CN102735260B - Determination method of star sensor on-orbit measurement errors - Google Patents

Abstract

The invention relates to a determination method of star sensor on-orbit measurement errors. First, a star sensor output quaternion sequence obtained through on-orbit measuring is subjected to polynomial fitting and a normalization process, such that a reference output quaternion sequence is obtained. The measured sequence and the reference sequence are subjected to quaternion multiplication, such that deviation quaternion is obtained; the deviation quaternion is converted into Euler angles, such that a total error of three axis of a coordinate system measured by star wounding sensor is obtained; digital filtering is carried out upon the total error, such that low frequency error (LFE) is obtained; a star sensor output quaternion sequence measured by using a static star simulator in a ground testing stage is used for replacing a star sensor output quaternion sequence measured on orbit; the previous steps are repeated, such that a temporal error (TE) of the three axis of the coordinate system measured by the star wounding sensor is obtained; and with a formula, high frequency error (HFE) is obtained by calculation. The method provided by the invention is especially suitable for star sensor error determination under a condition that a target attitude cannot be accurately predicted.

Description

A kind of star sensor inflight measurement method for determining difference by mistake

Technical field

The present invention relates to a kind of analysis of measurement errors method of star sensor, be applicable to satellite period and the determination of whole star ground test stage star sensor measuring error in orbit, particularly can not the error accurately in precognition situation determine at targeted attitude.

Background technology

Star sensor is the attitude measurement sensor that precision that current satellite uses is the highest, it is by measuring orientation in co-ordinates of satellite system of the measurement vector of some fixed star and brightness, recycling ephemeris obtains the orientation of these fixed stars in inertial coordinates system, calculate through attitude determination algorithm and can provide the attitude information of satellite in inertial system, precision can reach several rads of levels.The measuring error of star sensor can directly affect the precision of satellite attitude measurement, and error source when also contributing to ground-based mission process to the analysis of star sensor inflight measurement error decomposes and elimination, lifting tasks carrying effect.

Usually need one when calculating attitude measurement error with reference to attitude, or be referred to as true attitude, utilize and measure attitude and obtain measuring error with reference to the statistical of attitude.Chinese Academy of Sciences Nation Astronomical Observatory doctor Sun Caihong in the Ph.D. Dissertation's " miniature type star sensor method of production and Development Techniques " delivered on June 1st, 2002 when assessing the measuring accuracy of star sensor, star sensor is directly put in ground and optical axis is pointed to zenith, so the reference attitude tends of star sensor is just the rotation angle of the earth, then just can carry out statistical study to measuring error.

But, after satellier injection, be usually difficult to set up this real reference attitude, therefore can not with compared with attitude.To the method that the calculating of star sensor inflight measurement error adopts be at present: statistics star sensor exports the variable quantity of attitude quaternion, or using the expression amount of star sensor optical axis at inertial space---right ascension (RA), declination (DE), roll angle (ROLL) are analyzed as index.The Carl Christian Liebe in JPL laboratory just employs the inflight measurement precision of RA-DE methods analyst star sensor in the article " The new generation ofautonomous star trackers " of the article " Accuracy Performance of Star Trackers-A Tutorial " to deliver for 2002 and Allan Read Eisenman.

Although said method can analyze the fluctuation situation of change of star sensor measurement data, but all describe according to the measuring error around star sensor body three axle in common star sensor index definition and the ground integrated error of star decompose, said method is difficult to be converted to the measuring error around each axle of star sensor.On the other hand, the measuring error of star sensor is made up of several error term: low frequency aberration (LFE, Low Frequency Error), high frequency error (HFE, High Frequency Error), stochastic error (TE, Temporal Error), analytical error source is contributed to the decomposition of these error terms, and all these error terms is not separated in said method.

Summary of the invention

Technology of the present invention is dealt with problems and is: overcome the deficiencies in the prior art, provide a kind of with reference to the star sensor inflight measurement error defining method under attitude unknown situation, solve the analysis of measurement errors problem of star sensor in-orbit and in whole star test process, meanwhile, analysis result can provide the measuring error around star sensor three axle.

Technical solution of the present invention is: a kind of star sensor inflight measurement is method for determining difference by mistake, and step is as follows:

(1) the output Quaternion Sequence of inflight measurement star sensor, carries out fitting of a polynomial and normalized to the Quaternion Sequence of inflight measurement, and the reference obtaining star sensor exports Quaternion Sequence;

(2) reference of the star sensor output Quaternion Sequence of inflight measurement star sensor and step (1) obtained exports the inverse multiplication carrying out hypercomplex number of Quaternion Sequence, obtains deviation hypercomplex number;

(3) the deviation hypercomplex number that step (2) obtains is converted to Eulerian angle, obtains the departure δ φ of the Eulerian angle parametric form around the X-axis of star sensor surving coordinate system, Y-axis, Z axis thus _x.k, δ φ _{y, k}, δ φ _{z, k}as total error Total Error;

(4) digital filtering is carried out to the total error Total Error that step (3) obtains, obtain low frequency aberration LFE;

(5) the output Quaternion Sequence of the star sensor adopting ground test stage static star simulator to measure replaces the star sensor of inflight measurement to export Quaternion Sequence, and use rear sampling instant data of each element in the output Quaternion Sequence of star sensor to replace with reference to exporting Quaternion Sequence, repeat step (1) ~ (3), obtain the stochastic error TE of the Eulerian angle parametric form around the X-axis of star sensor surving coordinate system, Y-axis, Z axis thus;

(6) formula is passed through $\mathrm{Total\; Error}=\sqrt{{\mathrm{LFE}}^2+{\mathrm{HFE}}^2+{\mathrm{TE}}^2}$ Calculate high frequency error HFE.

The present invention's advantage is compared with prior art:

(1) the present invention adopts the reference attitude of relative attitude method establishment satellite, solves the analysis of measurement errors problem in the unknowable situation of the true attitude of satellite;

(2) error analysis result of the present invention can provide the measuring error around each coordinate axis of star sensor, defines corresponding with the error in star sensor measurement index, is conducive to the decomposition at the ground integrated error component of star;

(3) the inventive method has carried out systematic decomposition to each error term in measuring error, is extracted low frequency aberration, high frequency error and stochastic error from total error, can be further analyzed for the data of each error term to error source;

(4) present invention employs running mean method to extract the low frequency term in data, calculate simple, quick, be easy to the trend term in analytical error data;

(5) the inventive method make use of in-orbit from the quick measurement data of the star in ground test stage to calculate different error term, make result of calculation more accurate.

Accompanying drawing explanation

Fig. 1 is the FB(flow block) of the inventive method;

Fig. 2 is that moving window when carrying out running mean in the inventive method chooses schematic diagram.

Embodiment

The attitude data that star sensor exports is the attitude quaternion of star sensor surving coordinate system relative to inertial coordinates system.Wherein, the initial point of star sensor surving coordinate system is CCD linear array center, and X-axis points to the line direction of CCD, and Y-axis points to CCD column direction, and Z axis is the optical axis direction of star sensor; Inertial coordinate is J2000 coordinate system, and initial point is earth centroid, and X-axis points to first point of Aries during 1 day 12 January in 2000, and Z axis is vertical with X-axis, and point to the celestial sphere arctic, Y-axis and other diaxon are determined by right-hand rule.

The measuring error of star sensor is made up of many factors, comprise low frequency aberration (LFE, Low Frequency Error), high frequency error (HFE, High Frequency Error), stochastic error (TE, Temporal Error).Due to the complicacy of space environment, the result that these error components and ground experiment obtain has a great difference, also not easily carries out decomposing and qualitative assessment.The inventive method solves the problem analysis of star sensor inflight measurement error, decomposes each error term, gives a kind of error defining method.

Fig. 1 gives decomposition and the calculation process of star sensor measuring error.Utilize star sensor in-orbit data calculate total error Total Error, low frequency aberration LFE and noise equivalent angle NEA concrete calculation procedure as follows:

(1) suppose that the N group Quaternion Sequence that star sensor exports is wherein, m=0,1,2,3, corresponding to [the q of same k value ₀q ₁q ₂q ₃] be one group of hypercomplex number;

(2) according to hypercomplex number observed quantity sequence can be expressed as:

${\left\{q_{m,k}\right\}}_{k=1}^N={\left\{\begin{array}{c}{q}_{0,k}\\ q_{1,k}\\ q_{2,k}\\ q_{3,k}\end{array}\right\}}_{k=1}^N=\left\{\begin{array}{cccc}{q}_{\mathrm{0,1}}& q_{\mathrm{0,2}}& ...& q_{0,N}\\ q_{\mathrm{1,1}}& q_{\mathrm{1,2}}& ...& q_{1,N}\\ q_{\mathrm{2,1}}& q_{2,2}& ...& q_{2,N}\\ q_{\mathrm{3,1}}& q_{\mathrm{3,2}}& ...& q_{3,N}\end{array}\right\}$

Utilize polynomial fitting method in sequence extremely carry out matching respectively, construct a hypercomplex number reference sequences

${\left\{q_{\mathrm{rm},k}\right\}}_{k=1}^N={\left\{\begin{array}{c}{q}_{r0,k}\\ q_{r1,k}\\ q_{r2,k}\\ q_{r3,k}\end{array}\right\}}_{k=1}^N=\left\{\begin{array}{cccc}{q}_{r\mathrm{0,1}}& q_{r\mathrm{0,2}}& ...& q_{r0,N}\\ q_{r\mathrm{1,1}}& q_{r\mathrm{1,2}}& ...& q_{r1,N}\\ q_{r\mathrm{2,1}}& q_{r\mathrm{2,2}}& ...& q_{r2,N}\\ q_{r\mathrm{3,1}}& q_{r\mathrm{3,2}}& ...& q_{r3,N}\end{array}\right\}$

Using the attitude of this sequence as " truly " of star sensor, consider that quaternary digital-to-analogue is the constraint condition of 1, need to do normalized to the hypercomplex number in sequence:

${\left\{q_{\mathrm{rm},k}\right\}}_{k=1}^N={\left\{\begin{array}{c}\frac{q_{r0,k}}{\sqrt{1-q_{r0,k}^T{q}_{r0,k}}}\\ \frac{q_{r1,k}}{\sqrt{1-q_{r1,k}^T{q}_{r1,k}}}\\ \frac{q_{r2,k}}{\sqrt{1-q_{r2,k}^T{q}_{r2,k}}}\\ \frac{q_{r3,k}}{\sqrt{1-q_{r3,k}^T{q}_{r3,k}}}\end{array}\right\}}_{k=1}^N\left\{\begin{array}{cccc}\frac{q_{r\mathrm{0,1}}}{\sqrt{1-q_{r\mathrm{0,1}}^T{q}_{r\mathrm{0,1}}}}& \frac{q_{r\mathrm{0,2}}}{\sqrt{1-q_{r\mathrm{0,2}}^T{q}_{r\mathrm{0,2}}}}& ...& \frac{q_{r0,N}}{\sqrt{1-q_{r0,N}^T{q}_{r0,N}}}\\ \frac{q_{r\mathrm{1,1}}}{\sqrt{1-q_{r\mathrm{1,1}}^T{q}_{r\mathrm{1,1}}}}& \frac{q_{r\mathrm{1,2}}}{\sqrt{1-q_{r\mathrm{1,2}}^T{q}_{r\mathrm{1,2}}}}& ...& \frac{q_{r1,N}}{\sqrt{1-q_{r1,N}^T{q}_{r1,N}}}\\ \frac{q_{r\mathrm{2,1}}}{\sqrt{1-q_{r\mathrm{2,1}}^T{q}_{r\mathrm{2,1}}}}& \frac{q_{r\mathrm{2,2}}}{\sqrt{1-q_{r\mathrm{2,2}}^T{q}_{r\mathrm{2,2}}}}& ...& \frac{q_{r2,N}}{\sqrt{1-q_{r2,N}^T{q}_{r2,N}}}\\ \frac{q_{r\mathrm{3,1}}}{\sqrt{1-q_{r\mathrm{3,1}}^T{q}_{r\mathrm{3,1}}}}& \frac{q_{r\mathrm{3,2}}}{\sqrt{1-q_{r\mathrm{3,2}}^T{q}_{r\mathrm{3,2}}}}& ...& \frac{q_{r3,N}}{\sqrt{1-q_{r3,N}^T{q}_{r3,N}}}\end{array}\right\}$

It should be noted that, this step is when carrying out fitting of a polynomial, and matching order should be chosen according to the trend of hypercomplex number, range of choices is 1 ~ 15 rank, and when the fluctuation of hypercomplex number curve is less, exponent number can be set to less numerical value, under limiting case, exponent number is set to 1, and namely trend is straight line, when fluctuating larger, exponent number can suitably increase, but not easily more than 15 rank, because order increases again, the marginal portion of matching can be made to produce distortion, do not conform to the actual conditions;

(3) at each moment t _k(k=1 ..., N), utilize attitude transfer function between hypercomplex number represent taking advantage of of hypercomplex number, solve the deviation hypercomplex number δ q of observed quantity and reference quantity _k, this departure is indivisible, therefore can by this deviation hypercomplex number δ q _kbe converted to Eulerian angle by turning arbitrarily sequence, the present invention turns sequence by deviation hypercomplex number δ q by 3-1-2 _kbe converted to the departure δ φ of the Eulerian angle parametric form of the X, Y, Z axis around star sensor surving coordinate system _x.k, δ φ _{y, k}, δ φ _{z, k}, concrete computation process is as follows:

Attitude matrix A (q) is changed into Quaternion Matrix form:

$A\left(q\right)=\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]=\left[\begin{array}{ccc}{q_0}^2+{q_1}^2-{q_2}^2-{q_3}^2& 2(q_1{q}_2+q_0{q}_3)& 2(q_1{q}_3-q_0{q}_2)\\ 2(q_1{q}_2-q_0{q}_3)& {q_0}^2-{q_1}^2+{q_2}^2-{q_3}^2& 2(q_2{q}_3+q_0{q}_1)\\ 2(q_1{q}_3+q_0{q}_2)& 2(q_2{q}_3-q_0{q}_1)& {q_0}^2-{q_1}^2-{q_2}^2+{q_3}^2\end{array}\right]$

Utilize Eulerian angle x, y, z to describe the attitude tends of star sensor three axle, when turning sequence and being 3-1-2, be calculated as follows with the attitude matrix A that attitude angle is expressed:

$A=R_y\left(y\right)R_x\left(x\right)R_z\left(z\right)=\left[\begin{array}{ccc}\cos \left(y\right)& 0& -\sin \left(y\right)\\ 0& 1& 0\\ \sin \left(y\right)& 0& \cos \left(y\right)\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \left(x\right)& \sin \left(x\right)\\ 0& -\sin \left(x\right)& \cos \left(x\right)\end{array}\right]\left[\begin{array}{ccc}\cos \left(z\right)& \sin \left(z\right)& 0\\ -\sin \left(z\right)& \cos \left(z\right)& 0\\ 0& 0& 1\end{array}\right]$

$=\left[\begin{array}{ccc}\cos \left(y\right)\cos \left(z\right)-\sin \left(x\right)\sin \left(y\right)\sin \left(z\right)& \cos \left(y\right)\sin \left(z\right)+\sin \left(x\right)\sin \left(y\right)\cos \left(z\right)& -\cos \left(x\right)\sin \left(y\right)\\ -\cos \left(x\right)\sin \left(z\right)& \cos \left(x\right)\cos \left(z\right)& \sin \left(x\right)\\ \sin \left(y\right)\cos \left(z\right)+\sin \left(x\right)\cos \left(y\right)\sin \left(z\right)& \sin \left(y\right)\sin \left(z\right)-\sin \left(x\right)\cos \left(y\right)\cos \left(z\right)& \cos \left(x\right)\cos \left(y\right)\end{array}\right]$

Then by A (q)=A, the relation turning Eulerian angle and Direct cosine matrix under sequence according to 3-1-2 obtains utilizing the attitude angle of star sensor three axle of quaternion representation:

$x=\arctan [-\frac{A_{21}}{A_{22}}]\arctan [-\frac{2(q_1{q}_2-q_0{q}_3)}{{q_0}^2-{q_1}^2+{q_2}^2-{q_3}^2}]$

y＝arcsin[A ₂₃]＝arcsin[2(q ₂q ₃+q ₀q ₁)]

$z=\arctan [-\frac{A_{13}}{A_{33}}]\arctan [-\frac{2(q_1{q}_3-q_0{q}_2)}{{q_0}^2-{q_1}^2-{q_2}^2+{q_3}^2}]$

Then correspond to each moment t _k(k=1 ..., N) star sensor three axle departure δ φ _x.k, δ φ _{y, k}, δ φ _{z, k}, above formula and deviation hypercomplex number δ q can be utilized _kobtain accordingly;

(4) basis the standard deviation statistics index asking for whole measuring error of reflection star sensor is:

(φ i is the measurement data in each moment, average for whole measurement data) then each axle of star sensor measure 3 σ statistics of total error and be:

(5) basis carry out digital filtering and obtain low frequency LFE error sequence in the present invention, digital filtering adopts running mean method, and the method filtering is simple, practical, controllability is good, and specific formula for calculation is as follows:

$f\left[k\right]=\frac{1}{L}\underset{L-M}{\overset{L+M}{\mathrm{\Σ}}}\mathrm{\φ}\left[k\right]$

Wherein: L=2M+1 is counting of filtering running mean, i.e. length of window, integer, f is mean value, and k is data point position, and φ is the actual numerical value at k place, and M is integer.Will when calculating directly substitute into φ [k].

For choosing of length of window size, not easily select too little or too large numerical value, numerical value is too little, can not filtering low frequency term, too greatly, then can by other error in the lump filtering, the size of general moving window makes the LFE curve extracted can reflect the trend of ordered series of numbers curve, as shown in Figure 2.

Certainly, except running mean method can also carry out low frequency term extraction by some other digital filtering method, such as adopt Finite Impulse Response filter, but the design of this filter model is comparatively complicated, calculated amount is large.

After extracting LFE item, the statistical indicator that 3 times of standard deviations provide corresponding reflection LFE error is asked for LFE error sequence

(6) will from in ask difference eliminate, obtain all the other errors of star sensor comprise high frequency error item HFE and stochastic error TE, this part error forms noise equivalent angle NEA, asks for 3 times of standard deviations to these fractional error data namely the statistical error size of noise equivalent angle NEA can be obtained.

Because star sensor inflight measurement data change with the factor such as satellite motion, environmental change, and stochastic error is to the measured analysis and evaluation taking multiple measurements acquired results under repeated condition, therefore data in-orbit should not be selected when analyzing star sensor random meausrement error.This just needs the ground test data by star sensor to analyze the stochastic error of star sensor, after star sensor dress star, static star simulator can be adopted to test the performance of star sensor, now, the starry sky image of star sensor shooting can not change, celestial body does not also move, and the measurement data that ideally star sensor exports can be invariable, and variable quantity then can be regarded as the random meausrement error of star sensor.

The concrete steps that the star sensor data adopting static star simulator to measure according to the whole star ground test stage assesses random meausrement error TE are as follows:

(1) according to the star sensor hypercomplex number observed quantity of ground test phase sequence, utilizes the attitude transfer function of moment k and adjacent moment k+1 ( represent taking advantage of of hypercomplex number) obtain differential quaternary Number Sequence

(2) at each moment t _k(k=1 ..., N-1), turn sequence from differential quaternary number δ q by 3-1-2 _kbe converted to around star sensor surving coordinate system X, Y, Z tri-difference component of Eulerian angle parametric form of axle specific formula for calculation with calculate the identical of total error above;

(3) basis sequence, provides 3 times of standard deviation statistics indexs of the quick random meausrement error of reflection star because star sensor is separate at the measured value of each sampling instant, comprise the joint contribution of two quick measuring error of sampling instant star in this error statistics amount, therefore final stochastic error amount needs to be treated to

(4) assess the impact of high frequency error HFE according to total error Total Error, low frequency aberration LFE, stochastic error TE, the relational expression between whole error is:

$\mathrm{Total\; Error}=\sqrt{{\mathrm{LFE}}^2+{\mathrm{HFE}}^2+{\mathrm{TE}}^2}=\sqrt{{\mathrm{LFE}}^2+{\mathrm{NEA}}^2}$

Utilize the e that previous calculations obtains _total, e _nEA, e _lFE, e _tE, high frequency error item can be obtained:

$e_{\mathrm{HFE}}=\sqrt{E_{\mathrm{Total}}^2-e_{\mathrm{LFE}}^2-e_{\mathrm{TE}}^2}=\sqrt{e_{\mathrm{NEA}}^2-e_{\mathrm{TE}}^2}$

The content be not described in detail in instructions of the present invention belongs to the known technology of those skilled in the art.

Claims (1)

1. a star sensor inflight measurement method for determining difference by mistake, is characterized in that step is as follows:

(1) the output Quaternion Sequence of inflight measurement star sensor, carries out fitting of a polynomial and normalized to the Quaternion Sequence of inflight measurement, and the reference obtaining star sensor exports Quaternion Sequence; Described reference exports Quaternion Sequence expression formula as follows:

${\left\{q_{\mathrm{rm},k}\right\}}_{k=1}^N={\left\{\begin{array}{c}\frac{q_{r0.k}}{\sqrt{1-q_{r0,k}^T{q}_{r0,k}}}\\ \frac{q_{r1,k}}{\sqrt{1-q_{r1,k}^T{q}_{r1,k}}}\\ \frac{q_{r2,k}}{\sqrt{1-q_{r2,k}^T}{q}_{r2,k}}\\ \frac{q_{r3,k}}{\sqrt{1-q_{r3,k}^T{q}_{r3,k}}}\end{array}\right\}}_{k=1}^N=\left\{\begin{array}{cccc}\frac{q_{r0,1}}{\sqrt{1-q_{r\mathrm{0,1}}^T{q}_{r\mathrm{0,1}}}}& \frac{q_{r\mathrm{0,2}}}{\sqrt{1-q_{r\mathrm{0,2}}^T{q}_{r\mathrm{0,2}}}}& \·\·\·& \frac{q_{r0,N}}{\sqrt{1-q_{r0,N}^T{q}_{r0,N}}}\\ \frac{q_{r\mathrm{1,1}}}{\sqrt{1-q_{r\mathrm{1,1}}^T{q}_{r\mathrm{1,1}}}}& \frac{q_{r\mathrm{1,2}}}{\sqrt{1-q_{r\mathrm{1,2}}^T{q}_{r\mathrm{1,2}}}}& \·\·\·& \frac{q_{r1,N}}{\sqrt{1-q_{r1,N}^T{q}_{r1,N}}}\\ \frac{q_{r\mathrm{2,1}}}{\sqrt{1-q_{r\mathrm{2,1}}^T{q}_{r\mathrm{2,1}}}}& \frac{q_{r\mathrm{2,2}}}{\sqrt{1-q_{r\mathrm{2,2}}^T{q}_{r\mathrm{2,2}}}}& \·\·\·& \frac{q_{r2,N}}{\sqrt{1-q_{r2,N}^T{q}_{r2,N}}}\\ \frac{q_{r\mathrm{3,1}}}{\sqrt{1-q_{r\mathrm{3,1}}^T{q}_{r\mathrm{3,1}}}}& \frac{q_{r\mathrm{3,2}}}{\sqrt{1-q_{r\mathrm{3,2}}^T{q}_{r\mathrm{3,2}}}}& \·\·\·& \frac{q_{r3,N}}{\sqrt{1-q_{r3,N}^T{q}_{r3,N}}}\end{array}\right\}$

Wherein, the group number of the Quaternion Sequence of m=0,1,2,3, N star sensor output;

(2) reference of the star sensor output Quaternion Sequence of inflight measurement star sensor and step (1) obtained exports the inverse multiplication carrying out hypercomplex number of Quaternion Sequence, obtains deviation hypercomplex number;

(3) the deviation hypercomplex number that step (2) obtains is converted to Eulerian angle, obtains the departure δ φ of the Eulerian angle parametric form around the X-axis of star sensor surving coordinate system, Y-axis, Z axis thus _{x, k}, δ φ _{y, k}, δ φ _{z, k}as total error Total Error, concrete computation process is:

Attitude matrix A (q) is changed into Quaternion Matrix form:

$A\left(q\right)=\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]=\left[\begin{array}{ccc}{q_0}^2+{q_1}^2-{q_2}^2-{q_2}^2& 2(q_1{q}_2+q_0{q}_3)& 2(q_1{q}_3-q_0{q}_2)\\ 2(q_1{q}_2-q_0{q}_3)& {q_0}^2-{q_1}^2+{q_2}^2-{q_3}^2& 2(q_2{q}_3+q_0{q}_1)\\ 2(q_1{q}_3+q_0{q}_2)& 2(q_2{q}_3-q_0{q}_1)& {q_0}^2-{q_1}^2-{q_2}^2+{q_3}^2\end{array}\right]$

Utilize Eulerian angle x, y, z to describe the attitude tends of star sensor three axle, when turning sequence and being 3-1-2, be calculated as follows with the attitude matrix A that attitude angle is expressed:

$\begin{array}{c}A=R_y\left(y\right)R_x\left(x\right)R_z\left(z\right)=\left[\begin{array}{ccc}\cos \left(y\right)& 0& -\sin \left(y\right)\\ 0& 1& 0\\ \sin \left(y\right)& 0& \cos \left(y\right)\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \left(x\right)& \sin \left(x\right)\\ 0& -\sin \left(x\right)& \cos \left(x\right)\end{array}\right]\left[\begin{array}{ccc}\cos \left(z\right)& \sin \left(z\right)& 0\\ -\sin \left(z\right)& \cos \left(z\right)& 0\\ 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{ccc}\cos \left(y\right)\cos \left(z\right)-\sin \left(x\right)\sin \left(y\right)\sin \left(z\right)& c\left(y\right)\mathrm{som}\left(z\right)+\sin \left(x\right)\sin \left(y\right)\cos \left(z\right)& -\cos \left(x\right)\sin \left(y\right)\\ -\cos \left(x\right)\sin \left(z\right)& \cos (x\cos (z& \sin \left(x\right)\\ \sin \left(y\right)\cos \left(z\right)+\sin \left(x\right)\cos \left(y\right)\sin \left(z\right)& \sin \left(y\right)\sin \left(z\right)-\sin \left(x\right)\cos \left(y\right)\cos \left(z\right)& \cos \left(x\right)\cos \left(y\right)\end{array}\right]\end{array}$

Then by A (q)=A, the relation turning Eulerian angle and Direct cosine matrix under sequence according to 3-1-2 obtains utilizing the attitude angle of star sensor three axle of quaternion representation:

$x=\arctan [-\frac{A_{21}}{A_{22}}]=\arctan [-\frac{2(q_1{q}_2-q_0{q}_3)}{{q_0}^2-{q_1}^2+{q_2}^2-{q_3}^2}]$

y＝arcsin[A ₂₃]＝arcsin[2(q ₂q ₃+q ₀q ₁)]

$z=\arctan [-\frac{A_{13}}{A_{33}}]=\arctan [-\frac{2(q_1{q}_3-q_0{q}_2)}{{q_0}^2-{q_1}^2-{q_2}^2+{q_3}^2}]$

Then correspond to each moment t _kstar sensor three axle departure δ φ _{x, k}, δ φ _{y, k}, δ φ _{z, k}, above formula and deviation hypercomplex number δ q can be utilized _kobtain accordingly;

(4) carry out digital filtering to the total error Total Error that step (3) obtains, obtain low frequency aberration LFE, digital filtering adopts running mean method, and specific formula for calculation is as follows:

$f\left[k\right]=\frac{1}{L}\underset{L-M}{\overset{L+M}{\mathrm{\Σ}}}\mathrm{\φ}\left[k\right]$

Wherein L=2M+1 is counting of filtering running mean, and f is mean value, and k is data point position, and φ is the actual numerical value at k place, and M is integer, will when calculating directly substitute into φ [k];

(5) the output Quaternion Sequence of the star sensor adopting ground test stage static star simulator to measure replaces the star sensor of inflight measurement to export Quaternion Sequence, and use rear sampling instant data of each element in the output Quaternion Sequence of star sensor to replace with reference to exporting Quaternion Sequence, repeat step (1) ~ (3), obtain the stochastic error of the Eulerian angle parametric form around the X-axis of star sensor surving coordinate system, Y-axis, Z axis thus, concrete steps are as follows:

(51) according to the star sensor hypercomplex number observed quantity of ground test phase sequence, utilizes the attitude transfer function of moment k and adjacent moment k+1 obtain differential quaternary Number Sequence represent taking advantage of of hypercomplex number;

(52) at each moment t _k, turn sequence from differential quaternary number δ q by 3-1-2 _kbe converted to around star sensor surving coordinate system X, Y, Z tri-difference component of Eulerian angle parametric form of axle

(53) basis sequence, provides 3 times of standard deviation statistics indexs of the quick random meausrement error of reflection star final stochastic error amount TE is

(6) formula is passed through calculate high frequency error HFE.