector. It is the weighted vector of the th Subvectors, It is the number of fuzzy rules. It is a basis function vector. It is the first basis function vector Subvectors, It is a Gaussian function. It is a natural constant. It is the first The center vector of each follower yes The One element, It is the width of the Gaussian function.

[0031] The position adaptation rate estimate for the multi-robotic arm system is:

[0032] ;

[0033] The virtual control signal for the position of the multi-arm robotic system is:

[0034] ;

[0035] ;

[0036] ;

[0037] ;

[0038] ;

[0039] ;

[0040] In the formula, , For the parameters that can be determined, It is the first Position adaptation rate estimation of a multi-arm robotic system with one follower yes The derivative, , , These are three parameters with known ranges. It is the first Design parameters for the position of a multi-arm robotic system with a follower. It is the first The basis function vectors of the position of a multi-arm robotic system with followers.

[0041] The adaptive rate estimate for the speed of the multi-robotic arm system is:

[0042] ;

[0043] The virtual control signal for the speed of the multi-arm robotic system is:

[0044] ;

[0045] In the formula, Representing the Virtual control signals for the speed of a multi-arm robotic system with a follower. It is the first The adaptive rate estimation of the speed of a multi-arm robotic system with a follower. yes The derivative, It is the first Design parameters for the speed of a multi-arm robotic system with a follower. It is the first The basis function vector of the velocity of a multi-arm robotic system with a follower.

[0046] The event triggering mechanism includes the following triggering conditions:

[0047] ;

[0048] 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 for each follower Indicates the first A follower's controller Used for regulation and , Indicates a positive number parameter;

[0049] ;

[0050] ;

[0051] ;

[0052] ;

[0053] In the formula, yes The state of being maintained , Represents a positive number parameter. yes time ;

[0054] Update when the trigger condition is met. .

[0055] S5 includes:

[0056] ;

[0057] ;

[0058] ;

[0059] ;

[0060] ;

[0061] ;

[0062] In the formula, It is a differential operator. It is a Lyapunov function. It is the first The velocity of a multi-arm robotic system with a follower is a positive constant. It is the first The position of a multi-arm robotic system with a follower is a positive constant. , , It is a positive number. It is the first A positive constant for each follower. These are the actual parameters of the multi-robotic arm system.

[0063] Total Lyapunov function for:

[0064] ;

[0065] The predefined time of a multi-robotic arm system is considered stable if the following inequalities are satisfied:

[0066] ;

[0067] ;

[0068] ;

[0069] In the formula, It is all The sum of.

[0070] Compared with the prior art, the present invention has the following advantages: The present invention adopts an event triggering mechanism to reduce the update frequency of the controller, and combined with the consensus error of the experimental multi-manipulator system, it can converge to a small neighborhood within a predefined time of 1 second. Attached Figure Description

[0071] Figure 1 A flowchart illustrating the control method of the present invention is shown.

[0072] Figure 2 The diagram shown is a topology diagram of the multi-robotic arm system of the present invention.

[0073] Figure 3 The multi-robotic arm system output of the present invention is shown. And the reference signals of leaders A diagram illustrating the changes over time.

[0074] Figure 4 This demonstrates the consensus error of the multi-robotic arm system of the present invention. A diagram illustrating the changes over time.

[0075] Figure 5 The present invention is shown in the first part. The controller input of the follower A diagram illustrating the changes over time.

[0076] Figure 6 The present invention is shown. A diagram illustrating the changes over time.

[0077] Figure 7 The present invention is shown. A diagram illustrating the changes over time.

[0078] Figure 8 The diagram illustrates the change of the event triggering interval of follower one in the multi-robotic arm system of the present invention over time.

[0079] Figure 9 The diagram illustrates the change of the event triggering interval of follower 2 in the multi-robotic arm system of the present invention over time.

[0080] Figure 10 The diagram illustrates the change in the event triggering interval of follower three in the multi-robotic arm system of the present invention over time.

[0081] Figure 11 The diagram illustrates the change in the event triggering interval of follower four in the multi-robotic arm system of this invention over time. Detailed Implementation

[0082] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention are described clearly and completely below. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0083] Event-triggered multi-robotic arm systems employ random predefined time control methods, including:

[0084] S1, the dynamic equations of the multi-manipulator system;

[0085] S2. Introduce a trajectory error tracking system to obtain the consensus error of the multi-robotic arm system;

[0086] S3. Estimate the unknown functions in the dynamic equations of the multi-manipulator system;

[0087] S4. Based on the adaptive backstepping method, design the virtual control signal, adaptive rate, and event triggering mechanism;

[0088] S5. Perform stability analysis using Lyapunov stability theory.

[0089] S1 includes:

[0090] ;

[0091] ;

[0092] ;

[0093] ;

[0094] ;

[0095] In the formula, The differential symbol, For the first The position of a follower multi-arm robotic system For the first The speed of a multi-arm robotic system with a follower For time, yes The diffusion term function, For a standard one-dimensional Wiener process, For the first A follower's controller input, For the first The output of a multi-arm robotic system with a follower For the first The drift term function of each follower for The diffusion term function, For the first The state of a multi-arm robotic system with a follower. For the first The rotational inertia of the servo motor of the follower Representing the The mass of the link of the follower. Represents gravitational acceleration. Representing the The length of the link in the follower's link Indicates the first Damping coefficient of a follower.

[0096] S2 includes:

[0097] ;

[0098] In the formula, , It is the first Consensus error of a multi-arm robotic system with multiple followers , Corresponding to , consensus error, yes The maximum value, Representing the Virtual control signals for the position of a multi-arm robotic system with a follower Reference signals representing leaders, Indicates the first One follower This represents the parameters of a multi-arm robotic system, including position and velocity. For the first The position of a multi-arm robotic system with a follower;

[0099] if Receive from Information, variables ,otherwise ;

[0100] if Receive information from the leader, variables ,otherwise .

[0101] S3 includes using the approximation properties of fuzzy logic systems to estimate unknown functions:

[0102] ;

[0103] ;

[0104] ;

[0105] ;

[0106] ;

[0107] ;

[0108] ;

[0109] In the formula, It is an unknown function. It is the independent variable of the unknown function. It is a fuzzy logic system. It is an estimation error. It is a positive number. It is a weighted vector. It is the weighted vector of the th Subvectors, It is the number of fuzzy rules. It is a basis function vector. It is the first basis function vector Subvectors, It is a Gaussian function. It is a natural constant. It is the first The center vector of each follower yes The One element, It is the width of the Gaussian function.

[0110] The position adaptation rate estimate for the multi-robotic arm system is:

[0111] ;

[0112] The virtual control signal for the position of the multi-arm robotic system is:

[0113] ;

[0114] ;

[0115] ;

[0116] ;

[0117] ;

[0118] ;

[0119] In the formula, , For the parameters that can be determined, It is the first Position adaptation rate estimation of a multi-arm robotic system with one follower yes The derivative, , , These are three parameters with known ranges. It is the first Design parameters for the position of a multi-arm robotic system with a follower. It is the first The basis function vectors of the position of a multi-arm robotic system with followers.

[0120] The adaptive rate estimate for the speed of the multi-robotic arm system is:

[0121] ;

[0122] The virtual control signal for the speed of the multi-arm robotic system is:

[0123] ;

[0124] In the formula, Representing the Virtual control signals for the speed of a multi-arm robotic system with a follower. It is the first The adaptive rate estimation of the speed of a multi-arm robotic system with a follower. yes The derivative, It is the first Design parameters for the speed of a multi-arm robotic system with a follower. It is the first The basis function vector of the velocity of a multi-arm robotic system with a follower.

[0125] The event triggering mechanism includes the following triggering conditions:

[0126] ;

[0127] 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 for each follower Indicates the first A follower's controller Used for regulation and , Indicates a positive number parameter;

[0128] ;

[0129] ;

[0130] ;

[0131] ;

[0132] In the formula, yes The state of being maintained , Represents a positive number parameter. yes time ;

[0133] Update when the trigger condition is met. .

[0134] S5 includes:

[0135] ;

[0136] ;

[0137] ;

[0138] ;

[0139] ;

[0140] ;

[0141] In the formula, It is a differential operator. It is a Lyapunov function. It is the first The velocity of a multi-arm robotic system with a follower is a positive constant. It is the first The position of a multi-arm robotic system with a follower is a positive constant. , , It is a positive number. It is the first A positive constant for each follower. These are the actual parameters of the multi-robotic arm system.

[0142] Total Lyapunov function for:

[0143] ;

[0144] The predefined time of a multi-robotic arm system is considered stable if the following inequalities are satisfied:

[0145] ;

[0146] ;

[0147] ;

[0148] In the formula, It is all The sum of.

[0149] The derivation process of the formulas in this invention is as follows, assuming the following three formulas are (1), (2), and (3):

[0150] (1);

[0151] (2);

[0152] (3);

[0153] From formulas (1) and (2), we obtain formula (4):

[0154] (4);

[0155] ;

[0156] ;

[0157] In the formula, yes The sum of;

[0158] Constructing Lyapunov functions :

[0159] (5);

[0160] ;

[0161] ;

[0162] In the formula, It is an intermediate variable. These are design parameters. These are the actual positional parameters of the multi-arm robotic system;

[0163] According to Itoh's formula:

[0164] (6);

[0165] ;

[0166] In the formula, the superscript indicates differentiation. It is an intermediate variable.

[0167] According to Young's inequality, for ,have to:

[0168] (7);

[0169] (8);

[0170] Substitute formulas (7) and (8) into formula (6):

[0171] (9);

[0172] ;

[0173] In the formula, It is an intermediate variable.

[0174] set up Let be the unknown function to be estimated. Approximation using fuzzy logic systems ,for :

[0175] (10);

[0176] In the formula, It is the approximation error of the fuzzy logic system;

[0177] For a given :

[0178] (11);

[0179] ;

[0180] In the formula, It is the weight vector of the fuzzy logic system for the position of the multi-robotic arm system;

[0181] In conclusion:

[0182] (12);

[0183] Let the following two expressions be equation (13) and equation (14):

[0184] (13);

[0185] (14);

[0186] Substituting equations (13) and (14) into equation (12), we get:

[0187] (15);

[0188] We obtain the following from equations (1) and (2):

[0189] (16);

[0190] (17);

[0191] Indicates the first One follower.

[0192] Choose the following Lyapunov functions. :

[0193] (18);

[0194] Combining equations (16) and (18), we can derive the following using Itō's formula:

[0195] (19);

[0196] ;

[0197] In the formula, It is a diffusion term function;

[0198] According to Young's inequality, for :

[0199] (20);

[0200] Combining equations (15) and (20), inequality (19) can be rewritten as:

[0201] (twenty one);

[0202] ;

[0203] ;

[0204] In the formula, It is an intermediate variable. yes The second derivative;

[0205] Applying fuzzy logic systems to approximate unknown functions, for any :

[0206] (twenty two);

[0207] For a given get:

[0208] (twenty three);

[0209] ;

[0210] ;

[0211] In the formula, These are the actual speed parameters of the multi-arm robotic system. It is the weight vector of the fuzzy logic system for the velocity of the multi-robotic arm system;

[0212] Number the following formulas:

[0213] (twenty four);

[0214] (25);

[0215] (26);

[0216] (27);

[0217] (28);

[0218] For time-varying parameters and :

[0219] (29);

[0220] ;

[0221] .

[0222] for and ,inequality Established.

[0223] Based on equation (26), we can derive... .

[0224] because , ,get:

[0225] (30);

[0226] (31);

[0227] Based on equations (26), (30), and (31), we can derive:

[0228] (32);

[0229] Combining equations (24), (35), and (32), we can obtain:

[0230] (33);

[0231] Depend on:

[0232] (34);

[0233] According to Young's inequality, we can obtain:

[0234] (35);

[0235] when , When, inequalities Establishment, for We can obtain:

[0236] (36);

[0237] For any positive constant , , , , ,inequality If this holds true, then we can deduce that:

[0238] (37);

[0239] Substituting equations (34), (35), (36), and (37) into equation (33), we obtain:

[0240] (38);

[0241] Further deduction:

[0242] (39);

[0243] From this, the total Lyapunov function can be calculated.

[0244] ;

[0245] ;

[0246] In the formula, As expected.

[0247] To verify the effectiveness of the event-triggered adaptive practical predefined time control method for multiple robotic arms provided in this embodiment, simulation experiments were conducted using MATLAB, and detailed descriptions are provided with reference to the accompanying drawings.

[0248] The technical process of this invention is as follows: Figure 1As shown, considering random noise, the dynamic equations of the multi-robotic arm system are obtained; a trajectory error tracking system is introduced to obtain the consensus error (consensus error) of the multi-robotic arm system; using the approximation characteristics of the fuzzy logic system, the unknown functions in the dynamic equations of the multi-robotic arm system are estimated; based on the adaptive backstepping method, virtual control signals, event triggering mechanisms, actual control, and adaptive update rates are designed; and the stability analysis and simulation verification of the proposed control method are performed in conjunction with Lyapunov stability theory. Figure 2 Weight matrix of multi-arm robotic system for:

[0249] ;

[0250] In the simulation experiment, the parameters of the selected single-link robotic arm model are as follows:

[0251] ;

[0252] ;

[0253] ;

[0254] ;

[0255] .

[0256] The initial state of the selected system is:

[0257] ;

[0258] .

[0259] The key signals for leaders are:

[0260] .

[0261] The design parameters are:

[0262] , , , , , , , , , , .

[0263] 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 4It can be seen that the consensus error of the multi-robotic arm system can converge to a small neighborhood within a predefined time of 1 second. Figure 5 The demonstration shows the controller for the follower in a multi-arm robotic system. Only when the event triggering condition is met, It will then be updated. Figure 6 and Figure 7 The convergence of the two adaptive parameters is shown. Figure 8 , Figure 9 , Figure 10 , Figure 11 The event triggering intervals of the four followers of the multi-robotic arm system are shown.

[0264] 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 random predefined time control method for a multi-robotic arm system based on event triggering, characterized in that, include: S1, the dynamic equations of the multi-manipulator system; S2. Introduce a trajectory error tracking system to obtain the consensus error of the multi-robotic arm system; S3. Estimate the unknown functions in the dynamic equations of the multi-manipulator system; S4. Based on the adaptive backstepping method, design the virtual control signal, adaptive rate, and event triggering mechanism; S5. Perform stability analysis using Lyapunov stability theory; S2 include: ； In the formula, , It is the first Consensus error of a multi-arm robotic system with multiple followers , Corresponding to , consensus error, yes The maximum value, Representing the Virtual control signals for the position of a multi-arm robotic system with a follower Reference signals representing leaders, Indicates the first One follower This represents the parameters of a multi-arm robotic system, including position and velocity. For the first The position of a multi-arm robotic system with a follower; if Receive from Information, variables ,otherwise ; if Receive information from the leader, variables ,otherwise ; The position adaptation rate estimate for the multi-robotic arm system is: ； The virtual control signal for the position of the multi-arm robotic system is: ； ； ； ； ； ； In the formula, , For the parameters that can be determined, It is the first Position adaptation rate estimation of a multi-arm robotic system with one follower yes The derivative, , , These are three parameters with known ranges. It is the first Design parameters for the position of a multi-arm robotic system with a follower. It is the first The basis function vectors of the positions of a multi-arm robotic system with a follower; The adaptive rate estimate for the speed of the multi-robotic arm system is: ； The virtual control signal for the speed of the multi-arm robotic system is: ； In the formula, Representing the Virtual control signals for the speed of a multi-arm robotic system with a follower. It is the first The adaptive rate estimation of the speed of a multi-arm robotic system with a follower. yes The derivative, It is the first Design parameters for the speed of a multi-arm robotic system with a follower. It is the first The basis function vector of the velocity of a multi-arm robotic system with a follower; 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 for each follower Indicates the first A follower's controller Used for regulation and , Indicates a positive number parameter; ； ； ； ； In the formula, yes The state of being maintained , Represents a positive number parameter. yes time ; Update when the trigger condition is met. .

2. The event-triggered multi-robotic arm system random predefined time control method according to claim 1, characterized in that, S1 includes: ； ； ； ； ； In the formula, The differential symbol, For the first The position of a follower multi-arm robotic system For the first The speed of a multi-arm robotic system with a follower For time, yes The diffusion term function, For a standard one-dimensional Wiener process, For the first A follower's controller input, For the first The output of a multi-arm robotic system with a follower For the first The drift term function of a follower for The diffusion term function, For the first The state of a multi-arm robotic system with a follower. For the first The rotational inertia of the servo motor of the follower Representing the The mass of the link of the follower. Represents gravitational acceleration. Representing the The length of the link in the follower's link Indicates the first Damping coefficient of a follower.

3. The event-triggered multi-robotic arm system random predefined time control method according to claim 2, characterized in that, S3 includes using the approximation properties of fuzzy logic systems to estimate unknown functions: ； ； ； ； ； ； ； In the formula, It is an unknown function. It is the independent variable of the unknown function. It is a fuzzy logic system. It is an estimation error. It is a positive number. It is a weighted vector. It is the weighted vector of the th Subvectors, It is the number of fuzzy rules. It is a basis function vector. It is the first basis function vector Subvectors, It is a Gaussian function. It is a natural constant. It is the first The center vector of each follower yes The One element, It is the width of the Gaussian function.

4. The event-triggered multi-robotic arm system random predefined time control method according to claim 3, characterized in that, S5 include: ； ； ； ； ； ； In the formula, It is a differential operator. It is a Lyapunov function. It is the first The velocity of a multi-arm robotic system with a follower is a positive constant. It is the first The position of a multi-arm robotic system with a follower is a positive constant. , , It is a positive number. It is the first A positive constant for each follower. These are the actual parameters of the multi-robotic arm system.

5. The event-triggered multi-robotic arm system random predefined time control method according to claim 4, characterized in that, Total Lyapunov function for: ； The predefined time of a multi-robotic arm system is considered stable if the following inequalities are satisfied: ； ； ； In the formula, It is all The sum of.

Images