A control method of a mechanical arm under a markov jump cooperative competition network

Abstract

The application discloses a kind of Markov jump cooperative competition network under the control method of mechanical arm, belong to the field of mechanical arm control, method includes the following steps: obtaining the dynamics of 1 leader mechanical arm and N follower mechanical arm;Based on the dynamics of leader mechanical arm and N follower mechanical arm, the random variation of symbol weighted directed graph between the communication interaction of mechanical arm is described using double-layer Markov, and a non-homogeneous Markov jump cooperative competition network model is constructed;Based on the non-homogeneous Markov jump cooperative competition network model, a cost-preserving event-triggering mechanism is introduced, and a binary synchronization controller based on event triggering is designed;Based on the binary synchronization controller, the dynamics of N follower mechanical arms are controlled and adjusted, so that the dynamics of N follower mechanical arms are consistent with the dynamics of the leader mechanical arm.This application designs an event-triggered controller based on cost preservation to achieve the binary synchronization goal of the system, and ensures the dissipation performance of the system to external disturbance.

Description

Technical Field

[0001] This invention belongs to the field of robotic arm control and relates to a control method for a robotic arm under a Markov jumper cooperative competitive network. Background Technology

[0002] Synchronization problems in complex dynamic networks have received widespread attention and research in applications such as neural networks, multi-agent systems, and social networks. However, network nodes also exhibit both cooperative and competitive behaviors simultaneously. For example, consider an application scenario where multiple robotic arms on an assembly line need to collaboratively transport / assemble large components (such as car bodies). Due to spatial constraints or task priorities, some robotic arms need to move synchronously (cooperation), while others need to avoid each other or adjust their posture (competition). Therefore, cooperative-competitive networks represented by symbolic graphs have gained increasing attention because they can simultaneously characterize cooperative and competitive behaviors. Furthermore, due to environmental factors, the parameters and structure of complex dynamic networks may change randomly. Moreover, considering communication constraints and environmental interference, fixed-switching topology networks cannot effectively describe information exchange between nodes. To address this, researchers have used Markov transition theory to characterize such problems and proposed a stochastic switching model. Considering equipment limitations and economic effects, a practical problem is how to effectively save communication resources and alleviate communication pressure. To solve this problem, researchers have proposed different types of event-triggered control schemes to reduce the number of communications. In this case, how to ensure control performance is a worthy research question. Under the aforementioned circumstances, studying the binary synchronization problem of complex network systems with randomly switching topologies is a challenging and significant topic.

[0003] Most existing Markov processes considered are homogeneous stochastic processes with time-invariant transition probabilities, neglecting the time-varying transition probability phenomenon caused by disturbances. Furthermore, random topology switching only considers the randomness of synchronization with the system or being entirely influenced by external disturbances, without simultaneously analyzing the impact of both on the topology switching process, thus failing to comprehensively characterize real-world scenarios. Moreover, the mode-dependent event-triggered mechanisms proposed in existing technologies are not applicable to scenarios where the system modes are unknown, nor do they consider the need to guarantee the performance of quadratic cost control. Summary of the Invention

[0004] To address the challenges of designing novel event-triggered mechanisms to alleviate communication pressure while considering quadratic cost performance metrics, and of designing suitable control protocols to achieve stochastic binary synchronization control objectives under a switching topology modeled in a two-layer Markov process, this invention employs the following technical solution: a control method for a robotic arm under a Markov jump-cooperative competition network, comprising the following steps:

[0005] Obtain the dynamics of one leader robotic arm and N follower robotic arms;

[0006] Based on the dynamics of the leader robotic arm and N follower robotic arms, a non-homogeneous Markov jump cooperative competition network model is constructed by using a two-layer Markov to describe the random changes of the symbol-weighted directed graph when describing the communication interaction between the robotic arms.

[0007] Based on the non-homogeneous Markov jump cooperative competition network model, a cost-preserving event triggering mechanism is introduced to design an event-triggered binary synchronization controller.

[0008] The dynamics of N follower robotic arms are controlled and adjusted based on a binary synchronous controller, so that the dynamics of the N follower robotic arms are consistent with the dynamics of the leader robotic arm.

[0009] Furthermore, the expression for constructing the non-homogeneous Markov jump transition cooperative competitive network model is as follows:

[0010]

[0011] in: express Time-based robotic arm system state, express +1 indicates the state of the robotic arm system; express Time-lapse robotic arm Neighbor robotic arm state, express Time-lapse robotic arm Control input, Indicates process disturbance. express Time-lapse robotic arm Measurement output information; subscript This represents the index of the robotic arm, where N represents a natural number.

[0012] , , and It is the parameter matrix of the robotic arm system;

[0013] Parameter matrix , , and Non-homogeneous Markov processes Scheduling, Communication Weight By two-layer Markov process To depict, It is the coupling strength. Indicates the transformation parameters;

[0014] nonlinear functions It is a bounded odd function.

[0015] Furthermore, the process of obtaining the aforementioned cost-preservation event triggering mechanism is as follows:

[0016] Define the synchronization error between the leader robotic arm and the follower robotic arm. Leader's robotic arm status ,make The details are as follows:

[0017] ;

[0018] in: It is an intermediate variable, derived from the state variable. Transformed parameters Obtained after transformation;

[0019] make Define the event triggering error for the triggering sequence. :

[0020] ;

[0021] in: Indicates the time of triggering At that time, the latest transmitted error signal; Indicates time;

[0022] Integrating the quadratic performance function into the event-triggered strategy yields a cost-preserving event-triggered mechanism of the following form, i.e., the next triggering time. The calculation is as follows:

[0023]

[0024] in: yes The initial trigger time, the quadratic cost is , , Given a constant matrix, , It is a synchronization error The transpose of , yes The transpose of ;

[0025] intermediate variables Update in the following format:

[0026]

[0027] in: It is a matrix. It's a scalar, a scalar , , , These are the parameters to be designed for the event triggering mechanism, representing the scaling factor; It is an event triggering error. The transpose of .

[0028] Furthermore, the expression for the event-triggered binary synchronization controller is as follows:

[0029]

[0030] in: For the controller gain, determined by the Hidden Markov Process Scheduling, and satisfying conditional probability :

[0031] , , ,

[0032] in: Indicates the modality of the controller. Represents system modes, and Represents a finite set of integers. This indicates the signal received by the actuator from the event-triggered generation. It is a hidden Markov process The modes in Is it a homogeneous Markov process? In the modal representation, let (the text is incomplete and requires further context to be fully translated). Modal of the time controller The controller gain is expressed as .

[0033] Furthermore: the quadratic cost index of the global synchronization error system consisting of one leader robotic arm and N follower robotic arms. for:

[0034]

[0035] in: It is a control input transpose; Given a weight matrix,

[0036] The binary synchronization objective based on a non-homogeneous Markov jump-variant cooperative competitive network model ensures that the global synchronization error system is effective for any... , and ,satisfy:

[0037]

[0038] in: Is it a homogeneous Markov process? middle The value at time;

[0039] It is a hidden Markov process. In The value at time;

[0040] It is a two-layer Markov process. middle The value at time;

[0041] Global synchronization error system exist The value at time; It represents a finite set of integers.

[0042] Furthermore: the controller gain The solution process is as follows:

[0043] First, the controller gain is given. Existence criterion: Pre-defined dissipation index considering disturbances Given a matrix , , , , and scalar ,in , , , , It is the weight matrix of the dissipation inequality, and satisfies ; express The decomposition matrix, It is a decomposition matrix The transpose of the matrix;

[0044] If a matrix exists ,matrix ,matrix , Makes any , , as well as The following criteria hold true:

[0045]

[0046] in: It is a set of integers, with subscripts Only indicates the index symbol. It is a Lyapunov function. It is the transpose of the Lyapunov function; , It is an intermediate variable. Indicates the mode at the current moment. Indicates the mode at the next moment. The mode for the next moment;

[0047]

[0048] in: , , , , , , , , , , , It is an intermediate variable; Represents the identity matrix. , It is a nonlinear function Boundary-dependent intermediate constants; It is a matrix transpose; : Parameter matrix, It is the decomposition matrix of matrix Q. : Dissipative inequality weight matrix : Controller gain The global form; symbols This represents the symmetrical part of the lower triangle of the symmetric matrix;

[0049] Upper bound of the cost function of a global synchronization error system satisfy Therefore, the global synchronization error system is stochastically stable and satisfies strict dissipation; where: For Lyapunov functions exist The value at time;

[0050] For the nonlinear part of the above criteria, traceable solution conditions are given:

[0051] First, the controller gain is given. Solution criteria: Pre-defined dissipation index considering disturbances. Given a matrix , , , , and scalar If a scalar exists ,matrix ,matrix ,matrix ,matrix ,matrix , such that for any , , It is an index of a positive integer. as well as The following criteria hold true:

[0052]

[0053]

[0054] in: , , For intermediate variables; symbols This represents the symmetrical part of the lower triangle of the symmetric matrix; It is the transpose of the matrix;

[0055] Therefore, the global synchronization error system is stochastically stable and satisfies strict dissipation, and the gain of the binary synchronization controller is... It is given by the following formula:

[0056]

[0057] in: , It is an intermediate variable.

[0058] Furthermore, the leader robotic arm and the follower robotic arm are single-link robotic arms.

[0059] This invention provides a control method for a robotic arm under a Markov jump-type cooperative competitive network. It addresses a class of coupled single-link robotic arm cooperative competitive network systems with randomly switching communication topologies, studying the event-triggered binary synchronization problem based on cost-preserving control. To characterize the randomly switching communication topology caused by environmental and system factors, and to overcome the difficulty of simultaneously satisfying cost-preserving control performance and alleviating communication pressure, a two-layer Markov process is used to model the random topology. A novel event-triggered scheme integrating quadratic performance indices is proposed. Finally, a cost-preserving event-triggered controller is designed to achieve the system's binary synchronization objective while ensuring the system's dissipation performance against external disturbances.

[0060] Compared with the prior art, the advantages of the present invention are:

[0061] 1. Considering the impact of both external environment and system factors on communication topology, a two-layer Markov process is used to describe its more general switching properties; and a two-layer Markov process is used to model communication topology with random switching, which can simultaneously characterize the impact of external environment and system on communication topology, and is more in line with practical considerations.

[0062] 2. Design a new cost-preservation event triggering mechanism. By integrating secondary cost performance indicators into the dynamic event triggering mechanism and increasing the triggering threshold, control performance can be guaranteed while alleviating communication pressure. Attached Figure Description

[0063] To more clearly illustrate the technical solutions in the embodiments of the present invention or 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 some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0064] Figure 1 This is a schematic diagram of the single-link robotic arm of the present invention;

[0065] Figure 2 This is the framework of the binary synchronization control method proposed in this invention;

[0066] Figure 3 This is a schematic diagram of the switching topology of the present invention;

[0067] Figure 4 The diagram shows the angular and angular velocity trajectories of the robotic arm; where (a) represents the angular trajectory of the single-link robotic arm, and (b) represents the angular velocity trajectory of the single-link robotic arm.

[0068] Figure 5 The diagram shows the consistency error between the follower and leader robotic arms in terms of angle and angular velocity; where (a) represents the consistency error curve of the angle between the follower and leader robotic arms, and (b) represents the consistency error curve of the angular velocity between the follower and leader robotic arms. Detailed Implementation

[0069] It should be noted that, unless otherwise specified, the embodiments and features in the embodiments of the present invention can be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0070] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the present invention or its application or use. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0071] A control method for a robotic arm under a Markov jumper cooperative competitive network includes the following steps:

[0072] S1: Obtain the dynamics of 1 leader robotic arm and N follower robotic arms; N represents a natural number greater than zero;

[0073] S2: Based on the dynamics of the leader robotic arm and N follower robotic arms, a non-homogeneous Markov jump cooperative competition network model is constructed by using a two-layer Markov to describe the random changes of the symbol-weighted directed graph when describing the communication interaction between the robotic arms.

[0074] S3: Based on the non-homogeneous Markov jump cooperative competition network model, a cost-preserving event triggering mechanism is introduced to design an event-triggered binary synchronization controller.

[0075] S4: Based on the binary synchronization controller, the dynamics of N follower robotic arms are controlled and adjusted to keep the dynamics of the N follower robotic arms consistent with the dynamics of the leader robotic arm.

[0076] Steps S1 / S2 / S3 / S4 are executed sequentially;

[0077] Consider as Figure 1 The dynamic description of the single-link robotic arm shown is as follows:

[0078] (1.1)

[0079] in: , , , and These represent the angular acceleration, angular velocity, angle, control input, and disturbance of the robotic arm, respectively. The payload is affected by random disturbances. Total moment of inertia Random variations are generated by non-homogeneous Markov processes. Characterization, making the current time modality as The next time-state is Satisfying the transition probability , It is a finite set;

[0080] Time-varying transition probability matrix: ,in , , It is a finite set of integers;

[0081] Let the sampling period of the single-link robotic arm as shown in (1.1) be... , , Using the Euler approximation, the following discrete form can be obtained:

[0082] (1.2)

[0083] A communication topology is introduced to characterize the communication interactions between multiple robotic arms. Considering environmental interference, equipment limitations, and other random or sudden disturbances in network communication, the communication connectivity between robotic arms may experience random changes.

[0084] Two-level Markov process To characterize the random changes in a symbolically weighted directed graph, i.e. transition probability , For a finite set, These represent the current and next time-instance modes, respectively. This stochastic process is generated by a high-level Markov sequence. Adjustment. In a weighted directed graph, and Let the set of edges and the set of nodes be represented by the weighted adjacency matrix. , , satisfying when node and nodes When there is competition between them When node and nodes When the behavior between them is cooperative Define nodes The degree is The Laplace matrix of this topology It can be obtained from the following formula:

[0085]

[0086] Considering the cooperative and competitive behaviors among the robotic arms, and treating each robotic arm as a node in a communication network, the following compact form of the non-homogeneous Markov jump-cooperative competitive network model can be expressed:

[0087]

[0088] in: express Time-based robotic arm system state, express Constant monitoring of the robotic arm system status; express Time-lapse robotic arm Neighbor robotic arm state, express Time-lapse robotic arm Control input, Indicates process disturbance. express Time-lapse robotic arm Measurement output information; subscript Indicates the index of the robotic arm. Represent natural numbers;

[0089] , , and It is the parameter matrix of the robotic arm system;

[0090] Parameter matrix , , and Non-homogeneous Markov processes Scheduling, Communication Weight By two-layer Markov process To depict, It is the coupling strength. Indicates the transformation parameters;

[0091] The parameter matrix structure of the robotic arm system is represented as follows:

[0092] , , , ,

[0093] parameter , , and By a nonhomogeneous Markov process Scheduling. Communication weights By two-layer Markov process To depict, It is the coupling strength. Indicates the transformation parameters.

[0094] nonlinear functions It is a bounded odd function that satisfies ,in, It is a known constant.

[0095] To simplify the representation, let , Introducing the transformation matrix To effectively handle contention in communication networks, depending on its nature, there are... .

[0096] make Then, the non-homogeneous Markov jump cooperative competition network model, after transformation by the transformation matrix, can be expressed as:

[0097] (1.3)

[0098] The dynamics of the robotic arm are given as follows:

[0099] (1.4)

[0100] This invention aims to design a controller to ensure the quadratic performance indicators of the error system of the robotic arm. and its dissipation performance against disturbances, i.e., for any Under initial conditions of 0, we have:

[0101] (1.5)

[0102] in: Dissipation index The weight matrix of the dissipation inequality and Given in advance, and satisfying Furthermore, it achieves the binary synchronization objective of the robotic arm system (1.3), namely:

[0103] (1.6)

[0104] Considering communication bandwidth and equipment limitations, an event-triggered mechanism needs to be designed to alleviate communication pressure and reduce the likelihood of communication congestion.

[0105] The process of obtaining the protection value event triggering mechanism is as follows:

[0106] Define the synchronization error between the leader robotic arm and the follower robotic arm. Leader's robotic arm status ,make The details are as follows:

[0107] ;

[0108] in: It is an intermediate variable, derived from the state variable. Transformed parameters Obtained after transformation;

[0109] make Define the event triggering error for the triggering sequence. :

[0110] ;

[0111] in: Indicates the time of triggering At that time, the latest transmitted error signal, Indicates time;

[0112] Integrating the quadratic performance function into the event-triggered strategy yields a cost-preserving event-triggered mechanism of the following form, i.e., the next triggering time. The calculation is as follows:

[0113] (1.7)

[0114] in: yes The initial trigger time, the quadratic cost is , Given a constant matrix, It is a synchronization error The transpose of , yes The transpose of ;

[0115] intermediate variables Update in the following format:

[0116]

[0117] in: It is a matrix. It's a scalar, a scalar , , , These are the parameters to be designed for the event triggering mechanism, representing the scaling factor; It is an event triggering error. The transpose of .

[0118] The expression for the event-triggered binary synchronization controller is as follows:

[0119] (1.8)

[0120] in: For the controller gain, determined by the Hidden Markov Process Scheduling, and satisfying conditional probability :

[0121] , , ,

[0122] in: Indicates the modality of the controller. Represents system modes, and Represents a finite set of integers. This indicates the signal received by the actuator from the event-triggered generation. It is a hidden Markov process The modes in Is it a homogeneous Markov process? The modalities in [the context]. For simplicity, let [the following be an example]. Modal of the time controller The controller gain is expressed as .

[0123] Based on the above analysis, the global synchronization error system can be represented in the following compact form:

[0124] (1.9)

[0125] in: , , , , , ,

[0126] ,

[0127] ,

[0128] According to nonlinear functions Based on the properties of , the following inequality holds:

[0129]

[0130] in, It is a positive definite parameter matrix to be determined. , ,symbol This represents the symmetrical part of the lower triangle of the symmetric matrix.

[0131] The quadratic cost index of the global synchronization error system consisting of one leader robotic arm and N follower robotic arms for:

[0132] (1.10)

[0133] in: It is a control input transpose; It is a given weight matrix;

[0134] Based on the binary synchronization objective of the non-homogeneous Markov jump-variant cooperative competitive network model, the secondary synchronization control objective (1.6) can be equivalently transformed into such that the error system (1.9) for any , and ,satisfy:

[0135] (1.11)

[0136] in: Is it a homogeneous Markov process? middle The value at time;

[0137] It is a hidden Markov process. In The value at time;

[0138] It is a two-layer Markov process. middle The value at time;

[0139] Global synchronization error system exist The value at time; It represents a finite set of integers.

[0140] Based on the above analysis, the framework of the binary synchronization control method proposed in this invention is as follows: Figure 2 As shown; Figure 3 A schematic diagram of the switching topology of the present invention is shown;

[0141] The controller gain The solution process is as follows:

[0142] First, we define the Lyapunov function that represents the dual dependence of the topological modes and the robotic arm system modes, as follows:

[0143] (1.12)

[0144] in It is a Lyapunov matrix. For simplicity, let... , ,

[0145] make You can get

[0146]

[0147] in , , Indicates the index of the set of integers.

[0148] So when Substituting the event triggering condition (1.7) into the above formula yields:

[0149] (1.13)

[0150] in:

[0151]

[0152] Among the symbols This represents the symmetrical part of the lower triangle of the symmetric matrix.

[0153] Furthermore, the combined quadratic cost (1.10) can be obtained

[0154]

[0155] in ,

[0156]

[0157] in: ,symbol This represents the symmetric part of the lower triangular region of the symmetric matrix. Therefore, the error system (1.9) is stochastically stable.

[0158] Next, to ensure secondary performance, based on the above derivation, we have:

[0159]

[0160] From both sides of the above formula arrive Summing yields:

[0161] (1.14)

[0162] This means that the cost-preserving performance of the error system is satisfied.

[0163] Finally, considering the dissipation performance of the error system, when... When it is time, we can obtain:

[0164] (1.15)

[0165] in

[0166]

[0167] symbol This represents the symmetrical part of the lower triangle of the symmetric matrix.

[0168] Further analysis reveals that, from the two sides of the above equation... arrive Summing yields:

[0169]

[0170] Therefore, the error system (1.9) satisfies the dissipation performance (1.5).

[0171] To give the controller gain The specific solution method can be obtained by further deriving (1.15) using Schul's complement lemma:

[0172]

[0173] in ,

[0174]

[0175] symbol This represents the symmetrical part of the lower triangle of the symmetric matrix.

[0176] Based on the above analysis, controller gain The specific solution conditions are given as follows:

[0177] First, the controller gain is given. Existence criterion: Pre-defined dissipation index considering disturbances Given a matrix , , , , and scalar ,in ,

[0178] , , , It is the weight matrix of the dissipation inequality, and satisfies ; express The decomposition matrix, It is a decomposition matrix The transpose of the matrix;

[0179] If a matrix exists , , , Makes any , , as well as The following criteria hold true:

[0180] (1.16)

[0181] in: It is a set of integers, with subscripts Only indicates the index symbol. It is a Lyapunov function. It is the transpose of the Lyapunov function; , It is an intermediate variable; Indicates the mode at the current moment. Indicates the mode at the next moment. The mode for the next moment;

[0182] (1.17)

[0183] in: , , , , , , , , , , , It is an intermediate variable; Represents the identity matrix. , It is a nonlinear function Boundary-dependent intermediate constants; It is a matrix transpose; : Parameter matrix, It is the decomposition matrix of matrix Q. : Dissipative inequality weight matrix : Controller gain The global form; symbols This represents the symmetrical part of the lower triangle of the symmetric matrix.

[0184] , , ,

[0185] , , , , , , ,

[0186]

[0187] Furthermore, the upper bound of the cost function of the global synchronization error system. satisfy Therefore, the global synchronization error system (1.9) is stochastically stable and satisfies strict dissipation. Wherein: For Lyapunov functions exist The value at time;

[0188] For the nonlinear part of the solution criterion (1.17), the following traceable solution conditions are given:

[0189] First, the controller gain is given. Solution criteria: Pre-defined dissipation index considering disturbances. Given a matrix , , , , and scalar If a scalar exists ,matrix ,matrix ,matrix ,matrix , such that for any , , It is an index of a positive integer. as well as The following criteria hold true:

[0190] (1.18)

[0191] (1.19)

[0192] in: , , For intermediate variables; symbols This represents the symmetrical part of the lower triangle of the symmetric matrix. It is the transpose of the matrix;

[0193] , , , , , ;

[0194] ,

[0195] ,

[0196] Therefore, the error system (1.9) is stochastically stable and satisfies strict dissipation. The gain of the binary synchronous controller is given by the following equation:

[0197] (1.20)

[0198] in: , It is an intermediate variable.

[0199] The leader and follower robotic arms are single-link robotic arms.

[0200] To demonstrate the effectiveness of the method proposed in this invention, a competitive network of a system consisting of six single-link robotic arms is considered. The simulation results are as follows: Figure 4 and Figure 5 As shown;

[0201] Figure 4 The angular and angular velocity trajectories of the robotic arm are represented; where (a) represents the angular trajectory of the single-link robotic arm, and (b) represents the angular velocity trajectory of the single-link robotic arm.

[0202] Figure 5 The following represents the consistency error between the follower and leader robotic arms in terms of angle and angular velocity; where (a) represents the consistency error curve of the angle between the follower and leader robotic arms, and (b) represents the consistency error curve of the angular velocity between the follower and leader robotic arms.

[0203] 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. A control method for a robotic arm under a Markov jump transition cooperative competition network, characterized in that: Includes the following steps: Obtain the dynamics of one leader robotic arm and N follower robotic arms; Based on the dynamics of the leader robotic arm and N follower robotic arms, a non-homogeneous Markov jump cooperative competition network model is constructed by using a symbolically weighted directed graph with two-layer Markov to describe the random changes of the communication interaction between the robotic arms. Based on the non-homogeneous Markov jump cooperative competition network model, a cost-preserving event triggering mechanism is introduced to design an event-triggered binary synchronization controller. The process of obtaining the protection value event triggering mechanism is as follows: Define the synchronization error between the leader robotic arm and the follower robotic arm. Leader's robotic arm status ,make The details are as follows: ； in: It is an intermediate variable, derived from the state variable. Transformed parameters Obtained after transformation; make Define the event triggering error for the triggering sequence. : ； in: Indicates the time of triggering At that time, the latest transmitted error signal; Indicates time; Integrating the quadratic performance function into the event-triggered strategy yields a cost-preserving event-triggered mechanism of the following form, i.e., the next triggering time. The calculation is as follows: in: yes The initial trigger time, the quadratic cost is , , Given a constant matrix, , It is a synchronization error The transpose of , yes The transpose of ; intermediate variables Update in the following format: in: It is a matrix. It's a scalar, a scalar , , These are the parameters to be designed for the event triggering mechanism, representing the scaling factor; , It is an event triggering error. The transpose of ; The dynamics of N follower robotic arms are controlled and adjusted based on a binary synchronous controller, so that the dynamics of the N follower robotic arms are consistent with the dynamics of the leader robotic arm.

2. The control method for a robotic arm under a Markov jump transition cooperative competition network according to claim 1, characterized in that: The expression for constructing the non-homogeneous Markov jump transition cooperative competitive network model is as follows: in: express Time-based robotic arm system state, express +1 moment robotic arm system robotic arm The state; express Time-lapse robotic arm Neighbor robotic arm state, express Time-lapse robotic arm Control input, Indicates process disturbance. express Time-lapse robotic arm Measurement output information; subscript This represents the index of the robotic arm, where N represents a natural number. , , and It is the parameter matrix of the robotic arm system; Parameter matrix , , and Non-homogeneous Markov processes Scheduling, Communication Weight By two-layer Markov process To depict, It is the coupling strength. Indicates the transformation parameters; nonlinear functions It is a bounded odd function that satisfies ,in, It is a known constant.

3. The control method for a robotic arm under a Markov jump transition cooperative competition network according to claim 2, characterized in that: The expression for the event-triggered binary synchronization controller is as follows: in: For the controller gain, determined by the Hidden Markov Process Scheduling, and satisfying conditional probability. : ， ， ， in: Indicates the modality of the controller. Represents system modes, and Represents a finite set of integers. Indicates the time of triggering At that time, the latest transmitted error signal; It is a hidden Markov process The modes in Is it a homogeneous Markov process? In the modal representation, let (the text is incomplete and requires further context to be fully translated). Modal of the time controller The controller gain is expressed as .

4. The control method for a robotic arm under a Markov jump transition cooperative competition network according to claim 3, characterized in that: The quadratic cost index of a global synchronization error system consisting of one leader robotic arm and N follower robotic arms. for: in: It is a control input Transpose of; Given a weight matrix, The binary synchronization objective based on a non-homogeneous Markov jump-variant cooperative competitive network model ensures that the global synchronization error system is effective for any... , and ,satisfy: in: Is it a homogeneous Markov process? middle The value at time; It is a hidden Markov process. In The value at time; It is a two-layer Markov process. middle The value at time; Global synchronization error system exist The value at time; It represents a finite set of integers.

5. The control method for a robotic arm under a Markov jump transition cooperative competition network according to claim 4, characterized in that: The controller gain The solution process is as follows: First, the controller gain is given. Existence criterion: Pre-defined dissipation index considering disturbances Given a matrix , , , , and scalar ,in , , , , It is the weight matrix of the dissipation inequality, and satisfies ; express The decomposition matrix, It is a decomposition matrix The transpose of the matrix; If a matrix exists ,matrix ,matrix , Makes any , , as well as The following criteria hold true: in: It is a set of integers, with subscripts Only indicates the index symbol. It is a Lyapunov function. It is the transpose of the Lyapunov function; , It is an intermediate variable. Indicates the mode at the current moment. Indicates the mode at the next moment. The mode for the next moment; in: , , , , , , , , , , , It is an intermediate variable; Represents the identity matrix. , It is a nonlinear function Boundary-dependent intermediate constants; It is a matrix Transpose of; : Parameter matrix, It is a matrix The decomposition matrix, : Dissipative inequality weight matrix : Controller gain The global form; symbols This represents the symmetrical part of the lower triangle of a symmetric matrix; Upper bound of the cost function of a global synchronization error system satisfy Therefore, the global synchronization error system is stochastically stable and satisfies strict dissipation; where: For Lyapunov functions exist The value at time; For the nonlinear part of the above criteria, traceable solution conditions are given: First, the controller gain is given. Solution criteria: Pre-defined dissipation index considering disturbances. Given a matrix , , , , and scalar If a scalar exists ,matrix ,matrix ,matrix ,matrix ,matrix , such that for any , , It is an index of a positive integer. as well as The following criteria hold true: in: , , For intermediate variables; symbol This represents the symmetrical part of the lower triangle of a symmetric matrix; It is a matrix The transpose of the matrix; ， ， ， ， ， ； ， ， Therefore, the global synchronization error system is stochastically stable and satisfies strict dissipation, and the gain of the binary synchronization controller is... It is given by the following formula: in: , It is an intermediate variable.

6. The control method for a robotic arm under a Markov jump transition cooperative competition network according to claim 1, characterized in that: Both the leader and follower robotic arms utilize single-link robotic arms.

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