Abstract
With the increasing requirements for accuracy of pendulous integrating gyroscopic accelerometer (PIGA), the key to the development of accuracy of PIGA is the advanced calibration methods. This paper focuses on the calibration method of the main error coefficients of PIGA on the indexing head table. The precise input accelerations and angular velocities are deduced and the complete error calibration model is established based on the corresponding coordinate systems. Then, the orthogonal 4-pose calibration method of dual PIGAs is proposed to identify the harmonic term coefficients of angle error. The calibration uncertainty and efficiency of equal angle sequency and equal acceleration sequency calibration schemes are analyzed. Then, the optimal calibration method is proposed and the complete test process is designed, which can accurately and efficiently calibrate PIGA by compensating with the angle error and combining the two different sequency schemes. Simulation results show that 22-position equal acceleration sequency scheme can calibrate bias and scale factor more accurately and the test cost of equal angle sequency scheme is lower. After compensation with angle error, the magnitudes of calibration uncertainties are decreased from 10−6 to 10−7 and the maximum value of relative fitting accuracy is decreased from 6.6 × 10−5 to 2.7 × 10−5 g by the proposed 22-position optimal calibration method.
1. Introduction
The introduction should be succinct, as the key devices in inertial navigation systems, accelerometer can accurately measure the input acceleration of aircraft and motor vehicle to provide the motion and attitude information in real time [1]. It is also widely applied to robotics, identification, spots and health aspects to offer a practical method for researchers to determine position, detect tire pressure, and quantify an athlete’s workload [2–4]. In order to accurately measure acceleration, the main error coefficients of linear accelerometer including bias, scale factor, and nonlinear error terms should be calibrated before being used. The accuracy of calibration directly determines the actual accuracy and performance of accelerometer. Therefore, it is becoming more and more important to effectively improve instrument accuracy and performance by utilizing advanced calibration technology.
The calibration methods of accelerometer can be categorized into gravity field calibration, high-g calibration, and dynamic calibration [5]. Since the gravity field test is carried out on the earth surface, the related physical quantities such as local gravitational acceleration, earth rate, and geomagnetic field can be accurately measured, which can be used as a high-precision reference to calibrate the bias and scale factor of accelerometer. Although the high-g calibration and dynamic calibration can calibrate the nonlinear coefficients and testing the dynamic performance of accelerometer by utilizing centrifuge and vibration table [6, 7]. The test cost is much higher and the calibration accuracy of linear coefficients is lower than that of gravity field calibration.
Generally, high-precision accelerometers are mainly tested by the high-precision indexing head, multi-axis position turntable, tilting table, and other positioning equipment in gravity field [8–11]. The multiposition methods are widely used to calibrate the linear coefficients of accelerometers based on the least squares, Kalman filter, differential evolution (DE), and neural networks algorithms [8, 12–15]. Two-position tests are adopted to calibrate two-axial and triaxial accelerometers without precision turntable [15, 16]. After calibration and compensation with installed direction errors in the error model of accelerometers, the measurement accuracy is less than 1 mg at ± 1 g positions [17]. Xu et al. [18] proposed three-position method for calibrating the scale factor of accelerometer without considering the misalignments and the relative error is less than 0.07%. In order to improve the calibration accuracy, Xu et al. [19] and Pan et al. [20] proposed 6-position methods for triaxial accelerometer calibration on the low-cost three-axis turntable. However, 2-position and 6-position calibration methods without high-precision testing equipment cannot meet the accuracy requirement of high-precision accelerometers such as pendulous integrating gyro accelerometer (PIGA). Thus, the more test positions such as 12-position and 18-position methods are adopted to improve the magnitude of calibration accuracy to less than 10−5 [21, 22]. By using the genetic algorithm, Marinov et al. [23] proposed 24-position static calibration method to ensure the standard deviations of error coefficients are 10−5 g. The magnitude of standard deviations of error coefficients can be less than 10−6 by using a 20-position calibration method on two-axis table based on D-optimal experimental design [24]. In order to restrain the influence of angular position errors on the calibration test, a 24-position combinational measurement method is designed and the practical measurement results show that the magnitude of standard deviations is 10−6 [25].
Although the aforementioned multiposition methods can accurately calibrate accelerometers, the error sources including equipment error, installation error, and alignment error during the test are few comprehensively considered and deeply analyzed. Meanwhile, the uncertainty and efficiency of different calibration coefficients, at different positions and for different calibration schemes also should be analyzed to further improve the accuracy and test cost. Moreover, the angular velocity should be considered in the error calibration model because the angular velocity about the input axis of PIGA has a great influence on the output accuracy of PIGA.
In this paper, an optimal calibration method for gravity field testing of orthogonal PIGAs on an indexing head table is proposed to improve the calibration uncertainty and reduce the test cost. The precise error calibration model of PIGA is deduced based on the established coordinate systems. The orthogonal 4-pose calibration method of PIGAs is designed to separate and identify the harmonic term coefficients of the angular error. The uncertainty and efficiency of different calibration schemes are analyzed, and then the optimized calibration process is proposed. Thus, the main error coefficients of PIGA can be accurately and efficiently calibrated after compensation with the angle error and combination different calibration schemes. The main contributions of this paper are as follows.(1)The precise acceleration and angular velocity components of gravity and earth rate about the three input reference axes of PIGA are obtained based the calibration system. The main error sources are systematically summarized and analyzed. Then, the complete error calibration model of PIGA is deduced.(2)The orthogonal 4-pose calibration method of dual PIGAs is proposed. The harmonic term coefficients of the angle error of indexing head table can be mostly separated and identified. Thus, the error coefficients including bias, scale factor, second order, and gravity effect error can be accurately calibrated after compensation.(3)The optimized calibration method is proposed by combining the features of the two calibration schemes (equal angle sequence and equal acceleration sequence) according to the analysis of calibration uncertainty and the proposed sensitivity function. Moreover, the detailed test process including self-alignment method are designed to further improve the calibration accuracy.
The rest of the paper is organized as follows. In Section 2, the coordinate systems of calibration are established and the main error sources are analyzed. The precise inputs of acceleration and angular velocity of PIGA are calculated and the complete error calibration model of PIGA are obtained in Section 3. The orthogonal 4-pose calibration method of PIGAs is proposed in Section 4. Based on the analyses of uncertainty and efficiency of different calibration schemes, the optimal and detailed calibration process are represented in Section 5. In Section 6, the simulations are conducted to demonstrate the feasibility of the proposed method. Finally, the conclusions are drawn in Section 7.
2. Calibration System
As shown in Figure 1, the calibration system mainly includes indexing head table, fixture, and PIGA. The indexing head table is mainly composed of foundation, drive mechanism, and rotation axis. The rotation mode of rotation axis can be divided into manual driven by utilizing rotation handles as shown in Figure 1 and stepper motor driven. Compared with multi-axis turntable, the positioning accuracy of high-precision indexing head table can be less than 0.1″, namely, 5 × 10−7 rad. Thus, the magnitude of standard deviation of PIGA’s calibration can theoretically reach 10−7.
Based on the calibration system and the installation pose of PIGA, the coordinate systems in Figure 1 are established as follows:(1)Geographic coordinate system o0-x0y0z0. The three axes o0x0, o0y0, and o0z0, respectively, coincide with local horizontal east, horizontal north, and vertical upward.(2)Foundation coordinate system o1-x1y1z1. It is necessary to adjust the pose of foundation to ensure that the three axes o1x1, o1y1, and o1z1, respectively, coincide with o0x0, o0y0, and o0z0. Due to the limit to the measurement and alignment accuracy, the main error sources include horizontal errors ∆θx0 and ∆θy0, and direction error ∆θz0. Its homogeneous transformation matrix (H-matrix) with respect to geographic coordinate system can be expressed as follows:where Rot represents the rotation of attitude around the corresponding axis.(1)Rotation axis coordinate system o2-x2y2z2. It is formed by rotating angle θi around the axis o1x1. The main error sources include initial misalignment θ0, angular error ∆θ, axial wobbles αy(θi), and αz(θi). The angular error and axial wobbles are the periodic function of period 2π based on the generation mechanism of these errors. Thus, the Fourier series can be utilized to express these errors as follows:
Its H-matrix with respect to rotation axis coordinate system can be expressed as follows:(1)Fixture coordinate system o3-x3y3z3. The fixture is mounted on the top of the rotation axis. The main error sources are installation pose errors ∆θx1、∆θy1, and ∆θz1. Its H-matrix with respect to rotation axis coordinate system can be expressed as follows:(1)PIGA coordinate system o4-x4y4z4. The three reference axes of PIGA IA, PA, and OA, respectively, coincide with o4x4, o4y4, and o4z4 as shown in Figure 1. The main error sources are installation pose errors of PIGA ∆θx2, ∆θy2, and ∆θz2 about the three axes o3x3, o3y3, and o3z3. Its H-matrix with respect to rotation axis coordinate system can be expressed as follows:
According to the established coordinate systems, the main error sources of PIGA testing on the indexing head table including horizontal errors, installation pose errors, initial misalignment, and angular error are explained.
3. Error Calibration Model
PIGA is different with the linear accelerometers such as quartz accelerometer and MEMS accelerometer, it is sensitive to both acceleration and angular velocity. Thus, the precise accelerations and angular velocities about the three reference axes should be calculated first. Then, the error calibration model of PIGA on the indexing head table can be obtained.
3.1. Precise Inputs Deduction
First, the gravitational acceleration components about the three reference axes of PIGA can be calculated as follows:where is the gravitational acceleration components about IA axis, is the gravitational acceleration components about PA axis, is the gravitational acceleration components about OA axis, and is the gravitational acceleration.
The gravitational acceleration g has been unified into a constant value in our paper based on the following equation [26]:where λ is the latitude and h is the altitude. When λ = 39.94° and h = 2.7 m, the calculation result of g is 9.8016093 m/s2.
According to Equations (1)–(5), the detailed expression of Equation (6) can be calculated as follows:where , , and .
According to Equation (8), when the axial wobbles of the high-precision indexing head table are all limited within 2″, they will not affect the input acceleration along the input reference axis of PIGA. The input acceleration errors of PIGA mainly compose of horizontal error, initial misalignment θ0, angular error ∆θ, and installation pose errors.
Then, the precise angular velocity components about the three reference axes of PIGA can be calculated as follows:where is the earth rate and its value is .
Since the magnitude of the earth rate is 10−5, when the horizontal errors and direction error is less than 1.4 × 10−4 rad, that is, less than 29″, the magnitude of the error terms related to the earth rate is less than 10−8. Therefore, the detailed expressions of can be calculated as follows:
It is noted that the angular velocity component of earth rate needs to be compensated in the error calibration model of PIGA on the indexing head table. Meanwhile, it is necessary to ensure that the value of horizontal errors and direction error are restrained less than 0.5′.
Based on the previous calculations, the set values of each major error sources and the generated input errors are shown in Table 1.
The error ∆θz0 can be ignored since the input angular velocity error is less than 1 × 10−8 rad/s. The main errors affecting the input acceleration accuracy of PIGA include horizontal errors, installation errors, and initial misalignment. In order to ensure the calibration accuracy, these error terms must be suppressed, separated, and compensated. The installation error and the initial misalignment can strongly affect the input acceleration and the generated input errors are more than 1 × 10−5 g (g is unit constant of acceleration which equals to 9.8016093 m/s2). The angular error ∆θi varies with the different angular positions of rotation axis and its value could be ranged from 0.1″ to 5″ during the calibration test. Thus, it will not only affect the accuracy of input accelerations, but also twining around the main error coefficients.
3.2. Error Calibration Model of PIGA
According to the calculation results of the input accelerations and angular velocities, the input acceleration and angular velocity are too small. Thus, the error terms related to the accelerations and angular velocities about the OA axis and PA axis of PIGA can be ignored. Then, the average output angular velocity of PIGA iswhere is the test time of PIGA (s), is bias (rad/s), is scale factor (rad/s/g), is the second-order coefficient (rad/s/g2), is odd-quadratic coefficient (rad/s/g2), and is random error (rad/s).
In the gravity field, the pendulum length of PIGA varies with the different positions because the gravity will affect the position of the rotor motor’s center of mass. Thus, a new error terms are generated and the scale factor can be rewritten as follows:where is the gravity interference coefficient (rad/s/g2).
Finally, the complete error calibration model of PIGA on the indexing head table is
Although the gravity effect will affect the test accuracy of PIGA in the gravity field, the influence of gravity effect on pendulum length can be greatly improved by updating the design and manufacturing process. It is noted that a stable storage environment and pose should be ensured to restrain the impact on the pendulum length.
When the initial misalignment and the pose error are less than 30″, the input error is less than 1 × 10−7 g. Thus, the corresponding error terms could be ignored. Then, according to Equation (12), the error calibration model can be simplified as follows:
By only considering with the first three order of the expression of ∆θi in Equation (2), that is,
Equation (14) can be rewritten when the angle position of rotation axis is θi () as follows
Obviously, the common multiposition calibration method cannot accurately and directly calibrate PIGA on the indexing head table. The harmonic terms of angular error caused by the indexing positioning, such as and , cannot be separated and will affect the calibration results of the coefficients in the PIGA’s error model.
4. Error Calibration Model
In order to solve the aforementioned problems, the orthogonal poses method of PIGA calibration is proposed in this section. Pose 1 is the initial installation pose of PIGA A as shown in Figure 1 (the average output angular velocity of PIGA is ), Pose 2 of PIGA B is installed as shown in Figure 2 (the average output angular velocity of PIGA is ).
According to the calculation method in Section 3, the error calibration model of PIGA A in Pose 1 is same with the Equation (16), and the error calibration model of PIGA B in Pose 2 is
In order to identify the main error coefficients in the error calibration model, the trigonometric calculation method by combining Equations (16) and (17) is given as follows:
Thus, the expression of can be obtained as follows:
The Equation (19) can be rewritten as a matrix form as follows:where
Then, the least square method can be used to identify coefficients in Equation (20):
When the identification results of and are and , the scale factors of PIGA A and PIGA B can be obtained as follows:
Compared with Equations (16) and (17), it is obviously shown that the scale factors of PIGAs can be accurately and directly calibrated by Equation (27). Moreover, the angle errors related to the scale factors are automatically separated.
However, the other error terms in error model of PIGA still cannot be identified by the orthogonal dual-pose calibration method. In order to identified the angle error coefficients, the function is constructed as follows based on the identification results of scale factor of PIGAs in Equation (28) as follows:
Then, the function of and can be given as follows:
The expressions of Equations (30) and (31) can be combined and rewritten as a matrix form as follows:where
In order to improve the calibration uncertainties of the harmonic term coefficients of error angle in the error calibration model, the third and fourth installation poses of PIGAs is designed as shown in Figure 3.
Then, the function of and can be given as follows:where and is the average output angular velocities of PIGA A and PIGA B in the Pose 4 and 3, respectively.
Finally, the expressions of Equations (30), (31), (40), and (41) can be combined and rewritten as a matrix form as follows:where the calculation processes and detailed expressions of , and are similar to Equation (32).
According to the Equation (42), the error coefficients of angle error , and can be accurately identified by LS method for the orthogonal 4-pose calibration testing of dual PIGAs.
After compensation with these errors in the error calibration model of PIGA, the other main error coefficients of PIGA can be calibrated by LS method as follows:where
is the random matrix,
and are the identification results by the orthogonal 4-pose calibration method.
The least squares estimation of iswhere is the information matrix.
Then, the residual matrix r and the value of standard deviation can be obtained as follows:
Based on the definition of Type A evaluation of measurement uncertainty in [27], the calibration uncertainty of the main error coefficients can be expressed as the standard deviation of the main error coefficients. Thus, the calibration uncertainties of the main error coefficients of PIGA can be calculated as follows:where () is the main diagonal element of matrix .
According to Equations (47) and (49), It should be noted that the calibration uncertainty could be decreased with the increasing of the number of test position n. However, too many test positions will increase the test cost and reduce the test efficiency. Moreover, the arrangement of test positions will also affect the value of and . Therefore, it is necessary to optimize the calibration process and make reasonable arrangement of the test positions to meet the requirement of calibration uncertainty and test efficiency.
5. Error Calibration Model
At present, the common calibration schemes of accelerometer test on the indexing head table, turntable, and centrifuge are equal angle sequence calibration scheme and equal acceleration sequence calibration scheme.
The equal angle sequence calibration scheme is that the rotation angles of the main rotation axis of indexing head table between the two adjacent test positions of PIGA are equal during the calibration testing. Namely, the expression of () can be given as .
The equal acceleration sequence calibration scheme is that the absolute difference value of the input acceleration of PIGA between the two adjacent test positions are equal during the calibration testing. Namely, , when PIGA is installed as shown in Figure 1 and the angle position of the rotation axis is (, m > 1). Thus, the expression of can be given as follows:
In order to calibrate PIGA more efficiently and accurately, it is necessary to analyze calibration uncertainty of the main error coefficients of PIGA in different test positions and different test schemes.
5.1. Uncertainty Analysis of The Two Calibration Schemes
According to the computational equations of calibration uncertainty Equations (48) and (49), the value of for the two different calibration schemes can be obtained as shown in Figure 4. It is shown that there is a certain difference in the calculation results of by using the two calibration schemes. However, with the increasing of the number of test positions, the difference between them becomes smaller.
(a)
(b)
(c)
(d)
(e)
(f)
As shown in Figure 4(a), the calculation results of d11 for equal acceleration sequence calibration are all less than those for equal angle sequence calibration.
It is demonstrated that the bias of PIGA can be calibrated more accurately by adopting the equal acceleration sequence calibration. As shown in Figure 4(b), the calculation results of d22 show that the equal acceleration sequence calibration scheme can identify more accurately when i > 6. In addition, the value of d22 of 7 and 8 positions calibration are larger than that of 6 positions for equal acceleration sequence calibration. It is indicated that 7-position and 8-position equal angle sequence scheme are not suitable for calibrating . The calculation results of d33 are similar to those of d11 as shown in the Figure 4(c). As shown in Figure 4(d), when the number of test position is less than 9, can be calibrated more accurately by using equal acceleration sequence scheme. When the number of test position is more than 9, the equal angle sequence calibration scheme is better. The calculation results of d55 are similar to those of d22 when the number of test position is less than 20, as shown in the Figure 4(e). However, when the number of test position is more than 20, the values of d66 is larger by using equal acceleration sequence scheme. It is shown that can be calibrated more accurately for equal angle sequence calibration. As shown in Figure 4(f), when the number of test position is less than 18, can be calibrated more accurately by using equal angle sequence calibration scheme. When the number of test position is more than 18, the difference of d66 between the two sequence calibration schemes are small.
5.2. Efficiency Analysis of The Two Calibration Schemes
At present, the D-optimal algorithm is used to design the calibration test of accelerometers. Actually, the optimal design based on D-optimal algorithm is the optimization problem of the information matrix Q in Equation (47). In order to accurately identify the coefficients of PIGA by using the LS method, the matrix Q should be a full rank matrix and the rank of Q should be larger than the number of coefficients, which need to be identified. Thus, the value of the determinant of Q cannot be zero, that is, . Since the larger value of means the smaller calibration uncertainty, the structure of Q should be considered. Although the more test positions will effectively increase the value of , the test efficiency will decrease simultaneously. Thus, the sensitivity function is constructed to evaluate the sensitivity of number increment of test position:
The value of ES(n) of two different calibration schemes are calculated as shown in Figure 5. The calculation results of equal angle sequence scheme show that sharply increase with the increase of the value of n when 6 < n ≤ 8, and the maximum value of is 0.476 when n = 8. Then, the value of begins to decrease when n > 9. It means that the influence of the increment of n on the calibration uncertainty has become less and less. Meanwhile, the sensitivity is decreasingly lower when n > 31 (< 0.02). The calculation results of equal acceleration sequence scheme show that sharply decrease with the increase of the value of n when 6 < n ≤ 24. Similarly, the value of equal acceleration sequence scheme decrease below 0.02 when n > 50. It is indicated that both the calibration uncertainty and test efficiency should be taken into account for the design of calibration.
In addition to ensuring the calibration uncertainty and test efficiency, it is also necessary to consider the test time and cost. Thus, the simulations are conducted to analysis the test cost. The main simulation parameters of PIGA are designed as follows: = 4 × 10−4rad/s, = 0.55 rad/s/g, = 5 × 10−6 rad/s/g2, = 3 × 10−6 rad/s/g2, and = 3 × 10−6/g. When the precession angle of PIGA is 10 π, the total test time ttotal of two different calibration schemes are shown in Figure 6.
As shown in Figure 6, the variation tendencies of ttotal are severely fluctuant with the increase of n. When the number of test position is 4k (), the total test time of both two calibration schemes exceed 2,500 min. The main reason is that the precession angular velocities of PIGA are quite small when . According to Equation (14), the output of PIGA can be approximately estimated as follows:
The average precession angular velocities of PIGA at these test positions are all less than 5 × 10−4 rad/s. Then, the test time are all exceed 1,000 min at these positions. Obviously, it is not suitable for the efficient calibration of PIGA, nor can it accurately calculate the average precession angular velocity of PIGA. Therefore, the 4k-position () calibration method should be excluded for calibrating PIGA on the indexing head table.
When the number of test position is , the simulation results of ttotal are shown in Figure 7. The growth trends of ttotal of the two calibration schemes are generally consistent. However, the ttotal of equal angle sequency schemes still presents fluctuation. The main reason is that there are some test positions close to the angular position and which the input accelerations about the input axis of PIGA are approximately equal to zero. As shown in Figure 7, the test time of equal angle sequency calibration scheme is shorter than that of the equal acceleration sequency calibration scheme. It is means that the test cost of equal angle sequency calibration scheme is lower than that of equal acceleration sequency calibration scheme.
The advantages of two sequence calibration schemes are listed as shown in Table 2. In summary, when the number of test position is less than 20, the bias and scale factor of PIGA can be calibrated more accurately by using equal acceleration sequence calibration scheme, and the nonlinear error coefficients of PIGA can be calibrated more accurately by using equal angle sequence calibration scheme. When the number of test position is more than 40, the equal angle sequence calibration scheme can both ensure the calibration accuracy and improve the test efficiency.
5.3. Optimized Calibration Method
Based on the aforementioned analyses, the optimized calibration process of orthogonal 4-pose of PIGAs on the indexing head table is designed as shown in Figure 8.
The complete calibration process includes the following detailed steps:(1)The two PIGAs are installed on the fixture as shown in Figure 2.(2)According to the error analysis of θ0 in Table 1 and the error calibration model Equation (13), the misalignment and the installation errors should be restrained less than 30ʺ. Thus, the alignment test is designed before calibration test. Without considering the nonlinear error terms in the error calibration model, the alignment model of PIGA A of Pose 1 can be obtained as follows:
When = 90° and 270°, the outputs of PIGA are
Then, the error terms can be estimated as follows:where is the initial value of scale factor of PIGA A.
Finally, can be repeatedly adjusted and calculated according to the calculation results of Equation (55) until the calculated misalignment meets the requirement of test accuracy (<30ʺ).
It is assumed that the uncertainty of average output angular velocity of PIGA is = 6.8 × 10−6 rad/s, the measurement uncertainty of the latitude is = 0.1°, the uncertainty of is = 10−4 rad/s/g. The uncertainty of by using the proposed self-alignment method is calculated as follows:
It is shown that the misalignment and installation error can be effectively suppressed by the propose self-alignment test without other test equipment. Thus, the uncertainty of the final misalignment completely depends on the accuracy of PIGA itself.(1)According to the advantage analysis of two calibration schemes, the equal acceleration sequence calibration scheme is used to accurately identify the scale factors of the two orthogonal PIGAs. According to Equations (17)–(27), the scale factors could be identified at Pose 1 and Pose 2.(2)After the calibration test of PIGAs in Pose 1 and Pose 2, the two PIGAs are installed in Pose 3 and Pose 4 on the fixture as shown in Figure 3 and the misalignment is adjusted by using the proposed self-alignment method.(3)According to the advantage analysis of two calibration schemes, the equal angle sequence calibration scheme is used to accurately identify the angle error coefficients. According to Equation (42), the coefficients , and can be identified by the orthogonal installation poses tests.(4)In order to calibrate the coefficients of PIGA, the pose of PIGA A should be adjusted to the initial pose which . The parameters relate to the calibration test should be designed such as the number of test position and PIGA’s precession period.(5)The data acquisition system mainly collects the test time and the output pulses of PIGA . Then, the average output angular velocity of PIGA is calculated by counting the number of output pulses: .(6)The main rotation axis rotates to the next test position when the number of reaches set value.(7)After completing all position tests of PIGA, the angle error coefficients , and should be compensated into the error calibration model as shown in Equation (43).(8)Then, the error coefficients of PIGA are calibrated by LS method and the uncertainties of these coefficients are calculation according to Equation (48).(9)If the calibration uncertainties meet the requirement of calibration accuracy, the calibration test of PIGA A is over, or the calibration test should be reset.
Summarily, the orthogonal 4-pose of dual PIGAs is designed to accurately identify the harmonic coefficients of angle error first. After self-alignment test, the two calibration schemes are flexibly utilized to improve the calibration uncertainties of these error coefficients based on the analyses of uncertainty and efficiency. Then, the multiposition calibration tests are adopted to calibrate the main error coefficients of PIGA. Meanwhile, the number of test position of PIGA are optimized. After compensation with the angle error, the calibration uncertainty of these coefficients can be further improved.
6. Simulations
In order to verify the validity of the proposed calibration method and further analyze the effects of different influence factors on the calibration result. The simulations are constructed and the main parameters are designed as follows: = 4 × 10−4 rad/s, = 0.55 rad/s/g, = 5 × 10−6 rad/s/g2, = 3 × 10−6 rad/s/g2, = 3 × 10−6/g, = 2 × 10−5, = 2 × 10−5, = 1 × 10−5, = 2 × 10−5, = 1 × 10−5, = 5 × 10−6, and = 5 × 10−6, and the magnitude of random error is 10−6.
The 10-position (n = 10) and 22-position (n = 22) calibration results of equal angle sequency scheme and equal acceleration sequency scheme without compensating with angle error are given in Tables 3 and 4.
Compared with the simulation results of equal angle sequency calibration for 10-position and 22-position test in Table 3, the absolute errors and calibration uncertainties of these error coefficients of PIGA are significantly decreased by 22-position calibration. It is verified that the main coefficients of PIGA can be calibrated more accurately by a large number of n.
Although the magnitudes of the calibration uncertainties reach 10−6, the magnitudes of the absolute errors of and are over 1.0 × 10−6. Thus, the relative errors of the second-order error coefficients are greater than 40%. It is illustrated that the multiposition calibration method without optimization and compensation cannot accurately calibrate the high-order error terms of PIGA due to the interferences from angle error as shown in Equation (16).
Compared with the simulation results of equal acceleration sequency calibration for 10-position and 22-position test in Table 4, the calibration uncertainty of is improved from 1.4 × 10−6 to 9.3 × 10−7 rad/s by increasing the number of test position. Compared with the calibration results of and in Tables 3 and 4, it is shown that the equal acceleration sequency scheme can calibrate these error coefficients more accurately. Comparatively, the second-order error coefficients can be calibrated more accurately by using the equal angle sequency scheme. The maximum value of the calibration uncertainties reaches 6.6 × 10−6 rad/s/g2 as shown in Table 4. Thus, the relative error of are greater than 200% by using the equal acceleration sequency scheme. Meanwhile, compared with the value of ttotal in Tables 3 and 4, it is shown that the total test time of equal acceleration sequency calibration test is longer for the same number of test position, which means that the test cost of equal angle sequency scheme is lower.
The residual errors of the two different sequency calibration schemes for 22-position test are shown in Figure 9. The difference between two calibration schemes is small and the value of residuals range from −5 × 10−6 to 5 × 10−6 rad/s. It is illustrated that the error calibration model without compensation with angle error terms cannot accurately fit the actual input–output relationship of PIGA because the residuals contain the related components of angle error terms.
In order to further analyze the calibration performance, the relative fitting accuracy can be calculated as follows:
The calculation results of relative fitting accuracy of two different sequency calibration schemes for 22-position test are shown in Figure 10. It is shown that most of are less than 5 × 10−6 g. However, when i = 6, 7, 17, and 18, the value of is greater than 3 × 10−6 g. The main reason is that the error terms in the error calibration of PIGA cannot be fully excited because the input accelerations of PIGA are less than 0.15 g at these test positions.
Simulation results of the two calibration schemes are shown in Table 5, when the number of test position is increased to 42. It is shown that there is little difference between the two different sequency calibration schemes in the calibration uncertainties of PIGA error coefficients. However, the total test time is shortened nearly 1 hr by using equal angle sequence calibration scheme. Therefore, when the number of test position is over 40, the equal angle sequence calibration scheme can ensure the calibration uncertainty of PIGA and effectively improve the test efficiency and reduce the test cost.
According to the above simulation results, it is verified that bias and scale factor of PIGA can be calibrated more accurately by the equal angle sequence calibration test. Then, the nonlinear error coefficients of PIGA can be calibrated more accurately and the test efficiency could be improved by the equal angle sequence calibration test. Meanwhile, too many test positions will significantly increase the test cost but have limited influence on calibration uncertainty. Moreover, in order to improve the calibration uncertainties of nonlinear error coefficients and the residuals, the harmonic terms of angle error must be compensated in the error calibration model of PIGA.
Thus, according to the proposed orthogonal calibration method of PIGAs, the harmonic coefficients of angle error should be identified by the 4-pose test. The identification results of the angle error harmonic coefficients of two different sequency calibration schemes for 22-position test are shown in Table 6. Compared with the calibration results of in Tables 3 and 4, the calibration uncertainties of scale factors of PIGA A and B are obviously decreased by the orthogonal 4-pose calibration testing. As expected, the harmonic term coefficients of angle error , and can be accurately identified by LS method. The magnitudes of calibration uncertainty of these coefficients are all less than 5.0 × 10−6.
Compared with the calibration uncertainties of different sequency schemes in Table 6, it is noted that equal acceleration sequence calibration scheme should be used to identify the scale factors when PIAGs are installed in Pose 1 and Pose 2. Then, in order to improve the calibration uncertainties of harmonic coefficients, the equal angle sequence calibration scheme should be used when PIGAs are installed in Pose 3 and Pose 4.
According to Equation (31), the main error coefficients of PIGA can be calibrated after compensating with the harmonic terms of angle error. Compared with the calibration results in Tables 3 and 4, the magnitudes of calibration uncertainty of , and are significantly decreased from 10−6 to 10−7 as shown in Table 7. Similarly, the absolute errors of these error coefficients are significantly decreased. Especially, the relative errors of the nonlinear error coefficients of and are decreased from 40% to 10%.
The calculation results of relative fitting accuracy of two different sequency calibration schemes for 22-position test are shown in Figure 11. It is shown that the relative fitting accuracies are significantly decreased by compensating with the angle error, the maximum value of has been reduced from 6.6 × 10−5 to 2.7 × 10−5 g, which means that the error calibration model Equation (43) can describe the input/output relationship more adequately.
7. Conclusions
This paper has proposed an optimal calibration method for calibrating PIGA’s main error coefficients by compensating with the harmonic terms of angle error. The precise error calibration model of PIGA is deduced based on the precise input accelerations and angular velocities of PIGA. The orthogonal 4-pose calibration method of the harmonic error coefficients are proposed by utilizing dual PIGAs test to identify the harmonic term coefficients of angle error. Thus, the main error coefficients of PIGA can be more precisely calibrated by compensating with these harmonic error terms. The optimized calibration processes are designed based on analyses of calibration uncertainty and efficiency for the equal angle sequency and the equal acceleration sequency calibration scheme. Thus, the calibration uncertainty can be further decreased. The simulation results show that the harmonic term coefficients of angle error can be accurately identified by the proposed orthogonal 4-pose calibration method. The proposed 22-position calibration method can calibrate the main error coefficients of PIGA more accurately and efficiently with compensation the angle error terms. Moreover, the calibration uncertainties and test cost can be further improved by combining the two different sequency schemes.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by baseline project of integrated management platform for smart industrial and commercial enterprises funded by Dahua Zhilian Information Technology Co. Ltd (DH3.RDA000561).