A false data injection attack method for a multi-elastic-joint robot system

Abstract

The application provides a false data injection attack method for a multi-elastic joint robot system, and belongs to the technical field of multi-agent network attacks based on computer data processing; firstly, a nonlinear dynamic mathematical model of the multi-elastic joint robot is constructed, and the model is converted into a high-order full-drive pseudo-linear multi-robot system dynamic model based on a full-drive system theory; then, a distributed consistency controller containing a linearization term and a neighbor error related term is designed, a closed-loop global dynamics is established in combination with model equivalent conversion, and a state analytical solution is derived. An attack strategy of false data injection is designed, a system dynamic model after being attacked is constructed, a closed-loop terminal state error with or without attack is taken as an objective function, an attack optimization problem is converted into a combination optimization problem, a greedy algorithm is designed by using a submodular optimization theory, a suboptimal solution in a polynomial time is obtained, and the suboptimal solution is used as an optimal scheme of the false data injection attack of the system. The application improves the applicability of a complex nonlinear multi-agent system and reduces the calculation complexity.

Description

Technical Field

[0001] This invention belongs to the field of multi-agent network attack technology based on computer data processing, and particularly relates to a method for injecting false data into a multi-elastic joint robot system. Background Technology

[0002] In recent years, multi-agent systems have been widely applied in various cutting-edge fields, with collaborative consensus being a core research foundation. However, the heavy reliance of these systems on communication networks exposes them to severe security threats, particularly highly covert spoofing attacks, which can maliciously tamper with control information, leading to severe performance degradation or even system collapse. Considering that attackers typically face resource constraints in real-world scenarios, how to accurately select attack nodes to maximize destructive effects within a limited budget is a pressing technical problem. Furthermore, existing attack and defense designs are mostly aimed at linear systems, while real-world agents often exhibit complex nonlinear dynamic characteristics. Therefore, this paper explores optimal spoofing attack node selection strategies under resource constraints for real-world nonlinear multi-agent systems, providing extreme-case test models and theoretical basis for security assessment and defense strategy design of multi-agent systems. Summary of the Invention

[0003] To address the above problems, this invention provides a method for injecting spoofed data into a multi-elastic joint robot system, comprising the following steps:

[0004] S1. Construct a dynamic mathematical model of a multi-elastic joint robot system with nonlinear characteristics;

[0005] S2. Based on the theory of all-drive systems, the dynamic mathematical model of the multi-elastic joint robot is transformed to construct a high-order all-drive quasi-linear multi-robot system dynamic model.

[0006] S3, based on the theory of high-order all-drive systems, constructs a distributed consensus controller that includes linearization terms and neighbor error correlation terms;

[0007] S4, combining the high-order all-drive quasi-linear multi-robot system dynamic model in S2 and the distributed consensus controller in S3, establishes a closed-loop global dynamic form through equivalent transformation, and then derives the state analytical solution of the closed-loop system.

[0008] S5, in response to the attacker's need to maximize convergence error, designs a fake data injection attack strategy and constructs a dynamic model of a high-order fully driven quasi-linear multi-robot system under attack conditions;

[0009] S6. Based on the attacker's objective, the error of the state analytical solution of the dynamic model of the high-order fully driven quasi-linear multi-robot system under attack and non-attack conditions is defined as the objective function.

[0010] S7 transforms the attacker's objective function maximization problem into a combinatorial optimization problem. Using submodular optimization theory, a greedy algorithm is designed to find the suboptimal solution of this combinatorial optimization problem in polynomial time, which serves as the best solution for the attacker to inject false data into the multi-elastic joint robot system.

[0011] Preferably, the dynamic mathematical model of the multi-elastic joint robot system in S1 is in the following specific form:

[0012] Determine the dynamic mathematical model of a multi-elastic joint robot system with nonlinear characteristics. This multi-elastic joint robot system consists of... Composed of a series of flexible joint robots, the first... The specific form of the dynamic mathematical model of an elastic joint robot is as follows:

[0013] ;

[0014] in:

[0015] The first A flexible joint robot at any time The angle between the two joints;

[0016] The first A flexible joint robot at any time Angular velocities of the two joints;

[0017] The first A flexible joint robot at any time The angular acceleration of the two joints;

[0018] For the joint rotational inertia of this type of robot;

[0019] For generalized joint damping of such robots;

[0020] For the spring stiffness of this type of robot, ;

[0021] For the first A flexible joint robot at any time The motor output torque;

[0022] For the first A flexible joint robot at any time The rate of change of motor output torque;

[0023] For the first A flexible joint robot at any time Piecewise continuous functions;

[0024] For the first A flexible joint robot at any time Motor voltage input.

[0025] Preferably, the specific form of the high-order all-drive quasi-linear multi-robot system dynamic model constructed in S2 is as follows:

[0026] Based on the theory of all-drive systems, the dynamic mathematical model of a multi-elastic joint robot system with nonlinear characteristics can be transformed into a high-order dynamic model of an all-drive quasi-linear multi-robot system. The specific form of the high-order all-drive quasi-linear system dynamic model of an elastic joint robot is as follows:

[0027] ;

[0028] in:

[0029] For the first The first joint angle of an elastic articulated robot relative to time The fifth derivative;

[0030] ;

[0031] ;

[0032] ;

[0033] ;

[0034] For the first A flexible joint robot, first joint angle relative to time The m-th derivative;

[0035] When state variables If all variables are measurable, then the state variables of a high-order fully driven multi-elastic joint robot system are... Both are available.

[0036] Preferably, the specific process of S3 is as follows:

[0037] S31, for the first A distributed consensus controller is constructed using a flexible joint robot, specifically in the following form:

[0038] ;

[0039] in:

[0040] It is the first A high-order all-drive controller that eliminates system nonlinearity and meets the requirements of a closed-loop system is constructed using an elastic articulated robot.

[0041] It is the first A controller is constructed for a flexible joint robot to achieve the goal of consistency and coordination;

[0042] S32, for the first A high-level all-drive controller is built using a flexible joint robot. The specific form is as follows:

[0043] ;

[0044] in:

[0045] The linearization parameters to be constructed;

[0046] For the first External input signals for a flexible joint robot;

[0047] S33, for the first A flexible joint robot constructs a consistent cooperative controller. The specific form is as follows:

[0048] ;

[0049] in:

[0050] For consistency controller parameters;

[0051] Recorded as the number A set of neighbors of an articulated robot ;

[0052] For adjacency matrix elements, when the elastic joint robot From flexible joint robots When receiving information ,otherwise ;

[0053] Preferably, the specific process of S4 is as follows:

[0054] S41, the distributed consensus controller is substituted into the dynamic model of the high-order fully driven quasi-linear multi-robot system, and the model is transformed into an equivalent closed-loop global dynamic form, specifically:

[0055] ;

[0056] in:

[0057] ;

[0058] for An identity matrix of order 1;

[0059] For Kronecker product;

[0060] ;

[0061] The graph is a Laplace matrix;

[0062] To control the augmented matrix;

[0063] This represents the global form of the external input signal;

[0064] S42 is generally set to external input. The general solution form of the state analytical solution of the nonhomogeneous differential equation in the closed-loop global form is derived, and its specific form is as follows:

[0065] ;

[0066] in:

[0067] This is the state transition matrix;

[0068] ;

[0069] This represents the initial state in a global context.

[0070] Preferably, the specific process of designing the fake data injection attack strategy in S5 is as follows:

[0071] Considering the scenario where a multi-joint robot system is subjected to a spoofing attack, the specific form of its high-order all-drive quasi-linear multi-robot system dynamic model is as follows:

[0072] ;

[0073] in:

[0074] The data is fake data injected by the attacker; since the elastic joint robot was also compromised by the attacker, the fake data is designed as follows:

[0075] ;

[0076] Represents a flexible joint robot Belongs to set ;

[0077] Represents a flexible joint robot Not a set ;

[0078] It is a fake data injection attack signal, which is arbitrarily set by the attacker and is the same for all the elastic joint robots that are attacked.

[0079] gather This is a subset of the elastically jointed robots that are being attacked.

[0080] Preferably, the specific process of S6 is as follows:

[0081] S61, obtain the global state of the dynamic model of the high-order fully driven quasi-linear multi-robot system under attack conditions, specifically in the form of:

[0082] ;

[0083] in For the instruction vector of the attacked elastic joint robot, If and only if the elastic joint robot When under attack ,otherwise ;

[0084] S62, the general solution of the nonhomogeneous differential equation in the global state form is obtained as follows:

[0085] ;

[0086] in:

[0087] This represents the time variable from the initial time 0 to the current time. The definite integral of the attack reflects the energy accumulation process of the attack's impact over time;

[0088] This represents the state transition matrix of a closed-loop system.

[0089] S63 is generally configured with external input. The attacker's goal, given a limited total attack cost, is to select the optimal subset. To maximize the convergence error; the convergence error is defined as the error of the closed-loop analytical solution of the dynamic model of a high-order fully driven quasi-linear multi-robot system under attacked and unattacked conditions, specifically in the form of:

[0090] ;

[0091] in:

[0092] The termination time;

[0093] Represents the norm.

[0094] Preferably, the specific process of S7 is as follows:

[0095] S71, the attacker's search for the optimal subset can be viewed as a combinatorial optimization problem:

[0096] . ;

[0097] in:

[0098] max represents the maximum value;

[0099] Represents a set For set A subset of;

[0100] st represents the constraints that the optimization problem must follow;

[0101] Attackers can successfully inject fake data, and the fake data is Bounded It is the upper bound of the injected fake data;

[0102] For flexible joint robots The cost of the attack;

[0103] Attackers have an upper bound on total attack cost. ,and ;

[0104] S72, this combinatorial optimization problem has been reduced to and proven to be NP-hard. The knapsack problem; to solve this combinatorial optimization problem, a greedy algorithm is designed using submodular optimization theory to find the suboptimal subset in polynomial time.

[0105] Compared with the prior art, the present invention has the following beneficial effects:

[0106] This invention enhances the applicability to complex nonlinear multi-agent systems: existing attack design research is mostly limited to linear systems, making it difficult to characterize the complex nonlinear dynamics of real-world agents. Specifically targeting multi-elastic joint robot systems, this invention transforms the system into a high-order omnidirectional model using omnidirectional system theory and designs a controller to actively counteract the inherent nonlinearity of the system, effectively filling the technical gap in accurately quantifying the convergence error of spurious data injection attacks within a nonlinear control framework.

[0107] Significantly reducing computational complexity, achieving near-optimal solution in polynomial time: Selecting the optimal attack node under resource constraints is essentially an NP-hard 0-1 knapsack combinatorial optimization problem. Using traditional methods for precise solutions would lead to unacceptable combinatorial explosion. The cost-constrained greedy node selection algorithm (C-GNSA) designed in this invention effectively avoids extremely high computational overhead and can accurately lock onto high-influence target nodes in polynomial time.

[0108] The attack's destructive effectiveness (system convergence error) is significantly superior to traditional algorithms: The algorithm proposed in this invention ensures optimal resource allocation under a limited budget by rigorously evaluating the "marginal benefit to cost ratio" in each iteration. The system convergence error caused by the strategy of this invention is stable and significantly higher than that of traditional random selection methods and maximum degree attack methods. Attached Figure Description

[0109] Figure 1 This is a schematic diagram of the overall process of the present invention.

[0110] Figure 2 This is an interactive topology diagram of a multi-elastic joint robot system.

[0111] Figure 3 This is a flowchart of a greedy algorithm based on submodular optimization theory.

[0112] Figure 4 This is a performance comparison chart of three algorithms under time-invariant attacks when the attack cost is set to a constant value.

[0113] Figure 5 This is a performance comparison chart of three algorithms under time-varying attacks when the attack cost is set to a constant value.

[0114] Figure 6 This is a performance comparison chart of three algorithms under time-invariant attacks when the attack cost is set to degrees.

[0115] Figure 7 This is a performance comparison chart of three algorithms under time-varying attacks when the attack cost is set to degrees.

[0116] Figure 8 This is a performance comparison chart of three topologies under time-varying attacks when the attack cost is set to a constant value.

[0117] Figure 9 This is a performance comparison chart of three topologies under time-varying attacks when the attack cost is set to degrees. Detailed Implementation

[0118] The present invention will be further described below with reference to embodiments. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0119] Example 1:

[0120] This embodiment provides a method for injecting false data into a multi-elastic joint robot system. The overall process is as follows: Figure 1 As shown, it includes the following steps:

[0121] Step 1: Determine the dynamic mathematical model of the multi-elastic joint robot system with nonlinear characteristics. This multi-elastic joint robot system consists of... Composed of a series of flexible joint robots, the first... The specific form of the dynamic mathematical model of an elastic joint robot is as follows:

[0122] ;

[0123] in:

[0124] The first A flexible joint robot at any time The angle between the two joints;

[0125] The first A flexible joint robot at any time Angular velocities of the two joints;

[0126] The first A flexible joint robot at any time The angular acceleration of the two joints;

[0127] For the joint rotational inertia of this type of robot;

[0128] For generalized joint damping of such robots;

[0129] For the spring stiffness of this type of robot, ;

[0130] For the first A flexible joint robot at any time The motor output torque;

[0131] For the first A flexible joint robot at any time The rate of change of motor output torque;

[0132] For the first A flexible joint robot at any time Piecewise continuous functions;

[0133] For the first A flexible joint robot at any time Motor voltage input.

[0134] Step 2: Based on the theory of all-drive systems, the dynamic mathematical model of a multi-elastic joint robot system with nonlinear characteristics can be transformed into a high-order dynamic model of an all-drive quasi-linear multi-robot system.

[0135] From the equation We can obtain:

[0136] ;

[0137] For equation Taking the second derivative, we get:

[0138] ;

[0139] For equation Differentiation yields:

[0140] ;

[0141] From the equation We can obtain:

[0142] ;

[0143] therefore Can be used express:

[0144] ;

[0145] From the above equations, the following dynamic model of a fourth-order fully driven quasi-linear multi-robot system can be obtained:

[0146] ;

[0147] in:

[0148] ;

[0149] For the first A flexible joint robot, first joint angle relative to time The m-th derivative;

[0150] ;

[0151] The dynamic model of the fourth-order all-drive quasi-linear multi-robot system is differentiated and substituted into the equation. We can obtain:

[0152] ;

[0153] in:

[0154] ;

[0155] When state variables If all variables are measurable, then the state variables of a high-order fully driven multi-elastic joint robot system are... Both are available.

[0156] Step three, for the first A distributed consensus controller is constructed using a flexible joint robot, specifically in the following form:

[0157] ;

[0158] in:

[0159] It is the first A high-order all-drive controller for a flexible articulated robot is designed to eliminate system nonlinearity and meet the requirements of a closed-loop system.

[0160] It is the first A controller is constructed for a flexible joint robot to achieve the goal of consistency and coordination;

[0161] For the first A high-level all-drive controller is built using a flexible joint robot. The specific form is as follows:

[0162] ;

[0163] in:

[0164] The linearization parameters to be designed;

[0165] For the first External input signals for a flexible joint robot;

[0166] For the first A flexible joint robot design with consistent collaborative controller The specific form is as follows:

[0167] ;

[0168] in:

[0169] For consistency controller parameters;

[0170] Recorded as the number A set of neighbors of an articulated robot ;

[0171] For adjacency matrix elements, when the elastic joint robot From flexible joint robots When receiving information ,otherwise ;

[0172] Step four: Substitute the distributed consensus controller into the dynamic model of the high-order fully driven quasi-linear multi-robot system to obtain the closed-loop form:

[0173] ;

[0174] The model transformation is equivalent to a closed-loop global dynamics form, specifically:

[0175] ;

[0176] in:

[0177] ;

[0178] for An identity matrix of order 1;

[0179] For Kronecker product;

[0180] ;

[0181] The graph is a Laplace matrix;

[0182] To control the augmented matrix;

[0183] This represents the global form of the external input signal;

[0184] Generally, an external input is set. The general solution form of the state analytical solution of the nonhomogeneous differential equation in the closed-loop global form is derived, and its specific form is as follows:

[0185] ;

[0186] in:

[0187] This is the state transition matrix;

[0188] ;

[0189] This represents the initial state in a global context.

[0190] Step 5: Consider the scenario where a multi-elastic joint robot system is subjected to a spoofing attack. The specific form of the attack model is as follows:

[0191] ;

[0192] in:

[0193] The data is fake data injected by the attacker; since the elastic joint robot was also compromised by the attacker, the fake data is designed as follows:

[0194] ;

[0195] Represents a flexible joint robot Belongs to set ;

[0196] Represents a flexible joint robot Not a set ;

[0197] It is a fake data injection attack signal, which is arbitrarily set by the attacker and is the same for all the elastic joint robots that are attacked.

[0198] gather This is a subset of the elastically jointed robots that are being attacked.

[0199] Step 6: Obtain the global state of the dynamic model of the high-order fully driven quasi-linear multi-robot system under attack conditions, specifically in the following form:

[0200] ;

[0201] in For the instruction vector of the attacked elastic joint robot, If and only if the elastic joint robot When under attack ,otherwise ;

[0202] The general solution of the nonhomogeneous differential equation in the global state form is obtained as follows:

[0203] ;

[0204] in:

[0205] This represents the time variable from the initial time 0 to the current time. The definite integral of the attack reflects the energy accumulation process of the attack's impact over time;

[0206] This represents the state transition matrix of a closed-loop system.

[0207] Generally, an external input is set. The attacker's goal, given a limited total attack cost, is to select the optimal subset. To maximize the convergence error; the convergence error is defined as the error of the closed-loop analytical solution of the dynamic model of a high-order fully driven quasi-linear multi-robot system under attacked and unattacked conditions, specifically in the form of:

[0208] ;

[0209] in:

[0210] The termination time;

[0211] Represents the norm.

[0212] Step seven, the attacker's search for the optimal subset can be viewed as a combinatorial optimization problem:

[0213] ;

[0214] in:

[0215] max represents the maximum value;

[0216] Represents a set For set A subset of;

[0217] st represents the constraints that the optimization problem must follow;

[0218] Attackers can successfully inject fake data, and the fake data is Bounded It is the upper bound of the injected fake data;

[0219] For flexible joint robots The cost of the attack;

[0220] Attackers have an upper bound on total attack cost. ,and ;

[0221] This combinatorial optimization problem has been reduced to and proven to be NP-hard. The knapsack problem; to solve this combinatorial optimization problem, a greedy algorithm is designed using submodular optimization theory to find the suboptimal subset in polynomial time.

[0222] The specific algorithm flow of the cost-constrained greedy node selection algorithm is as follows: Figure 3 As shown;

[0223] The specific implementation steps of the cost-constrained greedy node selection algorithm are as follows:

[0224] Input parameter: The set of attack costs for all elastic articulated robots in the system. Attacker's total cost budget Preset termination time and attack strategies ;

[0225] Output: The final selected subset of attacked nodes ;

[0226] Step 1 (Initialization):

[0227] Initialize the selected node subset The candidate node pool is empty; Initialized to include all nodes in the system The complete collection; at the same time, the accumulated attack costs will be... Initialize to 0;

[0228] Step 2 (Iterative Condition Judgment): Determine the current candidate node pool Is it non-empty, and what is the current cumulative attack cost? Is it strictly less than the total cost budget? If both of the above conditions are met, proceed to step 3 for a new round of node selection; if either condition is not met (i.e., the candidate pool is exhausted or the budget is depleted), terminate the iteration and directly output the currently selected subset of nodes. ;

[0229] Step 3 (Evaluation of Marginal Revenue per Unit Cost):

[0230] Traverse the current candidate node pool Each candidate node in For each node Calculate and add it to the currently selected subset. The marginal benefit per unit cost resulting from adding this node; specifically, the increment of the objective function (i.e., the system convergence error) after adding this node is calculated. and divide it by the cost of attacking that node. ;

[0231] Step 4 (Local Optimal Node Locking):

[0232] After evaluating all candidate nodes, the node that provides the maximum marginal revenue per unit cost is selected and designated as the optimal node. ;

[0233] Step 5 (Status and Budget Update):

[0234] Determine the optimal node Formal inclusion in the selected node subset In the middle; at the same time, this node From the candidate node pool Remove from the middle to avoid repeated selections later; recalculate the current subset. Total cumulative attack cost After the update is complete, return to step 2.

[0235] Example 2:

[0236] This embodiment considers a simulation of six elastic joint robots to verify the effectiveness of the proposed attack method. The interaction topology of the multi-elastic joint robot system is as follows: Figure 2 As shown;

[0237] Make the system poles as follows ;

[0238] After parameterization, the parameter matrix can be obtained:

[0239] ;

[0240] Consider the process of six flexible joint robots achieving cooperative consistency. In the absence of attacks, the system will progressively achieve cooperative consistency, and a system termination time is set. ;

[0241] Consider two attack cost options:

[0242] The cost of attacking each elastic joint robot is the same (e.g.) );

[0243] The cost of attacking each flexible joint of a robot depends on its degree (e.g.) );

[0244] Consider two attack schemes:

[0245] Time-invariant attack ;

[0246] Time-varying attack ;

[0247] In this embodiment, the settings are as follows: ;

[0248] This embodiment compares three algorithms: cost-constrained greedy node selection algorithm, random selection method, and maximum degree attack method.

[0249] The only difference between all the algorithms lies in the choice of the elastic joint robot being attacked;

[0250] In the random selection algorithm, a random subset of elastic joint robots is selected for attack;

[0251] In the maximum degree attack method, elastic joint robots are selected for attack from high to low degree.

[0252] Figure 4 and Figure 5 Demonstrates attack cost selection methods At the same time, regardless of the total attack cost, whether under time-invariant or time-varying attack conditions, the cost-constrained greedy node selection algorithm produces the largest convergence error compared to other algorithms.

[0253] Figure 6 and Figure 7 Demonstrates attack cost selection methods Despite the different total attack costs, the cost-constrained greedy node selection algorithm outperforms the random selection algorithm and the maximum degree attack algorithm in both time-invariant and time-varying attack scenarios.

[0254] Figure 8 and Figure 9 This paper demonstrates the impact of different network structures on convergence error. Performance is analyzed for linear, cyclic, and fixed topologies under two different attack selection methods. In this case, the convergence error of linear and cyclic graphs is greater than that of fixed graphs. In this case, linear graphs and fixed graphs significantly outperform cyclic graphs. The above descriptions are merely preferred embodiments of this application and are not intended to limit the application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

[0255] While the specific embodiments of the present invention have been described above, they are not intended to limit the scope of protection of the present invention. 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 the present invention are still within the scope of protection of the present invention.

Claims

1. A method for injecting spoofed data into a multi-elastic joint robot system, characterized in that, Includes the following processes: S1. Construct a dynamic mathematical model of a multi-elastic joint robot system with nonlinear characteristics; S2. Based on the theory of all-drive systems, the dynamic mathematical model of the multi-elastic joint robot is transformed to construct a high-order all-drive quasi-linear multi-robot system dynamic model. S3, based on the theory of high-order all-drive systems, constructs a distributed consensus controller that includes linearization terms and neighbor error correlation terms; S4 combines the high-order all-drive quasi-linear multi-robot system dynamic model in S2 with the distributed consensus controller in S3. Through equivalent transformation, a closed-loop global dynamic form is established, and then the state analytical solution of the closed-loop system is derived. S5, in response to the attacker's need to maximize convergence error, designs a fake data injection attack strategy and constructs a dynamic model of a high-order fully driven quasi-linear multi-robot system under attack conditions; S6. Based on the attacker's objective, the error of the state analytical solution of the dynamic model of the high-order fully driven quasi-linear multi-robot system under attack and non-attack conditions is defined as the objective function. S7 transforms the attacker's objective function maximization problem into a combinatorial optimization problem. Using submodular optimization theory, a greedy algorithm is designed to find the suboptimal solution of this combinatorial optimization problem in polynomial time, which serves as the best solution for the attacker to inject false data into the multi-elastic joint robot system.

2. The method for injecting false data into a multi-elastic joint robot system as described in claim 1, characterized in that: The specific form of the dynamic mathematical model of the multi-elastic joint robot system in S1 is as follows: Multi-elastic joint robot system Composed of a series of flexible joint robots, the first... The specific form of the dynamic mathematical model of an elastic joint robot is as follows: ； in: The first A flexible joint robot at any time The angle between the two joints; The first A flexible joint robot at any time Angular velocities of the two joints; The first A flexible joint robot at any time The angular acceleration of the two joints; For the joint rotational inertia of this type of robot; For generalized joint damping of such robots; For the spring stiffness of this type of robot, ; For the first A flexible joint robot at any time The motor output torque; For the first A flexible joint robot at any time The rate of change of motor output torque; For the first A flexible joint robot at any time Piecewise continuous functions; For the first A flexible joint robot at any time Motor voltage input.

3. The method for injecting false data into a multi-elastic joint robot system as described in claim 2, characterized in that: The specific form of the high-order all-drive quasi-linear multi-robot system dynamic model constructed in S2 is as follows: The dynamic mathematical model of a multi-elastic joint robot system with nonlinear characteristics is transformed into a high-order dynamic model of a fully driven quasi-linear multi-robot system. The specific form of the high-order all-drive quasi-linear system dynamic model of an elastic joint robot is as follows: ； in: For the first The first joint angle of an elastic articulated robot relative to time The fifth derivative; ； ； ； ； For the first A flexible joint robot, first joint angle relative to time The m-th derivative; When state variables If all variables are measurable, then the state variables of a high-order fully driven multi-elastic joint robot system are... Both are available.

4. The method for injecting false data into a multi-elastic joint robot system as described in claim 3, characterized in that: The specific process of S3 is as follows: S31, for the first A distributed consensus controller is constructed using a flexible joint robot, specifically in the following form: ； in: It is the first A high-order all-drive controller that eliminates system nonlinearity and meets the requirements of a closed-loop system is constructed using an elastic articulated robot. It is the first A controller is constructed for a flexible joint robot to achieve the goal of consistency and coordination. S32, for the first A high-level all-drive controller is built using a flexible joint robot. The specific form is as follows: ； in: The linearization parameters to be constructed; For the first External input signals for a flexible joint robot; S33, for the first A flexible joint robot constructs a consistent cooperative controller. The specific form is as follows: ； in: For consistency controller parameters; Recorded as the number A set of neighbors of an articulated robot ; For adjacency matrix elements, when the elastic joint robot From flexible joint robots When receiving information ,otherwise .

5. The method for injecting false data into a multi-elastic joint robot system as described in claim 4, characterized in that: The specific process of S4 is as follows: S41, the distributed consensus controller is substituted into the dynamic model of the high-order fully driven quasi-linear multi-robot system, and the model is transformed into an equivalent closed-loop global dynamic form, specifically: ； in: ； for An identity matrix of order 1; For Kronecker product; ； The graph is a Laplace matrix; To control the augmented matrix; This represents the global form of the external input signal; S42, Assume external input The general solution form of the state analytical solution of the nonhomogeneous differential equation in the closed-loop global form is derived, and its specific form is as follows: ； in: This is the state transition matrix; ； This represents the initial state in a global context.

6. The method for injecting false data into a multi-elastic joint robot system as described in claim 1, characterized in that: The S5 implementation details a fake data injection attack strategy as follows: Considering the scenario where a multi-joint robot system is subjected to a spoofing attack, the specific form of its high-order all-drive quasi-linear multi-robot system dynamic model is as follows: ； in: For the joint rotational inertia of this type of robot; For the first The first joint angle of an elastic articulated robot relative to time The fifth derivative; ； For the first A flexible joint robot, first joint angle relative to time The m-th derivative; For the spring stiffness of this type of robot, ; For the first A flexible joint robot at any time The motor output torque; For the first A flexible joint robot at any time A piecewise continuous function; For the first A flexible joint robot at any time Motor voltage input; For the first The first joint angle of an elastic articulated robot relative to time The fifth derivative; ； ； For generalized joint damping of such robots; It is fake data injected by an attacker; the fake data is designed as follows: ； Represents a flexible joint robot Belongs to set ; Represents a flexible joint robot Not a set ; It is a fake data injection attack signal, which is arbitrarily set by the attacker and is the same for all the elastic joint robots that are attacked. gather This is a subset of the elastically jointed robots that are being attacked.

7. The method for injecting false data into a multi-elastic joint robot system as described in claim 6, characterized in that: The specific process of S6 is as follows: S61, obtain the global state of the dynamic model of the high-order fully driven quasi-linear multi-robot system under attack conditions; S62, obtain the general solution of the nonhomogeneous differential equation in the global state form; S63, Set external input The attacker's goal, given a limited total attack cost, is to select the optimal subset. To maximize the convergence error; the convergence error is defined as the error of the closed-loop analytical solution of the dynamic model of a high-order fully driven quasi-linear multi-robot system under attack and non-attack conditions.

8. The method for injecting false data into a multi-elastic joint robot system as described in claim 7, characterized in that: The specific process of S7 is as follows: The attacker's search for the optimal subset is defined as a combinatorial optimization problem: ； in: max represents the maximum value; Represents a set For set A subset of; st represents the constraints that the optimization problem must follow; Represents the norm; The attacker successfully injected fake data, and the fake data is Bounded It is the upper bound of the injected fake data; For flexible joint robots The cost of the attack; Attackers have an upper bound on total attack cost. ,and ; A greedy algorithm is designed using submodular optimization theory to find suboptimal subsets in polynomial time.

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