A method for identifying error terms related to angular rate of a hemispherical resonator gyroscope

Abstract

The present application relates to the technical field of inertial test, and relates to a method for identifying error model coefficients related to angular velocity of a hemispherical resonator gyro on a three-axis turntable. The method comprises the following steps: Step 1: establishing an installation error model of the three-axis turntable and the gyro; Step 2: considering the earth rotation angular velocity, the three-axis turntable error, the gyro installation error, and accurately determining the angular velocity of three input reference axes input to the gyro relative to the inertial space; Step 3: substituting the angular velocity input components into the error model of the gyro to obtain a full error model; Step 4: according to the full error model, designing a calibration method for error model coefficients related to angular velocity high-order terms based on the single-axis velocity and double-axis position roll method of the three-axis turntable; Step 5: designing a method for identifying all error model coefficients related to angular velocity; Step 6: analyzing the test uncertainty of the error model coefficients. The scheme provided by the present application can improve the test accuracy of the gyro on the three-axis turntable.

Description

Technical Field

[0001] This invention relates to the field of inertial testing technology, and more particularly to a method for identifying the error model coefficients of a hemispherical resonator gyroscope related to angular rate on a three-axis turntable. Background Technology

[0002] The angular rate range of force feedback hemispherical resonator gyroscopes is generally less than 30° / s. The quadratic coefficient related to the angular rate affects the gyroscope's output, and the error model coefficients related to the angular rate are generally not measured. Currently, a research boom has emerged in China for rate integrating hemispherical resonator gyroscopes. The range of rate integrating hemispherical resonator gyroscopes can reach 500° / s, and the impact of their quadratic error coefficients on the gyroscope's output will increase significantly (50 / 3). 2 The error rate is approximately 278 times, therefore it is imperative to study an accurate testing method for the error model of the rate integral hemispherical resonator gyroscope.

[0003] Spin missiles can reach spin angular velocities of 1800° / s, and during high-maneuverability flight in offensive and defensive combat maneuvers, these angular velocities can reach 500° / s. On these high-angular-velocity vehicles, the quadratic term error of the gyroscope will be maximized in the output. During gyroscope testing, the identified error model coefficients related to angular rate are coupled with the turntable error in the gyroscope output, leading to inaccurate identified gyroscope error model coefficients and poor or even worse compensation results. Therefore, considering the turntable error is of great theoretical and practical significance for improving the attitude accuracy of high-speed aircraft.

[0004] Based on this, the present invention establishes an error calibration model for the gyroscope, including the error of the three-axis turntable of the test equipment and the alignment error of the gyroscope installation. A new identification method is used to separate and compensate the error coefficients, thereby improving the identification accuracy of the gyroscope error. Summary of the Invention

[0005] This invention provides a method for identifying the error term related to angular rate of a hemispherical resonant gyroscope, so as to improve the testing accuracy of the gyroscope on a three-axis turntable.

[0006] This invention provides a method for identifying error terms related to angular rate in a hemispherical resonant gyroscope. The gyroscope is mounted on the inner ring axis of a three-axis turntable, which includes an outer ring axis, a middle ring axis, and an inner ring axis. The method simultaneously considers the turntable error and the gyroscope mounting alignment error, establishing a full error calibration model that includes these errors. Appropriate error separation techniques are then employed to improve the calibration accuracy of the error model coefficients related to angular rate. The method includes:

[0007] Step 1: Establish the error model of the three-axis turntable and the installation and alignment error model of the gyroscope on the three-axis turntable;

[0008] Step 2: When the outer ring axis of the three-axis turntable rotates at a uniform angular rate, the Earth's rotation angular rate, the three-axis turntable error, and the gyroscope installation alignment error are considered simultaneously to accurately determine the angular rate of the relative inertial space on the three input reference axes input to the gyroscope.

[0009] Step 3: Substitute the three angular rate input components into the preset gyroscope error model to obtain a full error model containing the three-axis turntable error and the gyroscope installation alignment error;

[0010] Step 4: Based on the full error model, a single-axis rate dual-axis position roll method for calibrating the gyroscope on a three-axis turntable was designed, and an experimental plan was adopted to rotate the outer ring axis at two angular rates.

[0011] Step 5: Design a method for identifying the coefficients of the error model related to angular rate;

[0012] Step 6: The test uncertainty of the error model coefficients was analyzed.

[0013] As described above, the error model related to angular rate incorporates the errors of the three-axis turntable and the alignment error of the gyroscope on the turntable. Then, considering the uniform angular rate rotation of the outer ring axis of the turntable and the Earth's rotation angular rate, the components of these two factors are substituted into the preset gyroscope error model, resulting in a complete error model containing both the three-axis turntable error and the gyroscope alignment error. Finally, the least squares method is used to identify the coefficients of the error model. Because a complete error model is established and corresponding error separation techniques are employed, the influence of the three-axis turntable error and the gyroscope alignment error on the identification accuracy can be eliminated. Therefore, this technical solution can improve the testing accuracy of the gyroscope on the three-axis turntable. Attached Figure Description

[0014] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0015] Figure 1 This is a schematic diagram of the structure of a three-axis rotary table provided in an embodiment of the present invention, and the initial positions of each coordinate system on the three-axis rotary table.

[0016] In the attached diagram, 1 represents the outer ring axis, 2 represents the middle ring axis, and 3 represents the inner ring axis. Detailed Implementation

[0017] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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 some embodiments of the present invention, but 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.

[0018] Figure 1 This is a schematic diagram of a three-axis rotary table according to an embodiment of the present invention. Please refer to [link / reference]. Figure 1 The gyroscope is mounted on the inner ring axis 3 of the three-axis turntable, which includes the outer ring axis 1, the middle ring axis 2 and the inner ring axis 3, while taking into account both the error of the three-axis turntable and the alignment error of the gyroscope installation.

[0019] The following is combined with Figure 1 This paper provides a detailed description of the method for identifying the error model coefficients of gyroscopes and angular rates on a three-axis turntable, as provided in the embodiments of the present invention.

[0020] This invention provides a method for identifying error terms related to angular rate in a hemispherical resonant gyroscope, comprising:

[0021] Step 1: Establish the error model of the three-axis turntable and the installation and alignment error model of the gyroscope on the three-axis turntable;

[0022] Step 2: When the outer ring axis 1 of the three-axis turntable rotates at a uniform angular rate, the angular rate of the Earth's rotation, the error of the three-axis turntable, and the alignment error of the gyroscope installation are considered simultaneously to accurately determine the angular rate of the relative inertial space on the three input reference axes input to the gyroscope.

[0023] Step 3: Substitute the three angular rate input components into the preset gyroscope error model to obtain a full error model containing the three-axis turntable error and the gyroscope installation alignment error;

[0024] Step 4: Based on the full error model, a single-axis rate dual-axis position roll method for calibrating the gyroscope on a three-axis turntable was designed, and an experimental plan was adopted to rotate the outer ring axis 1 at two angular rates.

[0025] Step 5: Design a method for identifying the coefficients of the error model related to angular rate;

[0026] Step 6: The test uncertainty of the error model coefficients was analyzed.

[0027] In this embodiment, the errors of the three-axis turntable and the alignment error of the gyroscope on the turntable are introduced into the error model related to angular rate. Then, considering the uniform angular rate of rotation of the outer ring axis 1 of the three-axis turntable and the Earth's rotation angular rate, their components are incorporated into the preset error model of the gyroscope, resulting in a complete error model containing the errors of the three-axis turntable and the alignment error of the gyroscope. Finally, the least squares method is used to identify the coefficients of the error model. Therefore, the above technical solution can improve the testing accuracy of the gyroscope on the three-axis turntable.

[0028] like Figure 1 As shown, the following coordinate systems are established on the three-axis turntable to facilitate the analysis of the attitude relationships between the various coordinate systems. o0-x0y0z0 is the geographic coordinate system (northeast-sky coordinate system), o1-x1y1z1 is the outer ring axis 1 coordinate system, o2-x2y2z2 is the middle ring axis 2 coordinate system, o3-x3y3z3 is the inner ring axis 3 coordinate system, and o4-IXY is the gyroscope coordinate system, where X is the gyroscope X-axis, Y is the gyroscope Y-axis, and I is the gyroscope input axis I. Ideally, these coordinate systems coincide when the three-axis turntable is in the zero position. The main sources of error for the three-axis turntable are perpendicularity error, angular position error, zero-position error, and gyroscope installation alignment error.

[0029] A hemispherical resonator gyroscope is mounted on a three-axis turntable. The outer ring axis 1 of the turntable rotates at a uniform angular rate, while the inner ring axis 3 and the outer ring axis 1 are fixed at a precise position. When the three-axis turntable is at zero position, the outer ring axis 1 points to the sky, the middle ring axis 2 points horizontally to the east, and the inner ring axis 3 points horizontally to the north. At this time, the input axis I of the hemispherical resonator gyroscope points to the east, the X-axis points to the north, and the Y-axis points to the sky.

[0030] In some implementations, step one includes:

[0031] The error model related to angular rate is:

[0032]

[0033] In the formula, Y ω To compensate for static errors in the gyroscope output, k ω0 Here are the model coefficients independent of angular rate, and K is the scaling factor of the gyroscope. ωx k ωy k represents the coefficients of the first-order error model of the angular velocity. ωIx k ωIy k ωxy The coefficients of the cross-coupling term for angular velocity, k ωII k ωxx k ωyy ω is the coefficient of the quadratic term of the angular velocity. I ω x ω yε represents the angular velocity on each axis of the gyroscope, and ε is the random error.

[0034] Calculate the attitude matrix of the middle ring axis coordinate system relative to the outer ring axis coordinate system using the following formula:

[0035]

[0036] In the formula, y1 is the o1y1 axis. The perpendicularity between the middle ring axis and the outer ring axis is given by x1, which is the o1x1 axis. The zero-position error of the middle ring axis is Δθ. x (θ x ) represents the angular position error of the central ring axis, θ x This is the nominal rotation angle of the central ring shaft;

[0037] Calculate the attitude matrix of the inner ring axis coordinate system relative to the middle ring axis coordinate system using the following formula:

[0038]

[0039] In the formula, z2 is the o2z2 axis. y1 represents the perpendicularity between the inner ring axis and the middle ring axis, and y2 represents the o2y2 axis. The zero-position error of the inner ring shaft is Δθ. y (θ y θ represents the angular position error of the inner ring shaft. y This is the nominal rotation angle of the inner ring shaft;

[0040] Calculate the attitude matrix of the gyroscope coordinate system relative to the inner loop axis coordinate system using the following formula:

[0041]

[0042] In the formula, Δλ x , Δλ y , Δλ z The installation alignment error of the gyroscope is represented by x3, x3 is the o3x3 axis, y3 is the o3y3 axis, and z3 is the o3z3 axis.

[0043] Calculate the attitude matrix of the gyroscope coordinate system relative to the outer ring axis coordinate system using the following formula:

[0044]

[0045] Because the Earth's rotation rate is very small, the attitude error can be ignored during the multiple transmissions of its three components in the gyroscope coordinate system. Only the nominal attitude matrix of the gyroscope coordinate system relative to the geographic coordinate system needs to be calculated.

[0046]

[0047] In the formula, z1 is the o1z1 axis, ω T Let x2 be the angular velocity of the outer ring axis, x2 be the o2x2 axis, and θ be the angular velocity of the outer ring axis. x θ is the nominal rotation angle of the middle ring axis, y3 is the o3y3 axis, and θ is the nominal rotation angle of the middle ring axis. y This is the nominal rotation angle of the inner ring shaft.

[0048] In some implementations, step two includes:

[0049] The component of the Earth's rotation angular rate in the gyroscope coordinate system is: In the formula ω ie Let L be the magnitude of the Earth's rotational angular rate, and L be the local geographical latitude. The component of the uniform angular velocity of the outer ring axis in the gyroscope coordinate system is T1. 4 [0 0 ω T ] T And T1 4 =(T4) 1 ) T Therefore, the angular rates of the three input reference axes of the gyroscope relative to inertial space are:

[0050]

[0051] Calculating the three components of the above formula, we can obtain:

[0052]

[0053] ω x =ω T sinθ x +ω ie cos L cosθ x cos(ω T t)+ω ie sin L sinθ x (8)

[0054] ω y =ω T cosθ x cosθ y +ω ie cos L[sinθ y sin(ω T t)-sinθ x cosθ y cos(ω T t)]+ω ie sin Lcosθ x cosθ y

[0055] In this embodiment, due to the existence of three-axis turntable errors and gyroscope installation alignment errors, the calibration accuracy of the gyroscope on the three-axis turntable is affected. By performing error propagation and comprehensive analysis on the respective error sources of the three-axis turntable and the gyroscope, it is beneficial to identify the error model of the gyroscope and angular rate related on the three-axis turntable, thereby improving the testing accuracy of the gyroscope on the three-axis turntable.

[0056] In some implementations, step three includes:

[0057] Substituting the angular rate component into the preset gyroscope error model, a full error model containing the three-axis turntable error and the gyroscope installation alignment error is obtained:

[0058]

[0059] Where, k ω0 These are model coefficients that are independent of angular velocity.

[0060] In some implementations, step four includes:

[0061] Based on the full error model, the experimental plan shown in Table 1 was designed.

[0062] Table 1 shows the error model coefficients related to angular rate, the test points, and the gyroscope output. The first angular velocity of the outer ring axis, The second angular velocity of the outer ring axis is represented by the subscript i, which is the position number.

[0063]

[0064] When the middle and inner ring axes of the three-axis turntable are in different angular positions, the angular velocity vectors of the three axes of the gyroscope relative to the outer ring axis will have different directions, thus resulting in different components of the angular rate on the three axes of the gyroscope. These components are then substituted into the full error model to identify the error model coefficients related to the angular rate.

[0065] In some implementations, step five includes:

[0066] The following identification method was designed:

[0067] When the outer ring axis moves at an angular velocity During rotation, the structure matrix is

[0068]

[0069] In the formula:

[0070] Φ i.1 =1

[0071]

[0072]

[0073]

[0074]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080]

[0081]

[0082]

[0083] When the outer ring axis moves at an angular velocity During rotation, the structure matrix is

[0084]

[0085] In the formula:

[0086] Ψ i.1 =1

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096]

[0097]

[0098]

[0099] The error model coefficients related to angular rate are identified by the following formula:

[0100] x=(A T A) -1 A T y

[0101] Where x is the model coefficient vector; A is the structure matrix; and y is the gyroscope output vector.

[0102] In the formula:

[0103]

[0104]

[0105]

[0106] in, This represents the output of the gyroscope; the first digit 1 in the subscript indicates the first angular velocity on the outer ring axis. Below, 2 represents the second angular velocity on the outer ring axis. The second digit of the subscript represents the position number when the middle and inner ring axes of the three-axis turntable are in different angular positions.

[0107] To compare the impact of turntable error on the model coefficient identification results with and without considering turntable error, a model coefficient identification method without considering turntable error is designed below.

[0108] When the outer ring axis moves at an angular velocity During rotation, only the first 10 columns of the structure matrix Φ are considered, denoted as structure matrix Φ′:

[0109]

[0110] When the outer ring axis moves at an angular velocity During rotation, only the first 10 columns of the structure matrix Ψ are considered, denoted as structure matrix Ψ′:

[0111]

[0112] The error model coefficients related to angular rate are identified by the following formula:

[0113] x′=(A′ T A′) -1 A′ T y

[0114] Where x′ is the model coefficient vector; A′ is the structure matrix; and y is the output vector of the gyroscope.

[0115] In the formula:

[0116]

[0117]

[0118] x′=[k ω0 k ωII k ωxx k ωyy K k ωx k ωy k ωIx k ωxy k ωIy ] T

[0119] in, This represents the output of the gyroscope, and the subscript definition is the same as described above.

[0120] In some implementations, step six includes:

[0121] The uncertainty of the identification error coefficients was analyzed, and the identification results were compared with and without considering turntable errors. This verifies that the present invention can improve the testing accuracy of gyroscopes on a three-axis turntable.

[0122] set up The set values ​​and identification results of the error model coefficients are shown in Table 2.

[0123] Table 2 Identification results with and without considering turntable error.

[0124]

[0125] The identification results considering and not considering turntable error are analyzed below:

[0126] As shown in Table 2, the model coefficient k is affected by considering and not considering turntable error. ωII k ωyy k ωy k ωIx k ωxy The impact is significant, reaching 10 -6 Order of magnitude. By comparing the difference between considering and not considering turntable error with the set value, it can be known that the coefficient k of the quadratic term of the angular rate... ωII With k ωyy The differences between the two values ​​are all on the same order of magnitude as the set values, indicating that neglecting the turntable error would significantly impact the test accuracy; the coefficient k of the cross-coupling term of the angular rate. ωIx With kωxy The differences between each are all one order of magnitude lower than the set value; k ωy The difference is two orders of magnitude lower than the set value, where k is identified after considering the influence of turntable error. ωy In fact The coupling value of these three factors, namely the scaling factor K of the gyroscope and the zero-position error of the inner loop axis, is... The alignment error Δλ of the gyroscope installation y Phase coupling; the gyroscope's scaling factor K is related to the Earth's rotation angular rate ω. ie Coupled with the local geographical latitude L, due to ω ie sinL / ω T The difference compared to 1 is several orders of magnitude, which can be ignored; that is, the Earth's rotational angular rate ω. ie The influence of local latitude L on the gyroscope's scaling factor K is negligible. The difference between considering and not considering turntable error in the remaining model coefficients is very small and can be ignored in engineering.

[0127] The comparison shows that if turntable error and gyroscope alignment error are not considered during calibration, the calibration results of the error model coefficients will have significant errors, affecting the calibration accuracy. Therefore, in actual calibration and testing, turntable error and gyroscope alignment error must be considered; otherwise, the testing accuracy will be greatly affected. This paper identifies error model coefficients that are closer to reality after considering turntable error, thus effectively improving the calibration accuracy of the gyroscope error model coefficients.

[0128] It should be noted that, in this document, the term "comprising" or its variations are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Unless otherwise specified, an element defined by the phrase "comprising a..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0129] Finally, it should be noted that the above description is merely a preferred embodiment of the present invention and is only used to illustrate the technical solution of the present invention, and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention are included within the scope of protection of the present invention.

Claims

1. A method for identifying error terms related to angular rate of a hemispherical resonator gyroscope, characterized in that, The method comprises the following steps: Step one: establishing an error model of the three-axis turntable and an installation and alignment error model of the gyroscope on the three-axis turntable; Step two: when the outer ring shaft of the three-axis turntable rotates at a uniform angular velocity, the angular velocity of the earth rotation is considered, the three-axis turntable error and the installation and alignment error of the gyroscope are considered, and the angular velocity of the three input reference axes of the gyroscope relative to the inertial space is accurately determined; Step three: the three angular velocity input components are substituted into the preset error model of the gyroscope to obtain a full error model containing the three-axis turntable error and the installation and alignment error of the gyroscope; Step four: a single-axis rate double-axis position roll method for calibrating the gyroscope on the three-axis turntable is designed according to the full error model, and an outer ring shaft is rotated at two angular velocities in the test plan; Step five: an identification method of the error model coefficient related to the angular velocity is designed; Step six: the test uncertainty of the error model coefficient is analyzed; The step one comprises: The error model related to the angular velocity is (1) wherein is the output of the gyroscope after compensation of the static error, is the model coefficient independent of the angular rate, is the scale factor of the gyroscope, , is the model coefficient of the first order term of the angular rate, , , is the coefficient of the cross-coupling term of the angular rate, , , is the coefficient of the second order term of the angular rate, , , is the corresponding angular rate on each axis of the gyroscope, is the random error; According to the following formula, the attitude matrix of the inner ring shaft coordinate system relative to the outer ring shaft coordinate system is calculated: (2) wherein is axis, is the perpendicularity of the middle ring axis and the outer ring axis, is axis, is the zero error of the middle ring axis, is the angular position error of the middle ring axis, is the nominal rotation angle of the middle ring axis; According to the following formula, the attitude matrix of the inner ring shaft coordinate system relative to the outer ring shaft coordinate system is calculated: (3) wherein is axis, is the perpendicularity of the inner ring axis to the middle ring axis, is axis, is the zero error of the inner ring axis, is the angular position error of the inner ring axis, is the nominal rotation angle of the inner ring axis; According to the following formula, the attitude matrix of the inner ring shaft coordinate system relative to the outer ring shaft coordinate system is calculated: (4) wherein , , is the mounting alignment error of the gyroscope, is the axis, is the axis, is the axis; According to the following formula, the attitude matrix of the inner ring shaft coordinate system relative to the outer ring shaft coordinate system is calculated: (5) Because the angular velocity of the earth rotation is very small, the attitude error in the three components of the gyroscope coordinate system can be ignored in the multiple transmission processes, and only the nominal attitude matrix of the gyroscope coordinate system relative to the geographical coordinate system needs to be calculated: (6) wherein is axis, is the rate of rotation of the outer ring axis, is axis, is the nominal rotation of the middle ring axis, is axis, is the nominal rotation of the inner ring axis; The step two comprises: The components of the angular velocity of the earth in the gyroscope coordinate system are where is the magnitude of the angular velocity of the earth, is the local geographic latitude, and ; the components of the uniform angular velocity of the outer ring axis in the gyroscope coordinate system are and ; then the angular velocity of the three input reference axes relative to the inertial space, which is input to the gyroscope, is (7) The three components of the above formula are calculated, and the following formula is obtained: (8) 。 2. The method of claim 1, wherein, The step three comprises: The angular velocity components are substituted into the preset error model of the gyroscope to obtain a full error model containing the three-axis turntable error and the installation and alignment error of the gyroscope: wherein, is a model coefficient independent of the angular rate.

3. The method of claim 1, wherein, The step four comprises: According to the total error model, the test plan shown in Table 1 is designed, Table 1 Test points for identifying error model coefficients related to angular rate and the output of the gyroscope, is the angular rate of the outer ring shaft for the first time of testing the outer ring shaft, is the angular rate of the outer ring shaft for the second time of testing the outer ring shaft, the angular bracket is the position number; When the middle ring shaft and the inner ring shaft of the three-axis turntable are at different angular positions, the three axes of the gyroscope realize different pointing directions relative to the angular velocity vector of the outer ring shaft, so that the angular velocity has different components on the three axes of the gyroscope, which are substituted into the full error model to identify the error model coefficient related to the angular velocity.

4. The method of claim 1, wherein, The step five comprises: The following identification method is designed: When the outer ring shaft is rotated at an angular velocity The structure matrix is In the formula: When the outer ring shaft is rotated at an angular velocity The structure matrix is In the formula: The error model coefficient related to the angular velocity is identified as wherein, is a model coefficient vector; is a structure matrix; is an output vector of the gyroscope; In the formula: wherein represents the output of the gyroscope, the first number of the superscript 1 represents the first angular rate on the outer ring axis , 2 represents the second angular rate on the outer ring axis ; the second number of the superscript represents the position number when the middle ring axis and the inner ring axis of the three-axis turntable are at different angular positions, is an error term independent of the angular rate of the gyroscope; In order to compare the influence of the identification results of the model coefficients considering the turntable error and not considering the turntable error, the following method for identifying the model coefficients without considering the turntable error is designed: When the outer ring shaft is rotated at an angular velocity only the first 10 columns of the structure matrix are considered, denoted as the structure matrix : When the outer ring shaft is rotated at an angular velocity only the first 10 columns of the structure matrix are considered, denoted as the structure matrix : The error model coefficient related to the angular velocity is identified as wherein, is a model coefficient vector to be identified; is a structure matrix; is an output vector of the gyroscope; In the formula: ； wherein represents the output of the gyroscope, the first number of the subscript 1 represents the first angular rate on the outer ring axis , 2 represents the second angular rate on the outer ring axis ; the second number of the subscript represents the position sequence number when the middle ring axis and the inner ring axis of the three-axis turntable are at different angular positions.

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