Azimuth information processing optimization method for a magnetically suspended gyroscope

Abstract

A north-seeking data quality optimization method for a magnetically suspended gyroscope is proposed based on two-position characteristics combined with the north-seeking characteristic principle of the magnetically suspended gyroscope total station theodolite. First, a simple linear regression model is established based on the linear relationship satisfied by the rotor current of two precise seeking positions and the north-seeking azimuth. Then, the confidence interval of the corresponding rotor current is calculated from the a priori value of the north-seeking azimuth, and the rotor current data outside the confidence interval are removed. Lastly, the gyroscope azimuth is recalculated. Engineering test data were used to test the effectiveness of the method, and the results showed that the method is effective at testing the north-seeking data quality of the gyroscope and correcting directional results.

Keywords

Magnetically suspended gyroscope data removal quality optimization

1 Introduction

A gyroscope is a direction-sensitive azimuth measurement instrument that relies on the Earth’s rotation. It is widely used in many large-scale mines, subways, tunnels, and other underground structures. The magnetically suspended gyroscope is a newly developed north-seeking instrument that uses magnetic bearing technology to replace the suspension strap support technology widely used in surveying gyroscopes. In addition, a series of key technologies, such as multi-position successive north-seeking technology, two-position rotary precise north-seeking technology, and massive data collection technology, have been developed to achieve high-precision and fully automated north-seeking directional measurement [1 –3].

However, in some extreme engineering environments, strong interference torque results in a great deal of noise in sampling data, which undermines the north-seeking stability of the magnetically suspended gyroscope. Figure 1 illustrates the influence of strong disturbance torque in measured data from six survey stations. Determining how to correct noise data effectively and optimize the quality of north-seeking data is critical to further improving the north-seeking stability and environmental adaptability of the magnetically suspended gyroscope total station theodolite system.

Fig.1

Time sequence diagrams of rotor sampling data with strong disturbance torque.

1.1 North-seeking principle

When the sensitive part of a gyroscope is in a suspended state, it generates a directive moment M that approaches the meridian due to the effect of the Earth’s rotation: $M = H \times ω_{e} cos φ sin α,$ where H is the moment of inertia of the gyroscope, ω_e is the angular velocity of the Earth’s rotation, φ is the geographic latitude, and α is the angle between the rotational axle of the gyroscope and the meridian direction.

As Fig. 2 shows, the suspension point support causes the suspension strap gyroscope to oscillate around the meridian direction under the directive moment. Figure 3 illustrates the time sequence curve of the gyroscope oscillation. A series of oscillatory north-seeking measurement methods, such as the reversal point, transit, difference, and integration methods, have been based on this time sequence curve [4 –6].

Fig.2

Force analysis diagram of a suspension strap gyroscope.

Fig.3

Oscillation curve of a suspension strap gyroscope in the ideal state.

However, a magnetically suspended gyroscope removes the constraint of the support at the point of suspension. In terms of the north-seeking mode, it no longer uses oscillatory north-seeking but instead applies a balance torque M′ to the sensitive portion of the gyroscope. This balance torque is equal in magnitude to and in the opposite direction of the directive moment, so that M′ = - M. In this state, the rotating axle of the gyroscope remains stationary relative to the Earth’s meridian. At this time, the directive moment can be measured by determine the current value inside the stator and rotor of the torquer: $M^{'} = k \cdot I_{R} \cdot I_{S},$ where k is a constant, I_R is the rotor current value of the torquer, and I_S is the stator current value of the torquer. The rotor current is determined from the analogue signal volume and recorded as a numerical value obtained from analogue-to-digital conversion. At the position of moment equilibrium, the measurement lasts approximately 100 s, which means that 20,000 stator and rotor current values can be collected. The following equation can be obtained from Equations (1 and 2) [2, 3]: $α = \arcsin \frac{{kI}_{R} I_{S}}{H ω_{e} cos φ} .$

1.2 Two-position north-seeking technology

In one round of north-seeking measurement, the system rotates 180° after it completes a precise north-seeking measurement at one position and then conducts a north-seeking measurement in the opposite direction to collect the corresponding stator and rotor data. As Fig. 4 shows, O is the vertical axis center of the gyroscope total station theodolite, OR is the fixed axis orientation of the internal gyroscope and zero position of the total station horizontal circle, ON is the true north azimuth, ON₁ and ON₂ are the two coarse north-seeking positions, ON₃ is the first precise north-seeking position, ON₄ is the gyroscope rotary position and second precise north-seeking position, and ∠NON₃ is the α angle. According to Equation (1), the observation data collected from the two precise north-seeking positions are equal in magnitude and opposite in sign. The differential method can eliminate the majority of the system errors and accurately calculate the north declination of the rotation axis of the gyroscope: $α = \arcsin \frac{{kI}_{R 3} I_{S 3} - {kI}_{R 4} I_{S 4}}{2 H ω_{e} cos φ} .$

Fig.4

Illustration of the two-position north-seeking principle of a magnetically suspended gyroscope.

On this basis, the true north azimuth value of ON and ∠NOR (i.e., the north azimuth angle of the zero position of the total station theodolite) can be determined from the azimuth value of ON₃, as shown in Equation (5). From there, the north azimuth angle [2, 3] of the sides can be determined from the total station targeting direction: $\begin{matrix} ∠ NOR & = & (OR - {ON}_{3}) \\ + [\frac{{ON}_{3} - ({ON}_{4} \pm 180^{\circ})}{2} + k \\ \times arcsin (\frac{M_{ON 3} - M_{ON 4}}{H \times ω_{e} cos φ})] . \end{matrix}$

2 Results

2.1 Characteristics of magnetically suspended gyroscope data

First, the north-seeking data characteristics of one position are considered. For the stator and rotor of the torquer, because the rotor is fixed to the bottom of the sensitive portion of the gyroscope, the internal current changes with the suspension state of the sensitive portion. Thus, it continuously reflects the situation of the sensitive portion of the gyroscope, which is affected by the disturbance torque. Because the stator of the torquer is fixed to the sensitive case and does not rotate, its internal current should remain constant. Based on the north-seeking principle of a magnetically suspended gyroscope, after the system completes the closing process, the sensitive portion of the gyroscope stays at the normal position of the reflecting prism in the rotation system. In the ideal state, the time sequence diagram of the current value for the stator and rotor should be a straight horizontal line. Because of the characteristics of the electronic components and current in the internal system, however, even in the absence of a significant disturbance torque impact, the stator and rotor sampling data also have some unavoidable random high-frequency noise, including error associated with the conversion of the analogue signal to a digital signal, error caused by instability of the current and voltage, and torquer measurement error. Therefore, the time sequence diagram of the sampling data exhibits a certain bandwidth distribution. Figure 5(a) and (b) show the time sequence diagrams of the sampling rotor and stator data from the gyroscope at a certain position in the ideal state in a laboratory. The width of the sampling data band reflects the stability of the data acquisition system.

Fig.5

Time sequence diagram of stator sampling data (a, b).

Figure 6 are histograms of the above stator sampling data. Hypothesis testing showed that when the influence of the disturbance torque is not significant, both the stator data and the rotor data of the torquer are normally distributed. Through parameter estimation, the mean of the rotor sampling data distribution was found to be 3.971×10^–5, with a 95% confidence interval of [3.960×10^–5, 3.982×10^–5]. The variance was found to be 8.1171×10^–6, with a 95% confidence interval of [8.038×10^–6, 8.197×10^–6]. The mean of the stator sampling data distribution was found to be –0.06323A, with a 95% confidence interval of [–0.0633, –0.0632]. The variance was found to be 7.4161×10^–4, with a 95% confidence interval of [7.344×10^–4, 7.490×10^–4].

Fig.6

Torquer stator sampling data histograms.

A number of similar statistical experiments showed that, in a laboratory and under ideal conditions or when the disturbance torque has no significant effect, the stator and rotor sampling data of the torquer are normally distributed. Based on this property, the influence of white noise in the data can be minimized by taking the average of a large amount of data. In practical applications, however, such ideal measurements are sometimes difficult to achieve.

The rotor and stator sampling data characteristics under ideal laboratory conditions and strong disturbance torque conditions were compared to assess the degree of environmental sensitivity of the rotor and stator of the torquer. Independent experiments were conducted three times for each condition, and statistical analyses were carried out to determine the distributions, means, and dispersion of the sampling data. Table 1 presents the results.

Table 1

Comparison of torquer stator and rotor sampling data under ideal state and disturbance torque impact conditions

Ideal laboratory state
Torque stator				Torque rotor
No.	Data distribution	Average value	Variance (×10^–4)	No.	Data Distribution	Average value (×10^–5)	Variance (×10^–6)
1	Normal	–0.0633	7.4161	1	Normal	–0.26735	9.1331
2	Normal	–0.0632	7.4615	2	Normal	–0.20895	8.6001
3	Normal	–0.0632	7.3910	3	Normal	3.9710	8.1171
Strong interference of disturbance torque
Torque stator				Torque rotor
No.	Data distribution	Average value	Variance (×10^–4)	No.	Data distribution	Average value (×10^–5)	Variance (×10^–4)
4	Normal	–0.0632	8.9771	4	Non-normal	–2.66 12	2.1273
5	Normal	–0.0632	9.1699	5	Non-normal	–4.0909	2.3109
6	Normal	–0.0632	11.0000	6	Non-normal	–2.7011	3.1237

The torquer rotor was found to be more sensitive to the external environment than the torquer stator. A strong disturbance torque had a slight effect on the dispersion of the stator data but did not significantly affect the results. There was no change in the normally distributed stator data, and the average value was always maintained at a stable value. However, the disturbance torque had a more obvious effect on the rotor or the torquer. It not only changed the distribution pattern of the rotor data but also significantly increased the degree of dispersion of the data. This indicates that the torquer rotor has strong sensitivity to interference from the external environment and can accurately reflect the north-seeking state of the sensitive portion of the gyroscope. Therefore, the impact of the external disturbance torque on the rotor current data can be concluded to be an important factor that undermines the north-seeking precision of the magnetically suspended gyroscope. Rotor data optimization is one of the most effective methods for improving the north-seeking stability of a magnetically suspended gyroscope [3 , 7–12].

2.2 Linear regression analysis of magnetically suspended gyroscope data

Because the angle between the equilibrium rest position of the north-seeking gyroscope and the radial direction is α < 10°, Equation (3) can be simplified to the following: $α = \frac{{kI}_{S}}{H ω_{e} cos φ} I_{R} .$

Because the angular momentum of the gyroscope H, the angular velocity ω_e of Earth’s rotation, the measurement station latitude φ, the coefficient k, and the stator current value I_s are all constant, the angle between the equilibrium rest position of the north-seeking gyroscope and the radial direction α can be inferred to linearly related to the rotor current value I_R in the ideal state.

Two-position sampling data of a magnetically suspended gyroscope at different orientations were experimentally investigated. The system-sensitive directive moment was obtained through a large number of repeated observations of the internal current value of the torque rotor (20,000 sets of current values were acquired for each north-seeking position). Twenty groups of different setup orientation conditions in the interval (N - 10°, N + 10°) were chosen for analysis of the north-seeking rotor current data for the two precise positions. For each dataset, the average values of the sampling data for the two precise positions ${\bar{I}}_{1}$ and ${\bar{I}}_{2}$ were calculated, and the overall average value of each north-seeking position $\bar{I} = \frac{{\bar{I}}_{1} + {\bar{I}}_{2}}{2}$ and the azimuth values β₁ and β₂ of the precise positions 1 and 2 were obtained based on the horizontal angle measurement system. Table 2 presents the results.

Table 2
Two-position north-seeking sampling data statistics

No. α _T ${\bar{I}}_{1}$ ×10^–5A ${\bar{I}}_{2}$ ×10^–5A $\bar{I}$ ×10^–5A β ₁ β ₂

° ′ ″ ° ′ ″ ° ′ ″

1 –7 03 11 –4.34 7.71 1.68 17 19 54 197 20 03

2 –6 21 16 –4.07 7.51 1.72 16 37 24 196 37 23

3 –5 15 00 –3.40 6.81 1.71 15 29 17 195 29 09

4 –3 51 50 –2.30 5.71 1.70 14 03 00 194 02 57

5 –2 24 24 –1.81 5.19 1.69 12 34 12 192 34 06

6 –0 50 49 –0.93 4.34 1.71 10 58 13 190 58 06

7 1 09 59 0.35 3.06 1.71 8 53 49 188 53 44

8 1 10 02 0.34 3.11 1.73 8 53 54 188 53 45

9 3 16 30 1.31 2.04 1.67 6 44 37 186 44 26

10 5 16 53 2.38 1.07 1.73 4 41 11 184 41 25

11 7 29 45 3.88 –0.43 1.72 2 24 17 182 24 14

12 8 59 18 5.07 –1.92 1.57 0 50 56 180 50 58

13 7 59 35 4.34 –1.05 1.65 1 52 57 181 52 51

14 6 00 53 3.36 –0.08 1.64 3 54 24 183 54 15

15 3 29 46 1.60 1.90 1.75 6 30 35 186 30 44

16 1 50 28 1.05 2.25 1.65 8 11 09 188 11 16

17 0 08 28 –0.33 3.65 1.66 9 57 07 189 57 02

18 –1 04 58 –0.89 4.26 1.69 11 12 10 191 12 05

19 –2 54 15 –2.11 5.53 1.71 13 04 52 193 04 52

20 –5 47 50 –3.50 6.95 1.73 16 02 24 196 02 18

21 –6 35 20 –4.17 7.62 1.72 16 52 27 196 52 30

22 –4 35 01 –3.00 6.41 1.71 14 40 52 194 40 49

23 –2 34 41 –1.87 5.26 1.69 12 39 41 192 39 46

24 –0 34 22 –9.30 4.34 1.71 10 30 27 190 30 35

25 1 25 56 0.45 3.01 1.73 9 04 11 189 04 20

26 3 26 15 1.48 2.02 1.75 6 23 57 186 24 05

27 5 26 35 2.45 1.14 1.73 4 56 30 184 56 42

28 7 26 54 3.67 –0.23 1.72 2 18 32 182 18 45

29 4 25 30 1.85 1.57 1.71 5 52 38 185 52 51

30 5 20 32 2.42 1.00 1.70 4 45 28 184 45 40

No.	α _T	${\bar{I}}_{1}$ ×10^–5A	${\bar{I}}_{2}$ ×10^–5A	$\bar{I}$ ×10^–5A	β ₁	β ₂
1	–7	03	11	–4.34	7.71	1.68	17	19	54	197	20	03
2	–6	21	16	–4.07	7.51	1.72	16	37	24	196	37	23
3	–5	15	00	–3.40	6.81	1.71	15	29	17	195	29	09
4	–3	51	50	–2.30	5.71	1.70	14	03	00	194	02	57
5	–2	24	24	–1.81	5.19	1.69	12	34	12	192	34	06
6	–0	50	49	–0.93	4.34	1.71	10	58	13	190	58	06
7	1	09	59	0.35	3.06	1.71	8	53	49	188	53	44
8	1	10	02	0.34	3.11	1.73	8	53	54	188	53	45
9	3	16	30	1.31	2.04	1.67	6	44	37	186	44	26
10	5	16	53	2.38	1.07	1.73	4	41	11	184	41	25
11	7	29	45	3.88	–0.43	1.72	2	24	17	182	24	14
12	8	59	18	5.07	–1.92	1.57	0	50	56	180	50	58
13	7	59	35	4.34	–1.05	1.65	1	52	57	181	52	51
14	6	00	53	3.36	–0.08	1.64	3	54	24	183	54	15
15	3	29	46	1.60	1.90	1.75	6	30	35	186	30	44
16	1	50	28	1.05	2.25	1.65	8	11	09	188	11	16
17	0	08	28	–0.33	3.65	1.66	9	57	07	189	57	02
18	–1	04	58	–0.89	4.26	1.69	11	12	10	191	12	05
19	–2	54	15	–2.11	5.53	1.71	13	04	52	193	04	52
20	–5	47	50	–3.50	6.95	1.73	16	02	24	196	02	18
21	–6	35	20	–4.17	7.62	1.72	16	52	27	196	52	30
22	–4	35	01	–3.00	6.41	1.71	14	40	52	194	40	49
23	–2	34	41	–1.87	5.26	1.69	12	39	41	192	39	46
24	–0	34	22	–9.30	4.34	1.71	10	30	27	190	30	35
25	1	25	56	0.45	3.01	1.73	9	04	11	189	04	20
26	3	26	15	1.48	2.02	1.75	6	23	57	186	24	05
27	5	26	35	2.45	1.14	1.73	4	56	30	184	56	42
28	7	26	54	3.67	–0.23	1.72	2	18	32	182	18	45
29	4	25	30	1.85	1.57	1.71	5	52	38	185	52	51
30	5	20	32	2.42	1.00	1.70	4	45	28	184	45	40

When the gyroscope setup angle was changed from west to east, the means of the sampling data from the two precise positions both exhibited significant changes in regularity: ${\bar{I}}_{1}$ changed from negative to positive and gradually increased and ${\bar{I}}_{2}$ changed from positive to negative and gradually decreased, but the mean value $\bar{I}$ did not exhibit significant changes in its trend. In-depth quantitative analysis of this experimental dataset was conducted as follows.

Two straight lines were fitted with the gyroscope setup azimuth as the independent variable and the arithmetic mean of the sampling current value of the two precise positions as the dependent variable, as shown in Fig. 7.

Fig.7

Linear fitting of the instrument erection azimuth and the mean current value.

Regression equations were established to solve for the rotor current value and gyroscope setup azimuth based on the fitted lines: $\begin{matrix} y = (3 . 297 \times 10^{- 4}) x - 0 . 32515 \times 10^{- 5} \\ y = (- 3 . 3376 \times 10^{- 4}) x + 3 . 7097 \times 10^{- 5} \end{matrix}} .$

The coefficient of correlation R of each model was determined as follows: $\begin{matrix} R & = & \frac{\sum_{i = 1}^{n} [(α_{i} - \bar{α}) (I_{i} - \bar{I})]}{\sqrt{\sum_{i = 1}^{n} {(α_{i} - \bar{α})}^{2} \cdot \sum_{i = 1}^{n} {(I_{i} - \bar{I})}^{2}}} \\ Q & = & (1 - R^{2}) \cdot \sum_{i = 1}^{n} {(I_{i} - \bar{I})}^{2} . \end{matrix}$

The closer |R| is to 1, the closer Q is to 0, and the better the regression equation fits the two-position test data. If 0.7 ≤ |R| ≤ 1, α and I are highly linearly correlated, and the north-seeking data are in line overall with the two-position north-seeking characteristics, which confirms the accuracy and reliability of the regression equations [13–23 , 26]. As seen in Table 2 the calculated R value is 0.72, and the Q value is 0.02, which indicates that the data is in a good linear relationship.

2.3 Confidence intervals of the rotor current

Based on the interval prediction method of regression analysis, the forecast interval of the rotor current I is equal to the “point prediction value” ± “estimation error.” The point prediction value is obtained from the external reference orientation, and the estimation error can be determined from the confidence interval method.

In an underground engineering survey, the process for determining the gyroscope orientation typically consists of two parts: the ground directional measurement, which compares instrument constants of known positional edges, and underground directional measurement to determine the orientation of edges in the hole and correcting the orientation. Whether it is ground or underground measurement, external orientation information can always be used as a priori values. The precision of a priori values from ground-based measurements can be very high because they are obtained by measure-ment of the ground truth. Underground directional measurement has poorer orientation precision because of the accumulation of errors from connected surveys and wire transfers but can still be used to choose a priori values for gross error culling, as described below [27, 28].

First, based on statistical analysis of the two-position sampling data, the external orientation is converted into a gyroscope north-seeking azimuth and plugged into the regression equation I = a · α + b as an a priori value to calculate the north-seeking predictive value of the sampling data ${\hat{I}}_{1}$ and ${\hat{I}}_{2}$ . Next, the estimation error is calculated based on the sub-bit values of the confidence level and the standard deviation of the predicted value point $\hat{I}$ . S_ind1 and S_ind2 represent the standard deviation estimators of the two north-seeking positions, which are calculated as follows:

$\begin{matrix} {\begin{matrix} S_{ind 1} = \sqrt{\frac{\sum {(I_{1 i} - {\hat{I}}_{1})}^{2}}{n - 2}} \cdot \sqrt{1 + \frac{1}{n} \cdot \frac{{(α - {\bar{α}}_{1})}^{2}}{\sum {(α_{1 i} - {\bar{α}}_{1})}^{2}}} \\ S_{ind 2} = \sqrt{\frac{\sum {(I_{2 i} - {\hat{I}}_{2})}^{2}}{n - 2}} \cdot \sqrt{1 + \frac{1}{n} \cdot \frac{{(α - {\bar{α}}_{2})}^{2}}{\sum {(α_{2 i} - {\bar{α}}_{2})}^{2}}} \end{matrix} \\ {\begin{matrix} α_{1 i} = (OR - {ON}_{3}) + \arcsin (\frac{k \cdot I_{S 1} \cdot I_{1 i}}{H \times ω_{e} cos φ}) \\ α_{2 i} = (OR - {ON}_{4}) + \arcsin (\frac{k \cdot I_{S 2} \cdot I_{2 i}}{H \times ω_{e} cos φ}) \end{matrix} \\ {\begin{matrix} {\bar{α}}_{1} = \frac{1}{n} \sum α_{1 i} \\ {\bar{α}}_{2} = \frac{1}{n} \sum α_{2 i} \end{matrix} . \end{matrix}$

Here, I_1i and I_2i are the ith rotor current values of the first and second precise north-seeking positions, respectively, where i ranges from 1 to n. α_1i and α_2i are the gyroscope north-seeking azimuths calculated from the ith rotor current values from the first and second precise north-seeking equilibrium positions, respectively. OR - ON₃ is the angle between the marked north direction OR and the first precise north-seeking equilibrium position ON₃. k is a constant. I_S1 and I_S2 are the stator current values of the first and second precise north-seeking equilibrium positions, respectively. H is the moment of inertia of the gyroscope. ω_e is the angular velocity of the Earth’s rotation. φ is the station latitude.

The estimation errors E₁ and E₂ of the two-position rotor current values are calculated for two-position north-seeking data from a certain round of measurement at a given confidence level: ${\begin{matrix} E_{1} = t_{\frac{φ}{2}} \cdot S_{ind 1} \\ E_{2} = t_{\frac{φ}{2}} \cdot S_{ind 2} \end{matrix},$ where $t_{\frac{φ}{2}}$ is the t quantile at a confidence level of 1 - φ. Therefore, for two-position north-seeking data from a certain round of measurement at the same confidence level 1 - φ, the two-position rotor current data prediction interval for a given gyroscope azimuth a priori value α is as follows: $\begin{matrix} ({\hat{I}}_{1} - t_{\frac{φ}{2}} \cdot S_{ind 1}, {\hat{I}}_{1} + t_{\frac{φ}{2}} \cdot S_{ind 1}) \\ \cup ({\hat{I}}_{2} - t_{\frac{φ}{2}} \cdot S_{ind 2}, {\hat{I}}_{2} + t_{\frac{φ}{2}} \cdot S_{ind 2}) . \end{matrix}$

In the absence of a strong influence from a disturbance torque, in addition to falling into two separate intervals for the two equations, the two-position sampling data should also have mean values that satisfy the regression prediction interval of the symmetric line.

Using the above two-position rotor current data prediction interval as a criterion for gross error data prediction, rotor north-seeking data that fall outside this interval can be identified as gross errors that need to be removed. The rotor current data remaining after gross error culling are used, and the gyroscope azimuth is recalculated according to Equation (5) [23–26 , 29–36].

2.4 Engineering application examples

The six sets of measured data shown in Fig. 1 were used to confirm the effectiveness of the gross error culling method described previously. The results are presented in Table 3.

Table 3
Comparison of gyroscope azimuth values before and after correction

Round No. Azimuth outcome before correction Two-position rotor data confidence interval (A) Data rejection rate Azimuth outcome after correction

° ′ ″ ° ′ ″

a 35 29 10 (–0.7×10^–5, 2.0×10^–5) ∪ (0.5×10^–5, 3.2×10^–5) 47% 35 28 25

b 35 28 37 (–2.5×10^–5, 1.2×10^–5) ∪ (2.0×10^–5, 5.8×10^–5) 38% 35 28 20

c 35 28 50 (1.0×10^–5, 1.6×10^–5) ∪ (1.2×10^–5, 4.0×10^–5) 29% 35 28 15

d 35 27 50 (–1.5×10^–5, 3.8×10^–5) ∪ (–1.8×10^–5, 3.5×10^–5) 22% 35 28 17

e 35 28 42 (–2.3×10^–5, –1.0×10^–5) ∪ (4.0×10^–5, 6.0×10^–5) 25% 35 28 10

f 35 28 20 (2.5×10^–5, 4.5×10^–5) ∪ (1.5×10^–5, 2.2×10^–5) 20% 35 28 12

Round No.	Azimuth outcome before correction	Two-position rotor data confidence interval (A)	Data rejection rate	Azimuth outcome after correction
a	35	29	10	(–0.7×10^–5, 2.0×10^–5) ∪ (0.5×10^–5, 3.2×10^–5)	47%	35	28	25
b	35	28	37	(–2.5×10^–5, 1.2×10^–5) ∪ (2.0×10^–5, 5.8×10^–5)	38%	35	28	20
c	35	28	50	(1.0×10^–5, 1.6×10^–5) ∪ (1.2×10^–5, 4.0×10^–5)	29%	35	28	15
d	35	27	50	(–1.5×10^–5, 3.8×10^–5) ∪ (–1.8×10^–5, 3.5×10^–5)	22%	35	28	17
e	35	28	42	(–2.3×10^–5, –1.0×10^–5) ∪ (4.0×10^–5, 6.0×10^–5)	25%	35	28	10
f	35	28	20	(2.5×10^–5, 4.5×10^–5) ∪ (1.5×10^–5, 2.2×10^–5)	20%	35	28	12

Comparison of the “azimuth outcome before correction” and “azimuth outcome after correction” values indicates that the latter are significantly more precise. This suggests that the gross error culling method described in this paper can be used to effectively remove gross error data and improve the stability of the azimuth outcome of the gyroscope.

3 Discussion

The two-position north-seeking and mass data observation technologies are the main characteristics that differentiate the magnetically suspended gyroscope from the suspended strap gyroscope. Making full use of these technical characteristics is vital to improving the north-seeking precision and stability of magnetically suspended gyroscopes. The present study made the following contributions:

First, the stator and rotor data distributions under ideal conditions and a strong disturbance torque were analyzed, and the results showed that the stator data has better stability because it satisfies the normal distribution pattern under all conditions. The rotor data are extremely sensitive to the external environment and thus are another key factor that affects the gyroscope azimuth precision.

Second, a statistical experiment and regression analysis were conducted for the two-position rotor data under different setup orientation conditions and combined with the north-seeking theoretical model of a magnetically suspended gyroscope. A simple linear regression model was obtained to provide a necessary prior condition for the optimization of rotor data processing at a later stage.

Third, based on the regression model, a method for gross error culling from the confidence interval of the two-position rotor data was developed, and the effectiveness of this method was demonstrated using experimental data.

Author contributions

S.Z. and J.G. conceived and designed the experiments; S.Z. performed the experiments; S.Z. and H.K. analysed the data; Y.Z. contributed experimental tools; S.Z. wrote the paper.

Conflicts of interest

The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) [grant number 41504001] and the Scientific research service fee of the Central University of Chang‘an University [grant number 310826151047].

References

Zhiqiang

, Research Report on GAT High Precision Maglev Gyro Station, Institute of Surveying and Spatial Information, Chang’an University, 2008.

Zhiqiang

, Jianhua

, Zhen

and Shuai

, The new maglev gyro station. China Patent [ZL2010 1 0107217.4], filed [2010.02.05] and issued [2012.05.30].

Zhen

, Key Technologies and Application of the GAT High-precision Gyro Station. PhD Thesis, Chang’an University, Xi’an, 2011.

Guisuo

, Research on Key Technology of Automatic North Seeking of Gyro Theodolite. Thesis, College of Precision Instrument and Optoelectronics Engineering, Tianjin University, Tianjin, 2006. [PhD]

Mingchun

, Research on the Key Technology of Digital and Automation of Gyro Theodolite. PhD Thesis, College of Precision Instrument and Optoelectronics Engineering, Tianjin University, Tianjin, 2006.

Guoliang

, Mining Surveying; China Mining University Press: Beijing, China, 2001, pp. 75–100.

Xiuzhong

, Inertial Navigation System Gyroscope Theory, National Defense Industry Press, Beijing, China, 1996.

Kai

, Principle and Application of Gyroscope, National Defense Industry Press, Beijing, China, 1981.

Jianhua

, The Automation of the GAT Magnet Suspended Gyro-total-station and the Gyroscope’s Data Analysis, Thesis, Chang’an University, Xi’an, 2011.

10.

Manfei

, Research on GAT Magnetic Levitation Gyro Total Station Data Processing and Analysis Based on Kalman Filter. Master’s Thesis, Chang’an University, Xi’an, 2012.

11.

Xin

, Random Drift Modeling of Silicon Micro-gyroscope. Master’s thesis, Harbin Engineering University, Harbin, 2008.

12.

Ping

, Testing method of normal distribution of measurement data, Process Autom Instrum7 (1994), 34–36.

13.

Dajie

and Benzao

, Practical Measurement Data Processing Method; Surveying and Mapping Press: Beijing, China, 2000, pp. 79–81.

14.

Guangcheng

and Haiming

, Applicable analysis and exploration of stationarity test methods, Missile Test Technol5(3) (2011), 36–39.

15.

Peng

and Yan

, Shift estimation based on autocorrelation function of periodogram, Telecommun Eng5 (2004), 107–109.

16.

Bendat

J.S.

and Piersol

A.G.

, Random Data Analysis Method, edited by Fugen

. National Defense Industry Press: Beijing, China, 1976.

17.

Bucy

R.C.

and Renne

K.D.

, Digital synthesis of nonlinear filters, Automatica7(3) (1971), 287–289.

18.

Barlow

R.E.

and Proschan

, Mathematical Theory of Reliability, Wiley, New York, USA, 1965.

19.

Misra

K.B.

, New Trends in System Reliability Evaluation, Elsevier, New York, USA, 1993.

20.

Ross

S.M.

, Stochastic Processes, John Wiley & Sons, Hoboken, USA, 1983.

21.

Xiangding

, Zhonggang

and Yanan

, Theory and Method of Nonlinear Numerical Analysis; Wuhan University Press, Wuhan, China, 2004.

22.

Zhaoyong

, Huanyun

and Zongben

, Nonlinear Analysis, Xi’an Jiaotong University, Xi’an, China, 1991.

23.

Ketang

, Lecture Notes on Linear System Control Theory, Chinese Academy of Sciences, Beijing, China, 1991.

24.

Kalman

R.E.

, A new approach to linear filtering and prediction theory, Trans ASME J Basic Eng82(2) (1960), 35–46.

25.

Wisher

R.P.

and Tabaezynski

J.A.

, A comparison of three nonlinear filters, Automatica5 (1969), 457–496.

26.

Leondes

C.T.

, Peller

J.B.

and Stear

E.B.

, Nonlinear smoothing theory, IEEE Trans Syst Sci Cybern6(1) (1970), 63–71.

27.

Zhenlong

, Time Series Analysis; China Statistics Press: Beijing, China, 1999, pp. 29–33.

28.

Wuxiong

, Time Series Analysis: Single Variable and Multivariable Methods, 2th ed.; Renmin University of China Press, Beijing, China, 2009.

29.

Choi

J.W.

and Cho

N.I.

, Suppression of narrow-band interference in DS-spread spectrum systems using adaptive IIR notch filter, Signal Process82(12) (2002), 2003–2013.

30.

Cho

N.I.

and Lee

S.U.

, On the adaptive lattice notch filter for the detection of sinusoids, IEEE Trans Circuits Syst40(7) (1993), 405–416.

31.

Cho

N.I.

, Choi

C.H.

and Lee

S.U.

, Adaptive line enhancement by using an IIR lattice notch filter, IEEE Trans. Acoust. Speech Signal Process37(4) (1989), 585–589.

32.

Zhiyi

, Zhiqiang

, Fei

and Xiaoli

, Study on method for reducing noises in maglev gyro total station directional system based on wavelet transform theory, J Géod Geodyn33(5) (2013), 154–156.

33.

Zhiyi

, Zhiqiang

and Fukai

, Analysis and modeling of directional error of maglev gyro total station, J Géod Geodyn33(2) (2013), 155–159.

34.

Lizhi

and Hanwei

, Wavelet and Discrete Transform Theory and Engineering Practice, Tsinghua University Press, Beijing, China, 2005.

35.

Grewal

M.S.

and Andrews

A.P.

, Kalman Filtering: Theory and Practice; Prentice Hall: New Jersey, USA, 1993, pp. 1–10.

36.

Parkum

J.E.

, Poulsen

N.K.

and Holst

, Recursive forgetting algorithms, Int J Control55(1) (1992), 109–128.