Robust distributed game control method and system for multi-robot system

Abstract

The invention discloses a robust distributed game control method and system for a multi-robot system, relates to the technical field of multi-robot control, and aims to overcome the defect that a multi-robot optimization control algorithm based on Bayesian online estimation cannot ensure that a cost function of each robot realizes Nash equilibrium optimization in literatures. And kinetic model parameters of the system are required to be known. According to the technical key points, firstly, a multi-robot cluster dynamic model containing modeling uncertain parameters is constructed, and the modeling uncertain parameters in the model are decomposed based on a matrix decomposition theory; aiming at the constructed multi-robot cluster system dynamic model, designing a cluster tracking error system based on a navigation-tracking mechanism, and constructing a dynamic performance cost function of each robot based on a non-cooperative game optimization idea; aiming at the constructed cluster tracking error system and the dynamic performance cost function, based on the constructed cluster tracking error system and the dynamic performance cost function, solving a robust distributed game controller by utilizing an optimal control theory; deducing controller parameters capable of ensuring global Nash equilibrium of the dynamic performance cost function of the system by using a linear quadratic regulation control theory; the Lyapunov stability theory is utilized to prove that the designed robust distributed game controller can ensure the asymptotic stability of a cluster tracking error system.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-robot control, and in particular to a robust distributed game control method and system for a multi-robot system. Background Art

[0002] A multi-robot swarm is a network system consisting of multiple autonomous robots that can complete complex collaborative tasks through information exchange. In some special application contexts, non-cooperative game characteristics are reflected between robots, such as racing driving, task decision-making, pursuit and escape game problems, etc.

[0003] Many solutions are given in the prior art for the optimization control of multi-robot clusters. For example, the prior art document with the document number CN112394719B (authorized) discloses a multi-robot system formation control device and method based on sampled data. The device includes an image acquisition and processing module, a sensor module, a wireless communication module and a control module. The method is as follows: first, multiple robots are arranged in a preset multi-robot network and the robots are numbered, and the system communication topology is set; then, a motion model of the robot is established, formation information is given, and control targets are established; then, a formation control algorithm based on sampled data and an observer is constructed; finally, each robot periodically obtains the position information of the neighbor or leader through the wireless communication network, fuses the collected data, executes the formation algorithm, and realizes formation control. The invention is used for measuring robot speed information, which is convenient and easy, and the measurement results are accurate. Formation control is realized through sampled data, saving communication costs and network resources. The prior art document with the document number CN116149371A discloses a multi-robot three-dimensional tracking and control platform based on a visual sensor network, including four main modules: a multi-camera system, a ground control system, an airborne infrared reflective ball and a target drone. The multi-camera system includes multiple cameras with infrared filters, infrared supplementary light sources for enhancing light intensity, synchronization triggers for synchronizing image data of multiple cameras, and a navigation terminal computer to run the core visual algorithm. Multiple cameras interact with each other to form a visual sensor network. The ground control system receives navigation data from the multi-camera system and then uploads control commands to the target drone via a wireless LAN. Data transmission utilizes the topic subscription and publishing mechanism of the robot operating system ROS to process navigation data, calculate control instructions for the drone in combination with reference trajectories and controllers, and finally publish the instructions via a wireless LAN. The above-mentioned prior art does not mention non-cooperative game control of multi-robot clusters.

[0004] The document "Bayesian graphical games for synchronization in networks of dynamical systems, IEEE Transactions on Control of Network Systems, 2020, 7(2): 1028-1039" discloses a multi-robot cluster non-cooperative game control method based on the pilot-tracking mechanism. According to the "maximum-minimum" idea, the document designs a multi-robot optimization control algorithm based on Bayesian online estimation to ensure that multiple follower robots achieve the same state as the leader robot. However, the technical problem with the optimization control algorithm described in the document is that it cannot guarantee that the performance cost function of each robot achieves the optimal Nash equilibrium, and requires that the system's dynamic model parameters are known, which is very unfavorable for the application of the controller in actual systems. Summary of the invention

[0005] The technical problems to be solved by the present invention are:

[0006] In order to overcome the shortcomings of the multi-robot optimization control algorithm based on Bayesian online estimation in the literature that it cannot guarantee that the cost function of each robot achieves the optimal Nash equilibrium and requires the system's dynamic model parameters to be known, the present invention provides a robust distributed game control method for a multi-robot system.

[0007] The technical solution adopted by the present invention to solve the above technical problems is:

[0008] The method of the present invention first constructs a multi-robot cluster dynamics model containing modeling uncertain parameters, and decomposes the modeling uncertain parameters in the model based on the matrix decomposition theory; for the constructed multi-robot cluster system dynamics model, a cluster tracking error system based on the pilot-tracking mechanism is designed, and the dynamic performance cost function of each robot is constructed based on the non-cooperative game optimization idea; based on the constructed cluster tracking error system and the dynamic performance cost function, the robust distributed game controller is solved using the optimal control theory, and the controller parameters that can ensure the global Nash equilibrium of the system dynamic performance cost function are derived using the linear quadratic regulation control theory; and the Lyapunov stability theory is used to prove that the designed robust distributed game controller can ensure the asymptotic stability of the cluster tracking error system. The method provided by the present invention can ensure the optimal Nash equilibrium of the performance cost function of each robot in the case of modeling uncertain parameters, which is more conducive to the application of the controller in the actual system.

[0009] The present invention proposes a multi-robot system robust distributed game control method, which comprises the following steps:

[0010] Step 1: First, a multi-robot cluster dynamics model containing modeling uncertain parameters is constructed, and the modeling uncertain parameters in the model are decomposed based on the matrix decomposition theory;

[0011] Step 2: Based on the multi-robot swarm system dynamics model constructed in step 1, a swarm tracking error system based on the pilot-tracking mechanism is designed, and the dynamic performance cost function of each robot is constructed based on the non-cooperative game optimization idea;

[0012] Step 3: Based on the cluster tracking error system and dynamic performance cost function constructed in step 2, the robust distributed game controller is solved using optimal control theory, and the controller parameters that can ensure the global Nash equilibrium of the system dynamic performance cost function are derived using linear quadratic regulation control theory;

[0013] Step 4: Use Lyapunov stability theory to prove that the designed robust distributed game controller can ensure the asymptotic stability of the cluster tracking error system.

[0014] Step 1: First, build a multi-robot cluster dynamics model containing modeling uncertain parameters, and decompose the modeling uncertain parameters in the model based on matrix decomposition theory, specifically:

[0015] First, the following multi-robot cluster dynamics model is given:

[0016]

[0017] In the formula, is the state variable of the robot, For the control input, A and B represent the known system matrix parameters, ΔA is the norm-bounded modeling uncertainty parameter, and matrix decomposition technology is used to characterize it as ΔA = DΦE, where D and E are known matrix parameters, Φ is the unknown parameter matrix satisfying Φ T Φ≤I, I is the identity matrix with appropriate dimensions;

[0018] The dynamic model of the leader robot that provides the reference trajectory is defined as

[0019] Step 2: Based on the multi-robot swarm system dynamics model constructed in step 1, a swarm tracking error system based on the pilot-tracking mechanism is designed, and the dynamic performance cost function of each robot is constructed based on the non-cooperative game optimization idea, specifically:

[0020] The control goal of each robot is to achieve the same state as the leader robot in the presence of a decomposable modeling uncertainty parameter ΔA. The following cluster tracking error system based on the pilot-tracking mechanism is designed:

[0021]

[0022] In the formula, e ij It is used to describe the information interaction relationship between the i-th robot and the j-th robot, e ij =1 means that the i-th robot can directly obtain the status information of the j-th robot, otherwise e ij =0;

[0023] Select directed graph represents the communication topology between multiple robots, and it includes a directed spanning tree with the leader robot as the root node;

[0024] and The associated graph Laplacian matrix can be expressed as:

[0025]

[0026] In the formula, And l ij =-e ij ,i≠j;

[0027] Defining matrix variables According to graph theory, when a directed graph When there is a directed spanning tree in is a non-singular matrix, and The matrix Θ can be expressed as:

[0028]

[0029] In the formula, a and b are vectors whose elements are all positive;

[0030] Taking the derivative of equation (2), we have

[0031]

[0032] The global form of equation (5) is expressed as follows using the Kronecker product:

[0033]

[0034] in, because is a non-singular matrix, so the pilot-tracking state consistency of the multi-robot cluster system is lim if and only if the system (6) is stable. t→∞ x i =lim t→∞ x 0 , Established;

[0035] According to the non-cooperative game optimization idea, the dynamic performance cost function of the ith robot is defined as follows:

[0036]

[0037] In the formula, represents the control input of the neighboring robot of the ith robot, R i >0,Ω ij >0 is a positive definite matrix, Q i , Q ij and F ij is a weight matrix that satisfies the following conditions:

[0038]

[0039] Step 3: For the cluster tracking error system and dynamic performance cost function constructed in step 2, based on the constructed cluster tracking error system and dynamic performance cost function, the robust distributed game controller is solved using the optimal control theory, and the controller parameters that can ensure the global Nash equilibrium of the system dynamic performance cost function are derived using the linear quadratic regulation control theory, specifically:

[0040] First, according to equations (5) and (7), the following Hamiltonian function is selected:

[0041]

[0042] In the formula, represents the value function associated with equation (7);

[0043] Based on optimal control theory, the robust distributed game controller of the ith robot can be obtained by solving the equation

[0044]

[0045] Selecting the value function P i is a symmetric positive definite matrix, let the weight matrix in equation (7) be

[0046]

[0047] Among them, Q i >0 and R 0 >0 is a symmetric positive definite matrix, the variable c is the coupling gain parameter and satisfies Based on equation (11), the robust distributed game controller (10) can be rewritten as

[0048]

[0049] According to the linear quadratic control theory, if the matrix Pi is the unique positive definite solution to the following algebraic Riccati equation:

[0050]

[0051] Then the robust distributed game controller (12) can ensure that the dynamic performance cost function (7) has global Nash equilibrium characteristics;

[0052] In addition, in,

[0053]

[0054] Then the condition to ensure that inequality (8) holds can be expressed as selecting appropriate parameters so that as well as Established at the same time;

[0055] According to Schur's complement theorem in matrix theory, Equivalent to

[0056]

[0057] Established, substituting formula (11) into formula (14) yields

[0058]

[0059] Right now Therefore, when the weight matrix Ω ij Satisfy the conditions Sometimes, there is Established;

[0060] Further, conditional To ensure; In addition, when ΔA=DΦE, there is

[0061]

[0062] Right now Established.

[0063] Step 4: Use Lyapunov stability theory to prove that the designed robust distributed game controller can ensure the asymptotic stability of the cluster tracking error system. Specifically:

[0064] First, substitute the robust distributed game controller (12) into the system (6) and we have

[0065]

[0066] In the formula, The Lyapunov function associated with system (6) is selected as

[0067]

[0068] In the formula, I p represents the identity matrix with appropriate dimensions, and the derivative of equation (18) is

[0069]

[0070] Note that when When established, there were

[0071]

[0072] Substituting equation (20), equation (13), equation (16) and equation (20) into equation (19), we have

[0073]

[0074] In the formula, Representation Matrix The minimum singular value of σ max {P i} represents the matrix P i The maximum singular value of

[0075]

[0076] According to Lyapunov stability theory, lim t→∞ ξ i =0 holds, that is, lim t→∞ x i =lim t→∞ x 0 , holds; therefore, the robust distributed game controller in equation (12) can ensure that the multi-robot cluster system achieves the same state as the leader robot.

[0077] The beneficial effects of the present invention are:

[0078] The technical solution proposed in the present invention can fully ensure that the performance cost function of each robot achieves the optimal Nash equilibrium, and does not require the system's dynamic model parameters to be known, which is very conducive to the application and promotion of the controller in actual systems.

[0079] The method of the present invention first constructs a multi-robot cluster dynamics model containing modeling uncertain parameters, and decomposes the modeling uncertain parameters in the model based on the matrix decomposition theory; for the constructed multi-robot cluster system dynamics model, a cluster tracking error system based on the pilot-tracking mechanism is designed, and the dynamic performance cost function of each robot is constructed based on the non-cooperative game optimization idea; based on the constructed cluster tracking error system and the dynamic performance cost function, the robust distributed game controller is solved using the optimal control theory, and the controller parameters that can ensure the global Nash equilibrium of the system dynamic performance cost function are derived using the linear quadratic regulation control theory; and the Lyapunov stability theory is used to prove that the designed robust distributed game controller can ensure the asymptotic stability of the cluster tracking error system. The method provided by the present invention can ensure the optimal Nash equilibrium of the performance cost function of each robot in the case of modeling uncertain parameters, which is more conducive to the application of the controller in the actual system.

[0080] Using the multi-robot cluster robust distributed game control method proposed in the present invention, each follower robot successfully tracks the state of the leader robot in about 1500 seconds, and the state component x i1 、x i2 、x i3 、x i4 、x i5 The tracking error norm converges to zero asymptotically, and it has good stability. Therefore, the robust distributed game controller proposed in the present invention can achieve the consistency of the states of multiple follower robots and the leader robot under the condition of network communication topology constraints and modeling uncertain parameters, and can optimize the control input of the multi-robot cluster system. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 is a schematic diagram of a network communication topology structure between a navigator robot (labeled as 0) and five follower robots (labeled as 1 to 5) in an embodiment of the present invention;

[0082] Figure 2 is an evolution curve of the state norm of the leader robot and five tracker robots over time under the action of the robust distributed game controller proposed in the present invention in an embodiment of the present invention;

[0083] Figure 3 In the embodiment of the present invention, under the action of the robust distributed game controller proposed in the present invention, the state component x i1 The tracking error norm curve of

[0084] Figure 4 In the embodiment of the present invention, under the action of the robust distributed game controller proposed in the present invention, the state component x i2 The tracking error norm curve of

[0085] Figure 5 In the embodiment of the present invention, under the action of the robust distributed game controller proposed in the present invention, the state component x i3 The tracking error norm curve of

[0086] Figure 6 In the embodiment of the present invention, under the action of the robust distributed game controller proposed in the present invention, the state component x i4 The tracking error norm curve of

[0087] Figure 7 In the embodiment of the present invention, under the action of the robust distributed game controller proposed in the present invention, the state component x i5 The tracking error norm curve of

[0088] Figure 8 In the embodiment of the present invention, under the action of four groups of non-Nash equilibrium controllers and the robust distributed game controller designed by the present invention, the dynamic performance cost function J 2 Evolution curve over time.

[0089] It can be clearly seen from these figures that the robust distributed game control method for multi-robot systems described in the present invention has good stability and practicality. DETAILED DESCRIPTION

[0090] Combined with Figure 1-8 , the implementation of a multi-robot system robust distributed game control method of the present invention is described as follows:

[0091] Step 1: First, construct a multi-robot cluster dynamics model containing modeling uncertain parameters, and decompose the modeling uncertain parameters in the model based on matrix decomposition theory. First, give the following multi-robot cluster dynamics model:

[0092]

[0093] In the formula, is the state variable of the robot, For the control input, A and B represent the known system matrix parameters, ΔA is the norm-bounded modeling uncertainty parameter, and matrix decomposition technology is used to characterize it as ΔA = DΦE, where D and E are known matrix parameters, Φ is the unknown parameter matrix satisfying Φ T Φ≤I, I is a unit matrix with appropriate dimensions. In addition, the dynamic model of the leader robot that provides the reference trajectory is defined as For convenience, the time stamp (t) will be omitted in the subsequent description.

[0094] Step 2: Design a cluster tracking error system based on the pilot-tracking mechanism for the multi-robot cluster system dynamics model constructed in step 1, and construct the dynamic performance cost function of each robot based on the non-cooperative game optimization idea. In the present invention, the control goal of each robot is to achieve the state consistency of the multi-robot cluster system with the pilot robot when there is a decomposable modeling uncertainty parameter ΔA. Therefore, the following cluster tracking error system based on the pilot-tracking mechanism is designed:

[0095]

[0096] In the formula, e ij It is used to describe the information interaction relationship between the i-th robot and the j-th robot, e ij =1 means that the i-th robot can directly obtain the status information of the j-th robot, otherwise e ij =0.

[0097] Select directed graph represents the communication topology between multiple robots, and it contains a directed spanning tree with the leader robot as the root node. The associated graph Laplacian matrix can be expressed as:

[0098]

[0099] In the formula, And l ij =-e ij ,i≠j;

[0100] Defining matrix variables According to graph theory, when a directed graph When there is a directed spanning tree in is a non-singular matrix, and The matrix Θ can be expressed as:

[0101]

[0102] In the formula, a and b are vectors whose elements are all positive;

[0103] Taking the derivative of equation (2), we have

[0104]

[0105] The global form of equation (5) is expressed as follows using the Kronecker product:

[0106]

[0107] in, because is a non-singular matrix, so the pilot-tracking state consistency of the multi-robot cluster system is lim if and only if the system (6) is stable. t→∞ x i =lim t→∞ x 0 , Established.

[0108] According to the non-cooperative game optimization idea, the dynamic performance cost function of the ith robot is defined as follows:

[0109]

[0110] In the formula, represents the control input of the neighboring robot of the ith robot, R i >0,Ω ij >0 is a positive definite matrix. Q ij and F ij is a weight matrix that satisfies the following conditions:

[0111]

[0112] Step 3: For the cluster tracking error system and dynamic performance cost function constructed in step 2, the robust distributed game controller is solved by using the optimal control theory, and the controller parameters that can ensure the global Nash equilibrium of the system dynamic performance cost function are derived by using the linear quadratic regulation control theory. First, the following Hamiltonian function is selected according to equations (5) and (7):

[0113]

[0114] In the formula, represents the value function associated with equation (7).

[0115] Based on optimal control theory, the robust distributed game controller of the ith robot can be obtained by solving the equation

[0116]

[0117] Selecting the value function P i is a symmetric positive definite matrix, let the weight matrix in equation (7) be

[0118]

[0119] Among them, Q i >0 and R 0>0 is a symmetric positive definite matrix, the variable c is the coupling gain parameter and satisfies Based on equation (11), the robust distributed game controller (10) can be rewritten as

[0120]

[0121] According to the linear quadratic control theory, if the matrix P i is the unique positive definite solution to the following algebraic Riccati equation:

[0122]

[0123] Then the robust distributed game controller (12) can ensure that the dynamic performance cost function (7) has global Nash equilibrium characteristics.

[0124] In addition, in,

[0125]

[0126] Then the condition to ensure that inequality (8) holds can be expressed as selecting appropriate parameters so that as well as Established at the same time.

[0127] According to Schur's complement theorem in matrix theory, Equivalent to

[0128]

[0129] Established, substituting formula (11) into formula (14) yields

[0130]

[0131] Right now Therefore, when the weight matrix Ω ij Satisfy the conditions Sometimes, there is Established.

[0132] Further, conditional To ensure. In addition, when ΔA=DΦE,

[0133]

[0134] Right now Established.

[0135] Step 4: Use Lyapunov stability theory to prove that the designed robust distributed game controller can ensure the asymptotic stability of the cluster tracking error system. First, substitute the robust distributed game controller (12) into the system (6) to obtain

[0136]

[0137] In the formula, The Lyapunov function associated with system (6) is selected as

[0138]

[0139] In the formula, I p represents the identity matrix with appropriate dimensions. The derivative of equation (18) is

[0140]

[0141] Note that when When established, there were

[0142]

[0143] Substituting equation (20), equation (13), equation (16) and equation (20) into equation (19), we have

[0144]

[0145] In the formula, Representation Matrix The minimum singular value of . Further, let σ max {P i} represents the matrix P i The maximum singular value of

[0146]

[0147] According to Lyapunov stability theory, lim t→∞ ξ i =0 holds, that is, lim t→∞ x i =lim t→∞ x 0 , Therefore, the robust distributed game controller in equation (12) can ensure that the multi-robot cluster system achieves the same state as the leader robot.

[0148] The following examples are used to verify the beneficial effects of the present invention (such as Figures 1 to 8 ):

[0149] Consider a cluster system consisting of one leader robot (labeled as 0) and five follower robots (labeled as 1 to 5). The communication topology between the robots is as follows: Figure 1 As shown, the Laplace matrix corresponding to the communication topology is expressed as follows:

[0150]

[0151] The dynamic model parameters of each robot system are selected as follows:

[0152]

[0153]

[0154] Modeling uncertainty parameter ΔA=DΦE, where:

[0155]

[0156]

[0157] α 1 , α 2 , α 3 , α 4 and α 5 is a bounded time-varying component, β 1 , β 2 , β 3 , β 4 and β 5 are their corresponding upper bounds. Let a =

[11111] T , b = [10.10.10.10.1] T , we can calculate Θ = diag{5,3.64,2.5,1.54,0.71}. According to the inequality It can be calculated that the upper limit of the coupling gain parameter is c ≥ 9.98, so c = 10 is selected. 0 =1×10 3 I 3 , Q i =1×10 -6 I 6 , i=1,...,5, the initial states of each robot are

[0158] x 0 (0) = [-2000, 2000, -2000, -20, 20, -20] T ,

[0159] x 1 (0) = [500, 600, 700, 5, 10, 10] T ,

[0160] x 2 (0) = [1000, 1000, 1000, 20, 10, 10] T ,

[0161] x 3 (0) = [-1000,-1000,1000,-10,-10,10] T ,

[0162] x 4 (0) = [-1000, 1200, 1500, -10, 20, 20] T ,

[0163] x 5 (0) = [-2000, 2000, -2000, -20, 20, -20] T .

[0164] Then, under the action of the robust distributed game controller proposed in the present invention, the evolution curve of the state norm of the leader robot and the five follower robots over time and the state component x can be obtained. i1 、x i2 、x i3 、x i4 、x i5 The evolution curve of the tracking error norm over time. From the simulation curve, we can see that the five follower robots successfully tracked the state of the leader robot at about 1500 seconds, and the state component x i1 、x i2 、x i3 、x i4 、x i5 The tracking error norm converges to zero asymptotically, which has good stability. Therefore, the robust distributed game controller proposed in the present invention can achieve the consistency of the states of multiple follower robots and the leader robot when the system has modeling uncertain parameters. In addition, in order to further verify the advantages of the control method proposed in the present invention, the dynamic performance cost function J under the action of multiple groups of distributed game controllers is also given. 2 The time evolution curve shows that the four groups of non-Nash equilibrium controllers can only ensure the performance cost function J 2 The final values ​​of are 4079, 2747, 2981 and 3628, and the Nash equilibrium robust distributed game controller proposed in the present invention can ensure that J 2 The final value of is 2617, so the method proposed in the present invention can minimize (optimize) the dynamic performance cost function. In the above formula, each row vector represents the position and posture of each robot.

[0165] The contents not described in detail in the present invention (such as graph theory, optimal control theory, Lyapunov stability theory, matrix theory) are common knowledge in the field.

[0166] The above embodiments are simulation experiments conducted by taking a space robot as an example.

[0167] It has been verified that the method proposed in the present invention solves the technical problem proposed in the present invention. The method described in the present invention has been verified through simulation tests and practical applications, and the technical effects described in the present invention have been verified.

[0168] The algorithm (method) proposed in the present invention is the underlying technical core of the present invention, and various products can be derived based on the algorithm.

[0169] Based on the algorithm (method) proposed in the present invention, a robust distributed game control system for a multi-robot system is developed using a programming language. The system has program modules corresponding to the steps of the above-mentioned technical solution, and executes the steps in the above-mentioned robust distributed game control method for a multi-robot system during operation.

[0170] The computer program of the developed system (software) is stored on a computer-readable storage medium, and the computer program is configured to implement the steps of the above-mentioned method for robust distributed game control of a multi-robot system when called by a processor, that is, the present invention is materialized on a carrier to become a computer program product.

[0171] A robust distributed game control device for a multi-robot system, the device comprising at least one processor and a memory communicatively connected to the at least one processor, wherein the memory stores instructions executable by the at least one processor, and the instructions are executed by the at least one processor so that the at least one processor can execute the above-mentioned robust distributed game control method for the multi-robot system to achieve robust distributed game control of the multi-robot system.

[0172] Various implementations of the systems and techniques described herein can be realized in digital electronic circuit systems, integrated circuit systems, dedicated ASICs (application specific integrated circuits), computer hardware, firmware, software, and / or combinations thereof. These various implementations can include: being implemented in one or more computer programs that can be executed and / or interpreted on a programmable system including at least one programmable processor, which can be a special purpose or general purpose programmable processor that can receive data and instructions from a storage system, at least one input device, and at least one output device, and transmit data and instructions to the storage system, the at least one input device, and the at least one output device.

[0173] The computer programs (also referred to as programs, software, software applications, or codes) of the present invention include machine instructions for programmable processors, and these computer programs can be implemented using high-level procedural and / or object-oriented programming languages, and / or assembly / machine languages. As used in the present invention, the terms "machine-readable medium" and "computer-readable medium" refer to any computer program product, device, and / or device (e.g., disk, optical disk, memory, programmable logic device PLD) for providing machine instructions and / or data to a programmable processor, including a machine-readable medium that receives machine instructions as machine-readable signals. The term "machine-readable signal" refers to any signal for providing machine instructions and / or data to a programmable processor.

[0174] It should be understood that although the present invention is described according to a limited number of embodiments, those skilled in the art, with the benefit of the above description, understand that other embodiments can be envisioned within the scope of the present invention described herein. Modifications, partial replacements and application expansions made to the description and drawings of the present invention, or direct or indirect application of the present invention in other related technical fields, should all be included in the scope of protection of the present invention patent.

Claims

1. A robust distributed game control method for a multi-robot system, characterized in that: The implementation process of the method is: Step 1: This method first constructs a multi-robot cluster dynamics model containing modeling uncertain parameters, and decomposes the modeling uncertain parameters in the model based on matrix decomposition theory; Step 2: Based on the constructed multi-robot swarm system dynamics model, a swarm tracking error system based on the pilot-tracking mechanism is designed, and the dynamic performance cost function of each robot is constructed based on the non-cooperative game optimization idea; Step 3: Based on the constructed cluster tracking error system and dynamic performance cost function, the optimal control theory is used to solve the robust distributed game controller, and the linear quadratic regulation control theory is used to derive the controller parameters that can ensure the global Nash equilibrium of the system dynamic performance cost function; Step 4: Use Lyapunov stability theory to prove that the designed robust distributed game controller can ensure the asymptotic stability of the cluster tracking error system.

2. A multi-robot system robust distributed game control method according to claim 1, characterized in that: Step 1: First, build a multi-robot cluster dynamics model containing modeling uncertain parameters, and decompose the modeling uncertain parameters in the model based on matrix decomposition theory, specifically: First, the following multi-robot cluster dynamics model is given: In the formula, is the state variable of the robot, For the control input, A and B represent the known system matrix parameters, ΔA is the norm-bounded modeling uncertainty parameter, and matrix decomposition technology is used to characterize it as ΔA = DΦE, where D and E are known matrix parameters, Φ is the unknown parameter matrix satisfying Φ T Φ≤I, I is the identity matrix with appropriate dimensions; The dynamic model of the leader robot that provides the reference trajectory is defined as 3. A multi-robot system robust distributed game control method according to claim 2, characterized in that: Step 2: Based on the multi-robot swarm system dynamics model constructed in step 1, a swarm tracking error system based on the pilot-tracking mechanism is designed, and the dynamic performance cost function of each robot is constructed based on the non-cooperative game optimization idea, specifically: The control goal of each robot is to achieve the same state as the leader robot in the presence of a decomposable modeling uncertainty parameter ΔA. The following cluster tracking error system based on the pilot-tracking mechanism is designed: In the formula, e ij It is used to describe the information interaction relationship between the i-th robot and the j-th robot, e ij =1 means that the i-th robot can directly obtain the status information of the j-th robot, otherwise e ij =0; Select directed graph represents the communication topology between multiple robots, and it includes a directed spanning tree with the leader robot as the root node; and The associated graph Laplacian matrix can be expressed as: In the formula, And l ij =-e ij ,i≠j; Defining matrix variables According to graph theory, when a directed graph When there is a directed spanning tree in is a non-singular matrix, and The matrix Θ can be expressed as: In the formula, a and b are vectors whose elements are all positive; Taking the derivative of equation (2), we have The global form of equation (5) is expressed as follows using the Kronecker product: in, because is a non-singular matrix, so the pilot-tracking state consistency of the multi-robot cluster system is lim if and only if the system (6) is stable. t→∞ x i =lim t→∞ x0, Established; According to the non-cooperative game optimization idea, the dynamic performance cost function of the ith robot is defined as follows: In the formula, represents the control input of the neighboring robot of the ith robot, R i >0,Ω ij >0 is a positive definite matrix. Q ij and F ij is a weight matrix that satisfies the following conditions:

4. A multi-robot system robust distributed game control method according to claim 3, characterized in that: Step 3: For the cluster tracking error system and dynamic performance cost function constructed in step 2, based on the constructed cluster tracking error system and dynamic performance cost function, the robust distributed game controller is solved using the optimal control theory, and the controller parameters that can ensure the global Nash equilibrium of the system dynamic performance cost function are derived using the linear quadratic regulation control theory, specifically: First, according to equations (5) and (7), the following Hamiltonian function is selected: In the formula, represents the value function associated with equation (7); Based on optimal control theory, the robust distributed game controller of the ith robot can be obtained by solving the equation get: Selecting the value function P i is a symmetric positive definite matrix, let the weight matrix in equation (7) be Among them, Q i >0 and R0>0 are symmetric positive definite matrices, and the variable c is the coupling gain parameter and satisfies Based on equation (11), the robust distributed game controller (10) can be rewritten as According to the linear quadratic control theory, if the matrix P i is the unique positive definite solution to the following algebraic Riccati equation: Then the robust distributed game controller (12) can ensure that the dynamic performance cost function (7) has global Nash equilibrium characteristics; In addition, in, Then the condition to ensure that inequality (8) holds can be expressed as selecting appropriate parameters so that as well as Established at the same time; According to Schur's complement theorem in matrix theory, Equivalent to Established, substituting formula (11) into formula (14) yields Right now Therefore, when the weight matrix Ω ij Satisfy the conditions Sometimes, there are Established; Further, Conditions available To ensure; In addition, when ΔA=DΦE, there is Right now Established.

5. A multi-robot system robust distributed game control method according to claim 4, characterized in that: Step 4: Use Lyapunov stability theory to prove that the designed robust distributed game controller can ensure the asymptotic stability of the cluster tracking error system. Specifically: First, substitute the robust distributed game controller (12) into the system (6) and we have In the formula, The Lyapunov function associated with system (6) is selected as In the formula, I p represents the identity matrix with appropriate dimensions, and the derivative of equation (18) is Note that when When established, there were Substituting equation (20), equation (13), equation (16) and equation (20) into equation (19), we have In the formula, Representation Matrix The minimum singular value of σ max {P i } represents the matrix P i The maximum singular value of According to Lyapunov stability theory, lim t→∞ ξ i =0 holds, that is, lim t→∞ x i =lim t→∞ x0, holds; therefore, the robust distributed game controller in equation (12) can ensure that the multi-robot cluster system achieves the same state as the leader robot.

6. A multi-robot system robust distributed game control system, characterized by: The system has a program module corresponding to the steps of any one of claims 1 to 5 above, and executes the steps in the robust distributed game control method for a multi-robot system when running.

7. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program, and the computer program is configured to implement the steps of a robust distributed game control method for a multi-robot system according to any one of claims 1 to 5 when called by a processor.

Images