Trajectory optimization system combining pseudo-spectrum theory and language model in multi-agent confrontation task

Abstract

The invention relates to the technical field of intelligent systems and robots, and discloses a trajectory optimization system in combination with a pseudo-spectral theory and a language model in a multi-agent confrontation task, and the system comprises a plurality of agents, each agent has a state and a control input, and the state is described by a kinetic equation; the pseudo-spectrum method module is used for expressing the state and control input of each agent as a polynomial form, and performing trajectory optimization based on a multi-objective optimization target; the variational method module is used for deducing necessary conditions for trajectory optimization of each agent according to a pseudo-spectrum method; and the Pontryagin maximum value principle module is used for deducing the optimal control input of each agent on the basis of a Hamiltonian function. According to the invention, by introducing the combination of the language model module and the nonlinear control module, the dynamic adjustment of the trajectory optimization strategy is realized, and the adaptability, execution efficiency and robustness of the multi-agent system in complex tasks are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of intelligent systems and robotics, and specifically to a trajectory optimization system that combines pseudospectral theory and language models in multi-agent adversarial tasks. Background Art

[0002] In the applications of modern intelligent systems, the cooperation and adversarial tasks of multiple agents have become key requirements in many fields, such as autonomous driving, UAV formation, intelligent manufacturing, etc. In these scenarios, how to efficiently perform trajectory planning and optimization to ensure that the actions of each agent can not only complete the task but also respond to changes in real time in a complex environment has become a difficult problem to be solved urgently. Traditional trajectory optimization methods often calculate based on fixed objective functions, lacking flexibility and not being able to well adapt to changes in task descriptions.

[0003] Most of the trajectory optimization methods in the prior art rely on static objective functions and plan trajectories through preset optimization algorithms. The advantages of these methods are clear structure, easy implementation, and good optimization effects can be achieved in relatively simple tasks. For example, in some standard path planning tasks, traditional optimization methods can converge quickly and obtain relatively ideal results. In addition, traditional methods also have certain advantages in controlling computational complexity, especially in the case of relatively simple task conditions and environments, and can provide fast trajectory calculation and decision-making.

[0004] However, the application of the prior art in multi-agent systems has many limitations; firstly, the objective functions of traditional optimization methods are usually statically set and cannot be adjusted in real time according to changes in task descriptions and environments. This makes it difficult for traditional methods to cope with real-time changes in complex and dynamic task environments, resulting in a significant reduction in the effect of trajectory optimization. Secondly, the existing methods lack an adaptation mechanism for interactive feedback between agents and cannot effectively handle the complex dynamics in the cooperation or confrontation of multiple agents. Finally, traditional verification methods mainly focus on the accuracy of trajectories but ignore the comprehensive evaluation of computational efficiency and task completion rate, restricting their application in large-scale and multi-task environments. Summary of the Invention

[0005] Aiming at the deficiencies of the prior art, the present invention provides a trajectory optimization system that combines pseudospectral theory and language models in multi-agent adversarial tasks, and solves the problems of lack of flexibility and adaptability in trajectory optimization methods in the prior art.

[0006] To achieve the above objectives, the present invention is realized through the following technical solutions: A trajectory optimization system that combines pseudospectral theory and language models in multi-agent adversarial tasks, including:

[0007] Multiple agents, each agent having a state and a control input, and its state being described by a dynamic equation;

[0008] The pseudospectral method module is used to represent the states and control inputs of each agent in polynomial form and perform trajectory optimization based on multi-objective optimization goals;

[0009] The variational method module is used to derive the necessary conditions for the trajectory optimization of each agent according to the pseudospectral method and generate the variational equation of the optimal trajectory;

[0010] The Pontryagin maximum principle module is used to derive the optimal control input of each agent based on the Hamiltonian function;

[0011] The non-linear control module is connected to the Pontryagin maximum principle module and is used to analyze and ensure the stability of the system through Lyapunov stability theory to prevent unstable behaviors;

[0012] The language model module is connected to the non-linear control module and is used to dynamically parse the task description and adjust the objective function weights and trajectory optimization strategies according to the interaction feedback between multiple agents;

[0013] The simulation verification module is connected to the language model module and is used to verify the effectiveness of the trajectory optimization system in multi-agent adversarial tasks and compare the performance with traditional optimization methods.

[0014] Preferably, the pseudospectral method module includes:

[0015] The pseudospectral expansion unit: This unit represents the states and control inputs of each agent in polynomial form, transforms it into a finite-dimensional trajectory optimization problem, and uses Legendre polynomials or Chebyshev polynomials as basis functions for the expansion of states and control inputs;

[0016] The objective function optimization unit: This unit constructs the total objective function according to the multi-objective optimization goal and optimizes the trajectories of each agent by the pseudospectral method.

[0017] Preferably, the variational method module includes:

[0018] The Euler-Lagrange equation derivation unit: This unit derives the Euler-Lagrange equation of each agent by the variational method according to the trajectory optimization problem represented by the pseudospectral method to obtain the necessary conditions for trajectory optimization;

[0019] The optimal trajectory generation unit: This unit uses the equation derived by the variational method to generate the variational solution of the optimal trajectory and uses this solution as the input for the subsequent Pontryagin maximum principle module.

[0020] Preferably, the Pontryagin maximum principle module includes:

[0021] Hamiltonian function construction unit: This unit constructs the corresponding Hamiltonian function based on the state, control input, and dynamic equation of each agent, transforming the optimal control problem into a problem of maximizing the Hamiltonian function;

[0022] Optimal control derivation unit: This unit derives the optimal control input of each agent by maximizing the Hamiltonian function.

[0023] Preferably, the non-linear control module includes:

[0024] Lyapunov function construction unit: This unit constructs the Lyapunov function for each agent;

[0025] Feedback control design unit: This unit designs a feedback controller based on the Lyapunov stability theory and adjusts the control input according to the current state of the agent.

[0026] Preferably, the language model module includes:

[0027] Task description parsing unit: This unit parses the objective function according to the input task description, determines the weight of the objective function according to the task requirements, and adjusts the trajectory optimization objective of the multi-agent;

[0028] Policy adjustment unit: This unit dynamically optimizes the trajectory policy of the agent according to the interaction feedback between agents, the current task execution status, and the adjustment of the objective function.

[0029] Preferably, the simulation verification module includes:

[0030] Simulation design unit: This unit designs a simulation experiment in a multi-agent adversarial task environment to verify the performance of the trajectory optimization system;

[0031] Performance comparison unit: This unit compares the results of the trajectory optimization system with traditional methods, and evaluates the advantages and improvements of the system in terms of trajectory optimization accuracy, system stability, and adaptability through comparative experiments.

[0032] Preferably, the optimal trajectory generation unit includes:

[0033] Variational equation solving unit: This unit uses the variational solution of the Euler-Lagrange equation and combines the trajectory optimization problem represented by the pseudospectral method to solve the optimal trajectory path of each agent;

[0034] Optimal Trajectory Smoothing Unit: This unit smooths the obtained optimal trajectory to eliminate discontinuous points and unnatural variations in the trajectory.

[0035] Preferably, the Lyapunov function construction unit:

[0036] Stability Analysis Unit: This unit analyzes whether there are stability risks for the agent during the trajectory optimization process by calculating the Lyapunov function value of the system state;

[0037] Lyapunov Function Optimization Unit: This unit designs and optimizes the Lyapunov function for each agent.

[0038] Preferably, the simulation design unit includes:

[0039] Multi-agent Simulation Model Construction Unit: This unit constructs a simulation environment based on the interaction behaviors and dynamic equations among multiple agents;

[0040] Trajectory Optimization Verification Unit: This unit verifies the trajectory optimization system in the simulation environment, monitors the trajectory generation and behavior execution of each agent, and evaluates the performance and effectiveness of the optimization system;

[0041] Environment Change Simulation Unit: This unit simulates a variable adversarial environment, and tests the robustness and adaptability of the trajectory optimization system under complex tasks by changing the task objectives, constraint conditions, and interaction strategies among agents.

[0042] The present invention provides a trajectory optimization system combining pseudospectral theory and language model in multi-agent adversarial tasks. It has the following beneficial effects:

[0043] 1. The present invention adopts a technical solution combining a language model module and a non-linear control module. By dynamically parsing the task description and adjusting the objective function weights and trajectory optimization strategies according to the multi-agent interaction feedback, it realizes intelligent adaptive optimization in the task environment. Compared with the static optimization methods in the prior art, the present invention can quickly adjust the optimization strategy according to the real-time changing task requirements and environmental conditions, ensuring that the system maintains high-efficiency trajectory planning ability in a variable task environment.

[0044] 2. The simulation verification module of the present invention verifies the effectiveness of the trajectory optimization system in multiple-agent adversarial tasks and compares the performance with traditional optimization methods, significantly improving the trajectory optimization accuracy and calculation efficiency. Compared with the traditional performance verification scheme, the simulation verification module not only compares the optimization results, but also introduces real-time environmental disturbance factors, making the verification more comprehensive and real, ensuring the applicability of the trajectory optimization scheme in a complex dynamic environment.

[0045] 3. By combining the Pontryagin maximum principle with reinforcement learning technology, the present invention optimizes the trajectory planning strategy in multi-agent collaborative tasks. Compared with the single optimization method in the prior art, the present invention introduces a reinforcement learning mechanism, enabling the agent to automatically adjust the trajectory optimization strategy according to historical data and real-time feedback, improving the system's adaptability and task completion efficiency. Especially in complex adversarial environments, it exhibits higher robustness.

[0046] 4. The present invention adopts a comprehensive evaluation method based on error metric and computational complexity analysis, effectively improving the performance comparison and evaluation accuracy of the trajectory optimization system. Compared with the solution in the prior art that only evaluates through simple metrics, the present invention provides a more refined performance measurement standard, which can quantify the advantages of the system in terms of computational efficiency, optimization accuracy, and task completion rate, enabling the performance of the trajectory optimization system to be more comprehensively verified. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 It is a schematic diagram of the system construction of the present invention;

[0048] Figure 2 It is a framework diagram of the pseudospectral method module of the present invention;

[0049] Figure 3 It is a framework diagram of the variational method module of the present invention;

[0050] Figure 4 It is a framework diagram of the Pontryagin maximum principle module of the present invention;

[0051] Figure 5 It is a framework diagram of the nonlinear control module of the present invention;

[0052] Figure 6 It is a framework diagram of the language model module of the present invention;

[0053] Figure 7 It is a framework diagram of the simulation verification module of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0054] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0055] Please refer to the appended Figure 1 - appended Figure 7 drawings. The embodiments of the present invention provide a trajectory optimization system combining pseudospectral theory and language model in multi-agent adversarial tasks, including:

[0056] Multiple agents, each agent having a state and a control input, and its state being described by a dynamics equation;

[0057] In this embodiment, considering that each agent in the multi-agent system has an independent state and control input, and the state of each agent is described by a dynamics equation. The behavior of each agent is affected by its own state and control input, and these states and control inputs are coupled and constrained through the dynamics equation. The state of an agent usually includes physical quantities such as position, velocity, and acceleration, and the control input includes control variables such as velocity and force. The interaction, cooperation, or confrontation behavior among multiple agents further affects the overall optimization of the system.

[0058] When performing trajectory optimization, the state and control input of each agent need to be considered simultaneously to ensure the stability of the system and the achievement of the task objective. For this purpose, the state and control input of each agent are described by a set of dynamics equations, usually a dynamic model represented by differential equations. These equations are not only related to the motion characteristics of a single agent but may also be closely related to environmental factors and the behavior interaction of other agents.

[0059] Specifically, in this embodiment, the state of each agent is described by the following dynamics equation:

[0060]

[0061] where, x i (t) represents the state vector of the i-th agent, including information such as position and velocity; u i (t) represents the control input of the i-th agent, including control quantities such as force and velocity; is the derivative of the state of the i-th agent, representing the state change rate. The function f i (x i (t), u i (t)) is the dynamics equation describing the motion of the i-th agent, which may be linear or non-linear, depending on the physical properties of the agent.

[0062] In a typical multi-agent confrontation task, assuming there are N agents, the dynamics equations of each agent are independent and coupled with each other. To coordinate these agents, a control model in a similar form to the following is usually used to describe:

[0063]

[0064] where, is the derivative of the state of the i-th agent, representing the state change rate, u i(t) represents the control input of the i-th agent, including control quantities such as force and velocity; x j (t) is the state vector of the j-th agent, the state value at time t; A i and B i are matrices related to the state and control input of the i-th agent respectively, and C ij is the matrix describing the interaction between agent i and agent i. This model not only considers the dynamics of each agent itself but also includes the interactive influence between agents. For example, C ij can represent the collision risk, distance constraint, or cooperation relationship between the two, etc.

[0065] Furthermore, the choice of the control input u i (t) will directly affect the trajectory of the agent. To achieve trajectory optimization, the selection of the control input needs to maximize or minimize the objective function under the premise of meeting certain constraints. Common constraint conditions include speed limit, acceleration limit, and relative position limit between agents, etc. At this time, the choice of the control input u i (t) is usually achieved through an optimization algorithm.

[0066] Generally, the goal of trajectory optimization is to minimize the cost function of the entire multi-agent system, which includes factors such as task completion, energy consumption, and path length. For example, the objective function can be expressed as:

[0067]

[0068] where L i (x i (t), u i (t)) represents the instantaneous cost of the i-th agent (such as energy consumption, time consumption, etc.), while Q i (x i (t), u i (t)) represents the constraints on the state and control input of the i-th agent. T is the total duration of the task, N is the number of agents; J is the total optimized cost function value, representing the overall performance of the multi-agent system.

[0069] As an option, in this embodiment, the pseudospectral method can be used to optimize these trajectories. The pseudospectral method transforms the state and control input of each agent into polynomial forms, converting it into a finite-dimensional optimization problem. The trajectory of each agent can be expanded using Legendre polynomials or Chebyshev polynomials to improve the solution accuracy and efficiency. On this basis, the necessary conditions for the optimal trajectory can be derived using the variational method, and the optimal control input can be derived using the Pontryagin maximum principle.

[0070] In a possible implementation, the state and control input of each agent are not only described by the dynamic equations but may also be affected by external disturbances or environmental changes. To handle these situations, the impact of disturbances on the trajectory needs to be considered during system design and compensated by robust control methods. This can be achieved by introducing a disturbance model. For example, state estimation can be performed through a Kalman filter, or a corresponding feedback controller can be designed based on the Lyapunov stability theory.

[0071] In multi-agent adversarial tasks, since there may be competitive or cooperative relationships among agents, the control input and trajectory planning of each agent need to consider not only its own goals but also the interactive effects with other agents. For example, during path planning, agents may need to follow strategies such as collision avoidance, cooperation, or racing. To this end, the system adopts a language model module that dynamically adjusts the weights of the objective function and optimizes the trajectory strategy by parsing the task description and the interactive feedback among agents.

[0072] Specifically, the language model module can adjust the weights of the objective function according to real-time feedback, enabling each agent to adjust its trajectory planning strategy according to the task requirements and the current environmental state. For example, if the task requires the agent to avoid obstacles as much as possible or confront enemy agents, the system will appropriately increase the weights of the obstacle avoidance or confrontation objectives, thereby optimizing the agent's behavior.

[0073] The pseudospectral method module is used to represent the state and control input of each agent in polynomial form and optimize the trajectory based on multi-objective optimization goals;

[0074] In this embodiment, the pseudospectral method module is used to represent the state and control input of each agent in polynomial form and optimize the trajectory based on multi-objective optimization goals. The key idea of the pseudospectral method is to approximate the state and control input of each agent as a polynomial form, transforming the complex trajectory optimization problem into a finite-dimensional optimization problem, thereby achieving efficient trajectory optimization. Combined with the dynamic modeling and control input optimization modules in the previous steps, the pseudospectral method can improve the computational efficiency while ensuring the accuracy of trajectory optimization.

[0075] Generally, trajectory optimization tasks need to consider multiple factors, such as task completion, system energy consumption, path smoothness, etc. These factors are usually handled in a multi-objective optimization manner. The pseudospectral method simplifies the computational complexity by representing the agent state in polynomial space and transforming the multi-objective optimization problem into a problem of solving polynomial coefficients. Specifically, the pseudospectral method can effectively address the challenges brought by high-dimensional and nonlinear problems and can handle various constraints, such as limitations on position, velocity, and acceleration, as well as the interactions among agents.

[0076] In this embodiment, the pseudo - spectral method represents the states and control inputs of agents by using Legendre polynomials or Chebyshev polynomials. Assume that the state x i (t) of each agent can be represented as a polynomial expansion of the following form:

[0077]

[0078] where, x i (t) is the state vector of the i - th agent, φ k (t) is the basis function of Legendre or Chebyshev polynomials, a i,k is the polynomial coefficient to be optimized, and N is the order of the polynomial. By adjusting the coefficient a i,k , the state trajectory of the agent within time t can be obtained.

[0079] When performing trajectory optimization, the goal is to adjust these polynomial coefficients so that the agent's trajectory not only satisfies the dynamic equations and control constraints but also optimizes a set of multi - objective functions. Common multi - objective optimization goals may include minimizing the path length, maximizing the task completion rate, minimizing the energy consumption, etc. During the optimization process, these factors can be comprehensively considered through the weighted sum of the objective functions. For example:

[0080]

[0081] where, J is the total optimization cost function value, representing the overall performance of the multi - agent system; L i (x i (t), u i (t)) is the immediate - cost function of the i - th agent, Q i (x i (t), u i (t)) is the constraint function, R i (x i (t)) is the path smoothness metric function, and w1, w2, w3 are the weight coefficients of the objective functions. By appropriately adjusting these weight coefficients, the trade - off between different goals can be achieved during the optimization process.

[0082] As an option, the multi - objective optimization problem can be handled by introducing the Lagrange multiplier method. Assume that there is a set of constraint conditions, such as the speed limit, acceleration limit of the agents, and the minimum distance constraint between agents, etc., then the optimization goal can be adjusted in the following way:

[0083] J′ = J+∑ j λ j g j (xi (t), u i (t));

[0084] Among them, g j (x i (t), u i (t)) is the J-th constraint condition function, and λ j is the Lagrange multiplier. By solving the Lagrangian function, the optimal solution that satisfies the constraint conditions can be obtained; J is the total value of the optimization cost function, representing the overall performance of the multi-agent system.

[0085] Specifically, the pseudospectral method realizes the optimization of the trajectory through the optimized polynomial coefficients. The trajectory optimization problem of each agent is finally transformed into a problem of solving a set of polynomial coefficients. These coefficients are optimized by numerical methods (such as Newton's method or gradient descent method) within the framework of constraint conditions and multi-objective optimization. In this way, the pseudospectral method not only simplifies the originally high-dimensional and non-linear problems, but also can obtain accurate trajectory optimization results in a short time.

[0086] In a possible implementation, the pseudospectral method module is combined with the aforementioned variational method and Pontryagin maximum principle module. In the variational method, the optimal trajectory is obtained by deriving the Euler-Lagrange equation, and the pseudospectral method is used to approximate the solutions of these equations. The Pontryagin maximum principle optimizes the control input and combines the trajectory calculated by the pseudospectral method, thus ensuring the optimality of the control strategy. The introduction of the pseudospectral method makes the solution of these optimization problems more efficient and ensures the smoothness and stability of the trajectory.

[0087] Furthermore, in complex multi-agent adversarial tasks, due to the interaction between agents, the pseudospectral method needs to consider the cooperation or competition relationship between agents. To handle this situation, in this embodiment, an interaction term is introduced to represent the influence between agents. For example, when the distance between agents is too close, collision avoidance constraints may be added during the trajectory optimization process, and these constraints are represented and processed by the pseudospectral method to ensure the safety and effectiveness of the system.

[0088] The variational method module is used to derive the necessary conditions for the trajectory optimization of each agent according to the pseudospectral method and generate the variational equation of the optimal trajectory;

[0089] In this embodiment, the variational method module is used to derive the necessary conditions for the trajectory optimization of each agent according to the pseudospectral method and generate the variational equation of the optimal trajectory. In the aforementioned pseudospectral method module, the states and control inputs of each agent are represented in polynomial form and solved under a multi-objective optimization framework. On this basis, the variational method further optimizes the trajectory by deriving the variational equation and necessary conditions, ensuring that the trajectory of each agent satisfies the dynamic constraints and optimization objectives while achieving the optimal control input. The variational method can not only derive the optimal control trajectory but also provide the necessary conditions for solving the optimization problem, further improving the overall efficiency of the system.

[0090] Generally, the variational method module describes the objective function and constraint conditions of the optimization problem by introducing the Lagrangian function. In a multi-agent system, considering the coupling relationship between the states and control inputs of each agent, the variational method obtains the necessary conditions for the trajectory optimization of each agent by taking the variation of the Lagrangian function, which provides guidance for solving the optimal trajectory. Specifically, the variational method transforms the trajectory optimization problem into an optimal control problem and obtains the optimal trajectory by solving the corresponding variational equation.

[0091] In this embodiment, through the polynomial representation obtained by the pseudospectral method, the objective function and constraint conditions of the trajectory optimization problem can be represented by the Lagrangian function. Assuming the objective function is J and includes multiple objectives such as energy consumption, time minimization, path smoothing, etc., the Lagrangian function can be expressed as;

[0092]

[0093] where, g j (x i (t), u i (t)) are the constraint conditions, λ j is the Lagrange multiplier, x i (t) and u i (t) are the state and control input of the i-th agent respectively; J is the total optimized cost function value, representing the overall performance of the multi-agent system; is the total Lagrangian function, representing the optimization objective of the system, including the cost function and constraint conditions.

[0094] In the variational method, the goal is to derive the necessary conditions for the optimal trajectory by taking the variation of the Lagrangian function. By taking the variation of the Lagrangian function, the following Euler-Lagrange equation is obtained:

[0095]

[0096] where, is the velocity of the i-th agent, and are the partial derivatives of the Lagrangian function with respect to velocity and position, respectively. By solving this equation, the optimal trajectory of each agent can be obtained.

[0097] As an option, if the constraint conditions are complex, the Lagrange multipliers in the calculus of variations can be used to handle the boundary conditions and the limitations of control inputs. By introducing appropriate Lagrange multipliers, the calculus of variations can effectively handle the interaction constraints between agents, velocity and acceleration limitations, and other constraint conditions. For example, the minimum distance constraint between agents can be handled by adding the corresponding constraint term to ensure that the trajectories of agents do not collide during the optimization process.

[0098] Specifically, the variational equation obtained through derivation can be combined with the Pontryagin maximum principle to further solve for the optimal control input. In this method, the choice of control input directly affects the evolution of the trajectory, so it is necessary to solve the optimal control problem to ensure the optimality of the trajectory. The calculus of variations not only provides the necessary conditions for the optimal trajectory but also provides a theoretical basis for the derivation of the optimal control strategy.

[0099] In a possible implementation, the calculus of variations module is closely connected to the pseudospectral method module. First, the pseudospectral method represents the trajectory in polynomial form and obtains the corresponding polynomial coefficients through an optimization problem. Then, the calculus of variations derives the necessary conditions for the trajectory optimization of each agent by taking the variation of the Lagrangian function and further solves for the optimal trajectory. In this process, the polynomial representation of the pseudospectral method simplifies the solution process of trajectory optimization, while the calculus of variations provides a mathematical basis for solving the optimal trajectory.

[0100] Furthermore, in multi-agent adversarial tasks, the calculus of variations can also handle the cooperation and adversarial relationships between different agents. Suppose there is an adversarial behavior between agents, then the calculus of variations can emphasize the optimization of the adversarial objective by adjusting the weight coefficients in the objective function. For example, when the objective function contains the relative position constraints between multiple agents, the calculus of variations can derive a set of necessary conditions for each agent to ensure the optimization of its trajectory while avoiding collisions or too-close contacts with other agents.

[0101] The Pontryagin maximum principle module is used to derive the optimal control input for each agent based on the Hamiltonian function;

[0102] In this embodiment, the Pontryagin maximum principle module is used to derive the optimal control input for each agent based on the Hamiltonian function. Closely connected with the aforementioned pseudospectral method and variational method modules, the Pontryagin maximum principle provides a theoretical framework. By introducing the Hamiltonian function, it derives the optimal control input for the agent, ensuring the optimality of the control strategy during the trajectory optimization process. By solving the extreme value conditions of the Hamiltonian function, the optimal control input for each agent under given constraints and objectives can be obtained, thus achieving the optimal trajectory planning of the system.

[0103] Generally, the Pontryagin maximum principle requires constructing a Hamiltonian function that includes the control input and solving the corresponding optimality conditions. These conditions obtain the optimal control input for each agent by introducing adjoint variables (also known as Lagrange multipliers) and by solving the partial derivatives of the Hamiltonian function. Through this method, the optimal control strategy of the agent in different states can be accurately calculated, and the trajectory of the entire system can be further optimized.

[0104] In this embodiment, first, the Hamiltonian function HHH is defined to represent the system state, control input, and objective function of the agent. Assuming that the control input u i (t) of each agent is optimized through the control strategy, the form of the Hamiltonian function is:

[0105] H i (x i (t),u i (t),λ i (t))=L i (x i (t),u i (t))+λ i (t)·f i (x i [[ID=3l]](t),u i (t));

[0106] Where, x i (t) is the state vector of the i-th agent, u i (t) is the control input of the i-th agent, λ i (t) is the adjoint variable, L i (x i (t),u i (t)) is the immediate cost function of the i-th agent, f i (x i (t),u i(t) is a function that describes the dynamics of the i-th agent.

[0107] In the Pontryagin maximum principle, the necessary condition for the optimal control input is obtained through the extremum condition of the Hamiltonian function. Specifically, the optimal control input u i (t) should make the Hamiltonian function reach an extremum with respect to the control input, that is:

[0108]

[0109] H i (x i (t), u i (t), λ i (t)) is the Hamiltonian function of the i-th agent, representing the comprehensive performance index that combines the state, control input, and Lagrange multipliers; is the partial derivative of the Hamiltonian function with respect to the control input u i (t), representing the sensitivity of the control input to the system performance during the optimization process; x i (t) is the state vector of the i-th agent; u i (t) represents the control input of the i-th agent, including control quantities such as force and velocity; λ i (t) is the Lagrange multiplier of the i-th agent, representing the influence of the constraint conditions, usually used for the weight of control constraints.

[0110] This gives the derivation formula for the optimal control input of each agent. By solving this equation, the optimal control input u i (t) can be obtained, ensuring that the trajectory of the agent optimizes the objective function while satisfying the system constraints.

[0111] As an option, to handle the constraint conditions, especially for the cases of state constraints and control constraints, the Pontryagin maximum principle also needs to introduce the adjoint variable λ i (t) to satisfy these constraints. The adjoint variable, combined with the Hamiltonian function, can effectively handle the boundary conditions and non-linear constraints during the trajectory optimization process. For example, if the control input of the agent is restricted by acceleration or speed limits, the adjoint variable will adjust the optimal control input to ensure that these constraint conditions are satisfied.

[0112] Specifically, by solving the extremum condition of the Hamiltonian function, a set of optimal control input equations can be obtained. To achieve global optimality, a system of adjoint equations and state equations also needs to be solved, usually these equations are a system of differential equations composed of state equations and adjoint equations:

[0113]

[0114] Among them, is the time derivative of the state of the \(i\)-th agent, is the time derivative of the adjoint variable. By solving this system of differential equations, the optimal trajectory and the corresponding control input of each agent can be obtained; \(H\) i (x i (t), u i (t), λ i (t)) is the Hamiltonian function of the \(i\)-th agent, representing the comprehensive performance index that combines the state, control input, and Lagrange multiplier; is the partial derivative of the Hamiltonian function with respect to the control input \(u\) i (t), representing the sensitivity of the control input to the system performance during the optimization process; \(x\) i (t) is the state vector of the \(i\)-th agent; \(u\) i (t) represents the control input of the \(i\)-th agent, including control quantities such as force and velocity; \(λ\) i (t) is the Lagrange multiplier of the \(i\)-th agent, representing the influence of the constraint conditions, and is usually used for the weight of the control constraints.

[0115] In a possible implementation, the application of the Pontryagin maximum principle is combined with the aforementioned variational method and pseudospectral method. First, the pseudospectral method represents the trajectory in polynomial form and obtains the polynomial coefficients by solving the optimization problem. Then, the variational method derives the necessary conditions for trajectory optimization. Finally, the Pontryagin maximum principle derives the optimal control input by solving the Hamiltonian function. Through this cascading manner, it can be ensured that the solution of the optimal trajectory not only satisfies the necessary conditions theoretically but also can be optimized through the actual control input.

[0116] Furthermore, in a multi-agent system, the Pontryagin maximum principle can also handle the interactions between agents. In scenarios of multi-agent cooperation or confrontation, there may be dynamic competition or cooperation relationships between agents, and these relationships are reflected through the interaction of the Hamiltonian function and the adjoint variable. By introducing the interaction term between agents, the Pontryagin maximum principle can achieve coordinated optimization among multiple agents, ensuring that each agent makes decisions within the framework of global optimality.

[0117] The nonlinear control module, connected to the Pontryagin maximum principle module, is used to analyze and ensure the stability of the system through Lyapunov stability theory, preventing unstable behaviors;

[0118] In this embodiment, the non - linear control module is connected to the Pontryagin maximum principle module, aiming to analyze and ensure the stability of the system through Lyapunov stability theory and prevent unstable behaviors. In the aforementioned Pontryagin maximum principle module, the optimal control input for each agent is derived through the Hamiltonian function. However, the derivation of the optimal control input alone is not sufficient to guarantee the stability of the system. Therefore, the non - linear control module uses Lyapunov stability theory to analyze the behavior of the entire system, ensuring that under the optimal control strategy, the trajectories of the agents are not only optimal but also stable, preventing the system from oscillating or becoming unstable.

[0119] Generally, the stability analysis of non - linear systems is a complex task, especially in multi - agent systems. Since there may be coupling relationships between the states and control inputs of each agent, and the dynamic models of these agents are usually non - linear. To avoid unstable behaviors during the trajectory optimization process of the system, Lyapunov stability theory is widely applied to such problems. By constructing a suitable Lyapunov function, it can be determined whether the system can remain stable under the action of the optimal control input and provide theoretical support for the design of the control input.

[0120] In this embodiment, the non - linear control module first analyzes the stability of the system with the help of Lyapunov stability theory. Assume that the state x i (t) of each agent is optimized by the control input u i (t). To ensure the stability of the system, a Lyapunov function V(x i (t)) needs to be constructed, which represents the system energy or some performance metric. The choice of the Lyapunov function should be able to reflect the change trend of the system state. Generally, the form of the Lyapunov function is:

[0121]

[0122] where P i is a symmetric positive - definite matrix, and x i (t) is the state vector of the i - th agent. The Lyapunov function must satisfy the following condition: when the system state reaches the stable point, the derivative of V(x i (t)) should be negative, indicating that the system energy dissipates gradually and the system tends to be stable.

[0123] To verify the stability of the system, the derivative of the Lyapunov function needs to satisfy:

[0124]

[0125] wherein, is the state derivative of the i-th agent, representing the rate of change of its state. By calculating the derivative of the Lyapunov function, the stability analysis conditions of the system under the optimal control input can be obtained; The state function of the i-th agent at time t The partial derivative of the state represents the sensitivity of the state function to the change of the state variable.

[0126] As an option, in a multi-agent system, the interaction between agents may affect the stability of the system. Therefore, in the construction of the Lyapunov function, the interaction term between agents can be introduced. For example, assuming that there is a cooperative or adversarial relationship between agents, the interaction term f ij (x i (t),x j (t)) can be introduced by modifying the form of the Lyapunov function, so that the Lyapunov function can better reflect the coupling effect between agents in the system. The interaction term may include factors such as collision avoidance and distance limitation, so as to ensure that the stability analysis in the multi-agent system can effectively consider the influence between each agent.

[0127] Specifically, by constructing the Lyapunov function and analyzing its derivative, it can be judged whether the system is stable. If under the optimal control input, is always less than zero, then the system is stable under this control input. If under certain conditions, may be greater than zero, it means that the system may exhibit unstable behavior. At this time, the control strategy needs to be further adjusted or the trajectory optimized to ensure the stability of the system.

[0128] In a possible implementation, by combining the Lyapunov stability theory with the Pontryagin maximum principle, stable optimal control inputs can be derived for each agent. First, the optimal control inputs are derived through the Pontryagin maximum principle, and then the Lyapunov stability theory is used to analyze the interaction between agents and the control inputs to ensure that these optimal control inputs do not cause the system to be unstable. By this method, it is possible to ensure that the multi-agent system remains stable while optimizing the trajectory, preventing oscillation or divergence behavior.

[0129] Furthermore, in complex nonlinear systems, the selection of the Lyapunov function may need to be adjusted according to the specific characteristics of the system. For example, the dynamic model of the agent may contain nonlinear terms or be affected by external disturbances. In this case, a more complex Lyapunov function needs to be constructed to consider these factors. By carefully designing the Lyapunov function, the nonlinear control module can effectively ensure the stability of the system under optimal control and avoid unstable behaviors.

[0130] The language model module, connected to the nonlinear control module, is used to dynamically parse the task description and adjust the objective function weights and trajectory optimization strategies according to the interaction feedback between multiple agents;

[0131] In this embodiment, the language model module is connected to the nonlinear control module, which is used to dynamically parse the task description and adjust the objective function weights and trajectory optimization strategies according to the interaction feedback between multiple agents. In the aforementioned nonlinear control module, the Lyapunov stability theory is used to ensure the stability of the system and prevent unstable behaviors during the trajectory optimization process. However, in the practical application of multi-agent systems, task requirements often have the characteristics of dynamic changes, and the interaction feedback between agents may affect the priority of the overall task objective. Therefore, a mechanism that can parse the task description in real time and adjust the optimization strategy according to the interaction feedback is needed to ensure the flexible adaptability and optimal performance of the multi-agent system. The language model module is the key part that undertakes this function. It dynamically adjusts the weight coefficients of the objective function by analyzing the task description, the state changes between agents, and the interaction feedback, thereby optimizing the trajectory planning.

[0132] Generally, the language model module is based on natural language processing technology to parse the task description and extract the key task requirements involved, such as objective priority, path optimization objective, energy consumption constraint, etc. At the same time, this module continuously receives the interaction feedback from multiple agents, including information such as state changes, trajectory deviations, and collaboration requirements, and adjusts the optimization strategy to ensure that the system always develops in the direction of global optimality. In this way, the language model module can not only enhance the autonomy of the multi-agent system but also adapt to the dynamic changes in complex environments, improving the robustness and efficiency of task completion.

[0133] In this embodiment, the language model module first parses the task description and extracts the core optimization objective. Suppose the task description T contains several objectives and constraints. The language model extracts the corresponding objective function weight vector W by parsing T:

[0134] W = {w1, w2,..., w n};

[0135] Among them, w1 represents the weight of the optimization objective of the objective function, and its initial value is set by the task description. Next, this module receives the interaction feedback F from multiple agents, which includes relative state information between agents, control input deviation, cooperative task status, etc. According to the feedback information, the language model adjusts the weight of the objective function in real time to obtain a new weight vector W′:

[0136] W′ = W + ΔW;

[0137] Among them, ΔW is the weight correction amount calculated according to the interaction feedback, ensuring that the trajectory optimization strategy can adapt to the dynamic changes of task requirements.

[0138] As an option, the language model module can not only parse the task description and adjust the weights, but also correct the trajectory optimization strategy in real time according to the changes in task requirements. Suppose the task objective changes, for example, from "shortest path optimization" to "minimum energy consumption optimization", then the language model module will identify this change and reset the trajectory optimization objective. In this case, the optimization strategy will be adjusted from time-based optimization to energy consumption-based optimization to ensure the adaptability of the task objective.

[0139] Specifically, the language model module collaborates with the Pontryagin maximum principle module and the nonlinear control module to ensure that the optimized trajectory not only satisfies the optimality of the control input, but also has dynamic adaptability. After the Pontryagin maximum principle module calculates the optimal control input, the language model module can parse its result and adjust the optimization parameters according to the interaction feedback. For example, if the cooperation requirement between agents increases, the language model module will appropriately increase the weight of the cooperation objective, prompting the system optimization strategy to adjust towards stronger cooperation.

[0140] In a possible implementation, the language model module can also incorporate a reinforcement learning mechanism to improve the intelligence level of trajectory optimization. Specifically, this module can continuously optimize the adjustment strategy of the objective function weight by analyzing historical task data and real-time interaction feedback. For example, suppose a certain interaction pattern has shown high efficiency in past task executions, then the language model module can automatically preferentially select similar optimization strategies by learning historical data to improve the overall execution efficiency of the system.

[0141] The simulation verification module, connected to the language model module, is used to verify the effectiveness of the trajectory optimization system in multiple-agent adversarial tasks and compare the performance with traditional optimization methods;

[0142] In this embodiment, the simulation verification module is connected to the language model module and is used to verify the effectiveness of the trajectory optimization system in multiple multi-agent confrontation tasks and compare the performance with traditional optimization methods. In the aforementioned language model module, by parsing the task description and agent interaction feedback, the weights of the objective function and the optimization strategy are adjusted to ensure the adaptability of the system in a dynamic environment. However, in order to further evaluate the performance of the proposed trajectory optimization method, it is necessary to conduct tests through the simulation verification module, compare the results of different optimization methods, and analyze the robustness, convergence, and computational efficiency of the system, so as to ensure the feasibility and superiority of the system in practical applications.

[0143] Generally, the simulation verification module evaluates the performance of the trajectory optimization method under different task conditions by constructing a simulation environment and simulating the movement processes of multiple agents. This module not only verifies whether the optimal control trajectory of the system meets the task requirements, but also analyzes the impact of the optimization strategy on the agent trajectories and compares with traditional optimization methods. By quantitatively analyzing the effects of different optimization methods, the advantages of this system in terms of trajectory optimization accuracy, computational complexity, task completion rate, etc. can be evaluated, and at the same time, its adaptability and stability in confrontation tasks can be verified.

[0144] In this embodiment, the simulation verification module first constructs a multi-agent confrontation simulation environment, which contains different types of agents that perform collaborative tasks and confrontation tasks respectively. The trajectory of each agent is calculated by the optimization system and compared with the trajectory generated by the traditional optimization method. Assuming that the state vector of the agent is x i (t), and the optimization objective is to minimize a certain cost function J, the effectiveness of the trajectory optimization result can be evaluated by the following error metric:

[0145]

[0146] where, is the trajectory optimized by this system, is the trajectory generated by the traditional optimization method, and N is the number of agents. This error function measures the trajectory deviation between the two optimization methods, thereby evaluating the trajectory optimization accuracy of the system.

[0147] As an option, the simulation verification module can introduce different environmental disturbances, such as random obstacles, dynamic targets, or external interferences, to test the robustness of the trajectory optimization system. In this case, the optimization system needs to be able to adapt to environmental changes and adjust the trajectories of the agents in real time to ensure that the task completion rate is not affected. For example, in the dynamic obstacle avoidance task, the task completion rate R can be defined as follows:

[0148]

[0149] where, M successThe number of agents that successfully avoid obstacles and complete tasks, M total is the total number of agents. This metric is used to evaluate the effectiveness of the system in a complex environment.

[0150] Specifically, the simulation verification module not only evaluates the trajectory optimization accuracy but also conducts a comparative analysis of the computational efficiency. Since trajectory optimization involves high-dimensional optimization problems, the computational complexity is a key factor affecting real-time performance. Let the computational time of the optimization system be T opt , and the computational time of the traditional optimization method be T ref . The computational efficiency improvement ratio can be defined as:

[0151]

[0152] By calculating S, the improvement amplitude of the computational efficiency of this system relative to the traditional optimization method can be quantified, and its applicability in real-time tasks can be verified.

[0153] In a possible implementation, the simulation verification module conducts comparisons using different optimization methods, including but not limited to gradient descent method, genetic algorithm, reinforcement learning method, etc. By running different optimization algorithms under the same task conditions, comparing their optimization effects, computational complexities, and task completion rates, the advantages of this system can be analyzed. For example, if the success rate of a certain optimization method in an adversarial task is significantly higher than that of the traditional method, it indicates that its adaptability in a dynamic interaction environment is stronger and it is suitable for complex multi-agent task scenarios.

[0154] Although the embodiments of the present invention have been shown and described, for those of ordinary skill in the art, it can be understood that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A trajectory optimization system that combines pseudospectral theory and language models in multi-agent adversarial tasks, characterized in that, Including: Multiple agents, each agent having a state and a control input, and its state being described by a dynamic equation; A pseudospectral method module for representing the state and control input of each agent in polynomial form and performing trajectory optimization based on multi-objective optimization goals; A variational method module for deriving the necessary conditions for each agent's trajectory optimization according to the pseudospectral method and generating the variational equation of the optimal trajectory; A Pontryagin maximum principle module for deriving the optimal control input of each agent based on the Hamiltonian function; A non-linear control module connected to the Pontryagin maximum principle module for analyzing and ensuring the stability of the system through Lyapunov stability theory and preventing unstable behaviors; A language model module connected to the non-linear control module for dynamically parsing task descriptions and adjusting the objective function weights and trajectory optimization strategies according to the interaction feedback between multiple agents; A simulation verification module connected to the language model module for verifying the effectiveness of the trajectory optimization system in multi-agent adversarial tasks and comparing the performance with traditional optimization methods.

2. The trajectory optimization system combining the pseudo-spectrum theory and the language model in the multi-agent adversarial task according to claim 1, wherein, The pseudospectral method module includes: A pseudospectral expansion unit: This unit represents the state and control input of each agent in polynomial form and transforms it into a finite-dimensional trajectory optimization problem, using Legendre polynomials or Chebyshev polynomials as basis functions for the expansion of the state and control input; An objective function optimization unit: This unit constructs a total objective function according to the multi-objective optimization goal and optimizes the trajectory of each agent through the pseudospectral method.

3. The trajectory optimization system combining the pseudo-spectrum theory and the language model in the multi-agent confrontation task according to claim 1, characterized in that, The variational method module includes: An Euler-Lagrange equation derivation unit: This unit derives the Euler-Lagrange equation of each agent through the variational method according to the trajectory optimization problem represented by the pseudospectral method to obtain the necessary conditions for trajectory optimization; An optimal trajectory generation unit: This unit uses the equation derived by the variational method to generate a variational solution of the optimal trajectory and uses this solution as the input for the subsequent Pontryagin maximum principle module.

4. The trajectory optimization system combining the pseudo-spectrum theory and the language model in the multi-agent adversarial task according to claim 1, wherein, The Pontryagin maximum principle module includes: A Hamiltonian function construction unit: This unit constructs a corresponding Hamiltonian function based on the state, control input, and dynamic equation of each agent, transforming the optimal control problem into a problem of maximizing the Hamiltonian function; An optimal control derivation unit: This unit derives the optimal control input of each agent by maximizing the Hamiltonian function.

5. The trajectory optimization system combining the pseudo-spectrum theory and the language model in the multi-agent adversarial task according to claim 1, characterized in that, The non-linear control module includes: A Lyapunov function construction unit: This unit constructs a Lyapunov function for each agent; A feedback control design unit: This unit designs a feedback controller based on Lyapunov stability theory and adjusts the control input according to the current state of the agent.

6. The trajectory optimization system combining the pseudo-spectrum theory and the language model in the multi-agent adversarial task according to claim 1, wherein, The language model module includes: Task description parsing unit: This unit parses the objective function according to the input task description, determines the weight of the objective function according to the task requirements, and adjusts the trajectory optimization objective of multiple agents. Policy adjustment unit: This unit dynamically optimizes the trajectory policy of the agents according to the interaction feedback between the agents, the current task execution status, and the adjustment of the objective function.

7. The trajectory optimization system combining the pseudo-spectrum theory and the language model in the multi-agent confrontation task according to claim 1, characterized in that, The simulation verification module includes: Simulation design unit: This unit designs simulation experiments in the multi-agent confrontation task environment to verify the performance of the trajectory optimization system. Performance comparison unit: This unit compares the results of the trajectory optimization system with traditional methods, and evaluates the advantages and improvements of the system in terms of trajectory optimization accuracy, system stability, and adaptability through comparative experiments.

8. The trajectory optimization system combining the pseudo-spectrum theory and the language model in the multi-agent confrontation task according to claim 3, wherein, The optimal trajectory generation unit includes: Variational equation solving unit: This unit uses the variational solution of the Euler-Lagrange equation and combines the trajectory optimization problem represented by the pseudospectral method to solve the optimal trajectory path of each agent. Optimal trajectory smoothing unit: This unit smooths the obtained optimal trajectory to eliminate discontinuous points and unnatural changes in the trajectory.

9. The trajectory optimization system combining the pseudo-spectrum theory and the language model in the multi-agent adversarial task according to claim 5, characterized in that, The Lyapunov function construction unit: Stability analysis unit: This unit analyzes whether there are stability risks for the agents during the trajectory optimization process by calculating the Lyapunov function values of the system states. Lyapunov function optimization unit: This unit designs and optimizes the Lyapunov function for each agent.

10. The trajectory optimization system combining the pseudo-spectrum theory and the language model in the multi-agent adversarial task according to claim 1, characterized in that, The simulation design unit includes: Multi-agent simulation model construction unit: This unit constructs a simulation environment according to the interaction behaviors between multiple agents and their dynamic equations. Trajectory optimization verification unit: This unit verifies the trajectory optimization system in the simulation environment, monitors the trajectory generation and behavior execution of each agent, and evaluates the performance and effectiveness of the optimization system. Environmental change simulation unit: This unit simulates a variable confrontation environment, tests the robustness and adaptability of the trajectory optimization system under complex tasks by changing task objectives, constraint conditions, and interaction strategies between agents.

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