A Method for Preserving and Restoring the Reference Trajectory of a Hypersonic Vehicle

Abstract

The present invention discloses a method for preserving and restoring the reference trajectory of a hypersonic vehicle. First, considering the Gibbs phenomenon that appears at the node endpoints in the Lagrange interpolation method, Chebyshev sampling is used to eliminate this phenomenon. Secondly, aiming at the problem of limited computing accuracy of the on-board computer, an adaptive scaling method is adopted to reasonably scale the interpolation calculation process, so that the stage error of the calculation process is within an acceptable range. Finally, interpolation nodes are constructed by integrating the differential terms of the original data to ensure the smoothness of the trajectory information. The method of the present invention greatly compresses the number of nodes of the traditional linear interpolation algorithm, and the compression ratio is less than 1%. Moreover, the nominal command is smooth and the inner loop response is stable.

Description

Technical Field

[0001] The present invention belongs to the technical field of aircraft, and particularly relates to a method for saving and restoring a reference trajectory of a hypersonic aircraft. Background Technique

[0002] In order to reduce the demand for the storage space of an on-board computer and ensure the stability of the measurement, launch, control device in transmitting reference trajectory information, it is necessary to adopt high-fidelity compression and decompression of trajectory information to effectively realize the low cost of the on-board computer and the accuracy of element information.

[0003] According to the Nyquist sampling theorem, traditional reference trajectory information saving usually adopts the method of highly frequent sampling of trajectory points and then performs linear interpolation. Due to the poor smoothness of linear interpolation, a large number of trajectory nodes need to be saved to generate instructions for a smooth trajectory. When the on-board computer needs to store a large amount of reference trajectory information, it will occupy a large amount of storage space, thus increasing the cost of the on-board computer. In the test stage, the reference trajectory information is usually uploaded through the measurement, launch, control device. A large amount of reference trajectory information will also congest the channel, the upload speed is significantly reduced, and at the same time, it is easy to cause data packet loss and generate potential safety hazards. Summary of the Invention

[0004] In order to overcome the deficiencies of the prior art, the present invention provides a method for saving and restoring a reference trajectory of a hypersonic aircraft. First, considering the Gibbs phenomenon that appears at the node endpoints in the Lagrange interpolation method, Chebyshev sampling is used to eliminate this phenomenon. Secondly, aiming at the problem of limited calculation accuracy of the on-board computer, an adaptive scaling method is adopted to reasonably scale the interpolation calculation process, so that the stage error in the calculation process is within an acceptable range. Finally, interpolation nodes are constructed by performing integral operations on the differential terms of the original data to ensure the smoothness of the trajectory information. The method of the present invention greatly compresses the number of nodes of the traditional linear interpolation algorithm, the compression ratio is less than 1%, and the nominal instruction is smooth and the inner loop response is stable.

[0005] The technical solution adopted by the present invention to solve its technical problems includes the following steps:

[0006] Step 1: Lagrange interpolation time allocation;

[0007] Suppose only ranges and altitudes are saved as the reference trajectory, and the time interval is allocated as nodes, denoted as:

[0008] (1)

[0009] Suppose is the function to be approximated by Lagrange interpolation, which are the range and elevation respectively, expressed as time function;

[0010] According to the definition of Lagrange interpolation, minimizing the remainder is approximately calculated by minimizing the maximum remainder, that is ; Let be the th-order Chebyshev polynomial, then the sampling nodes are in the interval and are the zeros of the th-order Chebyshev polynomial, that is the roots of;

[0011] and have the following conversion relationship:

[0012] (2)

[0013] Step 2: Calculation of the Lagrange interpolation differential matrix;

[0014] The Lagrange polynomial expression is as follows:

[0015] (3)

[0016] where is the value to be interpolated by the Lagrange interpolation function at the point, is the basis function of the Lagrange polynomial;

[0017] Suppose:

[0018] (4)

[0019] Considering the requirements of the on-board computer for truncation error and division-by-zero protection, there is the following expression:

[0020] (5)

[0021] where is the division-by-zero protection limit of the denominator;

[0022] To adaptively scale the denominator of while sacrificing calculation accuracy, the scaling factor is:

[0023] (6)

[0024] Therefore:

[0025] (7)

[0026] Differentiate Equation (3) using matrix multiplication. The expression for the differential matrix is:

[0027] (8)

[0028] According to Equation (2), the differential matrix has a similar transformation relationship, that is, :

[0029] (9)

[0030] Step 3: Construction of range and elevation interpolation nodes;

[0031] The nodes in Equation (3) are constructed by integrating their differential terms:

[0032] (10)

[0033] where , are the initial range and initial elevation; , are the range and elevation point differentials, that is, the course and longitudinal flight speed; is the range and elevation generated by integrating the differential terms;

[0034] Perform correction compensation on Equation (10):

[0035] (11)

[0036] where and are respectively 's linear estimates, that is, evenly distributing the range and elevation integration errors to each node;

[0037] In summary, saving the range and elevation of the reference trajectory only requires the initial and final time, as well as range and elevation interpolation points, that is:

[0038] (12)

[0039] Step 4: Range and elevation interpolation calculation;

[0040] Use Step 1 to divide the time interval into time segments, and let ;

[0041] Scale the interpolation nodes and interpolation inputs:

[0042] (13)

[0043] Calculate the elevation and range at any time according to Equation (3):

[0044] (14)

[0045] Calculate the coefficient vector :

[0046] (15)

[0047] Suppose the calculation error is , then the interpolation calculation formula is as follows:

[0048] (16).

[0049] The beneficial effects of the present invention are as follows:

[0050] The method of the present invention greatly compresses the number of nodes of the traditional linear interpolation algorithm, the compression rate is less than 1%, and the nominal instructions are smooth and the inner loop response is stable. Description of the Drawings

[0051] Figure 1 It is a schematic diagram of interpolation differential comparison of an embodiment of the present invention.

[0052] Figure 2 It is a schematic diagram of trajectory data interpolation information of an embodiment of the present invention.

[0053] Figure 3 It is a schematic diagram of the cross-range elevation estimation error of an embodiment of the present invention.

[0054] Figure 4 It is a schematic diagram of the cross-range elevation speed estimation error of an embodiment of the present invention. Detailed Embodiments

[0055] The present invention will be further described below with reference to the drawings and embodiments.

[0056] To solve the problems of poor smoothness and large amount of stored data in the existing trajectory linear interpolation. The present invention provides a method for saving and restoring the reference trajectory of a hypersonic vehicle, which can fully compress the smooth trajectory information into fewer nodes. Greatly reduce the amount of information stored in the trajectory.

[0057] A method for saving and restoring the reference trajectory of a hypersonic vehicle is described as follows:

[0058] Step 1, Lagrange interpolation time allocation;

[0059] Suppose only ranges and altitudes are saved as the reference trajectory, and the time interval is allocated to nodes, denoted as:

[0060] (1)

[0061] Let be the function to be approximated by Lagrange interpolation, i.e., the range and altitude, which can be expressed as a function of time . To perform the best approximation by interpolation, it is obvious that the remainder of the polynomial interpolation needs to be minimized. According to the definition of Lagrange interpolation, minimizing the remainder can be approximately calculated by minimizing the maximum remainder, i.e., . Satisfying the minimization of the maximum remainder results in the minimum interpolation order when the approximation residuals are the same, which is obviously beneficial for reducing the number of nodes. Let be the -th order Chebyshev polynomial, then the sampling nodes are the zeros of the -th order Chebyshev polynomial in the interval , i.e., the roots of .

[0062] and have the following conversion relationship:

[0063] (2)

[0064] Step 2, calculation of the Lagrange interpolation differential matrix;

[0065] The Lagrange polynomial expression is as follows:

[0066] (3)

[0067] where is the value to be interpolated by the Lagrange interpolation function at the point, and is the basis function of the Lagrange polynomial. Let

[0068] (4)

[0069] For an on-board computer, has a very large span in the denominator value range. Therefore, considering the requirements of the on-board computer for truncation error and division-by-zero protection, there is the following expression:

[0070] (5)

[0071] where is the division-by-zero protection limit for the denominator, generally a very small number.

[0072] Loss of calculation accuracy is used to adaptively scale the denominator of , and the scaling factor is:

[0073] (6)

[0074] Therefore:

[0075] (7)

[0076] Differentiating equation (3) can be done using matrix multiplication. The expression for the differential matrix is:

[0077] (8)

[0078] According to equation (2), the differential matrix has a similar transformation relationship, that is, :

[0079] (9)

[0080] Step 3: Construction of range and altitude interpolation nodes;

[0081] In the design of the guidance control algorithm, usually the nominal trajectory uses a PD (Proportional-Differential) link to generate the overload command as the input reference signal for the inner-loop response. To ensure smooth control output response, the tracking reference signal must be smooth. Therefore, to eliminate the non-smoothness caused by the computational error of the interpolation algorithm, the nodes in equation (3) are constructed by integrating its differential terms:

[0082] (10)

[0083] Where , are the initial range and initial altitude; , are the differentials of the range and altitude points, that is, the course and longitudinal flight speed; is the range and altitude generated by integrating the differential terms. Therefore, its derivative is highly accurate and smooth. However, and the true range and altitude have a large deviation at the end section after long-term integration. To eliminate the inconsistency, equation (10) is corrected and compensated:

[0084] (11)

[0085] Where and are the linear estimates of respectively, that is, the range and altitude integration errors are evenly distributed to each node.

[0086] In summary, saving the range and altitude of the reference trajectory only requires the initial and end times, and range and altitude interpolation points, that is:

[0087] (12)

[0088] Step 4, range and altitude interpolation calculation;

[0089] First, use Step 1 to divide the time interval for time division, and let .

[0090] To reduce calculation errors, scale the interpolation nodes and interpolation inputs:

[0091] (13)

[0092] Calculate the altitude and range at any time according to Equation (3):

[0093] (14)

[0094] Calculate the coefficient vector :

[0095] (15)

[0096] Suppose the calculation error is , then the interpolation calculation formula is as follows:

[0097] (16)

[0098] Example:

[0099] As Figure 1 shown, to verify the effectiveness of this algorithm, combined with the hypersonic vehicle's benchmark bounce trajectory curve, this method is further explained. Assume the calculation error of the embedded system is , the trajectory flight time is , and the range and altitude of the trajectory interpolation points are 100 respectively.

[0100] Step 1, Lagrange interpolation time allocation;

[0101] First, allocate time for the time interval , and calculate 100 time difference nodes as:

[0102]

[0103] Step 2, Lagrange interpolation differential matrix calculation;

[0104] Calculate the differential matrix as:

[0105]

[0106] Step 3, range and altitude interpolation node construction;

[0107] Calculate 100 interpolation nodes to be saved:

[0108]

[0109] Step 4, Range and elevation interpolation calculation:

[0110] Calculate the Lagrange interpolation at 2000 equally spaced moments in the time interval from 0 to 465 to obtain the range and elevation The difference results are as shown in Figure 2 Let the relative error of the interpolation be:

[0111]

[0112] where , , , are the range, elevation, azimuth velocity, and longitudinal velocity in the original data respectively. , are the differences in azimuth velocity and elevation velocity respectively, that is, the Lagrange interpolations with interpolation coefficients and respectively.

[0113] Compared with the original data, the relative errors of the range and elevation interpolations of the trajectory data obtained by 100 interpolations compared with the reference trajectory are as shown in Figure 3 The relative error of the velocity interpolation calculation is as shown in Figure 4 The linear interpolation with similar calculation results requires no less than 10,000 points. Therefore, the trajectory data compression ratio exceeds 1%.

Claims

1. A method for preserving and restoring the reference trajectory of a hypersonic vehicle, characterized in that, It includes the following steps: Step 1: Lagrange interpolation time allocation; Set to save only the range and altitude as the reference trajectory, and the time interval is allocated to nodes, denoted as: （1） Let be the function to be approximated by Lagrange interpolation, which are the range and altitude, respectively, expressed as functions of time ; According to the definition of Lagrange interpolation, minimizing the remainder is approximated by minimizing the maximum remainder, that is ; Let be the -th order Chebyshev polynomial, then the sampling nodes are the zeros of the -th order Chebyshev polynomial in the interval , that is, the roots of ; and There is the following conversion relationship: （2） Step 2: Lagrange interpolation differential matrix calculation; The Lagrange polynomial expression is as follows: （3） wherein is the interpolated value of the Lagrange interpolation function at points, is the basis function of the Lagrange polynomial; Let: （4） Considering the requirements of the truncation error of the missile-borne computer and division-by-zero protection, there is the following expression: （5） Among them is the denominator divide-by-zero protection limit; The loss calculation accuracy adaptively scales the denominator of with a scaling factor of: （6） Therefore: （7） Differentiate Equation (3) using matrix multiplication. The differential matrix is expressed as follows: （8） According to Equation (2), the differential matrix has a similar transformation relationship, that is, : （9） Step 3: Range and altitude interpolation node construction; In Equation (3), the construction of the nodes is generated by integrating their differential terms: （10） Among them and are the initial range and initial elevation; and are the differentials of the range and elevation points, i.e., the course and longitudinal flight speed; are the range and elevation generated by integrating the differential terms Perform correction and compensation on Equation (10): （11） wherein and are respectively linear estimates, that is, the range and altitude integration errors are evenly distributed to each node; In summary, saving the reference trajectory range and elevation only requires the initial and final times, as well as range and elevation interpolation points, namely: （12） Step 4: Range and altitude interpolation calculation; Use Step 1 to perform time division on the time interval and let ; Scale the interpolation nodes and interpolation inputs: （13） Calculate the elevation at any time according to Equation (3). and the range : （14） Calculate the coefficient vector : （15） Assume the calculation error is , then the interpolation calculation formula is as follows: （16）。

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