Vibration model of Ziyuan3 satellites considering frequency changing

Abstract

Ziyuan3-01 and Ziyuan3-02, Chinese civilian high-resolution stereo mapping satellites, were designed to satisfy the requirements of 1:50,000 digital topographical mapping. Platform vibration detection plays an important role in the Ziyuan3 satellite project. Various methods have been used to detect the vibration of Ziyuan3-01, but none focuses on that the vibration frequency is not a fixed value. To detect and model the vibration, this study proposes a method based on a set of raw gyroscope data to detect long-term vibration and model it with considering frequency changing. Two experiments were conducted, and the results showed that (a) the proposed method can detect the long-term vibration frequency and model it; (b) there is a same vibration frequency of approximately 0.65 Hz in the Ziyuan3-01 and Ziyuan3-02; (c) the 0.65 Hz vibration suffered by the Ziyuan3 satellite series is not a fixed value, varying from 0.59 Hz to 0.7 Hz, and the period of the vibration change is a constant value of approximately 1 year, which we modeled with a sine function; and (d) the working state of loads on the platform, such as solar panels, has an influence on the vibration frequency certainly.

Keywords

Ziyuan3 satellite series vibration detection frequency changes vibration model raw gyroscope data

1. Introduction

A stable platform is essential for high-resolution Earth observation satellites (HREOSs) to obtain high-accuracy geospatial image products. However, HREOSs always suffer from vibration, which considerably influences geolocation accuracy when it exceeds a particular threshold. Vibration may be caused by operating loads on the platform, including such devices as control moment gyroscopes (CMGs), momentum wheels, and solar panels (Zine, 2017; Zine and Salah, 2013, 2018). Orbiting satellites are also influenced by the space environment, such as solar radiation pressure and the Earth’s gravity (Amberg et al., 2013; Johnston and Thornton, 2000; Tong et al., 2014). Therefore, satellite platform vibration detection is an important area of HREOS research.

Various methods have been proposed to detect satellite platform vibration, and different frequencies have been found among HREOSs. There are four main types of vibration detection methods. The first type involves the application of high-performance attitude sensors, such as angular displacement sensors (Barker and Seiferth, 1996; Takanori et al., 2009; Wang et al., 2017). The second uses satellite imagery and its various products, such as raw image (Pan et al., 2017; Tong et al., 2015a, 2015b, 2017a, 2017b; Wang et al., 2016). The third method detects vibration using ground control data, such as dense ground control points and digital elevation model (Delevit et al., 2012; Grodecki, 2003; Shin et al., 1997; Tong et al., 2014). The fourth method applies satellite auxiliary data and thematic data, such as laser communication information (Toyoshima, 2001) and star images (Amberg et al., 2013). Different vibration frequencies for different satellite platforms have been detected using these various methods. The frequency and amplitude of vibration detected among different satellites are listed in Table 1 (Tong et al., 2015a). In addition to the satellites listed in the table, there have also been vibrations detected in other satellites, such as the Cartosat-1 (Baltsavias et al., 2007), Landsat-7 (Lee et al., 2004), WorldView-2 (Jacobsen, 2017), and HJ-1-C (Li et al., 2014).

Table 1.

Vibration detected among various satellites.

Satellite/sensor	Year	Frequencya (Hz)	Amplitudeb (m)
ISS (Dong, 2015; Savino and Lappa, 1971)	1993	0.01–300	—
ETS-VI (Toyoshima, 2001)	1994	0.39–250	—
MOC-NA (Kirk et al., 2003)	1996	1–4	≈15
ASTER (Ayoub et al., 2008; Iwasaki, 2011; Yu and Iwasaki, 2007)	1999	1.5–1.6	6–7
QuickBird (Ayoub et al., 2008)	2001	1, 4.3	2.5, 0.1
SPOT 5 (Ayoub et al., 2008)	2002	≈0.003	≈20
ALSat-1 (Mumtaz and Palmer, 2013)	2002	0.5	—
Nigeria Sat (Mumtaz and Palmer, 2013)	2003	0.5	—
UK-DMC (Mumtaz and Palmer, 2013)	2003	0.6	—
MEX-HRSC (Gläser et al., 2013; Gwinner et al., 2010)	2003	0.1–0.2, 1.7	8
HiRISE (Ayoub et al., 2008; Mattson et al., 2009)	2005	≈1.6	≈1
Beijing-1 (Ran et al., 2010)	2005	200	3
ALOS (Tadono et al., 2014; Takaku and Tadono, 2010)	2006	6–7, 60–70	1000, 100
Kompsat-2 (Lee et al., 2011)	2006	210	0.14
LROC (Mattson et al., 2010, 2011)	2009	6	0.1–1
Mapping Satellite-1 (Sun et al., 2015)	2010	0.105, 0.635, 4	0.2, 0.1, 0.1
Pleiades-HR (Savino and Lappa, 1971)	2011	70.9–78.4	0.14
ZY3-01 (Pan et al., 2017; Tong et al., 2015a, 2015b, 2017a, 2017b; Wang et al., 2016)	2012	0.6–0.7	2.5–7.5
Yaogan-26 (Wang et al., 2017)	2014	100, 200, 300	0.1, 0.1, 0.1

Satellite attitude vibration frequency may be detected in roll, pitch, or both; the information listed here are from either or both.

The amplitude has been transformed from onboard arcsec or pixel to ground meter (from satellite to ground), which can show the influence of vibration intuitively. Amplitudes not listed in existing records and research are indicated by “—”.

Table 1 includes the current primary vibration detection results in this area of study. Three conclusions may be drawn as follows: (1) Vibration exists in nearly every satellite platform, although the frequencies and amplitudes vary from one platform to another. (2) A satellite platform is not stable during the initial stage following launch, resulting in a large amplitude and unstable frequency. Following this initial stage, amplitude and frequency tend to stabilize around a particular value within a small range. (3) Vibration influences the product accuracy of satellite loads if it becomes sufficiently large, especially for HREOSs.

Traditional platform vibration can be expressed as follows

y = \hat{a} \sin (\hat{ω} t + \hat{θ}) + \hat{l}

(1)

where

y

is the vibration result, and

\hat{a}

\hat{ω}

\hat{θ}

, and

\hat{l}

are the vibration parameters. For the different satellite series, such as

\hat{ω}

can be calculated based on the frequency column,

\hat{a}

is equal to the amplitude column in Table 1,

\hat{θ}

and

\hat{l}

also can be acquired from detection.

The Ziyuan3-01 (ZY3-01) satellite, launched on 9 January 2012 is China’s first civilian stereo surveying and mapping satellite and is part of the Ziyuan3 (ZY3) satellite series. Ziyuan3-02 (ZY3-02), launched on 30 May 2016 was the second satellite in the series (Xie et al., 2017). Most of the loads in ZY3-02 are identical to those in ZY3-01, and both satellites use the same platform. The vibration detected in ZY3-01 generally had a frequency of approximately 0.65 Hz, with an amplitude that gradually diminished and stabilized.

In fact, we found that vibration frequency was constantly changing when we processed the ZY3 satellite’s products, with this frequency change continuing through the entire life cycle of the satellite. That means the traditional satellite platform vibration model should be improved, and it is not a single sine function. This change adds indeterminacy to the platform vibration detection, thereby making it more difficult to correct the vibration and improve the satellite products. There have been no existing investigations of this frequency change phenomenon, either in ZY3-01 or in any other satellites. To detect and model the vibration frequency changes of the ZY3 satellite series, this study proposes a satellite platform vibration detection and vibration frequency change modeling strategy using ZY3 satellite series long-term raw gyroscope data. The proposed method takes advantage of the high precision and continuity of gyroscope data because the gyroscopes in the satellite run continuously and provide a rich source of data.

2. Study data

The ZY3 attitude measurement hardware system includes three star trackers and four groups of gyroscopes, each group consisting of three orthogonal gyroscopes. The star trackers measure the absolute attitude of the satellite platform, each measurement of which is discrete. The gyroscopes acquire the relative attitude, and the continuous gyroscope data show the angular changes of the platform. The installment schematic of the star trackers and gyroscopes is illustrated in Figure 1.

Figure 1.

Installment schematic of the Ziyuan3 satellite series. (a) Three star trackers and (b) four groups of three orthogonal gyroscopes.

As shown in Figure 1(a), O-XYZ is the satellite body coordinate system, and the three star trackers (ST1, ST2, and ST3) are installed on the platform with the relative relationship illustrated in the schematic. ST1 is an active pixel sensors (APS), which is a back-up test load. ST2 and ST3 are ASTRO 10, which operate and provide data for attitude determination. There are 12 gyros (4 groups of 3) installed on the ZY3-01 platform. As seen in Figure 1(b), the first group consists of G1, G3, and G5; the second has G2, G4, and G6; the third includes G7, G8, and G9; and the fourth consists of G10, G11, and G12. The first and second groups are made up of three 3-floated gyroscopes, the third group includes three 2-floated gyros, and the fourth group includes three fiber-optic gyroscopes. The 12 gyroscopes are distributed in a divergent shape, with the three in each group aligned orthogonally.

ZY3-01 has been orbiting for approximately 7 years and has acquired more than 12,000 data tracks. ZY3-02 has been aloft for approximately 3 years and has obtained more than 3000 data tracks. Given that there is such a large amount of satellite transmission data, we store historical data in disk arrays. To detect the platform vibration and vibration frequency changes of the ZY3 satellite series, we selected a large amount of gyroscope data spread evenly overtime. Approximately 2400 tracks of ZY3-01 gyroscope data and 2100 tracks of ZY3-02 gyroscope data were used as test data. Because the preparation of ZY3-01 gyroscope data is extremely tedious, we selected them among disk arrays on a tape drive, applying 1 data track per day. As the ZY3-02 gyroscope data were collected ever since the satellite was launched in 2016 up to the present, we used 3 data tracks per day. Given that the primary mission of the ZY3 satellite series is to acquire a 1:50,000 digital topographical map of China, the test data mainly focus on areas in and around China. This means that there is not large variation in latitude among the test data, which may decrease the influence of latitude in our findings.

There are four groups of gyroscopes on-board ZY3-01 and ZY3-02. Not all of them are activated, and only a single group is actually used to determine the attitude. In addition, a certain group is activated to serve as a back-up. G1, G3, G5, G7, G8, G10, G11, and G12 of ZY3-01 were functional before 15 October 2015. However, only G1, G3, G5, G7, and G8 were working after 15 October 2015. This was because of a problem with one side of the solar panels that occurred on that day, resulting in a lower supply of power. To ensure that the satellite would continue to function normally, G10, G11, and G12 were turned off. G1, G3, G5, G7, and G8 on ZY3-02 have been working continuously up to the present.

As the gyroscopes of ZY3-01 are the same as those of ZY3-02, we take the 00504th gyroscope data track of ZY3-01, recorded on 11 February 2012, and the 00738th gyroscope data track of ZY3-02, recorded on 17 July 2017, as examples. These tracks are illustrated in Figures 2 and 3, respectively.

Figure 2.

00504th raw gyroscope data track of Ziyuan3-01 measured by (a) G1, (b) G3, (c) G5, (d) G7, (e) G8, (f) G10, (g) G11, and (h) G12, G1-G8 are outputs in satellite body coordinates, G10-G12 are in J2000 coordinates of the international celestial reference frame.

Figure 3.

00738th raw gyroscope data track of Ziyuan3-02 measured by (a) G1, (b) G3, (c) G5, (d) G7, (e) G8, all outputs in satellite body coordinates.

3. Methodology

To detect platform vibration and model its change in frequency, this study proposes a strategy based on long-term raw gyroscope data, which consist of three steps: (1) Preprocessing, which is the preparatory work for processing the gyroscope data. This involves examination and selection of the long-term gyroscope data. The goal of this step is to obtain suitable experimental data. (2) Processing, which consists of applying polynomial and fast Fourier transform (FFT) analyses to acquire the two-dimensional frequency spectra. The three-dimensional VFCI—vibration frequency changing image is then derived using superposition. (3) Modeling, which involves applying Gaussian low-pass filtering (GLPF) to suppress the high-frequency noise and grading local energy based on energy distribution. Finally, the vibration change curve is modeled by a weighted sine function for the VFCI. The framework of the proposed method is shown in Figure 4.

Figure 4.

Framework of satellite platform vibration detection and frequency change modeling.

3.1. Preprocessing

The environment in space is different from that of the laboratory and influences the payloads carried by satellites. Downloaded raw gyroscope data may include some outliers or lost data, which will lead to the failure of the vibration detection and frequency change modeling. The time distribution of gyroscope data should be uniform during the life of the satellite. This makes preprocessing a necessary first step. To screen the gyroscope data, we used two constraint conditions as the judgment criteria: time consistency and platform design stability.

3.1.1. Excluding gyroscope data based on time consistency

The sampling frequency of a gyroscope is 4 Hz, and the time interval between two adjacent points is 0.25 s. There is some data loss and some repeat transmissions, in the raw data, and the constant time interval can be used to check for these discrepancies. The data checking process is illustrated in Figure 5.

Figure 5.

Time coherence of satellite gyroscope data.

The open circles $G_{1} - G_{N}$ represent the gyroscope data, and the total amount of data is $N$ . The time interval between $G_{k}$ and $G_{k + 1}$ is $Δ T$ , and the time span from data point $G_{1}$ to data point $G_{N}$ is $Δ T_{N}$ . $Δ T$ and $Δ T_{N}$ can be calculated using the following formula

{\begin{matrix} T_{k + 1} - T_{k} = Δ T \\ T_{N} - T_{1} = Δ T_{N} \end{matrix}

(2)

Valid data should meet the $Δ T = 0.25$ and $Δ T_{N} = (N - 1) \times Δ T$ criteria, and this can reveal data repetition and loss.

3.1.2. Selecting gyroscope data with platform attitude stability

Platform stability is a basic design requirement for all orbiting satellites. Because of such factors as satellite platform swing and the operation of CMGs, gyroscope data may include small errors, which can influence test results. To minimize measurement error, we apply platform design stability as a constraint. As a gyroscope measures angular changes, its data should meet angular change stability requirements. The angular change stability formula is

S_{a n g u l a r} = F \times s = F \times \sum_{τ = 1}^{N} \sqrt{\frac{{(x_{τ} - x)}^{2}}{N}}

(3)

where

F

is the gyroscope sampling frequency,

S

is the platform design attitude stability,

N

is the total number of samples,

τ

is the order of the attitude data,

x_{τ}

is the attitude data, and

x

is the average of the attitude data.

The design value of the ZY3 satellite series platform is 5×10⁻⁴ degree/s, and we set 1×10⁻³ degree/s as the threshold value. We discard all data tracks in which $G_{k + 1} - G_{k} > 1 \times 10^{- 3} degree / s$ .

3.2. Processing

3.2.1. Gyroscope data analysis based on polynomial and FFT analyses

Gyroscopes can precisely record platform angular changes. Detection resolution is dependent on the sampling frequency of the gyroscope, being equal to half its value. Gyroscope data in satellite body coordinates are better for detecting vibration, which can express the state of the satellite platform with high sensitivity. For this reason, all angular measurements should be transformed to satellite body coordinates.

Gyroscopes measure angular changes along a particular axis, so different installation relationships between a gyroscope and its platform will yield different results. There is also some constant drift noise and other linear variations in the gyroscope data. Linear variations influence vibration detection when FFT is applied to analyze the data in the frequency domain, so a general polynomial is first used to fit the data, thereby reducing linear variations and their influence. The general polynomial is as follows

G_{p o l y} (t) = \sum_{τ = 0}^{N} a_{τ} t^{τ}

(4)

where

G_{p o l y}

is the polyfit result,

t

is the time,

a

represents the coefficients, and

N

is the order of the polynomial.

Following general polynomial processing, we apply FFT to process the data. The FFT divides the gyroscope data into its frequency components, consisting of single sinusoidal oscillations at distinct frequencies, each with their own amplitude and phase. An FFT computes the discrete Fourier transform (DFT) and produces exactly the same result as evaluating the DFT directly; the most important difference is that an FFT is much faster. The FFT is a calculation algorithm, and the DFT is the theoretical foundation. The DFT is defined by the following formula

F_{D F T} (f) = \sum_{t = 0}^{N^{'} - 1} G_{D F T} (t) e^{- i 2 π f t / N^{'}} f = 0, 1, \dots, N^{'} - 1

(5)

where

N^{'}

is the total amount of data,

G_{D F T} (t)

is the data in a time domain, and

i

is the imaginary part of a complex value.

The gyroscope data can be fitted by the general polynomial and the DFT. The formula is as follows

G (t) = G_{p o l y} (t) + G_{D F T} (t) = \sum_{τ = 0}^{N} a_{τ} t^{τ} + \frac{1}{N^{'}} \sum_{f = 0}^{N^{'} - 1} F_{D F T} (f) e^{- i 2 π f t / N^{'}} t = 0, 1, \dots, N^{'} - 1

(6)

3.2.2. Deriving the three-dimensional VFCI based on superposition

We can obtain the frequency spectra of one track of gyroscope data after polynomial and FFT processing. These spectra include the frequency domain information. As the gyroscopes are activated only when the cameras are scanning, the frequency spectra only express the frequency information in approximately 10 min intervals. This short-term analysis does not sufficiently show changes in vibration. To detect and model vibration frequency changes in the long term, we add a set of gyroscope data frequency spectra to the VFCI. The schematic diagram of the VFCI is shown in Figure 6. The left diagram illustrates a selection of two-dimensional frequency spectra. Here, the horizontal axis represents frequency, and the vertical axis represents amplitude. The right illustration is the VFCI, with time as the horizontal dimension, frequency as the vertical dimension, and the various colors representing amplitude intensity.

Figure 6.

Plotting of vibration frequency change image.

In the VFCI illustration, we set the detection result of one gyroscope’s data, which is the frequency spectrum in the left portion of Figure 6, as a column in the right grid. The gray intensity value of the VFCI is the same as the vertical axis variable of the two-dimensional frequency spectra; thus, it expresses vibration amplitude. From the VFCI, the frequency changes of various vibrations in the satellite platform can be obtained.

3.3. Modeling

The VFCI shows the long-term vibration frequency of the satellite platform, with the gray pixels representing vibration amplitude. As the vibration amplitude is not a fixed value and may be influenced by various factors, there is noise in the VFCI. This noise may influence the modeling of the frequency change curve. To decrease this influence, it is necessary to use image processing technology to improve the VFCI before modeling.

The VFCI is an image composed of some two-dimensional frequency spectra. Two-dimensional frequency spectra express the frequency information of raw gyroscope data. Gyroscope data are influenced by various factors in space; in addition, the main frequencies of satellite platform vibrationinclude several constant values. Thus, the two-dimensional frequency spectra include bits of low-frequency vibration information and high-frequency noise. To decrease the high-frequency noise, we apply a GLPF to process the VFCI in the frequency domain. The GLPF is as follows

H (u, v) = e^{- D^{2} (u, v) / 2 D_{0}^{2}}

(7)

where

(u, v)

is the row and column of the Gaussian low-pass filter, respectively,

D (u, v)

is the distance between the frequency matrix center and point

(u, v)

, and

D_{0}

is the cut-off frequency.

Taking the VFCI as an image and applying a GLPF to process the noise cause the vibration data of the VFCI to become more prominent. To model the curve shown in the filtered VFCI, we need to apply grading to the remaining data to normalize it. We first add the three adjacent values within the same column. If the summation of the three values is the maximum, the three values of the pixels will be left, and all others are set to 0. If the values in the column are all 0 after GLPF, no action is taken and the state is retained. The grading model is as follows

G_{v} [\begin{matrix} E_{u + 3} \\ E_{u + 2} \\ E_{u + 1} \end{matrix}] = {\begin{matrix} MAX (\sum_{τ = 1}^{3} E_{(τ + u, v)}) & E_{a l l} \neq 0 \\ NULL & E_{a l l} = 0 \end{matrix}

(8)

where

(u, v)

is the row and column of VFCI, respectively,

G_{v} [\begin{matrix} E_{u + 3} \\ E_{u + 2} \\ E_{u + 1} \end{matrix}]

are the three grading values of VFCI in column

v

MAX

is the maximum function, and

E_{a l l}

is the total value in column

v

$G_{v} [\begin{matrix} E_{u + 3} \\ E_{u + 2} \\ E_{u + 1} \end{matrix}]$ are the values that remain after the above processing. We can obtain three values, $E_{u + 1}$ , $E_{u + 2}$ , and $E_{u + 3}$ , for every column. As these values are the vibration amplitudes, the values at different times vary. To decrease the influence of the different amplitudes, we normalize $G_{v} [\begin{matrix} E_{u + 3} \\ E_{u + 2} \\ E_{u + 1} \end{matrix}]$ in every column. We set the maximum in $G_{v} [\begin{matrix} E_{u + 3} \\ E_{u + 2} \\ E_{u + 1} \end{matrix}]$ as 2 and the other two as 1, thus implying that the grading includes 1 and 2.

The VFCI shows a regular change curve; therefore, we apply the sine function to model it.

F (t) = a \sin (ω t + θ) + l

(9)

where

F

is the frequency,

a

ω

θ

, and

l

are required parameters, and

t

is the time T in Figure 6.

The curve in the VFCI is modeled by a sine function, but the amplitude grading (gray value) varies. To avoid ignoring this amplitude information when we model the curve, we apply a weighted sine function to fit the curve, with the amplitude grading used as the weight. The adjustment formula is as follows

X = {(A^{T} P A)}^{- 1} A^{T} P f

(10)

where

X

is the required parameter, including

a

ω

θ

, and

l

A

is the matrix of the linearization result,

P

is the weighted matrix based on the amplitude grading, and

f

is the frequency in Figure 6.

The vibration frequency change model based on grading the VFCI with a weighted sine function is shown in Figure 7. Each set of three adjacent boxes in the same column are the local frequency maximum summation for a particular gyroscope’s data, the gray value is the amplitude grading, and the red curve is a best-fit of the result. Based on VFCI processing, the vibration change of a satellite platform can be modeled, and the vibration frequency change curve can be derived.

Figure 7.

Weighted sine function fitting based on the graded vibration frequency changing image.

4. Experiments and discussion

To detect platform vibration and model the changes in its frequency, we performed two separate experiments with ZY3-01 and ZY3-02.

4.1. Experiment 1: ZY3-01 raw gyroscope data

We processed the raw gyroscope data from the ZY3-01 satellite using the proposed method and obtained the two-dimensional frequency spectra after polyfit and FFT analysis. The spectra show the frequency information of the gyroscope data. The ZY3-01 data include measurements from eight gyroscopes before 15 October 2015 and measurements from five gyroscopes after that date. We selected the 03427th track, acquired on 5 May 2013, for frequency spectrum analysis and shown as example.

In Figure 8, it can be seen that 0.6–0.7 Hz is the main frequency range of the ZY3-01 platform based on data from the 03427th track. Different gyroscope frequency spectra have the same frequency, but varying amplitudes. The main reason for this has to do with variations in the installment relationships between each gyroscope and the platform; a gyroscope can only measure the angle change in one direction. Two-dimensional frequency spectra only show the frequency data of a single track for approximately 10 min. To detect the vibration and its frequency changes for longer time periods, we superpose these two-dimensional frequency spectra on a three-dimensional VFCI.

Figure 8.

Ziyuan3-01 two-dimensional frequency spectra. Result of (a) G1, (b) G3, (c) G5, (d) G7, (e) G8, (f) G10, (g) G11, and (h) G12.

After processing all of the ZY3-01 raw gyroscope data for approximately 2400 tracks, we superposed them onto the VFCI. The column of the VFCI is the amount of raw gyroscope data, and the row is its frequency, ranging from 0–2 Hz. The gray values of pixels in the VFCI represent the amplitude shown in the two-dimensional frequency spectra. ZY3-01 has a total of 12 gyroscopes, eight of these were activated before 15 October 2015, and five remained activated following this date. Each gyroscope can generate a series of two-dimensional frequency spectra, so we get one VFCI per gyroscope. The VFCIs of ZY3-01 are shown in Figure 9.

Figure 9.

Three-dimensional vibration frequency changing images of Ziyuan3-01, which consisted of two-dimensional frequency spectra of 2400 tracks. The vibration frequency changing image of (a) G1, (b) G3, (c) G5, (d) G7, (e) G8, (f) G10, (g) G11, and (h) G12. The analysis time ranged from February 2012 to July 2018. On 15 October 2015, one side of the solar panels broke, resulting in a frequency break of the vibration frequency changing image.

From the VFCIs shown in Figure 9, we can see that the frequency change approximates a sine function curve. Vibration amplitude is large during the first 3 months following the satellite’s launch, after which it tends to stabilize. In these first 3 months, the satellite was in the onboard testing stage when various satellite functions were activated to ensure that the satellite is functioning properly. Various types of loads were turned on, and the platform swaying function would cause the vibration amplitude to be abnormal. To decrease the influence of the onboard testing, we graded the amplitude in every column of the VFCI. This provided us with the relative normalized values, which were used for weighting in the sine function fitting.

After GLPF and energy grading, the image quality of the VFCI improved and the noise in the VFCI decreased. Given that one side of the solar panel broke on 15 October 2015, the satellite platform vibration rule changed, as seen in Figure 9. Thus, we should not use a single sine function for fitting. The fitting result using two curves is shown in Figure 10.

Figure 10.

Ziyuan3-01 vibration frequency change model. Fitting model of (a) G1, (b) G3, (c) G5, (d) G7, (e) G8, (f) G10, (g) G11, and (h) G12. The red curve is the fitting based on the gyroscope data before 15 October 2015, and the green curve is the fitting using data after that date.

The fitting curves shown in Figure 10 are based on formula (8). The parameters of the vibration frequency change model vary among the various gyroscopes, resulting in small differences. The parameters are listed in Table 2, which includes the parameters of the fitting curve before 15 October 2015; Table 3 includes the parameters after that date. The parameter

2 π / ω

is the vibration frequency change period, which is approximately 1 year.

Table 2.

Parameters of Ziyuan3-01 vibration frequency change model before 15 October 2015.

Gyroscope	Parameters of sine function
Gyroscope	$a$ (^″)	$2 π / ω$ (day)	$θ$ (rad)	$l$ (^″)
G1	0.048	375	2.454	0.655
G3	0.048	372	2.306	0.659
G5	0.048	374	2.284	0.659
G7	0.045	371	2.126	0.658
G8	0.035	371	2.121	0.653
G10	0.047	373	2.277	0.659
G11	0.036	371	2.283	0.657
G12	0.051	374	2.279	0.655

Table 3.

Parameters of Ziyuan3-01 vibration frequency change model after 15 October 2015.

Gyroscope	Parameters of sine function
Gyroscope	$a$ (^″)	$2 π / ω$ (day)	$θ$ (rad)	$l$ (^″)
G1	0.049	364	7.649	0.598
G3	0.043	364	8.221	0.602
G5	0.045	365	7.982	0.600
G10	0.036	366	7.986	0.602
G11	0.042	365	7.985	0.599

4.2. Experiment 2: ZY3-02 raw gyroscope data

Most of the loads carried by the ZY3-02 platform are the same as those on the ZY3-01 platform, as are the platforms themselves. To verify the proposed method and to detect the ZY3-02 vibration frequency changes, we used ZY3-02 long-term raw gyroscope data for testing. Similar to ZY3-01, ZY3-02 only activates 5 gyroscopes, including 1 group of 3 gyroscopes (G1, G3, and G5) and 2 back-up gyroscopes (G7 and G8). In our verification test, we used the raw angular measurement data from these 5 gyroscopes.

We processed the raw gyroscope data using the proposed method, and we obtained the frequency spectra following FFT analysis. These spectral curves show the frequency information from the gyroscope data. The frequency spectra for the 00738th track, as an example, are shown in Figure 11 as an example.

Figure 11.

Ziyuan3-02 two-dimensional frequency spectra. Results from (a) G1, (b) G3, (c) G5, (d) G7, and (e) G8.

From Figure 11, it can be seen that 0.5–0.7 Hz is the main frequency range of the ZY3-02 platform based on data from the 00738th track, similar to that of ZY3-01 (see Figure 8).

After processing all of the ZY3-02 raw gyroscope data for approximately 2100 tracks, we superposed them onto the VFCI. The column of the VFCI is the amount of raw gyroscope data, and the row is its frequency, ranging from 0 to 2 Hz. The gray pixels in the VFCI represent the amplitude shown in the two-dimension frequency spectra. The VFCIs of ZY3-02 are shown in Figure 12.

Figure 12.

Three-dimensional vibration frequency changing images of Ziyuan3-02, which consisted of the frequency spectra of approximately 2100 tracks. The vibration frequency changing image of (a) G1, (b) G3, (c) G5, (d) G7, and (e) G8.

From the VFCIs shown in Figure 12, we see that the ZY3-02 frequency changes are sine function curves, as was the case with those of ZY3-01. Vibration amplitude is large during the first 3 months following the satellite’s launch, after which it tends to stabilize. In the onboard testing stage of ZY3-02, in addition to routine functional testing, an onboard laser altimeter calibration test was also performed. During the laser altimeter calibration test of ZY3-02, from 9–29 August 2016, the satellite platform swayed several times, which impacted the stability of the platform. We can see that the vibration amplitude is large, which may be related to the swaying.

We graded the amplitude in every column of the VFCI and derived the relative normalized values, which were used for weighting in the sine function fitting. After GLPF and energy grading, the image quality of the VFCI improved, and the noise in the VFCI decreased. The sine function was used to fit the vibration frequency change curve; the fitting model is shown in Figure 13.

Figure 13.

Ziyuan3-02 vibration frequency change model. Fitting model of (a) G1, (b) G3, (c) G5, (d) G7, (e) and G8.

The fitting curves shown in Figure 13 are based on formula (8). The parameters of the vibration frequency change model vary among the different gyroscopes, resulting in small differences. The parameters are listed in Table 4.

Table 4.

Parameters of the Ziyuan3-02 vibration frequency change model.

Gyroscope	Parameters of sine function
Gyroscope	$a$ (^″)	$2 π / ω$ (day)	$θ$ (rad)	$l$ (^″)
G1	0.046	366	−1.484	0.640
G3	0.039	362	−1.491	0.642
G5	0.033	365	−1.601	0.639
G7	0.043	366	−1.427	0.632
G8	0.034	362	−1.423	0.644

From above two experiments, we can find that the ZY3 satellite platform vibration frequency change can be expressed with a model, and the traditional satellite platform vibration model, showed as equation (1), could be improved based on the frequency changing.

After that, the ZY3 platform vibration is not a single and simple sine function. Instead, vibration frequency changes, and the platform vibration is actually a combination of two sine functions. The traditional and proposed model is showed in Figure 14; the traditional one is a sine wave with a certain frequency, but the improved one is a sine wave whose frequency is changing.

Figure 14.

The satellite platform vibration model schematic map, (a) is traditional one and its period T is a fixed value, (b) is an improved one, and its period (T₀–T₅) is changing with a frequency.

Formula (9) shows the change in platform vibration frequency. This frequency change is added to the traditional vibration model formula (1) to obtain the improved vibration model

y = \hat{a} \sin {2 π [a \sin (ω t + θ) + l] t + \hat{θ}} + \hat{l}

(11)

where

a

ω

θ

, and

l

are the variables listed in equation (9). Based on this study, the parameter

\hat{ω}

in formula (1) is changing with a frequency of approximately 1 year, and the result is shown in the parameter

ω

in Table 4. The vibration frequency changing parameters

a

θ

, and

l

are related to the loads on satellite platform, space environment, and other factors.

5. Conclusions

In this study, we built a satellite platform vibration model with considering frequency changing using long-term raw gyroscope data. We found that the ZY3 satellite series platform vibration frequency changes with a period of approximately 1 year. Several conclusions can be drawn:

The proposed strategy can detect ZY3 satellite series platform vibration, which is approximately 0.65 Hz and ranges from 0.59 Hz to 0.7 Hz. This vibration also includes several other small-amplitude frequencies, including 0.2 Hz and 1.1 Hz. The vibration amplitude of ZY3-01 decreased when one of its solar panels broke. There is noise of varying frequencies in the satellite platform vibration as well.

The model showed that the vibration of ZY3-01 is same as that of ZY3-02. The vibration frequency change of the two satellites is also similar, with a period of approximately 1 year. The vibration frequency change model improved and expanded on the traditional vibration detection analysis, taking into account variations overtime.

The platform vibration of ZY3-01 is almost the same as that of ZY3-02, although there are some improved loads in ZY3-02 compared with ZY3-01, and vibration is influenced by the loads carried by the platform. The vibration frequency change period showed almost no variation after one of ZY3-01’s solar panels broke on 15 October 2015. As the vibration frequency change period difference between ZY3-01 and ZY3-02 is very small, it may be influenced by other outside factors.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: National Natural Science Foundation of China; 41571440.

ORCID iD

Fan Mo

References

Amberg

, Dechoz

, Bernard

, et al. (2013) In-flight attitude perturbances estimation: application to PLEIADES-HR satellites. Proceedings of SPIE 8866: 12.

Ayoub

, Leprince

, Binet

, et al. (2008) Influence of camera distortions on satellite image registration and change detection applications. In: IEEE international geoscience and remote sensing symposium, Boston, USA, 7–11 July 2008, pp. 1072–1075. NJ, USA: IEEE.

Baltsavias

, Kocaman

, Akca

, et al. (2007) Geometric and radiometric investigations of cartosat-1 data. Chembiochem a European Journal of Chemical Biology 12: 224–234.

Barker

Seiferth

(1996) Landsat thematic mapper band-to-band registration. In: IEEE international geoscience and remote sensing symposium, Lincoln, USA, 31 May 1996, pp. 1600–1602. USA: IEEE.

Delevit

, Greslou

, Amberg

, et al. (2012) Attitude assessment using pleiades-hr capabilities. ISPRS – International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XXXIX–B1: 525–530.

Dong

(2015) Spacecraft Microvibration. Beijing: China Aerospace Publishing House.

Gläser

, Haase

, Oberst

, et al. (2013) Co-registration of laser altimeter tracks with digital terrain models and applications in planetary science. Planetary and Space Science 89: 111–117.

Grodecki

Dial

(2003) Block adjustment of high-resolution satellite images described by rational polynomials. Photogrammetric Engineering and Remote Sensing 69: 59–68.

Gwinner

, Scholten

, Preusker

, et al. (2010) Topography of mars from global mapping by HRSC high-resolution digital terrain models and orthoimages: characteristics and performance. Earth and Planetary Science Letters 294: 506–519.

10.

Iwasaki

(2011) Detection and estimation satellite attitude jitter using remote sensing imagery. In: Jason Hall (ed) Advances in Spacecraft Technologies. InTech. ISBN: 978-953-307-551.

11.

Jacobsen

(2017) Problems and limitations of satellite image orientation for determination of height models. ISPRS – International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII–1/W1: 257–264.

12.

Johnston

Thornton

(2000) Thermally induced dynamics of satellite solar panels. Journal of Spacecraft and Rockets 37: 604–613.

13.

Kirk

, Howington-Kraus

, Redding

, et al. (2003) High‐resolution topomapping of candidate MER landing sites with mars orbiter camera narrow‐angle images. Journal of Geophysical Research Planets 108: 29.

14.

Lee

D-H

, Yang

, Seo

, et al.(2011) Image restoration of the asymmetric point spread function of a high-resolution remote sensing satellite with time-delayed integration. Advances in Space Research 47: 690–701.

15.

Lee

, Storey

, Choate

, et al. (2004) Four years of landsat-7 on-orbit geometric calibration and performance. IEEE Transactions on Geoscience and Remote Sensing 42: 2786–2795.

16.

, Zhang

, Li

, et al. (2014) Coherent performance analysis of the HJ-1-C synthetic aperture radar. Journal of Radars 3: 320–325.

17.

Mattson

, Bartels

, Boyd

, et al. (2011) Continuing analysis of spacecraft jitter in LROC-NAC. In: Lunar and planetary science conference, Houston, 7–11 March 2011, p. 2756. TX, USA: LPSC.

18.

Mattson

, Boyd

, Kirk

, et al. (2009) HiJACK: correcting spacecraft jitter in HiRISE images of mars. Health Management Technology 33: A162.

19.

Mattson

, Robinson

, Mcewen

, et al. (2010) Early assessment of spacecraft jitter in LROC-NAC. In: Lunar and planetary science conference, Houston, 1–5 March 2010, p. 1871. TX, USA: LPSC.

20.

Mumtaz

Palmer

(2013) Attitude determination by exploiting geometric distortions in stereo images of DMC camera. IEEE Transactions on Aerospace and Electronic Systems 49: 1601–1625.

21.

Pan

, Che

, Zhu

, et al. (2017) Satellite jitter estimation and validation using parallax images. Sensors 17: 83.

22.

Ran

, Chi

Wang

. (2010) Property and removal of jitter in beijing-1 small satellite panchromatic images. ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XXXVII: 929–934.

23.

Savino

Lappa

(1971) Assessment of thermovibrational theory: application to G-jitter on the space station. Journal of Spacecraft and Rockets 40: 201–210.

24.

Shin

, Pollard

Muller

J-P

. (1997) Accurate geometric correction of ATSR images. IEEE Transactions on Geoscience and Remote Sensing 35: 997–1006.

25.

Sun

, Long

, Liu

B-C

, et al. (2015) Application of attitude jitter detection based on short-time asynchronous images and compensation methods for chinese mapping satellite-1. Optics Express 23: 1395–1410.

26.

Tadono

, Ishida

, Oda

, et al. (2014) Precise global DEM generation by ALOS PRISM. ISPRS Annals of Photogrammetry, Remote Sensing and Spatial Information Sciences II–4: 71–76.

27.

Takaku

Tadono

. (2010) High resolution dsm generation from alos prism – processing status and influence of attitude fluctuation. In: IEEE international geoscience and remote sensing symposium, HI, USA, 25–30 July 2010, pp. 4228–4231. Honolulu, USA: IEEE.

28.

Takanori

, Tetsuo

, Noboru

, et al. (2009) High-bandwidth attitude determination using jitter measurements and optimal filtering. In: AIAA guidance, navigation, and control conference, Chicago, USA, 10–13 August 2009, p. 6311. Chicago, USA: AIAA.

29.

Tong

, Li

, Liu

, et al. (2015a) Detection and estimation of ZY-3 three-line array image distortions caused by attitude oscillation. ISPRS Journal of Photogrammetry and Remote Sensing 101: 291–309.

30.

Tong

, Xu

, Ye

, et al. (2015b) Attitude oscillation detection of the ZY-3 satellite by using multispectral parallax images. IEEE Transactions on Geoscience and Remote Sensing 53: 3522–3534.

31.

Tong

, Ye

, Li

, et al. (2017a) Detection and estimation of along-track attitude jitter from Ziyuan-3 three-line-array images based on back-projection residuals. IEEE Transactions on Geoscience and Remote Sensing 55: 4272–4284.

32.

Tong

, Ye

Liu

. (2017b) Essential technology and application of jitter detection and compensation for high resolution satellites. Acta Geodaetica et Cartographica Sinica 46: 1500–1508.

33.

Tong

, Ye

, Xu

, et al. (2014) Framework of jitter detection and compensation for high resolution satellites. Remote Sensing 6: 3944–3964.

34.

Toyoshima

(2001) In-orbit measurements of short term attitude and vibrational environment on the engineering test satellite VI using laser communication equipment. Optical Engineering 40: 827–832.

35.

Wang

, Fan

, Pan

, et al. (2017) Image jitter detection and compensation using a high-frequency angular displacement method for Yaogan-26 remote sensing satellite. ISPRS Journal of Photogrammetry and Remote Sensing 130: 32–43.

36.

Wang

, Zhu

, Pan

, et al. (2016) Satellite jitter detection and compensation using multispectral imagery. Remote Sensing Letters 7: 513–522.

37.

Xie

, Tang

, Mo

, et al. (2017) ZY3-02 laser altimeter footprint geolocation prediction. Sensors 17: 2165.

38.

Iwasaki

. (2007) Correction of attitude fluctuation of terra spacecraft using ASTER/SWIR imagery with parallax observation. IEEE Transactions on Geoscience and Remote Sensing 46: 222–227.

39.

Zine

. (2017) Study and analysis of the piezoresistive accelerometer stability and improvement of their performances. International Journal of System Assurance Engineering and Management 8(2): 1520–1526.

40.

Zine

Salah

(2013) Development of measurement precision of sensor vibration. Journal of Vibration and Control 19: 1480–1486.

41.

Zine

Salah

(2018) Piezoresistive accelerometer mathematical model development with experimental validation. IEEE Sensors Journal 18(7): 2690–2696.