An unmanned aerial vehicle network system phase point trajectory modeling accuracy analysis method, system and software product

Abstract

The present application relates to the technical field of unmanned aerial vehicle network system modeling, and particularly relates to a method, system and software product for analyzing the accuracy of a phase point trajectory model of an unmanned aerial vehicle network system. The method constructs a three-layer seven-node equilateral triangle topology model with an analytical solution, sets node following rules using a unified PI controller, derives analytical expressions for the time-domain trajectories of each node through Laplace transformation, and compares the errors with the results of multi-step numerical simulation to calculate the maximum absolute error, mean square error and L2 norm. When the error is lower than a preset threshold, it is determined that the model has simulation accuracy. The system can automatically output an error evaluation report, support precision evaluation under complex topologies and disturbance conditions, and improve the efficiency and credibility of unmanned aerial vehicle cluster modeling verification.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of unmanned aerial vehicle network system modeling, and particularly relates to a method, system and software product for analyzing the accuracy of a phase point trajectory model of an unmanned aerial vehicle network system. BACKGROUND

[0002] With the improvement of single unmanned aerial vehicle (UAV) performance and the continuous decline in cost, multi-vehicle formation is rapidly becoming the mainstream technical solution for search and rescue, disaster monitoring, material distribution, air communication relay, and counter-drill tasks. Compared with traditional single unmanned aerial vehicles, multi-unmanned aerial vehicle network systems (Multi-UAV Network System, referred to as "unmanned aerial vehicle network system") have significant advantages in coverage area, task redundancy, robustness, and real-time response. However, the increase in the number of nodes and the complexity of cooperative behavior also brings new challenges in modeling, simulation, and control.

[0003] Existing research generally uses a graph-based cooperative control model or a distributed consensus model to describe the network topology and formation keeping logic. For example, the Vicsek model and its improved Boids model can produce flocking behavior with simple speed matching, position keeping, and collision avoidance rules; consensus algorithms depict global convergence characteristics through Laplacian matrices; and formation control often uses leader-follower, virtual structure, or behavior hierarchy to construct mathematical models. These methods can reproduce group motion at a macro level, but they are difficult to uniformly depict multi-dimensional dynamic properties such as task phase switching, communication link failure, and node performance differences.

[0004] The applicant's previous Chinese patent application (application number: 2025109865683, filing date: 20250717) provides a simulation modeling method for a phase point trajectory of an unmanned aerial vehicle network system. The method implements an end-to-end closed-loop process from system analysis, rule matrix construction, state analysis, stable domain approximation, to limit impulse determination and disturbance injection simulation, and achieves high-quality, structured trajectory data set output through data segmentation and transition zone marking; it realizes simulation, control, and evaluation integration, and provides a new technical means for robustness design, verification, and real-time adjustment of multi-unmanned aerial vehicle systems.

[0005] However, the existing technology has not yet provided a solution to the accuracy of the above modeling process itself. SUMMARY

[0006] In order to solve the above technical problems, the present application provides a kind of unmanned aerial vehicle network system phase point trajectory modeling accuracy analysis method, this method constructs hierarchical triangle formation-PI control closed loop analytical model, with formation variable / position offset as core state, in combination with Laplace inversion deduces analytical solution, again by multi-step numerical simulation, error statistics and limit impulse disturbance experiment, system assesses modeling precision and robustness, solves the problem that prior art cannot quantitatively describe modeling error and lacks unified evaluation framework.

[0007] In order to achieve the above-mentioned purpose, the present application adopts the following technical solutions:

[0008] A kind of unmanned aerial vehicle network system phase point trajectory modeling accuracy analysis method, comprising the following steps:

[0009] 1) in simulation environment, build the three-layer network topology consisting of n unmanned aerial vehicle nodes, the topology is composed of multiple nested equilateral triangular units;

[0010] 2) the desired trajectory of the first layer node number #1 is set as x1 (t), y1 (t), t is time variable, and the initial position, velocity, acceleration of each node in x and y direction are set to zero;

[0011] 3) for all nodes except node #1, configure PI controller with proportional gain Kp=1 and integral gain Ki=1;According to the geometric relationship of equilateral triangle, the output of the upper node is taken as the target input of the lower node by weight matrix;

[0012] 4) the trajectory of node #1 is taken as system input, according to the formula of negative feedback system closed loop transfer function, the analytical solution of each node in x and y direction is obtained in turn by Laplace transform, x i (t), y i (t), i=2-n, n is a positive integer greater than 2;

[0013] 5) calculate the formation deformation variable of network system in x and y direction;

[0014] 6) at least two different time steps Δt are used to run discrete simulation model, and the numerical solution of each node is obtained;The numerical solution is compared with the analytical solution obtained in step 4), and the maximum absolute error, mean square error and L2 norm are calculated;When the error is lower than the preset threshold, it is determined that the modeling method has the required accuracy.

[0015] Preferably, in step 2), the desired trajectory of the first layer node number #1 is set as:

[0016] - .

[0017] As a preference, the weight matrix in step 3) takes values of ±2 in a triangle with side length of 4, and ±1 in a triangle with side length of 2.

[0018] As a preference, the transfer function in step 4) is as follows:

[0019] ,

[0020] The transfer function solving adopts a block negative feedback structure, and first, for each following node, a forward path transfer function G(s) and a feedback path transfer function H(s) are established:

[0021] ,

[0022] and the following formula is used:

[0023] ,

[0024] The closed-loop response is derived;

[0025] wherein s is a Laplace domain complex frequency variable, G(s) is a forward path transfer function, H(s) is a unit feedback path, Φ i (s) is a closed-loop transfer function of the system.

[0026] As a preference, x2(t) in step 4) can be regarded as an output of a negative feedback system; the input signal of the system is r(t)=t-4, the forward path transfer function is G(s)=(s+1) / s 2 , and the transfer function of the feedback path is H(s)=1.

[0027] According to the closed-loop transfer function formula of the negative feedback system, the Laplace transform of the input r(t)=t-4 is obtained:

[0028] ;

[0029] The Laplace transform of the output is:

[0030] ;

[0031] The inverse Laplace transform of the above formula is obtained, and the analytical expression of the output in the time domain is:

[0032] ;

[0033] According to the above method, the analytical expressions of x i (t), y i (t) in the x and y directions are sequentially obtained, i=2-n.

[0034] wherein, [.] is the Laplace transform operator, R(s) is the Laplace transform of the input signal r(t)=t-4, X2(s) is the Laplace domain position representation of node #2 in x direction, e -t / 2 : exponential decay term.

[0035] Preferably, the time step Δt in step 6) comprises at least 0.01s and 0.05s to evaluate the model accuracy in both high-fidelity and fast simulation scenarios.

[0036] Preferably, the error index calculated in step 6) is used to automatically generate a simulation accuracy report, which includes the error-step relationship curve.

[0037] Preferably, the preset threshold in step 6) is set by the user according to task requirements, and the threshold range is 10 -4 to 10 -2 .

[0038] Further, the present application also provides a UAV network system for implementing the method, which comprises:

[0039] a) a topology configuration module for generating a three-layer seven-node equilateral triangle network topology;

[0040] b) a trajectory input module for injecting a reference trajectory x1(t), y1(t) into node #1;

[0041] c) a controller management module for assigning a PI controller to each following node and writing Kp, Ki parameters;

[0042] d) an analytical calculation module for performing Laplace transform and inverse transform, and outputting analytical solutions x i (t), y i (t);

[0043] e) a simulation execution module for running discrete simulation under a given time step and outputting numerical solutions;

[0044] f) an error analysis module for comparing analytical solutions with numerical solutions and generating error indicators;

[0045] g) a result determination module for outputting modeling accuracy conclusions according to error indicators and threshold values.

[0046] Further, the present application also provides a computer readable storage medium having a computer program stored thereon, wherein the program is executed by a processor to enable a computer to implement the method.

[0047] Further, the present application also provides a computer program product comprising a computer program or instructions, which are executed by a processor to implement the method.

[0048] The present application has the following significant technical effects by analyzing and deducing the multi-node hierarchical formation model and multi-step error evaluation:

[0049] 1. Establishing an analytical benchmark and absolute quantitative error

[0050] By using a three-layer seven-node equilateral triangle topology and a unified PI control law, a network model with a closed-form analytical solution is constructed, which is the first time that a multi-UAV system obtains a full-node time-domain analytical solution under given input. The analytical solution is used as a "true value" reference to quantify the absolute error of numerical solutions under any simulation solver and any time step setting, avoiding the drawbacks of traditional empirical methods that can only make relative comparisons or rely on subjective threshold values.

[0051] 2. Step length and error evaluation curve to determine the credible simulation step length interval

[0052] Numerical solutions are calculated in parallel under multiple discrete step lengths, and "step length, error" curves are drawn through indicators such as maximum absolute error, mean square error, and L2 norm. The "high-fidelity zone, transition zone, and distortion zone" are clearly divided, providing an objective basis for engineers to select the smallest acceptable step length, which can significantly reduce simulation time and computational power consumption while ensuring accuracy.

[0053] 3. Robustness and accuracy evaluation under multiple disturbance conditions

[0054] Under typical disturbance scenarios such as communication link failure and external impulsive wind field, the upper bound of model error and recovery time are analyzed, and the error and disturbance intensity mapping are output. This provides a quantitative baseline for control law tuning and fault-tolerant strategy design, improving the reliability of the system in complex environments.

[0055] 4. Integration of simulation, control, and evaluation to shorten the development iteration cycle

[0056] The system modularly integrates topology generation, PI control configuration, analytical solution, simulation execution, and error analysis, allowing researchers to complete model construction and accuracy verification on the same platform, reducing cross-tool data migration and repeated modeling workload.

[0057] 5. Automatic generation of precision report to improve repeatability and traceability

[0058] The software product can output a PDF / HTML report containing error curves, statistical tables, and threshold determination results with one click, facilitating project archiving and third-party review. Through fixed input, fixed topology, and unified evaluation indicators, external teams can reproduce the test process and achieve cross-institutional model accuracy comparison.

[0059] In conclusion, the present application solves the problem that the prior art cannot quantitatively evaluate the modeling accuracy of the multi-UAV network by combining the analytical reference with the multi-dimensional error measurement, significantly improves the simulation efficiency, result reliability and model reproducibility, and has important value for the research and development and engineering application in the field of UAV cluster control. BRIEF DESCRIPTION OF DRAWINGS

[0060] Figure 1 The figure is a basic flow chart of the UAV network system phase point trajectory simulation modeling method.

[0061] Figure 2 The figure is a task execution process chart of the 3-UAV transport formation.

[0062] Figure 3 The figure is a stable topology structure chart of the verification example.

[0063] Figure 4 The figure is a PI controller block diagram of the follower member.

[0064] Figure 5 The figure is an input-output x-direction relationship block diagram of the whole network system of the example.

[0065] Figure 6 The figure is an input-output y-direction relationship block diagram of the whole network system of the example.

[0066] Figure 7 The figure is a negative feedback block diagram of x 2 . t

[0067] Figure 8 The figure is a desired position vector and an actual position vector.

[0068] Figure 9 The figure is an absolute error of the model results and the analytical solution at different simulation step lengths at integer time points.

[0069] Figure 10 The figure is a local enlarged view of the red box in the x-direction chart in Figure 9 .

[0070] Figure 11 The figure is a local enlarged view of the red box in the y-direction chart in Figure 9 .

[0071] Figure 12 The figure shows that there is a linear rule between the simulation step length and the absolute error in the logarithmic coordinate system. DETAILED DESCRIPTION

[0072] ​The technical solutions in the embodiments will be clearly and completely described below. Obviously, the described embodiments are only some of the embodiments of the present application, but not all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative work belong to the protection scope of the present application.

[0073] As Figure 1 shown, referring to the previous Chinese patent application of the applicant (application number: 2025109865683, application date: 20250717), a kind of unmanned aerial vehicle network system phase point trajectory simulation modeling method includes three basic contents: 1) system analysis and task analysis: target unmanned aerial vehicle network is divided into members, attribute collection is obtained by member attribute set and system attribute set, static attribute model;2) interactive rule matrix construction: according to task stage, the rule description, numbering and coding of the member interaction relationship of each stage are obtained, and the interactive rule matrix of the corresponding stage is obtained;3) state analysis: based on task stage division, the state variable capable of measuring network performance is extracted and its type is determined, and then the phase vector space is formed in each stage;4) stable domain approximation calibration: the original data of phase point trajectory is expanded by linear Gaussian white noise, the future trajectory of phase point is predicted by using NARX neural network time series model, and the stable domain center vector and radial vector are calculated by combining the observation window data;5) limit impulse determination: for the predetermined disturbance type, the maximum tolerable step is obtained by using the strength decreasing-step increasing principle measurer, and the limit impulse is output after determining the linear characteristic;6) disturbance impulse injection simulation: generate disturbance strength-step length matrix within the limit impulse range, fix the initial state of network system model and run to stable, record the phase point trajectory after each disturbance injection until the termination threshold is met;7) data segmentation and transition zone marking: according to the phase point-stable domain distance and threshold condition, the starting point and end point of transition zone are automatically marked and the labeled phase point trajectory data set is output, which is used for subsequent modeling or algorithm training. The relationship of each content in the process is shown in Figure 1 .

[0074] Four "analyses" are included in the modeling process, namely system analysis of network structure and attribute, task analysis of task stage division, state analysis of network state definition and disturbance analysis of disturbance type and intensity. Among them, the goal of system analysis and task analysis is to establish network system model, the goal of state analysis is to build network phase vector space, and the goal of disturbance analysis is to integrate disturbance impulse injection module. The results of the four analyses are related to each other, and the acquisition of network system phase point trajectory is the ultimate goal.

[0075] To explain in detail Figure 1The content introduces an example of a 3-UAV transport formation consisting of three UAVs. Assuming that the UAV formation is required to complete a transport task, the three UAVs constituting the formation must cooperate during the execution of the task and maintain a "triangle" formation at all times. Meanwhile, they also need to avoid obstacles during the transport process. Figure 2 The process of the UAV formation executing the entire transport task is shown.

[0076] In order to verify the accuracy of the proposed modeling method, the present application Figure 2 The structure shown as a basic unit, designed an example that can be calculated formation variable analytical solution. With the analytical solution as a reference value, by comparing the error between the numerical solution of the simulation model and the analytical solution under different simulation steps, the accuracy of the modeling method is verified.

[0077] The topology structure of the example under the stable state is a 3-layer network system consisting of 7 member nodes, which contains 3 Figure 2 The triangular structure shown as Figure 3 . In the figure, it can be seen that the member nodes from 1 to 7 correspond to the number.

[0078] The initial state of the network system is that the values of the position, velocity and acceleration of all member nodes in the [x, y] direction are all [0, 0]. The dynamic rule of member #1 is:

[0079] (1.1)

[0080] Members #2-1 and #2-2 follow member #1 to form an equilateral triangle structure with a side length of 4; members #3-1 and #3-2 and members #3-3 and #3-4 follow members #2-1 and #2-2 respectively to form an equilateral triangle structure with a side length of 2.

[0081] The following rules of each triangular structure are designed as PI controllers. The proportional gain and integral gain of the controller are both set to 1, then the input-output relationship of each following member can be represented as the block diagram shown in Figure 4 , Figure 4 where i=2,3…7.

[0082] Taking equation (1.1) as the input of the example network system, according to the motion rules of Figure 4 , the input-output relationship of the whole network system can be described by Figure 5 , Figure 6 . Figure 5 , Figure 6 where, x 1 ( t ) and y 1 ( t ),x 2 ( t ) with y 2 ( t ), x 3 ( t ) with y 3 ( t ), x 4 ( t ) with y 4 ( t ), x 5 ( t ) with y 5 ( t ), x 6 ( t ) with y 6 ( t ), x 7 ( t ) with y 7 ( t ) respectively represent the actual positions of the member nodes 1-7 in the x and y directions.

[0083] According to the input and output relationships shown in Figure 5 , Figure 6 , it can be deduced in turn that x 1 ( t ) with y 1 ( t ), x 2 ( t ) with y 2 ( t ), x 3 ( t ) with y 3 ( t ), x 4 ( t ) with y 4 ( t ), x 5 ( t ) with y5 ( t ), x 6 ( t )and y 6 ( t ), x 7 ( t )and y 7 ( t The analytical expression of ).

[0084] The following is based on x 2 ( t The derivation process will be explained in detail using an example. x 2 ( t This can be seen as... Figure 7 The diagram shows the output of a negative feedback system. The input signal of this system is r(t) = t - 4, and the forward path transfer function is G(s) = (s + 1) / s. 2 The transfer function of the feedback path is H(s)=1.

[0085] According to the formula for the closed-loop transfer function of a negative feedback system, the transfer function of this system is:

[0086] (1.2)

[0087] Taking the Laplace transform of the input r(t) = t - 4, we get:

[0088] (1.3)

[0089] Therefore, the Laplace transform of the output is:

[0090] (1.4)

[0091] Performing an inverse Laplace transform on equation (1.4), the time-domain analytical expression of the output is obtained as follows:

[0092] (1.5)

[0093] Using the same method, it can be found that... x i ( t ), y i ( t The analytical expression for i=2,3…7.

[0094] Referring to the formula for calculating the formation deformation given in Definition 1, the analytical expressions for the formation deformation in the x and y directions are respectively...

[0095] (1.6)

[0096] (1.7)

[0097] wherein, and are two column arrays, x(i) and y(i) denote the i-th element of the array from left to right; d x,i (t) and d y,i (t) denote the absolute distance between the actual position and the target position of each member node.

[0098] Definition 1 (formation deformation variable) establishes a Cartesian coordinate system with the UAV No. 1 as the coordinate origin. Then at any time t, let the expected position vector of the UAV i=2, 3…7 be P e,i (t) and the actual position vector be P a,i (t) then the deformation of the UAV i is trf i (t) defined as the 2-norm of the difference between the expected position vector and the actual position vector, that is,

[0099] (1.8)

[0100] The overall formation deformation variable of the UAV formation network trf(t) is defined as:

[0101] (1.9);

[0102] wherein the expected position vector is a vector from the actual position of the UAV No. 1 to the position of the UAV i in the formation, and the actual position vector is a vector from the actual position of the UAV No. 1 to the actual position of the UAV i at time t, as shown in Figure 8 .

[0103] By combining equation (1.6), equation (1.7) and the equation in Table 1, the solution of the example network system can be obtained.

[0104] The network system model of the example is established according to the modeling method of the Chinese invention patent application (application number: 2025109865683, application date: 20250717) as shown in Figure 1 . The simulation step stp is set to 10 0 s, 10-1 s, 10 -2 s, 10 -3 s and 10 -4 s, and the simulation data of the array shape variables of the example network system at 5 discrete time points were obtained. At the integer time points (1s, 2s, …, 50s), the absolute errors of each time point at each simulation step were obtained by subtracting the values obtained by the analytical expression from the simulation data values. Figure 9 The cases of 1s to 16s are shown. After 16s, the absolute errors of each group of data tend to 0.

[0105] The average absolute errors of the 50 integer time points were obtained x The average absolute errors of the 5 simulation step settings in the direction and the direction were obtained, as shown in Table 1. y Table 1 Relationship between step and average absolute error

[0106]

[0107] The logarithms with base 10 of the step and the absolute values of the average absolute errors in Table 1 were taken, and the 5 points in the logarithmic coordinate system of the direction and the direction were plotted, and linear curves were fitted, as shown in

[0108] x y Figure 12

[0109] It can be seen that in the logarithmic coordinate system, the simulation step and the absolute error present a linear relationship, and the slope of the linear curve represents the rate of increase of the simulation error with the order of magnitude of the simulation step. For example, in the direction, the rate of increase of the simulation error is about 1.078, and in the direction, it is about 1.112. Figure 12 x y

[0110] The above rule means that with the decrease of the simulation step, the absolute error of the simulation will also decrease. Therefore, by controlling the simulation step, the simulation result can reach the required precision, and the modeling method has controllable accuracy.

[0111] The above is the description of the embodiments of the present application, and through the above description of the disclosed embodiments, the person skilled in the art can implement or use the present application. Various modifications of these embodiments will be apparent to those skilled in the art. The general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present application. Therefore, the present application will not be limited to these embodiments shown herein, but will conform to the widest scope consistent with the principles and novel features disclosed herein.​​​​​​​

Claims

1. A method for analyzing the modeling accuracy of a point trajectory of a UAV network system, characterized in that, Comprising the following steps: 1) Building a three-layer network topology consisting of n unmanned aerial vehicle nodes in a simulation environment, the topology being composed of multiple nested equilateral triangular cells; 2) Setting the desired trajectory of the first layer node numbered #1 as x1(t), y1(t), t being the time variable, and uniformly setting the initial position, velocity, and acceleration of each node in the x and y directions to zero; 3) Configuring a PI controller with proportional gain Kp=1 and integral gain Ki=1 for all nodes except node #1; according to the geometric relationship of the equilateral triangle, the output of the upper layer node is taken as the target input of the lower layer node through a weight matrix; 4) Take the trajectory of node #1 as the system input, according to the negative feedback system closed-loop transfer function formula, through Laplace transform, the analytical solution of each node in x, y direction is obtained in turn x i (t),y i (t), i = 2-n, n is a positive integer greater than 2; 5) Calculating the formation deformation variables of the network system in the x and y directions; 6) Running the discrete simulation model with at least two different time steps Δt to obtain the numerical solution of each node; comparing the numerical solution with the analytical solution obtained in step 4), calculating the maximum absolute error, mean square error, and L2 norm; when the error is lower than the preset threshold, it is determined that the modeling method has the required accuracy.

2. The method of claim 1, wherein, In step 2), the desired trajectory of the first layer node numbered #1 is set as: 。 3. The method of claim 1, wherein, The weight matrix in step 3) takes values of ±2 in a triangle with a side length of 4 and values of ±1 in a triangle with a side length of 2.

4. The method of claim 1, wherein, In step 4), the transfer function is as follows: ， The transfer function is solved using a block negative feedback structure, first establishing for each following node: ， And using the following formula: ， Deriving the closed-loop response; where s is the Laplace domain complex frequency variable, G(s) is the forward path transfer function, H(s) is the unity feedback path, Φ i (s) is the closed loop transfer function of the system.

5. The method of claim 4, wherein, x2(t) is the output of a negative feedback system in step 4); the input signal of the system is r(t) = t - 4, the transfer function of the forward path is G(s) = (s + 1) / s 2 , and the transfer function of the feedback path is H(s) = 1; According to the closed-loop transfer function formula of the negative feedback system, the Laplace transform of the input r(t)=t-4 is obtained as: ； The Laplace transform of the output is: ； The time-domain analytical expression of the output is obtained by inverse Laplace transform of the above formula: ； According to the above method, the analytical expressions of x i (t),y i (t) in the x, y directions are sequentially obtained, i = 2-n; wherein [.] is the Laplace transform operator, R(s) is the Laplace transform of the input signal r(t) = t"4, X2(s) is the Laplace domain position representation of node #2 in the x direction, e -t / 2 : an exponential decay term.

6. The method of claim 1, wherein, In step 5), the formation deformation variables of the network system in the x and y directions are calculated according to the following formula: ， 。 7. The method of claim 1, wherein, The time step Δt in step 6) includes at least 0.01s and 0.05s to evaluate the model accuracy in both high-fidelity and fast simulation scenarios; and / or, the calculation result of the error index is used to automatically generate a simulation accuracy report, which includes an error-step relationship curve; and / or, the preset threshold is set by the user according to the task requirements, and the threshold range is 10 -4 to 10 -2 .

8. An unmanned aerial vehicle network system for implementing the method of any one of claims 1-7, characterized by The system comprises: a) Topology configuration module for generating a three-layer seven-node equilateral triangular network topology; b) Trajectory input module for injecting a reference trajectory x1(t), y1(t) into node #1; c) Controller management module for assigning PI controllers to each following node and writing Kp, Ki parameters; d) an analytical computation module for performing Laplace transform and inverse transform, outputting the analytical solution x i (t), y i (t); e) Simulation execution module for running discrete simulation at a given time step and outputting numerical solution; f) Error analysis module for comparing analytical solution and numerical solution and generating error indicators; g) Result determination module for outputting modeling accuracy conclusion according to error indicators and threshold.

9. A computer-readable storage medium having stored thereon a computer program, characterized in that the program is executed by a processor to cause the computer to implement the method of any one of claims 1-7.

10. A computer program product comprising computer programs or instructions, characterized in that, The computer program or instructions are executed by a processor to implement the method of any one of claims 1-7.

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