Cluster adaptive path planning method based on geometric PDE and PINN

Abstract

The invention discloses a cluster adaptive path planning method based on geometric PDE and PINN. The method comprises the following steps: firstly, modeling a dynamic model of a fault cluster with time-varying topology; secondly, through Riemannian manifold and gradient evolution based on heat flow, a fault cluster is constructed based on a parabolic PDE system, and a reference trajectory of a cluster system in an obstacle environment is obtained; and learning a steady-state solution of the cluster based on a parabolic PDE system by using a physical information neural network (PINN), and obtaining a fault-tolerant optimal control law of the cluster system based on geometric PDE and PINN under the conditions of time-varying topology, actuator fault and obstacle operation. According to the method, it is ensured that the intelligent agent navigates to the target position from the initial position along the dynamic reference trajectory, meanwhile, endogenous compensation of the fault effect is achieved, and unified solution of fault compensation and motion planning is achieved.

Description

Technical Field

[0001] This patent aims to solve the fault-tolerant control problem of cluster systems with time-varying topology and actuator failures in obstacle environments, and proposes an adaptive fault-tolerant path planning algorithm based on geometric partial differential equations (PDEs) and physical information networks (PINNs). Background Art

[0002] Swarm systems, comprised of multiple autonomous agents working collaboratively in a collective environment, demonstrate remarkable potential for performing complex tasks through coordinated interactions in densely populated obstacle scenarios. A key application of these systems lies in motion planning to find feasible paths, as seen in areas such as autonomous driving, warehouse logistics, and industrial automation. Existing path planning methods primarily fall into three categories: heuristic, optimization, and geometric approaches.

[0003] In particular, geometric approaches focus on the geometric properties of the workspace and the dynamic constraints. These constraints are often dynamically represented through PDEs, revealing the essential characteristics of motion planning problems. However, existing geometric PDE methods often focus on single-agent systems, neglecting the critical role of inter-agent communication and collaboration in swarm dynamics. This stems from the theoretical challenges of incorporating internal interaction information into global PDE models to generate feasible trajectories.

[0004] In swarm systems with time-varying interacting topologies, the time-varying behavior of agents, influenced by their mobility and energy constraints, can induce dynamic uncertainty and increase coordination difficulties. Actuator failures can further exacerbate these challenges, potentially leading to control efficiency degradation, motion deviations, or actuator jamming, ultimately causing fault cascades within the swarm, compromising mission completion. Therefore, fault-tolerant control (FTC) techniques designed to compensate for the effects of faults are crucial for ensuring system reliability. Numerous FTC studies have been conducted on multi-agent systems with time-varying topologies and actuator failures. Key approaches include replacing the nominal controller with a fault-adaptive controller and constructing a reconfigurable system by deploying virtual actuators. However, these approaches focus on local fault compensation while ignoring global trajectory reconstruction, failing to effectively coordinate fault-induced motion distortion with swarm motion planning.

[0005] Although geometric PDE methods can characterize path planning problems from the dynamical essence, their application in swarm systems faces two core challenges: the numerical complexity of solving high-dimensional PDEs and the real-time adaptability of dynamically interacting topologies. Traditional numerical methods (such as finite element methods and spectral methods) require discretization of the state equations of each agent when dealing with multi-agent coupled PDE models, leading to the curse of dimensionality. Previous studies have demonstrated that PINNs exhibit excellent performance in complex PDE problems, such as those involving multi-physics coupling and time-varying boundary conditions. For example, in fluid mechanics, PINNs have successfully solved the Navier-Stokes equations with dynamically varying Reynolds numbers; in materials science, they have been applied to strongly nonlinear problems such as interface evolution in crystal growth. These achievements have laid the theoretical foundation for the application of PINNs to fault-tolerant control of swarm systems. However, existing PINNs have focused on multiple static or continuously varying PDE systems and have yet to fully explore their potential in hybrid discrete-continuous dynamics (such as those with time-varying topology and actuator failures), a core challenge in fault-tolerant path planning for swarm systems. Summary of the Invention

[0006] This patent proposes a distributed adaptive fault-tolerant optimal control algorithm based on geometric partial differential equations (PDEs) and physical information networks (PINNs) for cluster systems with time-varying topologies, actuator failures, and operating in obstacle environments. To address the problem of coordinated imbalance caused by faults, this paper introduces a geometric metric of the trajectory to coordinate the coordination mechanism between the nominal controller and the virtual controller. Based on this, the fault-tolerant optimal control problem is transformed into a trajectory optimization problem, in which the partial differential equation based on heat flow provides the reference trajectory for the system. This geometric modeling method characterizes the cluster dynamics evolution by combining the Riemann length and the gradient representation based on heat flow, and transforms communication restrictions and external obstacles into state constraints to ensure the stability and robustness of the cluster system reference trajectory. In order to quickly and accurately obtain the reference trajectory, a PINN architecture with an independent network is adopted, and a parameter sharing mechanism is used to adaptively obtain the numerical solution of the reference trajectory and the fault-tolerant optimal control law, which is easy to apply in engineering practice.

[0007] To achieve the above object, the technical solutions adopted by the present invention are as follows:

[0008] S1. Dynamic modeling of fault clusters with time-varying topology.

[0009] Consider a cluster of n agents operating in an ideal environment without faults and obstacles, where each agent follows the dynamic equation:

[0010]

[0011] variable and They represent the plane position and control input of agent i at time t, respectively, where and They represent the displacement of agent i in the horizontal x-axis coordinate direction and the vertical y-axis coordinate direction, respectively. g(·) is a nonlinear mapping. is the set of all agents in the system.

[0012] Communication between agents is done by undirected time-varying topology Description, where the edge set The dynamic evolution is: It shows that the communication edge between agents i and j is formed if and only if the distance condition at the current moment ||X ij ||≤l d When satisfied, the distance is defined as Time-varying signals The right limit at time t is defined as:

[0013]

[0014] in, and Represents the communication status of agents i and j respectively The values before and after the transfer time t represent the instantaneous change of the communication edge (i, j) at time t. c <l d Define the interval (l c ,l d ] is the hysteresis identification area, which simulates the negotiation delay in the actual communication connection establishment process by introducing the hysteresis effect of edge addition / deletion.

[0015] For cluster systems in this type of constrained time-varying topology Under this condition, the evolution of the system needs to rely on the potential function based on the interaction between individuals to ensure the global stability. Such a potential function is characterized as the control input of the cluster system when there is no fault. Specifically:

[0016]

[0017] Among them, ψ ij (||X ij ||) represents the total potential function variable with the agent distance as the variable, They represent the auxiliary potential function to ensure continuity, the potential function to maintain connectivity, the potential function to be added, and the potential function to be deleted. The specific design is:

[0018]

[0019] Parameters a0,h ad ,h a ,h r The selection of potential function ψij (||X ij ||) at the threshold point l c ,l d There is C 2 Continuity, and for all agents j∈N i (t) and any time t∈(0,+∞) satisfies Lipschitz continuity. N i (t) is the neighbor set of agent i, which is defined as And the symmetry condition is satisfied: if and only if j∈N i (t), i∈N j (t); In addition, in the above potential function, is the connectivity-maintaining set of individual i in the cluster system under collaborative tasks, is the attraction set, defined as When the judgment condition ||X is met ij ||≤l d When -q2 (q2>0 is the hysteresis threshold), j is included in the attraction set of i and the edge addition behavior is activated; is the exclusion set, defined as Satisfy the judgment condition||X ij ||≥l c +q1 (q1>0 is the hysteresis threshold), j is included in the exclusion set of i, and the edge deletion behavior is activated. and It is a binary discrimination signal, which takes 1 when the discrimination condition is met, and 0 otherwise.

[0020] Considering an unknown actuator failure in an agent, which destroys the interaction between agents and causes the collaborative task to fail, the dynamics of the faulty agent can be described as follows:

[0021]

[0022] in, Indicates the i f The state of a faulty agent, function The mathematical model representing the fault, the set of fault agents is

[0023] Define the state vector as With initial state in,

[0024] represents the global state of the fault cluster system, q(t)∈R 2m is an auxiliary variable for generating synergy. According to the vector augmentation property, a dynamic augmentation model of fault cluster is constructed:

[0025]

[0026] In order to make the subsequent formula expression concise, let the augmented matrix be in, Ξ is the Laplace moment; M is 0 2m is a 2m-dimensional all-zero column vector, I 2n×2m is a 2n×2m dimensional matrix of all 1s. In addition, the variable represents the interaction force of all fault-free agents, that is, the nominal control force, where the force of a single fault-free agent is It is the input of the fault-tolerant control to be designed.

[0027] S2. Construct the reference trajectory of the fault cluster dynamic augmentation model based on geometric heat flow in step S1 through Riemannian manifold.

[0028] The trajectory length of the fault cluster system in step S1 is defined by the positive symmetric Riemann metric tensor G(t)

[0029]

[0030] Where μ(t) represents the trajectory of the agent over time t, and the Riemannian metric tensor G(t) is designed to suppress the under-actuated dynamics effect caused by faults and satisfy the system constraints:

[0031]

[0032] in precomputed by Gram-Schmidt orthogonalization, is a supplementary matrix that represents the auxiliary input direction of the under-driven part that is complementary to the original control input matrix M. represents the time-varying regulator, η i (t) represents the time-varying regularization weight parameter with the structure The adjustment gain satisfies and Actuator failure leads to unbalanced collaborative potential between agents, so the time-varying gain is introduced. Adjust the Riemann metric tensor G(t) to adaptively adjust the trajectory length of the system under the metric. Introduce gain By adjusting the trajectory length through the constraint G(t), the intelligent agent can avoid obstacles. It should be noted that when the parameter η i When (t)>0 and takes a large value, it means that the most influential fault and obstacle is taken as the under-actuated direction of the actuator, and a penalty term is applied to force the agent trajectory to converge to the allowable motion subspace of the cluster system control and be robust.

[0033] Consider As an additional constraint for local communication (where is the local communication neighbor number constraint for agent i, |N i (t)| represents the total number of other agents that have direct communication or perception interaction with agent i at time t), construct a structure that satisfies v i (z)≤0, which transforms such inequalities into uniform constraints (a total of n terms, i≤n). Specifically, we define like This indicates that the maximum number of adjacent constraints is violated. Constraints based on the maximum degree of the edge in Represents the number of virtual neighbors of agent i, and the time-varying function can be further designed as follows:

[0034]

[0035] Here, the Heaviside step function is obtained by the logistic function ζ>0 approximates that,

[0036] Obviously, when hour, when hour, Based on this, the variable δ is introduced w To characterize the cumulative error, it is defined as follows:

[0037]

[0038] Furthermore, we define the vector Under this definition, the fault cluster system dynamics model can be further expressed as a Riemann augmented system:

[0039]

[0040] Define the state vector in

[0041] and By introducing the augmented Riemann metric tensor It comprehensively integrates topological constraints and adjacent quantity constraints, and its specific form is given by the following formula:

[0042]

[0043] in, element γ i Represents a time-varying function When (that is, the number of neighbors of agent i exceeds the limit), the penalty coefficient for exceeding the limit behavior is set.

[0044] matrix Characterizing time-varying topology based on the constraints of the Riemannian metric, the essence of which is derived from the constraints of the topological time-varying edges in It is a dynamically adjusted penalty coefficient used to quantify the deviation of the edge (i, j) from the preset target distance l during the optimization process. d the intensity of punishment.

[0045] Using the fault cluster Riemann augmentation model, the trajectory length is specified as:

[0046]

[0047] Among them, the scalar is the total constraint penalty term,

[0048] Map the constraint violation of each agent into a vector form, where each component corresponds to a weighted violation of a constraint. represents the constraint function of the i-th agent. It should be noted that when the constraint condition for any agent i If not met, it will lead to and Then through γ i Penalize the trajectory length. Conversely, when γ i ≤0, the item Does not affect the trajectory length. This penalty mechanism also applies to constraints In addition, from the trajectory length formula of the Riemann augmented system, it can be seen that when seeking the minimum length trajectory, the adaptive adjustment parameter introduced in the Riemann metric G(t) The actuator uncertainty caused by the fault will be compensated to optimize the agent trajectory.

[0049] In the Riemann metric tensor with constraints Under the action of , based on the gradient theory of heat flow, an energy functional based on the Riemann length is established:

[0050]

[0051] in That is, the Lagrangian of a time-varying system.

[0052] The Euler-Lagrange equation is modified by the variational principle and the correction term Δ(s, t) is introduced to obtain the heat flow partial differential equation (PDE), which is the reference trajectory dynamics of the cluster system with actuator failure and time-varying topology in step S1. It is expressed as a parabolic PDE:

[0053]

[0054] z(s,T)=ξ

[0055] in:

[0056]

[0057] Where s is the homotopy variable at time t, t∈[0,T], and z(s,t) is the initial connection position of the fault cluster under the Riemannian metric. and the trajectory of the terminal position ξ. ▽ represents the measurement The related Levi-Civita connection, and ψ(s,t) represents the metric-adjusted gradient term. Describes the trajectory acceleration and nominal dynamics With the help of potential field reshaping, the curvature and trajectory length are minimized, and the gradient term ψ(s,t) is used to adjust the trajectory so that Time-varying metrics in The introduction generates an additional acceleration term Δ(s, t) to reflect the impact of the dynamic adjustment of the constraint conditions on the trajectory deformation rate.

[0058] According to the LaSalle invariant set principle, the Lyapunov function is non-increasing, and then the state of the parabolic PDE system converges to the steady-state value The status By decomposing the steady-state z ＊ (s, t) is obtained. According to the isomorphism theory, represents the ideal trajectory of the cluster system under actuator failure, time-varying topology and obstacles, that is, the reference trajectory.

[0059] S3. Based on the reference trajectory of the parabolic PDE system obtained in step S2, the steady-state value z of the parabolic PDE system is accurately approximated by the Physics-Informed Neural Networks (PINN). ＊ (s, t), overcome the problem of difficulty in solving PDE and large error based on finite difference method, especially when the parabolic PDE system contains unknown variables η i (t), Furthermore, we obtain the fault-tolerant optimal control law for cluster systems with time-varying topology and actuator failures based on geometric PDE and PINN:

[0060]

[0061] The specific contents are as follows:

[0062] S3.1. Solve the parabolic PDE equation in S2 by PINN

[0063] First, the input variables are defined as homotopy parameters s∈[0,+∞) and time t∈[0,T]; the output variables are divided into main output and auxiliary output, where the main output is the state z(s,t)∈R 2n , that is, a 2n-dimensional real vector, the auxiliary output is the unknown parameter

[0064] Then, two parallel neural networks are designed. Network 1 learns state z ＊ (s, t), network 2 calculates the parameterized time-varying parameters The state network 1 takes the time-space dual variable (s, t) as input and dynamically characterizes the system evolution trajectory; the parameter network 2 only takes time t as input and focuses on the time-varying parameters identification.

[0065] This separated structure can effectively improve model training efficiency and generalization performance: the state network is dedicated to processing the complex mapping relationship of spatiotemporal dual variables (s, t), while the parameter network focuses on the dynamic pattern identification of the time-varying feature t. This structural difference effectively avoids cross-dimensional feature confusion; at the same time, the network adopts an innovative design that combines explicit decoupling and implicit coupling. It not only avoids multi-task gradient conflicts and improves training stability through independent branch structures, but also uses the PDE residual loss term to construct implicit parameter-state associations under physical constraints. It can also enhance cross-network information interaction by sharing the underlying time domain feature extractor. While maintaining the lightweight characteristics of the model, it achieves a balance between accuracy and efficiency in the collaborative identification of fast and slow variables in complex coupled systems.

[0066] Finally, the PDE residual is calculated by automatic differentiation. The specific steps are as follows:

[0067] 1. Calculate the Riemannian metric G(t):

[0068]

[0069] in Precomputed by Gram-Schmidt orthogonalization, based on which we obtain

[0070] 2. Calculate the gradient term

[0071] First use automatic differentiation to calculate Then calculate the Levi-Civita connection

[0072] 3. Based on the Riemannian metric and gradient term, calculate the correction term Δ[s,t]:

[0073]

[0074] S3.2 Loss Function Design

[0075] The total loss function consists of four parts:

[0076] 1. PDE residual loss

[0077] In order to make the neural network output satisfy the PDE equation in S2:

[0078]

[0079] where N coll is the number of configuration points.

[0080] 2. Boundary condition loss

[0081] The initial and terminal conditions are met:

[0082]

[0083] 3. Parameter Regularization Loss

[0084] Constraint parameter η i (t)>0:

[0085]

[0086] where N t Indicates the total number of time sampling points after time discretization, t j At the time step point, λ>0 is a safety threshold (e.g., λ=1e-3).

[0087] 4. Data-driven losses

[0088] Through the offline training phase, reference data z is generated obs (s k ,t k ). Based on the fault-free scenario and consistent topology, a reference solution of the PDE is generated. The PDE model at this time is the penalty term η of the parabolic partial differential equation (PDE) in step S2 under the time-invariant Riemannian metric G(t) i The reference solution is used to train PINN to learn the spatiotemporal evolution of geometric PDEs.

[0089] Based on the reference data z obs (s k ,t k )Construct the data-driven loss as follows:

[0090]

[0091] where N data Represents the total number of observed data points used for supervised training.

[0092] Beneficial Effects of the Invention: This paper proposes a geometric partial differential equation fault-tolerant control framework for cluster systems in environments with time-varying topologies and actuator failures. This method ensures that the agent navigates from an initial position to a target position along a dynamic reference trajectory while also providing intrinsic compensation for the effects of failures. Key innovative contributions include:

[0093] 1. Problem Modeling: The fault-tolerant control problem is transformed into a distributed trajectory optimization problem through geometric PDEs, unifying the mathematical descriptions of faults, topology, and motion constraints. Coordination forces are autonomously generated through dynamic adjustment of Riemannian metric weights. Geometric heat flow theory is used to decouple kinematic constraints from communication restrictions, establishing an intrinsic connection between geometric constraints and cluster dynamics. An improved Riemannian metric coordinated nominal controller and virtual controller are constructed to address the imbalance of coordination forces caused by actuator failures. A reference trajectory that satisfies cluster interaction constraints is derived through isomorphic parabolic PDE equations, achieving a unified solution for fault compensation and motion planning.

[0094] 2. Solution method level: Utilizing the parameter sharing mechanism of PINN, efficient adaptive optimal solutions for complex PDEs are achieved, breaking through the bottlenecks of traditional numerical methods in terms of real-time performance and scalability. Through independent network division of labor and lightweight fine-tuning, theoretical algorithms can be directly embedded in actual systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0095] Figure 1 This is a schematic diagram of the PINN architecture;

[0096] Figure 2 This is the training number and loss graph in PINN;

[0097] Figure 3 is the optimal state solution based on PINN;

[0098] Figure 4 is the solution of L based on PINN and finite difference method 2 norm;

[0099] Figure 5 It is a multi-agent trajectory space diagram. DETAILED DESCRIPTION

[0100] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, not all of the embodiments. The components of the embodiments of the present invention generally described and shown in the drawings herein can be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the claimed invention, but merely represents selected embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without making creative work are within the scope of protection of the present invention.

[0101] The cluster adaptive path planning method based on geometric PDE and PINN includes the following steps:

[0102] S1. Dynamic modeling of fault clusters with time-varying topology.

[0103] S2. Through geometric heat flow and Riemannian manifold, a fault cluster with time-varying topology is constructed based on the parabolic system PDE model to generate reference trajectories in obstacle environments.

[0104] S3. According to the reference trajectory of the cluster parabolic PDE system, the physical information neural network PINN is used to quickly solve the established steady-state value of the cluster parabolic PDE system, and the fault-tolerant optimal control law based on geometric PDE and PINN for the cluster system with time-varying topology and actuator failure is obtained.

[0105] Example 1:

[0106] In the embodiment of this patent, an adaptive geometric partial differential equation (PDE) distributed fault-tolerant optimal control algorithm is provided for cluster systems with time-varying topology and actuator failures in an obstacle environment. The algorithm solves the parabolic partial differential equation (PDE) in S2 based on the physical information neural network (PINN). The core idea is to integrate the physical equation constraints and boundary conditions into a loss function through a neural network to achieve end-to-end learning of the equation solution. Specifically, for a formation system composed of 7 intelligent agents, considering that the intelligent agent has a connected topology at the initial time t=0, and an actuator bias failure occurs in the intelligent agent at t=10, the PDE-PINN control method can eventually ensure that the center position of the formation deployment reaches the target position of the xy axis coordinate (-2,25). The implementation process includes the following steps:

[0107] Step S100: In the augmented dynamic model Under this condition, the compensation value η is introduced i (t), i=1,...,7, obtain the parabolic PDE system

[0108]

[0109] Step S110: Define the domain s∈[0,40] for isomorphic parameter solution. The boundary conditions include z=2 when the lower boundary S=0 and z=0.3 when the upper boundary S=2. The initial condition z=sint when s=0. The collocation point sampling is uniformly sampled in the domain [0,40]×[0,2] N coll =10 3 Configuration points {(s i ,t i )}, N samples are taken at the boundary position BC = 200 points, and the sampling density is dynamically adjusted according to the residual distribution.

[0110] Step S120: Use a four-layer fully connected network (input layer 2 dimensions, hidden layer 32 dimensions × 3, output layer 1 dimension), and the activation function is Tanh. Input (s i ,t i ) to Network 1, output the predicted z(s i ,t i ) value. Input t i Go to Network 2 and get To ensure η i (t)>0 and changes smoothly over time, taking an exponential function for the output of Network 2 The network approximates the spatial distribution of PDE solutions through nonlinear mapping.

[0111] Step S130: Calculate by automatic differentiation and Force the network to satisfy the PDE equation in S2. The mean square error of the residual constitutes the PDE residual loss The key improvement is to use PyTorch's high-order automatic differentiation function to directly calculate At the same time, the automatic mixed precision training module is used to accelerate calculations and reduce video memory usage.

[0112] Step S140: Boundary condition loss Constraining the predicted value of z with S=0 and S=2 to be consistent with the fixed values of 2 and 0.3, data-driven loss Ensure that z = sint holds when s = 0. The total loss is the weighted sum of four terms:

[0113]

[0114] For the acquisition of actual data, in the actual physical system, sensors are deployed to record spatiotemporal data, and high-precision numerical simulation is achieved by generating reference solutions using the finite element method (FEM) or finite difference method (FDM). If there is no data at all, this can be omitted. Rely only on the physical equation residuals and boundary condition losses

[0115] Step S150: Minimize the total loss using the Adam optimizer (learning rate 0.001). Update the weights of Network 1 and Network 2 simultaneously via backpropagation. The learning rate decay strategy is: initially 1e-3, decaying by 0.5 every 1000 steps. Train for 20,000 epochs, outputting the loss every 1000 epochs to monitor convergence, and generating a uniform grid of points.

[0116] In response to the convergence difficulties that may occur during training, this patent implements a multi-dimensional diagnosis and correction mechanism: first, it introduces the gradient clipping technology, and constrains the parameter update amplitude through the gradient clipping function of PyTorch to effectively suppress the gradient explosion phenomenon; secondly, it constructs an adaptive balancing mechanism of the loss function, based on (physical equation residual) and The real-time convergence state of the (boundary condition) dynamically adjusts the weight coefficient to eliminate the dominant deviation in the multi-objective optimization; in addition, according to the physical constraint characteristics of the parameter network, the uniform distribution initialization function is used to asymmetric initialize the weight of the last layer of the network to ensure that the time-varying parameter η i The initial output of (t) strictly satisfies the positive definiteness requirement, laying the foundation for physically compliant optimization trajectories.

[0117] The full-field solution is predicted by the trained network to generate the solution z(s,t). The loss curve is saved as the training round changes to analyze the training stability and calculate the L of the PDE residual. 2 Norm changes over training epochs. Figure 1 Two independent networks are shown, but sharing common parameters. Figure 2 It shows that as the number of training times increases, the value of the loss function approaches zero. Figure 3 The state evolution of the parabolic PDE solved by PINN is shown, which eventually reaches a stable value, that is, the reference trajectory of the cluster under faults and time-varying topology is obtained. Figure 4 The L solution based on the finite difference method and the PINN method is shown. 2 The evolution of the norm over time. It can be seen from the figure that the PINN-based method is faster than the finite difference method in obtaining a stable solution. Figure 5 The trajectory diagram of the multi-agent from the initial moment to the target position is shown. It can be seen that the actuator failure causes the agent trajectory to fluctuate after t = 10. Under the action of the PDE-PINN controller, it reaches the target position at t = 35 and avoids obstacles during operation.

Claims

1. A cluster adaptive path planning method based on geometric PDE and PINN, characterized by: The following steps are involved: S1. Dynamic modeling of fault clusters with time-varying topology; S2. Construct fault clusters with time-varying topology based on the parabolic system PDE model through geometric heat flow and Riemannian manifolds to generate reference trajectories in obstacle environments; S3. Based on the reference trajectory of the cluster based on the parabolic PDE system, the physical information neural network (PINN) is used to quickly solve the established steady-state value of the cluster parabolic PDE system. The fault-tolerant optimal control law based on geometric PDE and PINN for the cluster system with time-varying topology and actuator failure is obtained to complete the path planning.

2. The cluster adaptive path planning method based on geometric PDE and PINN according to claim 1, characterized in that: The process of modeling the dynamic model of the fault cluster with time-varying topology is as follows: In a cluster system consisting of n agents, in an ideal environment without faults and obstacles, the dynamics of each agent is described as follows: variable and They represent the plane position and control input of agent i at time t, and They represent the displacement of agent i in the horizontal x-axis coordinate direction and the vertical y-axis coordinate direction, respectively. g(·) is a nonlinear mapping. is the set of all agents in the system; Communication between agents is done by undirected time-varying topology Description, where the edge set The dynamic evolution of It shows that the communication edge between agents i and j is formed if and only if the distance condition at the current moment ||X ij ||≤l d When satisfied, Time-varying signals The right limit at time t is defined as: in, and Represents the communication status of agents i and j respectively The value before and after the transfer time t represents the instantaneous change of the communication edge (i, j) at time t; condition l c <l d Define the interval (l c ,l d ] is the hysteresis identification area, which simulates the negotiation delay in the actual communication connection establishment process by introducing the hysteresis effect of edge addition / deletion; Cluster system in time-varying topology Under this situation, the evolution of the cluster system relies on the potential energy generated by the interaction between individuals, specifically the control power of the fault-free cluster system. The dynamics of the agent due to actuator failure is characterized as in, Indicates the i f Fault agent states, function K if (·): R4→R2 represents the mathematical model of the fault, is the set of faulty agents; Define the state vector as With initial state in, represents the global state of the fault cluster system, q(t)∈R 2m is an auxiliary variable for generating synergy; according to vector augmentation, the dynamic augmentation model of fault cluster is described as: Let the augmented matrix be in, Ξ is the Laplace moment; M is 0 2m is a 2m-dimensional all-zero column vector, I 2n×2m is a 2n×2m dimensional all-one matrix; in addition, the variable represents the interaction force of all fault-free agents, that is, the nominal control force, where the force of a single fault-free agent is It is the input of the fault-tolerant control to be designed.

3. The cluster adaptive path planning method based on geometric PDE and PINN according to claim 2, characterized in that: The specific implementation process of step S2, based on the parabolic system PDE model, to generate a reference trajectory in an obstacle environment is as follows: The trajectory length of the fault cluster system is defined by the positive symmetric Riemann metric tensor G(t) Among them, μ(t) represents the trajectory of the agent moving over time t; in Precomputed by Gram-Schmidt orthogonalization, M ◇ is a supplementary matrix that represents the auxiliary input direction of the under-driven part that is complementary to the original control input matrix M. represents the time-varying regulator, η i (t) represents the time-varying regularization weight parameter satisfying The adjustment gain is and For actuator faults, time-varying gain Adjust the Riemann metric tensor G(t) to adaptively adjust the trajectory length of the cluster system under metric; introduce gain By adjusting the trajectory length through the constraint G(t), the intelligent agent can avoid obstacles. i (t)>0 means that the most influential fault or obstacle is taken as the under-actuated direction of the actuator, and a penalty term is applied to force the agent trajectory to converge to the permissible motion subspace of the cluster system control and be robust; Will As an additional constraint for local communication, |N i (t)| represents the total number of other agents that have direct communication or perception interaction with agent i at time t, and is defined as like This indicates that the maximum number of adjacent constraints is violated, based on the constraints of the maximum degree edge Represents the number of virtual neighbors of agent i and designs a time-varying function: ζ>0 is the Heaviside step function, and the variable δ is introduced w Characterize the cumulative error and define the vector The fault cluster system dynamics model is represented as a Riemann augmented system: Define the state vector in and Using the Riemann length theory, the augmented Riemann metric tensor is introduced Convert the topological restrictions of the failure cluster into constraints: in, element γ i Represents a time-varying function When , the penalty coefficient for exceeding the limit; matrix Characterizing time-varying topology based on the constraints of the Riemannian metric, the essence of which is derived from the constraints of the topological time-varying edges in It is a dynamically adjusted penalty coefficient used to quantify the deviation of the edge (i, j) from the preset target distance l during the optimization process. d the intensity of punishment; According to the time-varying Riemann metric And the gradient evolution based on heat flow, considering the energy functional under the Riemann length is: in The Lagrangian of the time-varying system is obtained by modifying the Euler-Lagrange equation by the variational principle and introducing the correction term Δ(s, t). This is the reference trajectory dynamics of the cluster system with actuator failure and time-varying topology in step S2: Where s is the homotopy variable at time t, t∈[0,T], and z(s,t) is the initial connection position of the fault cluster under the Riemannian metric. and the trajectory of the terminal position ξ; ▽ represents and measures The related Levi-Civita connection, and ψ(s,t) reflects the gradient term after metric adjustment; the term Describing trajectory acceleration and nominal dynamics The deviation between them; by reshaping the potential field to minimize the curvature and trajectory length, the gradient term ψ(s,t) adjusts the trajectory so that Time-varying metrics in The introduction generates the acceleration term Δ(s,t) to reflect the impact of the dynamic adjustment of the constraint conditions on the trajectory deformation rate; According to the LaSalle invariant set principle, the Lyapunov function is non-increasing, and then the state of the parabolic PDE system converges to the steady-state value The status By decomposing the steady-state z ＊ (s, t) is obtained; according to the isomorphism theory, represents the ideal trajectory of the cluster system under actuator failure, time-varying topology and obstacles, that is, the reference trajectory.

4. The cluster adaptive path planning method based on geometric PDE and PINN according to claim 3 is characterized in that: The specific implementation process of the fault-tolerant optimal control law based on geometric PDE and PINN in step S3 is as follows: According to the established parabolic PDE system steady-state value z ＊ (s, t), to overcome the unknown variables in the parabolic PDE system And the fault-tolerant optimal control law based on geometric PDE and PINN for cluster systems with time-varying topology and actuator failures is obtained: S3.

1. Learn the solution of parabolic PDE equations in S2 through PINN First, define the input variables as homotopy parameters s∈[0,+∞) and time t∈[0,T]; Output variables are divided into main output and auxiliary output, where the main output is the state That is, n-dimensional real vector, the auxiliary output is the unknown parameter Then, two parallel neural networks are designed. Network 1 approximates the state z ＊ (s, t), network 2 calculates the parameterized time-varying parameters The state network 1 takes the time-space dual variable (s, t) as input and dynamically characterizes the system evolution trajectory; the parameter network 2 only takes time t as input and focuses on the time-varying parameters identification; Finally, the PDE residual is calculated by automatic differentiation; S3.

2. Design a loss function to train the physical information neural network PINN. The total loss function consists of four parts: PDE residual loss In order to make the neural network output satisfy the PDE equation in S2; boundary condition loss Satisfy initial and terminal conditions; parameter regularization loss Constraint parameter η i (t)>0; data-driven loss Through the offline training phase, reference data z is generated obs (s k ,t k ).

5. The cluster adaptive path planning method based on geometric PDE and PINN according to claim 4 is characterized in that: The PDE residual is calculated by automatic differentiation. The specific process is as follows: First, calculate the Riemann metric G(t): in Precomputed by Gram-Schmidt orthogonalization, based on which we obtain Second, calculate the gradient term First use automatic differentiation to calculate Then calculate the Levi-Civita connection Finally, based on the Riemannian metric and the gradient term, the correction term Δ[s,t] is calculated:

6. The cluster adaptive path planning method based on geometric PDE and PINN according to claim 5, characterized in that: The S3.2 is specifically implemented as follows: PDE residual loss In order to make the neural network output satisfy the PDE equation in S2, it is designed as where N coll is the number of configuration points; Boundary condition loss The initial and terminal conditions are met: Parameter regularization loss Constraint parameter η i (t)>0 N t is the total number of time sampling points after time discretization, t j At the time step point, λ>0 is the safety threshold; Data-driven losses Through the offline training phase, reference data z is generated obs (s k ,t k ); Generate a reference solution of the PDE based on a fault-free scenario and consistent topology; Based on the reference data, the data-driven loss is constructed as follows where N data Represents the total number of observed data points used for supervised training.

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