A method for rapid evaluation of aircraft failure flight envelope based on parameterized curve

Abstract

The application discloses a kind of parameterized curve-based aircraft failure flight boundary fast evaluation method, it is related to aircraft flight boundary evaluation technical field.The present application is by establishing the parameterized mathematical relationship of closed boundary, utilizes ground database training to obtain the neural network model of different flight conditions and failure parameter to parameterized curve, then the parameterized curve obtained by neural network prediction is converted into closed boundary, to realize the fast evaluation of flight boundary.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of aircraft flight boundary evaluation, and particularly relates to a flight boundary rapid evaluation method for aircraft failure based on parameterized curves. BACKGROUND

[0002] The statements in this section merely provide background information related to the present disclosure and can not constitute the prior art.

[0003] Flight boundary (also known as flight envelope) is a common method for evaluating the flight performance of an aircraft, and can be used to describe the safe flight interval of an aircraft, which is usually a set of allowable control ranges of given flight speed, attitude angle and height. In the event of flight failure or extreme adverse flight environment, real-time acquisition of flight boundary online evaluation is of great significance for realizing flight boundary protection and improving flight safety.

[0004] Flight boundary evaluation is essentially solving the equilibrium points of the dynamic equation, which is a multi-dimensional variable space optimization problem. When the state variable space dimension of the dynamic model is high, the difficulty and computational complexity of the problem will increase exponentially, so the online evaluation of flight boundary has always been a difficult problem for flight boundary control protection. In order to realize the online evaluation of flight boundary, a fast evaluation technology needs to be developed, and direct optimization iteration to solve the extreme value of multi-dimensional space cannot meet the requirement of fast evaluation. The method currently used is to establish an offline safe flight boundary database covering all possible flight conditions and failure forms, and then perform online interpolation for real flight conditions to obtain online evaluation of flight boundary. For flight closed boundary interpolation in multi-dimensional space, a large number of points need to be taken on the boundary in sequence, and the interpolation relationship of each data point needs to be established to be applied to online evaluation. The modeling efficiency is low by using this method.

[0005] Therefore, in view of these problems, a flight boundary rapid evaluation method for aircraft failure based on parameterized curves is proposed. The method establishes a parameterized mathematical relationship of the boundary, and realizes rapid evaluation of the flight boundary by establishing a model of the parameterized curve corresponding to different flight conditions and failure conditions. This method can greatly reduce the number of points taken on the boundary, and does not need to establish a model for each data point, so it can greatly improve the modeling and prediction efficiency of the flight boundary, and can be carried on the airborne computing device to realize rapid prediction from the flight boundary ground database to the real flight process. SUMMARY

[0006] The application aims at the problem of rapid evaluation of flight boundary when the aircraft fails during flight, and provides a flight boundary rapid evaluation method for aircraft failure based on parameterized curve.

[0007] The technical scheme of the application is as follows:

[0008] A flight boundary rapid evaluation method for aircraft failure based on parameterized curve, comprising:

[0009] Step S1: flight boundary data preparation; flight boundary data of the aircraft is generated through simulation;

[0010] Step S2: parameterized boundary data; the flight boundary data is converted by parameterization, the vector radius, included angle and normalized angle coordinate of each flight boundary data relative to the geometric center are calculated, and the parameterized curve conversion relationship of the Cartesian coordinate system to the polar coordinate system is established;

[0011] Step S3: boundary data regularization and uniform sampling; the flight boundary data after parameterization is regularized, the end point data is supplemented according to the normalized angle coordinate sorting, the normalized angle coordinate is sampled in the [0, 1] interval according to a certain sampling frequency, and the parameterized curve is interpolated according to the sampling value, so that the regularized and uniformly sampled parameterized curve is formed;

[0012] Step S4: neural network modeling; a neural network model is constructed with flight parameters and failure parameters as input and uniformly sampled parameterized curve as output; and the model is trained based on the parameterized curve formed in step S3;

[0013] Step S5: model prediction result processing; the uniformly sampled prediction curve output by the neural network model is processed by backtracking interpolation, the parameterized curve corresponding to the regularized normalized angle coordinate point containing the end point before uniform sampling in step S3 is reobtained, the end point vector radius consistency is corrected, and the Cartesian coordinate system is converted, which is used as the predicted closed flight boundary coordinate;

[0014] Step S6: result storage; the finally obtained boundary point coordinate is stored as the flight boundary evaluation result.

[0015] Further, the step S1 comprises:

[0016] Flight boundary data of the aircraft is simulated and generated under the condition of covering all possible flight Mach numbers, altitudes and failure parameter values.

[0017] Further, the step S2 comprises:

[0018] Step S21: Assuming the Cartesian coordinates of n-dimensional flight boundary data points are , if there are m flight boundary data, calculate the geometric center position of all flight boundary data ;

[0019] Step S22: Calculate the vector radius, included angle and normalized angle coordinate of each flight boundary data point relative to the geometric center position ; ; is the initial point vector radius, according to the calculation relationship between the initial point vector radius and the vector radius of any flight boundary data point, the parametric curve is obtained;

[0020] Step S23: Construct the conversion relationship from the Cartesian coordinates of the flight boundary data points to the polar coordinates , is the vector radius of any flight boundary data point relative to the geometric center position, is a set of azimuth angles to determine the direction of the vector radius in the n-dimensional space.

[0021] Further, the geometric center position of all flight boundary data is calculated, including:

[0022]

[0023] Wherein: represents the Cartesian coordinates of the i-th flight boundary data point in the j-th dimensional space, j represents the geometric center of all flight boundary data points in the j-th dimensional space. i i Further, the vector radius, included angle and normalized angle coordinate of each flight boundary data point relative to the geometric center position are calculated, including:

[0024]

[0025]

[0026] Wherein:

[0027] represents the modulus of the vector;

[0028] , is the x-coordinate and y-coordinate, respectively, which is expressed as =1~ i n ​​​​​​​​​discrete form;

[0029] The calculation relationship between the starting point vector and the vector of any flight boundary data point is as follows:

[0030]

[0031] The parameterized curve obtained in step S22 is represented as follows:

[0032] .

[0033] Further, The following recursive method is used for calculation:

[0034] .

[0035] Further, the step S3 comprises:

[0036] The parameterized flight boundary data is regularized, the vector r is sorted in ascending order of the normalized angle coordinate ε, and it is judged whether the left and right endpoints of the sorted ε sequence are 0 and 1. If not, the coordinate point [0, ] is added at the left end of the data sequence, and the coordinate point [1, ] is added at the right end, the ε is sampled in the interval [0, 1] at a certain sampling frequency, and the regularized parameterized curve is interpolated according to the sampling value, and finally a regularized and uniformly sampled parameterized curve is formed to represent the closed flight boundary.

[0037] Further, the step S4 comprises:

[0038] The uniformly sampled parameterized curve obtained after interpolation is used as the output of the neural network model, and the input of the neural network model is the flight height, the Mach number and the fault parameter;

[0039] The parameterized curve formed in step S3 is divided into training data set and test data set according to a certain proportion, the prediction error of the neural network model is evaluated according to the test data set, and when the error meets the requirement, the neural network modeling is completed.

[0040] Further, the step S5 comprises:

[0041] The uniformly sampled data curve obtained by the neural network model is interpolated with respect to the regularized ε. If the lengths of the left and right end vectors after interpolation are not equal, the average value of the left and right end vectors is replaced by the length of the end vector. The final m vectors are converted into Cartesian coordinates.

[0042] Further, the final m vectors are converted into Cartesian coordinates, comprising:

[0043] .

[0044] The beneficial effects of the present application compared with the prior art are:

[0045] 1. The present application provides a kind of aircraft failure flight boundary fast evaluation method based on parameterized curve, by establishing the conversion relationship from closed boundary to parameterized curve, parameterized curve can be modeled, without modeling all boundary points, therefore can simplify model complexity, improve model prediction efficiency.

[0046] 2, the flight boundary fast evaluation method established by the present application can obtain the fast prediction of flight boundary data from flight height, Mach number and failure parameter, the calculation time of generating a flight boundary on desktop computer is about 2ms on average;Under elevator drift, breakage and icing three different failures, the flight boundary fast evaluation method predicts the flight boundary under all flight conditions, and the maximum relative sample value error is not more than 10%. BRIEF DESCRIPTION OF DRAWINGS

[0047] Figure 1 It is a kind of aircraft failure flight boundary fast evaluation method based on parameterized curve block diagram;In the figure, r Indicates vector path, f It is a function expression, e It is normalized angle. DETAILED DESCRIPTION

[0048] It should be noted that the terms "first" and "second" and the like relational terms are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between the entities or operations. Moreover, the terms "include", "contain" or any other variant thereof are intended to cover non-exclusive inclusion, so that the process, method, article or equipment including a series of elements not only includes those elements, but also includes other elements not explicitly listed or inherent to such process, method, article or equipment. Without more limitations, the element defined by the statement "including a" does not exclude the presence of other identical elements in the process, method, article or equipment including the element.

[0049] The features and performance of the present application will be further described in detail below in conjunction with embodiments.

[0050] Example one

[0051] Please refer to Figure 1 A kind of aircraft failure flight boundary fast evaluation method based on parameterized curve, comprising:

[0052] Step S1: flight boundary data preparation: generating aircraft flight boundary data through simulation;

[0053] In this embodiment, specifically, step S1 includes:

[0054] The flight boundary data of the aircraft is generated by simulation while covering all possible values ​​of flight Mach number, altitude and fault parameters; generally, the data of the controllable attitude angle and angular velocity of the aircraft is used, and the data boundary points are taken as the flight boundary data.

[0055] Step S2: Parameterize the boundary data; perform parameter conversion on the flight boundary data, calculate the radius vector, angle, and normalized angle coordinates of each flight boundary data relative to the geometric center, and establish a parameterized curve conversion relationship from the Cartesian coordinate system to the polar coordinate system;

[0056] In this embodiment, specifically, step S2 includes:

[0057] Step S21: Assume that the Cartesian coordinates of the n-dimensional flight boundary data point are If there are m flight boundary data, calculate the geometric center position of all flight boundary data ;

[0058] Calculate the geometric center position of all flight boundary data, including:

[0059] (Formula 1)

[0060] in: Indicates the flight boundary j The data point in i Cartesian coordinates in dimensional space, Indicates the i The geometric center of all flight boundary data points in the dimensional space;

[0061] Step S22: Calculate the radius vector of each flight boundary data point relative to the geometric center position , angle and normalized angular coordinates ; For a custom starting point radius vector, a parameterized curve is obtained based on the calculated relationship between the starting point radius vector and the radius vector of any flight boundary data point, as shown below:

[0062] (Equation 2)

[0063] Calculate the radius vector of each flight boundary data point relative to the geometric center position , angle and normalized angular coordinates ,include:

[0064] (Formula 3)

[0065] wherein:

[0066] denotes the modulus of a vector;

[0067] , i.e. coordinates and coordinates, are expressed as i =1~ n in discrete form;

[0068] The calculation relationship between the starting point vector and the vector of any flight boundary data point is as follows:

[0069] (Formula 4)

[0070] Step S23: constructing the conversion relationship from the Cartesian coordinates of the flight boundary data point to the polar coordinates , wherein is the vector of any flight boundary data point relative to the geometric center position, is a set of azimuth angles to determine the direction of the vector in the n-dimensional space;

[0071] In this embodiment, specifically, the following recursive method is used for calculation:

[0072] (Formula 5)

[0073] Step S3: boundary data normalization and uniform sampling; normalizing the parameterized flight boundary data, sorting the normalized angle coordinates and supplementing the endpoint data, sampling the normalized angle coordinates in the [0, 1] interval at a certain sampling frequency, and then interpolating the parameterized curve according to the sampling values to form a normalized and uniformly sampled parameterized curve;

[0074] In this embodiment, specifically, the step S3 comprises:

[0075] The parameterized flight boundary data is normalized, the vector r is sorted in the order of the normalized angle coordinates ε from small to large, and it is judged whether the left and right endpoints of the sorted ε sequence are 0 and 1, respectively. If not, the coordinate point [0, ] is added at the left end of the data sequence, and the coordinate point [1, ] is added at the right end, ε is sampled in the [0, 1] interval at a certain sampling frequency, and then the normalized parameterized curve is interpolated according to the sampling values, and finally a normalized and uniformly sampled parameterized curve is formed to represent the closed flight boundary.

[0076] Step S4: neural network modeling; constructing a neural network model with flight parameters and failure parameters as inputs and uniformly sampled parameterized curves as outputs; and training the model based on the parameterized curves formed in step S3;

[0077] In this embodiment, specifically, the step S4 comprises:

[0078] The uniformly sampled parameterized curve formed after interpolation is taken as the output of the neural network model, and the flight altitude, Mach number and failure parameters are taken as the inputs of the neural network model;

[0079] The parameterized curve formed in step S3 is divided into a training data set and a test data set according to a certain proportion, the prediction error of the neural network model is evaluated according to the test data set, and when the error meets the requirements, the neural network modeling is completed.

[0080] Step S5: model prediction result processing; performing back interpolation processing on the uniformly sampled prediction curve output by the neural network model, reobtaining the parameterized curve corresponding to the normalized angle coordinate point containing the end point before uniform sampling in step S3, converting back to the Cartesian coordinate system after correcting the end point vector length consistency, and taking the closed flight boundary coordinates obtained by prediction as the result;

[0081] In this embodiment, specifically, the step S5 comprises:

[0082] The uniformly sampled data curve predicted by the neural network model is interpolated with respect to the normalized ε, if the vector length of the left and right end points after interpolation is not equal, the average value of the left and right end vector length is replaced by the vector length of the end point, and the finally obtained m vector lengths are converted into Cartesian coordinates;

[0083] In this embodiment, specifically, converting the finally obtained m vector lengths into Cartesian coordinates comprises:

[0084] (Formula 6)

[0085] Step S6: result storage;

[0086] The finally obtained boundary point coordinates are stored as the flight boundary evaluation result.

[0087] This embodiment also proposes a flight vehicle failure flight boundary rapid evaluation device based on parameterized curves, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of the above-mentioned flight vehicle failure flight boundary rapid evaluation method based on parameterized curves when executing the computer program; preferably, the computer program can be run on a terminal device, such as a personal computer.

[0088] The embodiment also provides a computer readable storage medium storing a computer program, the computer program being executed by a processor to implement the steps of the method for quickly evaluating a failure flight boundary of an aircraft based on a parameterized curve. However, the computer readable storage medium of the present application is not limited thereto, and in the present document, the readable storage medium can be any tangible medium containing or storing a program, which can be used by or in conjunction with an instruction execution system, apparatus or device.

[0089] The readable medium can be a readable signal medium or a readable storage medium. The readable storage medium, for example, can be, but is not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus or device, or any suitable combination of the above. More specific examples (a non-exhaustive list) of the readable storage medium include an electrical connection having one or more wires, a portable disc, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), an optical fiber, a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the above.

[0090] The computer readable storage medium can include a data signal carried in a baseband or propagated as a carrier wave in a propagated data signal, in which the readable program code is carried. Such a propagated data signal can take any of a variety of forms, including but not limited to electro-magnetic, optical, or any suitable combination thereof. The readable storage medium can also be any readable medium that can send, propagate or transfer a program for use by or in connection with an instruction execution system, apparatus or device. The program code contained on the readable storage medium can be transmitted using any suitable medium, including but not limited to wireless, wired, optical, RF, etc., or any suitable combination of the above.

[0091] The program code for carrying out operations of the present application can be written in any combination of one or more programming languages, including an object-oriented programming language such as Java, C++, etc., and conventional procedural programming languages, such as the "C" programming language or similar programming languages. The program code can execute entirely on the user's computing device, partly on the user's device, as a stand-alone software package, partly on the user's computing device and partly on a remote computing device or entirely on the remote computing device or server. In the latter scenario, the remote computing device can be connected to the user's computing device through any kind of network, including a local area network (LAN) or a wide area network (WAN), or the connection can be made to an external computing device (for example, through the Internet using an Internet Service Provider).

[0092] Example 2

[0093] Using the parameterized curve-based rapid flight boundary assessment method for aircraft faults proposed in Example 1, a rapid flight boundary assessment is performed for elevator drift, damage, and icing faults that occur during the flight of an aircraft. The specific implementation steps of this example are as follows:

[0094] Step S1: Prepare flight boundary data. The aircraft's flight Mach number range is [0.4, 0.7] Mach, the flight altitude range is [0, 10] km, the elevator drift angle range is [-10, 10] degrees, the damage coefficient range is [0.2, 1.0], and the icing parameter range is [0.6, 1.2]. Sampling is performed within these parameter ranges. Ground simulation is used to generate longitudinal flight boundary data corresponding to each combination of flight conditions and fault parameters. This data is a two-dimensional closed curve formed by the angle of attack and pitch angle boundary points.

[0095] Step S2: Parameterize the boundary data. Let the boundary data dimension be n =2, and each set of flight boundary data is converted into a parameterized curve form as shown in (Formula 2) according to (Formula 1), (Formula 3), and (Formula 4). For this two-dimensional boundary case, the azimuth angle or The angle i .

[0096] Step S3: regularize and evenly sample boundary data. e In order from small to large r Sort the coordinates and judge after sorting e Are the left and right endpoints of the sequence 0 and 1? If not, add coordinate points [0, r 0], and add coordinate points [1, r 0], forming a regularized parameterized curve. e Sampling is performed, and then the parameterized curve is interpolated according to the sampled values ​​to obtain a regular and evenly sampled parameterized curve.

[0097] Step S4: Neural Network Modeling. The uniformly sampled data curve is used as the output of the neural network model, and the model inputs are flight altitude, Mach number, and fault parameters. For each fault type, the simulated database is divided into training data and test data at a ratio of 0.6 and 0.4. A three-layer fully connected feedforward neural network model is trained on the training dataset, with 200 neurons per layer and a ReLU activation function. The model's prediction error is evaluated based on the test dataset, and the average prediction error for all test samples is approximately 15%. Neural Network Modeling is complete.

[0098] Step S5: Model prediction result processing. The data curve obtained by the neural network model prediction is processed with respect to the normalized data in step three e Interpolation is performed. If the left and right endpoint vector lengths after interpolation are not equal, the average of the left and right end vector lengths is replaced with the endpoint vector length. The final obtained m The individual vector lengths are combined with the azimuth angle calculated by equation (5) or The Cartesian coordinates are converted according to equation (6).

[0099] Step S6: Store all the obtained boundary point coordinates as the flight boundary evaluation results under the flight and fault conditions.

[0100] For this flight simulation example, the above steps are used, and finally the relative sample value error of the flight boundary evaluation model prediction of the boundary enclosing area is not more than 10% for a total of 3138 groups of data under three fault modes. The time consumption of the model prediction flight boundary evaluation is evaluated on a desktop computer with a main frequency of 2.6 GHz, a memory of 32 G, a 64-bit Window operating system, and a code execution platform of Matlab 2021a. For the three fault prediction models, the calculation time of each flight boundary is about 2 ms on average, indicating that the evaluation method has high calculation efficiency.

[0101] The above-described embodiments only express the specific implementation of the present application, and the description is more specific and detailed, but it should not be understood as a limitation on the protection scope of the present application. It should be noted that for ordinary skilled persons in the art, without departing from the technical concept of the present application, some modifications and improvements can be made, which are within the protection scope of the present application.

[0102] This background section is provided to generally present the context of the application, the work of the current named inventors, the work described in this background section to the extent that it is described, and the descriptions in this section at the time of filing, neither expressly nor implicitly, are recognized as prior art of the present application.

Claims

1. A method for rapid assessment of aircraft fault flight boundaries based on parameterized curves, characterized in that: include: Step S1: flight boundary data preparation: generating aircraft flight boundary data through simulation; Step S2: parameterizing boundary data; Perform parameterized transformation on the flight boundary data, calculate the radius vector, angle and normalized angle coordinates of each flight boundary data relative to the geometric center, and establish the parameterized curve transformation relationship from the Cartesian coordinate system to the polar coordinate system; Step S3: regularizing and uniformly sampling boundary data; The parameterized flight boundary data is regularized, sorted by normalized angular coordinates and supplemented with endpoint data. The normalized angular coordinates are sampled in the interval [0, 1] at a preset sampling frequency, and the parameterized curve is interpolated based on the sampled values ​​to form a regular and evenly sampled parameterized curve. Step S4: Neural network modeling; constructing a neural network model with flight parameters and fault parameters as input and a uniformly sampled parameterized curve as output; and performing model training based on the parameterized curve formed in step S3; Step S5: Processing the model prediction results; performing back-interpolation processing on the uniformly sampled parameterized curve output by the neural network model to re-obtain the parameterized curve corresponding to the regularized normalized angular coordinate points containing the endpoints before uniform sampling in step S3, correcting the endpoint radial vector consistency and converting it back to the Cartesian coordinate system as the predicted closed flight boundary coordinates; Step S6: Result storage: The final boundary point coordinates are stored as the flight boundary assessment results.

2. The method for rapid assessment of aircraft fault flight boundaries based on parameterized curves according to claim 1 is characterized in that: The step S1 includes: The flight boundary data of the aircraft is generated by simulation while covering all possible flight Mach numbers, altitudes and fault parameter values.

3. The method for rapid assessment of aircraft fault flight boundaries based on parameterized curves according to claim 2 is characterized in that: The step S2 includes: Step S21: Assume that the Cartesian coordinates of the n-dimensional flight boundary data point are If there are m flight boundary data, calculate the geometric center position of all flight boundary data ; Step S22: Calculate the radius vector of each flight boundary data point relative to the geometric center position , angle and normalized angular coordinates ; For a customized starting point radius vector, a parameterized curve is obtained based on the calculated relationship between the starting point radius vector and the radius vector of any flight boundary data point; Step S23: Construct Cartesian coordinates from flight boundary data points To polar coordinates The conversion relationship, is the radius vector of any flight boundary data point relative to the geometric center position, is a set of azimuth angles to determine the direction of the radius vector in n-dimensional space.

4. The method for rapid assessment of aircraft fault flight boundaries based on parameterized curves according to claim 3 is characterized in that: Calculate the geometric center position of all flight boundary data, including: in: Indicates the flight boundary j The data point in i Cartesian coordinates in dimensional space, Indicates the i All flight boundary data points in the dimensional space are at the geometric center of all coordinate points in the dimensional space.

5. The method for rapid assessment of aircraft fault flight boundaries based on parameterized curves according to claim 4 is characterized in that: Calculate the radius vector of each flight boundary data point relative to the geometric center position , angle and normalized angular coordinates ,include: in: represents the magnitude of a vector; , That is Coordinates and Coordinates, expressed as i =1~ n Discrete form of The calculation relationship between the radius vector of the starting point and the radius vector of any flight boundary data point is as follows: The parameterized curve obtained in step S22 is expressed as follows: 。 6. The method for rapid assessment of aircraft fault flight boundaries based on parameterized curves according to claim 5, characterized in that: The calculation is done using the following recursive method: 。 7. The method for rapid assessment of aircraft fault flight boundaries based on parameterized curves according to claim 6, characterized in that: The step S3 comprises: The parameterized flight boundary data is regularized, and the radius vector r is sorted in the order of the normalized angle coordinate ε from small to large. It is judged whether the left and right endpoints of the sorted ε sequence are 0 and 1. If not, the coordinate points [0, ], and add coordinate points [1, ], sample ε in the interval [0, 1] according to the preset sampling frequency, and then interpolate the regularized parameterized curve according to the sampling values, and finally form a regular and evenly sampled parameterized curve to represent the closed flight boundary.

8. The method for rapid assessment of aircraft fault flight boundaries based on parameterized curves according to claim 7 is characterized in that: The step S4 comprises: The uniformly sampled parameterized curve formed after interpolation is used as the output of the neural network model, and the input of the neural network model is the flight altitude, Mach number and fault parameters; The parameterized curve formed in step S3 is divided into a training data set and a test data set according to a preset ratio. The prediction error of the neural network model is evaluated based on the test data set. When the error meets the requirements, the neural network modeling is completed.

9. The method for rapid assessment of aircraft fault flight boundaries based on parameterized curves according to claim 8, characterized in that: The step S5 comprises: The uniformly sampled parameterized curve predicted by the neural network model is interpolated about the regularized ε. If the lengths of the left and right endpoint radius vectors after interpolation are not equal, the average of the left and right endpoint radius vectors is replaced by the endpoint radius vector; the m radius vectors finally obtained are converted into Cartesian coordinates.

10. The method for rapid assessment of aircraft fault flight boundaries based on parameterized curves according to claim 9, characterized in that: The final m radius vectors are converted into Cartesian coordinates, including: 。

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