Distributed cooperative control method and system for multi-arm robotic arms with state constraints

Abstract

This disclosure relates to the field of robotic arm control technology, proposing a distributed cooperative control method and system for state-constrained multi-single-arm robotic arms. First, an equivalent unconstrained system is obtained through one-to-one state mapping. Then, a distributed observer is used to estimate the state of the leader system, decoupling the cooperative consistency problem of the multi-agent system into a tracking control problem involving multiple independent agents. An augmented system is constructed, and the optimal cooperative control strategy is obtained with the goal of minimizing the cost function. This disclosure transforms the system control problem into an unconstrained tracking problem, enabling performance optimization without violating state constraints.

Description

Technical Field

[0001] This disclosure relates to the field of robotic arm control technology, specifically to a distributed collaborative control method and system for multiple single-arm robotic arms with state constraints. Background Technology

[0002] The statements in this section are merely background information relating to this disclosure and do not necessarily constitute prior art.

[0003] Robotic arms are the most widely used automated mechanical devices in the field of robotics and are core equipment in intelligent manufacturing processes. Because single robotic arms cannot meet the increasingly complex manufacturing demands, multi-robotic arm systems, capable of performing more complex tasks, are playing an increasingly important role in intelligent manufacturing. Compared to single robotic arms, multi-robotic arm systems are more complex, and research on cooperative control theory is relatively weak. Therefore, the cooperative consistency control of multi-single-arm robotic arm systems is a challenging problem.

[0004] Optimal control can optimize system performance while ensuring the stability of the controlled system. However, for single-arm manipulators, solving the optimal control problem for such complex nonlinear systems is extremely difficult. Furthermore, to ensure the manipulator's safe operation, its state variables need to meet the constraints of the external environment. Violating these constraints will lead to a decline in system performance and may even damage the controlled system. The inventors have discovered that current methods for cooperative and consistent optimal control of multiple single-arm manipulators are difficult, and optimizing performance indicators without violating state constraints remains an unsolved technical problem. Summary of the Invention

[0005] To address the aforementioned issues, this disclosure proposes a distributed collaborative control method and system for multiple single-arm manipulators with state constraints. This method transforms the system control problem into an unconstrained tracking problem, enabling performance optimization without violating state constraints.

[0006] To achieve the above objectives, the present disclosure adopts the following technical solution:

[0007] One or more embodiments provide a distributed cooperative control method for multiple single-arm manipulators with state constraints, including the following steps:

[0008] The dynamic characteristics of multiple single-arm manipulators are modeled, and the time-varying asymmetric state constraint problem is handled by one-to-one state mapping to obtain an equivalent unconstrained system.

[0009] Based on a deterministic unconstrained system, and according to the communication topology between each single-arm manipulator, a distributed observer is designed to estimate the state of the leader in the multi-single-arm manipulator system and establish a simplified unconstrained tracking control task for each follower.

[0010] For each unconstrained tracking control task, an adaptive dynamic algorithm with a single evaluation network is used to approximate the optimal control strategy, and each single-arm manipulator is coordinated and controlled according to the optimal control strategy.

[0011] One or more embodiments provide a distributed collaborative control system for multiple single-arm robotic arms with state constraints, including:

[0012] Mapping and Transformation Module: Configured to model the dynamic characteristics of multiple single-arm manipulators, and use one-to-one state mapping to handle time-varying asymmetric state constraint problems to obtain equivalent unconstrained systems;

[0013] Unconstrained tracking control task establishment module: It is configured to design a distributed observer based on a deterministic unconstrained system and the communication topology between each single-arm manipulator, based on the obtained communication topology between the manipulators, to estimate the state of the leader in the multi-manipulator system and establish a simplified unconstrained tracking control task for each follower.

[0014] The solution module is configured to approximate the optimal control strategy for each unconstrained tracking control task using a single-evaluation network adaptive dynamic algorithm, and then perform coordinated control of each single-arm manipulator based on the optimal control strategy.

[0015] An electronic device includes a memory and a processor, as well as computer instructions stored in the memory and running on the processor, wherein the computer instructions, when executed by the processor, perform the steps described in the above method.

[0016] A computer-readable storage medium for storing computer instructions, which, when executed by a processor, perform the steps described in the above method.

[0017] Compared with the prior art, the beneficial effects of this disclosure are as follows:

[0018] In this disclosure, the cooperative control problem under state constraints is transformed into a traditional unconstrained tracking problem through a systematic transformation of the nonlinear mapping function and the cooperative design of distributed observers. Furthermore, the output constraints considered are asymmetric and time-varying, satisfying a more general range of constraints. Unlike existing adaptive dynamic programming algorithms based on execution and evaluation, this embodiment requires only one evaluation network, eliminating the approximation error of the execution network, and resulting in a simpler algorithm implementation structure.

[0019] The advantages of this disclosure, as well as its additional advantages, will be described in detail in the following specific embodiments. Attached Figure Description

[0020] The accompanying drawings, which form part of this disclosure, are used to provide a further understanding of this disclosure. The illustrative embodiments of this disclosure and their descriptions are used to explain this disclosure and do not constitute a limitation thereof.

[0021] Figure 1 This is a simulation example of the communication topology of a single-arm robotic arm cluster according to Embodiment 1 of this disclosure;

[0022] Figure 2(a) is a simulation example of the first state change trajectory of each single-arm manipulator in Embodiment 1 of this disclosure;

[0023] Figure 2(b) is a simulation example of the second state change trajectory of each single-arm manipulator in Embodiment 1 of this disclosure;

[0024] Figure 3(a) shows Embodiment 1 of this disclosure. Figure 1 Control input trajectory diagram of single-arm robotic arm 1;

[0025] Figure 3(b) shows Embodiment 1 of this disclosure. Figure 1 Control input trajectory diagram of the single-arm robotic arm 2;

[0026] Figure 3(c) shows Embodiment 1 of this disclosure. Figure 1 Control input trajectory diagram of a single-arm robotic arm 3;

[0027] Figure 3(d) shows Embodiment 1 of this disclosure. Figure 1 Control input trajectory diagram of the single-arm robotic arm 4;

[0028] Figure 4 This is a flowchart of the distributed collaborative control method according to Embodiment 1 of this disclosure. Detailed Implementation

[0029] The present disclosure will be further described below with reference to the accompanying drawings and embodiments.

[0030] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of this disclosure. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure pertains.

[0031] It should be noted that the terminology used herein is for descriptive purposes only and is not intended to limit the exemplary embodiments according to this disclosure. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof. It should be noted that, without conflict, the various embodiments and features within those embodiments can be combined with each other. The embodiments will now be described in detail with reference to the accompanying drawings.

[0032] This disclosure proposes a distributed cooperative consensus optimization control method for multi-agent robotic arm systems with state constraints. First, an equivalent unconstrained system is obtained through one-to-one state mapping. Then, a distributed observer is used to estimate the state of the leader system, decoupling the cooperative consensus problem of the multi-agent system into a tracking control problem of multiple independent agents. An augmented system is constructed, and the optimal cooperative control strategy is obtained with the goal of minimizing the cost function. Specific embodiments are described below.

[0033] Example 1

[0034] In one or more of the technical solutions disclosed in the embodiments, such as Figures 1 to 4 As shown, a distributed cooperative control method for multiple single-arm manipulators with state constraints includes the following steps:

[0035] Step S1: Model the dynamic characteristics of the multi-arm manipulator, use one-to-one state mapping to handle the time-varying asymmetric state constraint problem, and obtain an equivalent unconstrained system.

[0036] Step S2: Based on the defined unconstrained system, according to the obtained communication topology between each single-arm manipulator, design a distributed observer to estimate the state of the leader in the multi-single-arm manipulator system and establish a simplified unconstrained tracking control task for each follower.

[0037] Step S3: For each unconstrained tracking control task, the optimal control strategy is approximately solved using a single evaluation network adaptive dynamic algorithm, and each single-arm manipulator is coordinated and controlled according to the optimal control strategy.

[0038] In this embodiment, the cooperative control problem under state constraints is transformed into a traditional unconstrained tracking problem through a systematic transformation of the nonlinear mapping function and the cooperative design of distributed observers. Furthermore, the output constraints considered are asymmetric and time-varying, satisfying a more general range of constraints. Unlike existing adaptive dynamic programming algorithms based on execution and evaluation, this embodiment only requires an evaluation network, eliminating the approximation error of the execution network, resulting in a simpler algorithm implementation structure.

[0039] In step S1, the construction of the equivalent unconstrained system includes the following steps:

[0040] Step 11: Establish the kinematic equations of the multi-arm manipulator, and transform the kinematic equations into corresponding state-space equations with state constraints based on the physical characteristics of the single-arm manipulator.

[0041] Step 12: Establish a one-to-one state mapping relationship. Combine the state constraint information to map the constrained state variables one-to-one into new variables, thereby obtaining the equivalent unconstrained state space equation.

[0042] In step 11, the motion model of the single-arm manipulator is established, which is the kinematic equation of the single-arm manipulator, and the equation is as follows:

[0043]

[0044] Among them, g i It's the position of the robotic arm, u i It represents the control force of the robotic arm, where P represents the moment of inertia and Q represents the viscous friction force. For the mass of the load, It is the acceleration due to gravity. This is the length of the robotic arm.

[0045] Let x i1 =g i , The motion model of a single-arm robotic arm can be transformed into the following state-constrained state-space equations:

[0046]

[0047] Where, x i =[x i1 ,x i2 ] T , and q = [0, 1 / P] T State x ik Satisfying the time-varying asymmetric restricted range -b k (t) <x ik k (t), k = 1, 2; b k (t), B k (t) are time-varying functions of the upper and lower boundaries of the constraint, respectively;

[0048] For a multi-arm robotic system, the dynamic equations of the leader arm are assumed to be:

[0049]

[0050] Where, x0 = [x 01 ,x 02 ] T Let f0(x0) represent the leader state, which is a known nonlinear function.

[0051] In step 12, the position of the single-arm robotic hand is expressed as x in the state-space equation. ik With the introduced mapped variable s ik Establish a one-to-one state mapping relationship, specifically as follows:

[0052] s ik ​=A(x ik ,b k B k (4)

[0053] in,

[0054] b k B k These are the time-varying functions constraining the upper and lower boundaries, respectively, which represent the state-constrained information;

[0055] The system state variable x ik Mapped to a new state variable s ik .

[0056] At the same time, based on the one-to-one mapping relationship, we can obtain:

[0057] x ik =A -1 (s ik ,b k B k (5)

[0058] in,

[0059] Based on the transformed constrained state variables and mapping relationships, the state-constrained state-space equation of the multi-arm manipulator, i.e., formula (2), is transformed into an equivalent unconstrained state-space equation, as follows:

[0060]

[0061] Among them, s i =[s i1 ,s i2 ] T ;

[0062]

[0063]

[0064]

[0065] r i (s i )=diag{r i1 (s i1 ),r i2 (s i2 )};

[0066] in:

[0067]

[0068]

[0069] b = [b1, b2] T ;

[0070] B = [B1, B2] T ;

[0071] k = 1, 2;

[0072] b k B k Let be the time-varying functions constraining the upper and lower boundaries, respectively, and k = 1, 2 indicate that the robotic arm system is a second-order system.

[0073] Similarly, for the leader of a multi-arm robotic system, the one-to-one mapping relationship is defined as follows:

[0074] s 0k =A(x 0k ,b k B k ), k=1,2; (7)

[0075] The leader system of a multi-arm robotic system is transformed into an equivalent unconstrained state-space equation. The transformed equation is as follows:

[0076]

[0077] in:

[0078] s0=[s 01 ,s 02 ] T s0 is the newly constructed state vector;

[0079]

[0080]

[0081] r0(s0)=diag{r 01 (s 01 ),r 02 (s 02 )};

[0082]

[0083]

[0084] In step 2, the communication topology between each single-arm robot is determined, and the communication connection relationship between each single-arm robot is described using a directed graph G = (Γ,Θ,Λ).

[0085] Optionally, in the directed graph G=(Γ,Θ,Λ), each single-arm robot is taken as a node, the connection between nodes is taken as an edge, and the communication relationship between the single-arm robots is taken as the edge weight. The relationship between the leader and the follower in multiple single-arm robots is represented by a Laplace matrix. The weight of the edge that can establish a communication connection with the node is set to 1, otherwise it is set to 0.

[0086] Specifically, a directed graph G = (Γ,Θ,Λ) is used to describe the communication connections between each single-arm robotic arm. Where Γ = {μ1,...,μ...} N} represents the set of nodes in graph G. Let Λ represent the set of directed edges in the graph. im ]∈R N×N Let v represent the weight matrix of the directed graph G. If robot i can receive information from robot j, then v ij =1 (i≠j), otherwise, v ij =0; The relationship between the leader and follower (i.e., manipulator i) in a multi-armed robot can be represented by a Laplace matrix. Defined as Where Π=diag{κ1,...,κ} N}, The connection between the robotic arm i and the leader is represented by a diagonal matrix M = diag{ξ1,...,ξ}. N}, where ξ i =1 means that follower i can obtain information from the leader; otherwise, ξ = 1. i =0.

[0087] In step 2, a simplified unconstrained tracking control task is established for each follower in the multi-arm robotic system, including the following steps:

[0088] Step 21: Based on the communication topology of each robot, design a distributed observer to provide estimated leader state information for each follower, simplifying the consistency control problem into an independent tracking control problem for each robot.

[0089] Specifically, for the i-th single-arm manipulator, a distributed observer is designed based on the communication topology between the single-arm manipulators, as follows:

[0090]

[0091] Among them, consistency error This is an estimate of s0; In equation (8), F0(s0) is replaced by s0. C is the design parameter, N i Let m be the set of neighboring smart agents m of smart agent i;

[0092] Step 22: Construct an augmented system, building an independent cost function for each manipulator based on tracking error and control energy cost.

[0093] Specifically, the current position s of the single-arm robotic arm i i With estimated location The difference is used as the tracking error signal, as follows:

[0094]

[0095] Differentiate the error signal:

[0096]

[0097] Define augmented state The augmented system dynamic equations can be obtained as follows:

[0098]

[0099] in, and

[0100] To converge the tracking error while optimizing the control cost of each agent, the cost function for the i-th manipulator is defined as follows:

[0101]

[0102] Where, γ i >0 is the discount factor. Q i ∈R n×n and R i ∈R m×m U is a positive definite matrix. i It is the control force of the robotic arm, augmented state.

[0103] In step 3, for each independent cost function of the robot obtained in step 2, the optimal control strategy is approximately solved using a single-evaluation network adaptive dynamic programming algorithm. Based on the defined cost function, the HJB equation is derived, which is the Hamilton-Jacobi-Bellman equation. Then, the single-evaluation network adaptive dynamic programming algorithm is used to solve the HJB equation to obtain the optimal control strategy, including the following process:

[0104] Step 31: Construct a neural network as a single evaluation network, set the approximate optimal cost function of the evaluation neural network, and the weight update law of the evaluation neural network.

[0105] Optionally, the Bellman optimality principle can be used to determine the optimal control strategy, as follows:

[0106]

[0107] in, Ω u It is a set of allowed controllers.

[0108] By constructing a neural network as the evaluation network and using the evaluation network to approximate the optimal cost function, we can obtain:

[0109]

[0110] in, It is the ideal weight vector. It is an incentive condition, n ci ε is the number of nodes in the neural network. ci (X i ) represents the reconstruction error.

[0111] The approximate optimal control strategy is:

[0112]

[0113] in, It is an approximation of the ideal weight.

[0114] Optionally, the weight update law of the neural network is designed as follows:

[0115]

[0116] Where: α ci The learning rate;

[0117]

[0118]

[0119] Step 32: For each robot's independent cost function, the optimal control strategy is approximated by a single evaluation network adaptive dynamic algorithm. This involves using the evaluation network to approximate the optimal cost function, thereby obtaining the approximate optimal control strategy, and then designing the evaluation network weight update law.

[0120] The solution method of this embodiment is different from the existing execution-evaluation-based adaptive dynamic programming algorithm. This embodiment only requires one evaluation network, eliminating the approximation error of the execution network, and the algorithm implementation structure is simpler. The execution-evaluation adaptive dynamic programming algorithm approximates the optimal controller through the execution network and approximates the optimal cost function through the evaluation network. The single-evaluation network adaptive dynamic programming algorithm adopted in this embodiment only needs to evaluate the approximate optimal cost function. Since there is no execution network, the approximation error of the execution network is eliminated.

[0121] To prove the effectiveness of this embodiment, the following simulation verification is carried out:

[0122] In this simulation experiment, the control objective is to design a distributed cooperative consensus optimization control method so that the single-arm manipulator with state constraints can achieve consensus with the leader. The virtual leader dynamic equation considered in this example is:

[0123]

[0124] where the initial state x0(0) = [0, 0.5] T . The state of the i-th manipulator is restricted by -b1(t) < x i1 < B1(t) and -b2(t) < x i2 < B2(t), where b1(t) = 1 + e -0.3t , B1(t) = 0.85 + e -0.3t , b2(t) = 0.85 + e -0.3t and B2(t) = 1 + e -0.3t .

[0125] The manipulator parameters are specifically selected as P = 1, Q = 1 and The design parameters of the distributed observer are selected as C = 9. The design parameters of the cost function are selected as Q i = diag{10, 10}, R i = 0.1 and γ i = 0.5. For the evaluation network weight update law, the learning rate α ci = 2 is set.

[0126] Result analysis:

[0127] Select the Lyapunov function to analyze the stability of the results:

[0128] Taking the derivative, we can get . According to the Lyapunov stability theorem, all signals of the system are semi-globally uniformly bounded, and the multi-single-arm manipulator consensus error can converge to the neighborhood centered at the origin.

[0129] Figure 1This is a communication topology diagram of an example multi-single-arm robotic system, where 0 is the leader and the rest are followers;

[0130] In Figure 2, x 01 It's the leader's trajectory, the rest are x 11 x 21 x 31 With x 41 It is the follower trajectory, η 11 and η 21 Figure 2 shows that each single-arm manipulator can track the leader's trajectory well without violating the state constraint requirements. Figures 3(a)-3(d) The trajectory of the control input and the control force of the robot arm is given.

[0131] Example 2

[0132] Based on Embodiment 1, this embodiment provides a distributed collaborative control system for multiple single-arm robotic arms with state constraints, including:

[0133] Mapping and Transformation Module: Configured to model the dynamic characteristics of multiple single-arm manipulators, and use one-to-one state mapping to handle time-varying asymmetric state constraint problems to obtain equivalent unconstrained systems;

[0134] Unconstrained tracking control task establishment module: It is configured to design a distributed observer based on a deterministic unconstrained system and the communication topology between each single-arm manipulator, based on the obtained communication topology between the manipulators, to estimate the state of the leader in the multi-manipulator system and establish a simplified unconstrained tracking control task for each follower.

[0135] The solution module is configured to approximate the optimal control strategy for each unconstrained tracking control task using a single-evaluation network adaptive dynamic algorithm, and then perform coordinated control of each single-arm manipulator based on the optimal control strategy.

[0136] It should be noted that each module in this embodiment corresponds one-to-one with each step in embodiment 1, and their specific implementation process is the same, so it will not be repeated here.

[0137] Example 3

[0138] This embodiment provides an electronic device, including a memory and a processor, as well as computer instructions stored in the memory and running on the processor. When the processor executes the computer instructions, it performs the steps described in the method of Embodiment 1.

[0139] Example 4

[0140] This embodiment provides a computer-readable storage medium for storing computer instructions, which, when executed by a processor, complete the steps described in the method of Embodiment 1.

[0141] The above description is merely a preferred embodiment of this disclosure and is not intended to limit this disclosure. Various modifications and variations can be made to this disclosure by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this disclosure should be included within the scope of protection of this disclosure.

[0142] While the specific embodiments of this disclosure have been described above in conjunction with the accompanying drawings, this is not intended to limit the scope of protection of this disclosure. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of this disclosure are still within the scope of protection of this disclosure.

Claims

1. A distributed collaborative control method for multiple single-arm robotic arms with state constraints, characterized in that, Includes the following steps: The dynamic characteristics of multiple single-arm manipulators are modeled, and the time-varying asymmetric state constraint problem is handled by one-to-one state mapping to obtain an equivalent unconstrained system. Based on a deterministic unconstrained system, and according to the communication topology between each single-arm manipulator, a distributed observer is designed to estimate the state of the leader in the multi-single-arm manipulator system and establish a simplified unconstrained tracking control task for each follower. For each unconstrained tracking control task, an adaptive dynamic algorithm with a single evaluation network is used to approximate the optimal control strategy, and each single-arm manipulator is coordinated and controlled according to the optimal control strategy. The construction of an equivalent unconstrained system includes the following steps: Establish the kinematic equations of the multi-arm manipulator, and transform the kinematic equations into corresponding state-space equations with state constraints based on the physical characteristics of the single-arm manipulator. Establish a one-to-one state mapping relationship, and combine it with state constraint information to map the constrained state variables one-to-one into new variables, thereby obtaining the equivalent unconstrained state space equations. The state-space equations of a multi-arm robotic arm with constrained states are transformed into equivalent unconstrained state-space equations, as follows: ； ； ； ； ； ； ； ； ； ； in, P represents the moment of inertia; These are the time-varying functions constraining the upper and lower boundaries, respectively. k=1,2 This indicates that the robotic arm system is a second-order system; It is the control force of the robotic arm; The leader system of a multi-arm robotic system is transformed into an equivalent unconstrained state-space equation. The transformed equation is as follows: ； in: , For the newly constructed state vector, i=0 represents the leader; ； ； ； ； 。 2. The distributed collaborative control method for multiple single-arm robotic arms with state constraints as described in claim 1, characterized in that: For the expression of the position of a single-arm robotic arm in the state-space equation and the introduced mapped variables, a one-to-one state mapping relationship is established; Based on the one-to-one mapping relationship, the restricted state variables are transformed into new variables; Based on the transformed constrained state variables and mapping relationships, the state-constrained state-space equations of the multi-arm manipulator are transformed into equivalent unconstrained state-space equations.

3. The distributed collaborative control method for multiple single-arm robotic arms with state constraints as described in claim 1, characterized in that: Determine the communication topology between each single-arm robot and use a directed graph to describe the communication connection relationship between each single-arm robot.

4. The distributed collaborative control method for multiple single-arm manipulators with state constraints as described in claim 3, characterized in that: In the directed graph, each single-arm robot is considered as a node, the lines connecting the nodes are considered as edges, and the communication relationship between the single-arm robots is considered as the edge weight. The relationship between the leader and the follower among multiple single-arm robots is represented by a Laplace matrix, which serves as the weight of the edges between the leader and the follower. The edge weight is set to 1 for nodes that can establish a communication connection, and 0 otherwise.

5. The distributed collaborative control method for multiple single-arm robotic arms with state constraints as described in claim 1, characterized in that, Establish a simplified, unconstrained tracking control task for each follower in a multi-arm robotic system, including the following steps: Based on the communication topology of each robot, a distributed observer is designed to provide each follower with estimated leader state information, simplifying the consistency control problem into an independent tracking control problem for each robot. An augmented system is constructed, and an independent cost function is built for each manipulator based on tracking error and control energy cost.

6. The distributed collaborative control method for multiple single-arm robotic arms with state constraints as described in claim 1, characterized in that: A neural network is constructed as a single evaluation network. The HJB equation is derived through a defined cost function. The evaluation neural network is used to approximate the optimal cost function. The weight update law of the evaluation neural network is set. The HJB equation is solved by the single evaluation network adaptive dynamic programming algorithm to obtain the optimal control strategy.

7. A distributed collaborative control system for multiple single-arm robotic arms with state constraints, characterized in that, include: Mapping and Transformation Module: Configured to model the dynamic characteristics of multiple single-arm manipulators, using one-to-one state mapping to handle time-varying asymmetric state constraint problems, and obtain an equivalent unconstrained system; Unconstrained tracking control task establishment module: It is configured to design a distributed observer based on a deterministic unconstrained system and the communication topology between each single-arm manipulator, based on the obtained communication topology between the manipulators, to estimate the state of the leader in the multi-manipulator system and establish a simplified unconstrained tracking control task for each follower. The solution module is configured to approximate the optimal control strategy for each unconstrained tracking control task using a single-evaluation network adaptive dynamic algorithm, and then perform coordinated control of each single-arm manipulator based on the optimal control strategy. The state-space equations of a multi-arm robotic arm with constrained states are transformed into equivalent unconstrained state-space equations, as follows: ； ； ； ； ； ； ； ； ； ； in, P represents the moment of inertia; These are the time-varying functions constraining the upper and lower boundaries, respectively. k=1,2 This indicates that the robotic arm system is a second-order system; It is the control force of the robotic arm; The leader system of a multi-arm robotic system is transformed into an equivalent unconstrained state-space equation. The transformed equation is as follows: ； in: , For the newly constructed state vector, i=0 represents the leader; ； ； ； ； 。 8. An electronic device, characterized in that, It includes a memory and a processor, as well as computer instructions stored in the memory and running on the processor, which, when executed by the processor, perform the steps of any one of claims 1-6.

9. A computer-readable storage medium, characterized in that, Used to store computer instructions, which, when executed by a processor, perform the steps of any one of claims 1-6.

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