A method for improving the accuracy of inertial guidance based on polynomial recursive least squares.

Abstract

This invention discloses a method for improving the accuracy of inertial guidance based on polynomial recursive least squares, comprising: constructing a guidance tool error model based on the difference between inertial guidance telemetry observations and a flight environment function, wherein the guidance tool error model satisfies a linear relationship; constructing a recursive formula based on polynomial recursive least squares based on the guidance tool error model; determining the estimated value of the guidance tool error coefficient at the current recursive time based on the difference between guidance telemetry observations at the current recursive time and the recursive formula of least squares, and using the estimated value of the guidance tool error coefficient to compensate for the guidance telemetry observations at the current recursive time. This invention can significantly improve estimation accuracy and efficiency, and has a wider range of applications and higher engineering value.

Description

Technical Field

[0001] This invention relates to a method for improving the accuracy of inertial guidance using the polynomial recursive least squares method, belonging to the field of inertial navigation technology. Background Technology

[0002] Currently, inertial navigation in spacecraft primarily employs strapdown systems or platform systems composed of gyroscopes and accelerometers. Before live-fire flights, the error coefficients of the gyroscopes and accelerometers need to be calibrated on the ground. Error compensation based on the calibration results can effectively improve the accuracy of inertial navigation. However, even after ground calibration, in actual flight navigation tests, the theoretical values ​​of velocity and position calculated from telemetry data still deviate significantly from the actual flight velocity and position values ​​obtained from external measurements, resulting in a so-called "ground-to-ground inconsistency." Analysis reveals that this inconsistency is caused by insufficient precision in the ground calibration and data processing methods, leading to error accumulation during actual flight and consequently, decreased flight accuracy. Therefore, it is necessary to correct the error model and data processing methods used in ground calibration.

[0003] The commonly used method is to identify parameters using the least squares method, which has the advantage of providing the sum of squares of the output error. The minimum value is obtained, and the estimated value of the parameter X can be obtained in a single calculation. However, the downside is that as the amount of data accumulates, the amount of computation increases.

[0004] To reduce computational load, recursive least squares method is used in engineering, the core idea of ​​which is to use the estimated value from the previous time step. With the current observation y n+1 Solve for the estimated time of occurrence. The advantage is reduced computational load, meeting real-time requirements. The formula for recursive least squares is as follows:

[0005] K n+1 =Γ n c n+1 T [I k +c n+1 Γ n c n+1 T ] -1

[0006]

[0007] Γ n+1 =Γ n -K n+1 c n+1 Γ n

[0008]

[0009] Among them, I k It is a k-dimensional identity matrix.

[0010] However, it should be noted that there is a problem with the above recursive least squares method: It's not the smallest. For example, when the noise is zero, according to the formula of the recursive least squares method, we have...

[0011]

[0012] This means that the estimated value is biased, making thereby Not the minimum. To overcome the bias... The information inverse matrix Γ can be made n The value of approaches infinity, but no matter what value it takes, there will always be .

[0013] Therefore, a new method is needed to eliminate this bias in order to improve the accuracy of inertial guidance. Summary of the Invention

[0014] The purpose of this invention is to overcome the above-mentioned defects and provide a method for improving the accuracy of inertial guidance based on the polynomial recursive least squares method. This invention solves the technical problem that the traditional least squares method has unavoidable biases that lead to low accuracy in inertial guidance. This invention can significantly improve estimation accuracy and efficiency, and has a wider range of applications and higher engineering value.

[0015] To achieve the above-mentioned objectives, the present invention provides the following technical solution:

[0016] A method for improving the accuracy of inertial guidance based on polynomial recursive least squares method includes:

[0017] Based on the difference between inertial guidance telemetry and observations and the flight environment function, a guidance tool error model is constructed, which satisfies a linear relationship.

[0018] Based on the guidance tool error model, the recursive formula based on the polynomial recursive least squares method is constructed as follows:

[0019] K n+1 =Γ n c n+1 T [I k +c n+1 Γ n c n+1 T ] -1

[0020]

[0021]

[0022]

[0023] Γ n+1 =Γ n -K n+1,p c n+1 Γ n

[0024] Among them, K n+1 Let Γ be the feedback matrix at time n+1 of the recursion. n ,Γ n+1 These are the information inverse matrices at recursion time n and recursion time n+1, respectively. k For a k-dimensional identity matrix, c n+1 Let n+1 be the flight environment function matrix at recursive time n. The estimated values ​​of the guidance tool error coefficients at recursion time n and recursion time n+1 are respectively, K n+1,p Let y be the p-order polynomial feedback matrix at time n+1 of the recursion. n+1 y is the difference between the guided telemetry observations at time n+1 in the recursive time step. n+1 =y 遥,n+1 -y 外,n+1 y 遥,n+1 For the k-dimensional guided telemetry observations at time n+1, y 外,n+1 For the k-dimensional guided external observation at time n+1; n is an integer ≥ 0, and k is an integer ≥ 1;

[0025] Based on the difference between the guidance telemetry observations at the current recursive time and the recursive formula of the least squares method, the estimated value of the guidance tool error coefficient at the current recursive time is determined, and the guidance telemetry observations at the current recursive time are compensated using the estimated value of the guidance tool error coefficient.

[0026] Furthermore, the guidance tool error model is as follows:

[0027] y n+1 =c n+1 X+v

[0028] Among them, c n+1 Let X be the flight environment function matrix at time n+1, X be the constant error coefficient of the guidance tool, and v be the noise.

[0029] Furthermore, the guidance tool error model includes a telemetry velocity error model or a telemetry position error model composed of gyroscope error and accelerometer error.

[0030] Furthermore, the order p can take values ​​from 1 to 8.

[0031] Furthermore, the order p takes the value of 3.

[0032] Furthermore, n = 0 represents the initial time, Γ0 and For the given known quantities.

[0033] Furthermore, the k-dimensional guidance telemetry observations at the current recursive time are compensated using the estimated error coefficients of the guidance tool according to the following compensation formula:

[0034]

[0035] in, For the compensated guidance telemetry observations.

[0036] Furthermore, at each recursive time step, the binding value of the tool error is calculated using a determined tool error coefficient. Corrections are made to compensate for inertial guidance telemetry observations.

[0037] Furthermore, before determining the tool error coefficient at the current recursive time, it is determined whether the difference between the inertial guidance telemetry observations at the current recursive time is abnormal. If there is no abnormality, processing continues; otherwise, the telemetry observations from the previous recursive time are used to replace the telemetry observations at the current recursive time for subsequent processing.

[0038] Furthermore, before determining the tool error coefficient at each recursive time step, it is determined whether the difference between the current guidance telemetry observations is abnormal. If there is no abnormality, the subsequent processing continues. Otherwise, it is further determined that the dimension of the abnormal data is located, and the relevant recursion of that dimension is isolated from the recursion formula of the least squares method at the current recursive time step, while the other dimensions continue to be processed.

[0039] Compared with the prior art, the present invention has at least one of the following advantages:

[0040] (1) This invention creatively proposes a method to improve the accuracy of inertial guidance based on polynomial recursive least squares. By adding a polynomial feedback matrix and updating the error coefficient estimate and information inverse matrix of the guidance tool according to the polynomial feedback matrix, the estimation accuracy and estimation efficiency can be significantly improved.

[0041] (2) The method of the present invention can select an appropriate polynomial order to find the best match between real-time performance and accuracy based on the accuracy requirements of inertial navigation.

[0042] (3) The polynomial recursive least squares method given in this invention is the traditional recursive least squares method when p=1. In other words, the traditional recursive least squares method is a special case of this invention. This invention has a wider range of applications and engineering value. Attached Figure Description

[0043] Figure 1 This is a flowchart of the method of the present invention;

[0044] Figure 2 This refers to the output fitting residual calculated by the polynomial least squares method when p=1 in this embodiment of the invention.

[0045] Figure 3 This refers to the output fitting residual calculated by the polynomial least squares method when p=2 in this embodiment of the invention.

[0046] Figure 4 This refers to the output fitting residual calculated by the polynomial least squares method when p=3 in this embodiment of the invention.

[0047] Figure 5 The output of the fitted residual sum of squares Q in this embodiment of the invention p The relationship with the order p;

[0048] Figure 6 The sum of squared residuals Q is the output fitting residual of the method of the present invention and the method of method two. p A comparison diagram of the relationship between order p and order p. Detailed Implementation

[0049] The features and advantages of the present invention will become clearer and more apparent from the following detailed description.

[0050] The term “exemplary” as used herein means “serving as an example, embodiment, or illustration.” Any embodiment illustrated herein as “exemplary” is not necessarily to be construed as superior to or better than other embodiments. Although various aspects of embodiments are shown in the accompanying drawings, the drawings are not necessarily drawn to scale unless specifically indicated otherwise.

[0051] This invention provides a method for improving the accuracy of inertial guidance based on the polynomial recursive least squares method. It is a method to reduce the estimation error of the recursive least squares method, aiming to improve the accuracy of inertial guidance as much as possible while satisfying real-time requirements. The invention specifically includes the following steps:

[0052] (1) Based on the difference between inertial guidance telemetry observations and the flight environment function, a guidance tool error model is constructed. The guidance tool error model satisfies the linear relationship y n+1 =c n+1 X+v; where y n+1 Let X be the difference between k-dimensional guided telemetry observations at time n+1, and let X be the tool constant error coefficient. n+1 Let v be the flight environment function matrix, and v be the noise; the error compensation model is...

[0053] (2) Perform recursive initialization, that is, set the information inverse matrix Γ. n Initial values ​​and estimated values ​​of parameter X The initial value of parameter X is the tool constant error coefficient in the guidance tool error model;

[0054] (3) Process the error model of the guidance tool to determine the polynomial order p. Then the recursive formula of the polynomial recursive least squares method is as follows:

[0055] K n+1 =Γ n c n+1 T [I k +c n+1 Γ n c n+1 T ] -1

[0056]

[0057]

[0058]

[0059] Γ n+1 =Γ n -K n+1,p c n+1 Γ n

[0060] Among them, I k It is a k-dimensional identity matrix; K n+1 For the feedback matrix; K n+1,p It is a p-order polynomial feedback matrix; A new estimate for parameter X; Γ n+1 This is the new information inverse matrix.

[0061] (4) Based on the current difference y between inertial guidance telemetry observations and the aforementioned recursive formula, determine the tool error coefficient corresponding to each recursive time step, and use this tool error coefficient to evaluate the guidance telemetry observation y. 遥 Compensation will be provided, and the compensation formula is as follows: Used for subsequent guidance.

[0062] Preferably, the guidance tool error model includes a telemetry velocity error model or a telemetry position error model composed of gyroscope error and accelerometer error.

[0063] Preferably, the initialization in step (2) is performed at time zero Γ0. Given a known quantity, and at the remaining times Γn and It is necessary to pass through the Γ of the previous moment. n-1 and Perform recursive calculations.

[0064] Preferably, the compensation in step (4) is to correct the binding value of the tool error using the determined tool error coefficient at each recursive time, thereby achieving compensation for the inertial guidance telemetry observation.

[0065] Preferably, before determining the tool error coefficient at each recursive time step, it is determined whether the difference of the current inertial guidance telemetry observation is abnormal. If there is no abnormality, the processing continues; otherwise, the telemetry observation of the previous recursive time step is used to replace the current telemetry observation for subsequent processing.

[0066] Preferably, before determining the tool error coefficient at each recursive time, it is determined whether the difference between the current inertial guidance telemetry observations is abnormal. If there is no abnormality, the subsequent processing continues. Otherwise, the dimension of the abnormal data is further determined, and the relevant recursion of that dimension is isolated from the recursion formula in step (2) at the current recursive time. The remaining dimensions continue to be processed.

[0067] Example:

[0068] When a ballistic missile uses telemetry data to separate tool errors, the model structure of the velocity error equation in the x-axis direction is as follows:

[0069]

[0070] In the formula, δv x k represents the telemetry velocity error in the x-axis direction. 0x k 0y k 0z Zero bias of the accelerometers along the x, y, and z axes; δk y k yz D represents the first-order error coefficients of the y- and z-axis accelerometers. Fz This refers to the zero-order drift of the z-axis gyroscope; This is the environment function for the x-axis velocity error.

[0071] For ease of analysis, the errors were normalized to make k 0x =1, k 0y =1, k 0z =1, δk y =1, k yz =1, D Fz =1, meaning the tool constant error coefficient is

[0072]

[0073] Because there is a set of numbers in each external measurement cycle, namely the velocity error δv in the x-axis direction. x And the flight environment function of the x-axis velocity error during the external measurement period. Then the first 10 sets of remote telemetry observations y = δv x and flight environment function matrix

[0074]

[0075] The corresponding value is

[0076] (1) Let c1 = [7 26 6 28 13 60] and y1 = c1X = 140 when n+1 = 1;

[0077] (2) Suppose that when n+1=2, c2=[1 29 15 4 16 52] and y2=c2X=117;

[0078] (3) When n+1=3, c3=[11 56 8 44 19 20] and y3=c3X=158;

[0079] (4) When n+1=4, c4=[11 31 8 44 19 47] and y4=c4X=160;

[0080] (5) When n+1=5, c5=[7 52 6 28 13 33] and y5=c5X=139;

[0081] (6) When n+1=6, c6=[11 55 9 44 20 22] and y6=c6X=161;

[0082] (7) When n+1=7, c7=[3 71 17 12 20 6] and y7=c7X=129;

[0083] (8) When n+1=8, c8=[1 31 22 4 23 44] and y8=c8X=125;

[0084] (9) When n+1=9, c9=[2 54 18 8 20 22] and y9=c9X=124;

[0085] (10) Let c be the value when n+1 = 10. 10 =[21 47 4 84 25 26] and y 10 =c 10 X = 207.

[0086] Let the initial value of the tool constant error coefficient X be...

[0087]

[0088] The initial value of the information inverse matrix is ​​Γ0 = 10. 4 I6, below according to the polynomial (Ic) n+1 K n+1 ) p The order p is calculated using the fast polynomial recursive least squares method to obtain the output fitting residual with convergence accuracy.

[0089]

[0090] And the sum of squared residuals of the output fitting Q = δY T δY. In the formula, This is the estimated result when the calculation is repeated up to n+1=1, ..., n+1=10.

[0091] According to such Figure 1 The flowchart shown illustrates the specific method for improving inertial guidance accuracy based on the polynomial recursive least squares method of this invention:

[0092] (1) p = 1

[0093] The recurrence formula for the fast polynomial recursive least squares method is the same as that for the traditional recursive least squares method, as follows:

[0094] K n+1 =Γ n c n+1 T [I k +c n+1 Γ n c n+1 T ] -1

[0095]

[0096] Γ n+1 =Γ n -K n+1 c n+1 Γ n

[0097]

[0098] The calculated result for δY1 is -1.1 × 10⁻⁶. -5 ~6×10 -6 Magnitude, such as Figure 2 As shown. The output is the sum of squared residuals Q1 = δY1. T δY1=1.98347×10 -10 .

[0099] (2) p = 2

[0100] The recurrence formula for the fast second-order polynomial recursive least squares method is as follows:

[0101] K n+1 =Γ n c n+1 T [I k +c n+1 Γ n c n+1 T ] -1

[0102]

[0103]

[0104] Γ n+1 =Γ n -K n+1,2 c n+1 Γ n

[0105]

[0106] The calculated δY² result is -3.0 × 10⁻⁶. -11 ~1.7×10 -11 Magnitude, such as Figure 3 As shown. The output is the sum of squared residuals Q2 = δY2. T δY2=1.8735×10 -21 .

[0107] (3) p = 3

[0108] The recursive formula for the least squares method of recursion for a third-order polynomial is as follows:

[0109] K n+1 =Γ n c n+1 T [I k +c n+1 Γ n c n+1 T ] -1

[0110]

[0111]

[0112] Γ n+1 =Γ n -K n+1,3 c n+1 Γ n

[0113]

[0114] The calculated δY3 result is -2.9 × 10⁻⁶. -14 ~1.5×10 -14 Magnitude, such as Figure 4 As shown. The output is the sum of squared residuals Q3 = δY3. T δY3=1.81753×10 -27 .

[0115] (4) 4≤p≤8

[0116] The recursive formula for the p-order polynomial recursive least squares method is as follows:

[0117] K n+1 =Γ n c n+1 T [I k +c n+1 Γ n c n+1 T ] -1

[0118] K n+1,p =K n+1 [I-(Ic n+1 K n+1 ) 4 ](c n+1 K n+1 ) -1

[0119]

[0120] Γ n+1 =Γ n -K n+1,p c n+1 Γ n

[0121]

[0122] Calculated δY p The result is the same as δY3, and the output is the sum of squared residuals Q. p Same as Q3, for Q p =δY p T δY p =1.81753×10 -27 .

[0123] Plot Q1, Q2, ..., Q8 as follows Figure 5 As shown in the figure, when the polynomial order p≤3, the output fitted residual sum of squares Q pThe polynomial order p satisfies an exponential decay relationship; and when the polynomial order p ≥ 4, the output fit residual sum of squares Q p The result remains unchanged. This verifies that the fast polynomial recursive least squares method proposed in this invention has high accuracy and fast convergence speed. In engineering implementation, to strike a trade-off between computational complexity and accuracy, a polynomial order of p=3 is chosen.

[0124] In the method of this invention, the estimated error coefficients and information inverse matrix of the guidance tool are updated according to a polynomial feedback matrix, which can maximize the estimation efficiency and accuracy. Figure 6 The diagram shows a comparison between the convergence results obtained using the method of the present invention (denoted as Method 1) and the convergence results obtained by updating only the estimated values ​​of the guidance tool error coefficients according to the polynomial feedback matrix (denoted as Method 2). As can be seen from the diagram, compared with the values ​​calculated by Method 2, the method of the present invention (Method 1) is not only more accurate but also has a faster convergence speed, which also means that high-precision estimation can be achieved with only p=3.

[0125] The present invention has been described in detail above with reference to specific embodiments and exemplary examples; however, these descriptions should not be construed as limiting the present invention. Those skilled in the art will understand that various equivalent substitutions, modifications, or improvements can be made to the technical solutions and embodiments of the present invention without departing from the spirit and scope of the invention, and all such modifications and improvements fall within the scope of the present invention. The scope of protection of the present invention is defined by the appended claims.

[0126] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A method for improving the accuracy of inertial guidance based on polynomial recursive least squares method, characterized in that, include: Based on the difference between inertial guidance telemetry and observations and the flight environment function, a guidance tool error model is constructed, which satisfies a linear relationship. Based on the guidance tool error model, the recursive formula based on the polynomial recursive least squares method is constructed as follows: K n+1 =C n c n+1 T [I k +c n+1 C n c n+1 T ] -1 K n+1,p =K n+1 [I k -(I k -c n+1 K n+1 ) p ](c n+1 K n+1 ) -1 C n+1 =C n -K n+1,p c n+1 C n Among them, K n+1 Let Γ be the feedback matrix at time n+1 of the recursion. n ,Γ n+1 These are the information inverse matrices at recursion time n and recursion time n+1, respectively. k For a k-dimensional identity matrix, c n+1 Let n+1 be the flight environment function matrix at recursive time n. The estimated values ​​of the guidance tool error coefficients at recursion time n and recursion time n+1 are respectively, K n+1,p Let y be the p-order polynomial feedback matrix at time n+1 of the recursion. n+1 y is the difference between the guided telemetry observations at time n+1 in the recursive time step. n+1 =y 遥,n+1 -y 外,n+1 y 遥,n+1 For the k-dimensional guided telemetry observations at time n+1, y 外,n+1 For the k-dimensional guided external observation at time n+1; n is an integer ≥ 0, and k is an integer ≥ 1; Based on the difference between the guidance telemetry observations at the current recursive time and the recursive formula of the least squares method, the estimated value of the guidance tool error coefficient at the current recursive time is determined, and the guidance telemetry observations at the current recursive time are compensated using the estimated value of the guidance tool error coefficient.

2. The method for improving inertial guidance accuracy based on polynomial recursive least squares method according to claim 1, characterized in that, The guidance tool error model is as follows: y n+1 =c n+1 X+v Among them, c n+1 Let X be the flight environment function matrix at time n+1, X be the constant error coefficient of the guidance tool, and v be the noise.

3. The method for improving inertial guidance accuracy based on polynomial recursive least squares method according to claim 1, characterized in that, The guidance tool error model includes either a telemetry velocity error model or a telemetry position error model, which consists of gyroscope error and accelerometer error.

4. The method for improving inertial guidance accuracy based on polynomial recursive least squares method according to claim 1, characterized in that, The order p can take values ​​from 1 to 8.

5. The method for improving inertial guidance accuracy based on polynomial recursive least squares method according to claim 1, characterized in that, The order p takes the value of 3.

6. The method for improving inertial guidance accuracy based on polynomial recursive least squares method according to claim 1, characterized in that, n = 0 represents the initial time, Γ0 and For the given known quantities.

7. The method for improving inertial guidance accuracy based on polynomial recursive least squares method according to claim 1, characterized in that, The following compensation formula is used to compensate for the k-dimensional guidance telemetry observations at the current recursive time using the estimated error coefficient of the guidance tool: in, For the compensated guidance telemetry observations.

8. The method for improving inertial guidance accuracy based on polynomial recursive least squares method according to claim 1, characterized in that, At each recursive time step, the binding value of the tool error is calculated using a determined tool error coefficient. Corrections are made to compensate for inertial guidance telemetry observations.

9. A method for improving inertial guidance accuracy based on polynomial recursive least squares method according to claim 1, characterized in that, Before determining the tool error coefficient at the current recursive time, it is determined whether the difference between the inertial guidance telemetry observations at the current recursive time is abnormal. If there is no abnormality, processing continues; otherwise, the telemetry observations at the previous recursive time are used to replace the telemetry observations at the current recursive time for subsequent processing.

10. A method for improving inertial guidance accuracy based on polynomial recursive least squares method according to claim 1, characterized in that, Before determining the estimated value of the tool error coefficient at each recursive time step, it is determined whether the difference of the current inertial guidance telemetry observation is abnormal. If there is no abnormality, the subsequent processing continues. Otherwise, it is further determined that the dimension of the abnormal data is located, and the relevant recursion of that dimension is isolated from the recursion formula of the least squares method at the current recursive time step. The remaining dimensions continue to be processed.

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