A gyro flywheel calibration reference error traceability analysis method

Abstract

This invention belongs to the field of inertial instrument calibration and testing, and provides a method for error source analysis of gyroscope flywheel calibration references. Step 1: Analyze all error factors of the test turntable-gyroscope flywheel calibration system, and define a coordinate system to mathematically describe them based on the analyzed error factors. Step 2: Based on the coordinate system defined in Step 1, establish the pose transformation matrix of the gyroscope flywheel relative to the inertial coordinate system. Step 3: Based on the pose transformation matrix in Step 2, establish an error propagation model of the calibration system. Step 4: Based on the error propagation model in Step 3, analyze the error sensitivity of the calibration system. Step 5: Based on the error sensitivity analyzed in Step 4, trace the source of key error factors in the calibration system. This invention addresses the problems of numerous error factors, nonlinear error propagation, and large error variation range in calibration systems, achieving source tracing of key errors through quantitative analysis.

Description

Technical Field

[0001] This invention relates to the field of inertial instrument calibration and testing technology, specifically to a method for tracing and analyzing the source of calibration reference errors in a gyroscope flywheel. Background Technology

[0002] A gyroscope flywheel is an integrated actuator / sensor device used for attitude control of miniature spacecraft. During operation, the gyroscope flywheel experiences drift errors due to disturbance torques. Multi-position calibration tests are crucial for identifying and compensating for these drift errors. In calibration tests, a calibration system is built using testing equipment (a two-axis test turntable) and the gyroscope flywheel to provide a calibration reference. However, various non-ideal factors in the calibration system directly lead to calibration reference errors, thus affecting calibration accuracy and limiting the improvement of the gyroscope flywheel's attitude sensitivity. Therefore, tracing the impact of various error factors in the calibration system on the calibration reference error is essential for optimizing the calibration scheme, ensuring calibration accuracy, and improving the gyroscope flywheel's attitude sensitivity.

[0003] Currently, there is a lack of research on the source analysis of calibration reference errors for gyroscope flywheels. Related research in this field mainly focuses on traditional inertial instruments, with less attention paid to novel inertial instruments like gyroscope flywheels. Furthermore, existing methods are mostly limited to discussions of single or partial error factors, failing to reveal the error propagation mechanism under the combined effect of multiple factors, and lacking general quantitative analysis methods and source tracing methods for key error factors. Summary of the Invention

[0004] This invention provides a method for tracing and analyzing the source of errors in gyroscope flywheel calibration references, which solves the problems of multiple error factors, nonlinear error propagation, and large error variation range in calibration systems. It achieves source tracing of key errors through quantitative analysis.

[0005] This invention is achieved through the following technical solution:

[0006] A method for tracing and analyzing the source of calibration reference error of a gyroscope flywheel, the method comprising the following steps:

[0007] Step 1: Analyze all error factors of the test turntable-gyroscope flywheel calibration system, and define a coordinate system to mathematically describe the error factors based on the analysis.

[0008] Step 2: Based on the coordinate system defined in Step 1, establish the pose transformation matrix of the gyroscope flywheel relative to the inertial coordinate system;

[0009] Step 3: Based on the pose transformation matrix from Step 2, establish the error propagation model of the calibration system;

[0010] Step 4: Based on the error propagation model of the calibration system in Step 3, analyze the error sensitivity of the calibration system;

[0011] Step 5: Based on the error sensitivity analysis in Step 4, trace the source of key error factors in the calibration system.

[0012] A method for tracing and analyzing the source of calibration benchmark error of a gyroscope flywheel, wherein step 1 specifically involves dividing the system into four sub-links based on the composition structure of the test turntable-gyroscope flywheel calibration system: the outer ring of the turntable, the inner ring of the turntable, the connection between the inner and outer rings of the turntable, and the connection between the inner ring of the turntable and the gyroscope flywheel, and refining the analysis of each link to determine the error factors of the calibration system.

[0013] A method for tracing and analyzing the source of calibration reference errors in a gyroscope flywheel, wherein the coordinate system defined in step 1 is specifically as follows: the base coordinate system F0: O0X0Y0Z0 is fixedly connected to the turntable base, and its Y-axis is consistent with the average axis of the outer ring; the outer ring coordinate system F1: O1X1Y1Z1 rotates by φ1 around the Y-axis from F0, and then rotates by Δφ around its Y-axis and the intermediate Z-axis and X-axis in sequence. y1 , Δα z1 and Δα x1 We obtain the inner ring reference coordinate system F2: O2X2Y2Z2, which is translated by Δη along the X-axis from F1, and then rotated by Δε around the intermediate Y-axis and X-axis in sequence. y , Δε x We obtain the inner ring coordinate system F3: O3X3Y3Z3, which is rotated by φ2 around the Z-axis from F2, and then by Δα around the Y-axis and the intermediate Z-axis and X-axis in sequence. y2 , Δφ z2 and Δα x2 We obtain the following coordinate system: F4: O4X4Y4Z4, which rotates by Δθ around the Y-axis and the intermediate Z-axis and X-axis sequentially from F3. y , Δθ z and Δθ x get.

[0014] A method for tracing and analyzing the source of calibration reference error of a gyroscope flywheel, wherein step 2 specifically involves establishing adjacent coordinate systems F based on multi-rigid-body kinematics theory. i and F i+1 Relative pose transformation matrix between i T i+1 As shown in the formula, i∈{0,…,3}:

[0015]

[0016] in

[0017]

[0018] Let φ = [φ1, φ2] T Let χ be the turntable rotation angle vector, and let χ be the error parameter vector that causes the attitude error at the load end of the turntable. χ = [Δφ] y1,Δφ z2 ,Δα x1 ,Δα z1 ,Δα x2 ,Δα y2 ,Δε x ,Δε y ,Δθ x ,Δθ y ,Δθ z ] T The pose transformation matrix of the gyroscope flywheel relative to the inertial coordinate system under ideal conditions is calculated according to formula (3):

[0019]

[0020] The pose transformation matrix of the gyroscope flywheel relative to the inertial coordinate system under the influence of multiple error factors is calculated according to formula (4):

[0021]

[0022] A method for tracing the source of calibration reference error of a gyroscope flywheel, wherein step 3 specifically involves calculating the gyroscope flywheel attitude error matrix caused by multiple error factors according to the formula:

[0023]

[0024] In the formula

[0025] By substituting the rotation angle position and error parameter values ​​into the formula, the errors of five calibration reference quantities can be calculated, including the three-axis gravitational acceleration components and the two-axis Earth rotation angular velocity components:

[0026]

[0027] In the formula ω e λ is the Earth's rotational angular velocity, and λ is the latitude of the test location; Formula (6) is the error propagation model of the calibration system.

[0028] A method for tracing the source of calibration reference error in a gyroscope flywheel, wherein step 4 specifically involves performing a global error sensitivity analysis of the calibration system based on the established calibration system error propagation model and the Sobol method; Let n χ and n r n represents the number of error factors and calibration reference quantities, respectively; χ =11, n r =5. For i∈{1,2,…,n χ}, j∈{1,2,…,n r The i-th error factor is obtained by Monte Carlo simulation, and the error transfer function corresponding to the j-th calibration reference is simplified as f.j (χ).

[0029] A method for tracing and analyzing the source of calibration reference error of a gyroscope flywheel, wherein step 4 specifically includes the following steps:

[0030] Step 4.1: Set the number of samples N and the turntable angle φ;

[0031] Step 4.2: Perform Monte Carlo random sampling within the error parameter space. The parameter space is set according to the actual engineering requirements. Generate N×n χ 3D sampling matrix A and resampling matrix B;

[0032] Step 4.3: Initialize i = 1, j = 1;

[0033] Step 4.4: Calculate according to the formula

[0034]

[0035] In the formula f jk This means substituting the k-th sampling point into the error transfer function f. j The obtained output value, This means replacing the i-th column of A with the i-th column of B. This means replacing the i-th column of B with the i-th column of A;

[0036] Step 4.5: Calculate the sensitivity coefficient of the i-th error factor to the j-th calibration reference error according to the formula.

[0037]

[0038] Step 4.6: If i < n χ If i = n, then let i = i + 1 and jump to step 4.4; if i = n χ And j < n r If i = 1 and j = j + 1, then jump to step 4.4; if i = n χ And j = n r The operation ends and the sensitivity coefficient is output.

[0039] A method for tracing the source of calibration reference error of a gyroscope flywheel, wherein step 5 specifically involves, for i∈{1,2,…,n χ}, j∈{1,2,…,n r The error sensitivity calculation results for all corner positions are averaged to obtain the first-order contribution value of the i-th error factor to the j-th calibration reference error. and overall contribution value The contribution of the interaction between the i-th error factor and other factors to the j-th calibration reference error is calculated using the overall contribution value and the first-order contribution value. Determine the degree of influence of the interaction of error factors on the calibration reference error; for j∈{1,2,…,n r Based on the contribution value of each error factor, the degree of influence of each error factor on the error of the j-th calibration reference is determined, and the key error set κ affecting the j-th calibration reference is obtained. j ;

[0040] The interaction contribution value Specifically:

[0041]

[0042] The beneficial effects of this invention are:

[0043] The error propagation model of the turntable-gyroscope flywheel calibration system established in this invention comprehensively considers the various error sources of the calibration system and their interactions. Moreover, the model uses the calibration reference quantity error as the terminal output error, which effectively avoids including parameters to be calibrated, and has better applicability and scalability.

[0044] The gyroscope flywheel calibration reference error tracing analysis method proposed in this invention can effectively handle the problems of multiple error factors, nonlinear error propagation, and large error variation range in calibration systems. It can quantitatively provide the contribution values ​​of each error factor and their interactions to the calibration reference error before calibration testing, thus achieving source tracing of key errors. The analysis and tracing results provide ideas and directions for determining key error indicators and optimizing calibration schemes, which is beneficial for ensuring calibration accuracy and improving the attitude sensitivity performance of the gyroscope flywheel.

[0045] This invention is also applicable to traditional inertial instruments. Attached Figure Description

[0046] Figure 1 This is a flowchart of the method of the present invention.

[0047] Figure 2 This is a schematic diagram showing the contribution values ​​of various error factors to the x-axis calibration reference error of this invention.

[0048] Figure 3 This is a schematic diagram illustrating the contribution values ​​of various error factors to the y-axis calibration reference error, as presented in this invention.

[0049] Figure 4 This is a schematic diagram illustrating the contribution values ​​of various error factors to the z-axis calibration reference error, as presented in this invention. Detailed Implementation

[0050] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0051] A method for tracing and analyzing the source of calibration reference error of a gyroscope flywheel, the method comprising the following steps:

[0052] Step 1: Analyze all error factors of the test turntable-gyroscope flywheel calibration system, and define a coordinate system to mathematically describe the error factors based on the analysis.

[0053] Step 2: Based on the coordinate system defined in Step 1, establish the pose transformation matrix of the gyroscope flywheel relative to the inertial coordinate system;

[0054] Step 3: Based on the pose transformation matrix from Step 2, establish the error propagation model of the calibration system;

[0055] Step 4: Based on the error propagation model of the calibration system in Step 3, analyze the error sensitivity of the calibration system;

[0056] Step 5: Based on the error sensitivity analysis in Step 4, trace the source of key error factors in the calibration system.

[0057] A method for tracing and analyzing the source of errors in gyroscope flywheel calibration references, wherein step 1 specifically involves dividing the test turntable-gyroscope flywheel calibration system into four sub-components based on its structural composition: the outer ring of the turntable, the inner ring of the turntable, the connection between the inner and outer rings, and the connection between the inner ring and the gyroscope flywheel. The error factors of the calibration system are then analyzed and determined in detail for each sub-component. These error factors include the turntable outer ring shaft positioning error Δφ. y1 Rotational error Δα of the outer ring shaft of the turntable x1 and Δα z1 Positioning error Δφ of the inner ring shaft of the turntable z2 Rotational error Δα of the inner ring shaft of the turntable x2 and Δα y2 The verticality error Δε at the connection between the inner and outer rings of the turntable x and Δε y The intersection error Δη at the connection between the inner and outer rings of the turntable, and the installation error Δθ of the gyroscope flywheel at the connection between the inner ring of the turntable and the gyroscope flywheel. x , Δθ y and Δθ z .

[0058] A method for tracing and analyzing the source of calibration reference errors in a gyroscope flywheel, wherein the coordinate system defined in step 1 is specifically as follows: the base coordinate system F0: O0X0Y0Z0 is fixedly connected to the turntable base, and its Y-axis is consistent with the average axis of the outer ring; the outer ring coordinate system F1: O1X1Y1Z1 rotates by φ1 around the Y-axis from F0, and then rotates by Δφ around its Y-axis and the intermediate Z-axis and X-axis in sequence. y1 , Δα z1 and Δα x1 We obtain the inner ring reference coordinate system F2: O2X2Y2Z2, which is translated by Δη along the X-axis from F1, and then rotated by Δε around the intermediate Y-axis and X-axis in sequence. y , Δε x We obtain the inner ring coordinate system F3: O3X3Y3Z3, which is rotated by φ2 around the Z-axis from F2, and then by Δα around the Y-axis and the intermediate Z-axis and X-axis in sequence. y2 , Δφ z2 and Δα x2 We obtain the following coordinate system: F4: O4X4Y4Z4, which rotates by Δθ around the Y-axis and the intermediate Z-axis and X-axis sequentially from F3. y , Δθ z and Δθ x get.

[0059] A method for tracing the source of calibration reference error of a gyroscope flywheel, wherein step 2 specifically involves establishing adjacent coordinate systems F based on multi-rigid-body kinematics theory, considering multiple error factors. i and F i+1 Relative pose transformation matrix between i T i+1 As shown in the formula, i∈{0,…,3}:

[0060]

[0061] in

[0062]

[0063] Let φ = [φ1, φ2] T Let χ be the turntable rotation angle vector, and let χ be the error parameter vector that causes the attitude error at the load end of the turntable. χ = [Δφ] y1 ,Δφ z2 ,Δα x1 ,Δα z1 ,Δα x2 ,Δα y2 ,Δε x ,Δε y ,Δθ x ,Δθ y ,Δθ z ] TThe pose transformation matrix of the gyroscope flywheel relative to the inertial coordinate system under ideal conditions is calculated according to formula (3):

[0064]

[0065] The pose transformation matrix of the gyroscope flywheel relative to the inertial coordinate system under the influence of multiple error factors is calculated according to formula (4):

[0066]

[0067] A method for tracing the source of calibration reference error of a gyroscope flywheel, wherein step 3 specifically involves calculating the gyroscope flywheel attitude error matrix caused by multiple error factors according to the formula:

[0068]

[0069] In the formula

[0070] Assuming the initial orientation of the gyroscope flywheel is ideally north-west-sky, substituting the angular position and error parameter values ​​into the formula, we can calculate the errors of five calibration reference quantities: the three-axis gravitational acceleration components and the two-axis Earth rotation angular velocity components.

[0071]

[0072] In the formula ω e λ is the Earth's rotational angular velocity, and λ is the latitude of the test location; Formula (6) is the error propagation model of the calibration system.

[0073] A method for tracing the source of calibration reference error in a gyroscope flywheel, wherein step 4 specifically involves performing a global error sensitivity analysis of the calibration system based on the established calibration system error propagation model and the Sobol method; Let n χ and n r n represents the number of error factors and calibration reference quantities, respectively; χ =11, n r =5. For i∈{1,2,…,n χ}, j∈{1,2,…,n r The first-order error sensitivity of the i-th error factor to the j-th calibration reference error was obtained using Monte Carlo simulation. and overall error sensitivity For ease of description, the error transfer function corresponding to the j-th calibration reference is abbreviated as f. j (χ).

[0074] A method for tracing and analyzing the source of calibration reference error of a gyroscope flywheel, wherein step 4 specifically includes the following steps:

[0075] Step 4.1: Set the number of samples N and the turntable angle φ;

[0076] Step 4.2: Perform Monte Carlo random sampling within the error parameter space. The parameter space is set according to the actual engineering requirements. Generate N×n χ 3D sampling matrix A and resampling matrix B;

[0077] Step 4.3: Initialize i = 1, j = 1;

[0078] Step 4.4: Calculate according to the formula

[0079]

[0080] In the formula f jk This means substituting the k-th sampling point into the error transfer function f. j The obtained output value, This means replacing the i-th column of A with the i-th column of B. This means replacing the i-th column of B with the i-th column of A;

[0081] Step 4.5: Calculate the sensitivity coefficient of the i-th error factor to the j-th calibration reference error according to the formula.

[0082]

[0083] Step 4.6: If i < n χ If i = n, then let i = i + 1 and jump to step 4.4; if i = n χ And j < n r If i = 1 and j = j + 1, then jump to step 4.4; if i = n χ And j = n r The operation ends and the sensitivity coefficient is output.

[0084] A method for tracing the source of calibration reference error of a gyroscope flywheel, wherein step 5 specifically involves, for i∈{1,2,…,n χ}, j∈{1,2,…,n r The error sensitivity calculation results for all corner positions are averaged to obtain the first-order contribution value of the i-th error factor to the j-th calibration reference error. and overall contribution value The contribution of the interaction between the i-th error factor and other factors to the j-th calibration reference error is calculated using the overall contribution value and the first-order contribution value. Determine the degree of influence of the interaction of error factors on the calibration reference error; for j∈{1,2,…,n rBased on the contribution value of each error factor, the degree of influence of each error factor on the error of the j-th calibration reference is determined, and the key error set κ affecting the j-th calibration reference is obtained. j ;

[0085] The interaction contribution value Specifically:

[0086]

[0087] Figures 2-4 Contribution value and The distribution diagram shows that, since χ has the same error propagation relationship with the coaxial calibration reference, the contribution values ​​of each factor to the coaxial calibration reference quantity are consistent. A contribution threshold S is set. κ =0.1, if the contribution value is less than S κ This indicates that the influence of the corresponding factor is relatively small and can be ignored. The magnitude of the value represents the degree of influence of the interaction between the i-th error factor and other factors on the calibration reference error. According to... Figures 2-4 The distribution of contribution values ​​shown indicates that the interaction between various error factors has a relatively small impact on the calibration reference error.

[0088] For j∈{1,2,…,n r}, based on the contribution value of each error factor Determine the magnitude of the influence of each error factor on the error of the j-th calibration reference, and determine the critical error set κ affecting the j-th calibration reference according to the formula. j :

[0089]

[0090] according to Figures 2-4 The contribution value distribution and formula shown yield the error source tracing results for each calibration benchmark, i.e.

[0091]

[0092] This result indicates that the key error source for the x-axis calibration datum is Δα. y2 ,Δθ y The key error source for the y-axis calibration datum is Δα. x2 ,Δθ x The key error source for the z-axis calibration datum is Δφ. y1 ,Δε y .

[0093] This invention reveals the error propagation mechanism of a gyroscope flywheel calibration system under the combined influence of multiple factors, effectively addressing the problems of numerous error factors, nonlinear error propagation, and large error variation range in the calibration system. Quantitative analysis results enable the tracing of key errors. Based on the tracing results, higher performance requirements can be set for key error factors, and the calibration scheme can be optimized to reduce their impact. These measures effectively ensure calibration accuracy and improve the attitude sensitivity of the gyroscope flywheel. The method proposed in this invention has good generalizability and can be applied to error source analysis of other inertial instruments.

[0094] Table 1 Error Factors of Calibration System

[0095]

Claims

1. A method of traceability analysis of a calibration reference error of a gyroscopic flywheel, characterized in that, The source tracing analysis method includes the following steps: Step 1: Analyze all error factors of the test turntable-gyroscope flywheel calibration system, and define a coordinate system to mathematically describe the error factors based on the analysis. Step 2: Based on the coordinate system defined in Step 1, establish the pose transformation matrix of the gyroscope flywheel relative to the inertial coordinate system; Step 3: Based on the pose transformation matrix from Step 2, establish the error propagation model of the calibration system; Step 4: Based on the error propagation model of the calibration system in Step 3, analyze the error sensitivity of the calibration system; Step 5: Based on the error sensitivity analysis in Step 4, trace the source of key error factors in the calibration system; Specifically, step 1 involves dividing the test turntable-gyroscope flywheel calibration system into four sub-sections based on its composition: the outer ring of the turntable, the inner ring of the turntable, the connection between the inner and outer rings of the turntable, and the connection between the inner ring of the turntable and the gyroscope flywheel. Each sub-section is analyzed in detail to determine the error factors of the calibration system. Step 3 specifically involves calculating the gyroscope flywheel attitude error matrix caused by multiple error factors according to formula (1): （1） In the formula ; is the pose transformation matrix of the gyro flywheel relative to the inertial coordinate system in an ideal case, is the pose transformation matrix of the gyro flywheel relative to the inertial coordinate system under the action of multiple error factors; Substituting the rotation position and error parameter value into formula (2), the errors of five calibration reference quantities can be calculated, including the three-axis gravitational acceleration components and the two-axis Earth rotation angular velocity components: （2） In the formula , is the earth rotation angular velocity, is the latitude of the test site; formula (2) is the error transfer model of the calibration system; Step 4 specifically involves performing a global error sensitivity analysis of the calibration system based on the established calibration system error propagation model and the Sobol method; (Note: The original text contains some inconsistencies and unclear grammatical structures. A more accurate translation would require the full context.) and These represent the error factor and the number of calibration reference quantities, respectively. , ;for , Monte Carlo simulation was used to calculate the first... i The error factor will correspond to the first error factor. j The error transfer function of a calibration reference is abbreviated as: ; Step 4 specifically includes the following steps: Step 4.1: Set the number of samples to sample N , set the rotation angle of the turntable ; Step 4.2: Monte Carlo random sampling is performed in the error parameter space, and the parameter space is set according to engineering practice to generate a A dimensional sampling matrix B and a resampling matrix B ; Step 4.3: Initialization ; Step 4.4: Calculate according to formula (3) : （3） In the formula Indicates the first k Substituting each sampling point into the error propagation function The obtained output value, Indicates will A The i Column replacement B The i List, Indicates will B The i Column replacement A The i List; Step 4.5: Calculate the sensitivity coefficient of the first i error factor to the first j calibration reference error according to formula (4) （4） Step 4.6: If Then let Proceed to step 4.4; if ,and Then let , Proceed to step 4.4; if ,and The operation ends and the sensitivity coefficient is output. Specifically, step 5 involves... , The error sensitivity calculation results for all corner positions are averaged to obtain the first... i The error factor affects the first j The first-order contribution value of each calibration reference error and overall contribution value The first-order contribution value is calculated from the total contribution value and the first-order contribution value. i The interaction between the error factor and other factors affects the first error factor. j Contribution value of each calibration reference error To determine the extent to which the interaction of error factors affects the calibration reference error; for Based on the contribution value of each error factor, determine the impact of each error factor on the first error. j The degree of influence of the calibration reference error is obtained, thus affecting the degree of influence of the first calibration reference error. j Key error set of a calibration benchmark ; Interaction contribution value Specifically: （5）。 2. The method of claim 1, wherein: The coordinate system defined by the step 1 is specifically a base coordinate system Fixedly connected with the base of the turntable, the outer ring Y The shaft is consistent with the average axis of the outer ring. outer ring coordinate system Depend on Around Axis rotation Then, circle around it in turn. Shaft and intermediate system axis, Axis rotation and Obtain; Inner ring reference coordinate system Depend on along Axis translation Then, in sequence, around the middle system shaft and Axis rotation Obtain; Inner Loop Coordinate System Depend on Around Axis rotation Then, circle around in sequence. Shaft and intermediate system axis, Axis rotation and Obtain; gyroscope flywheel body coordinate system Depend on Circling in sequence Shaft and intermediate system axis, Axis rotation and get, This refers to the positioning error of the outer ring shaft. This refers to the positioning error of the inner ring shaft. This refers to the rotational error of the outer ring shaft. This refers to the rotational error of the outer ring shaft. This refers to the rotational error of the inner ring shaft. This refers to the rotational error of the inner ring shaft. For verticality error, For verticality error, To account for the installation error of the gyroscope flywheel, To account for the installation error of the gyroscope flywheel, To account for the installation error of the gyroscope flywheel, Intersection error at the connection between the inner and outer rings of the turntable.

3. The method of claim 2, wherein: Step 2 specifically involves establishing adjacent coordinate systems based on multi-rigid-body kinematics theory. and Relative pose transformation matrix between , As shown in formula (6): （6） in （7） remember Let be the turntable angle vector. The error parameter vector that causes attitude error at the load end of the turntable. The pose transformation matrix of the gyroscope flywheel relative to the inertial coordinate system under ideal conditions is calculated according to formula (8): （8） The pose transformation matrix of the gyroscope flywheel relative to the inertial coordinate system under the influence of multiple error factors is calculated according to formula (9): （9）。

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