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Research on Multi update rate Method of Precise Satellite Attitude Determination Based on Gyro and Star-Sensor

Guo Kang1 and Du Xiaojing2 School of Aerospace Engineering,Beijing institute of technology,Beijing,China,100081

Mao Xinyuan3 School of Aerospace Engineering,Beijing institute of technology,Beijing,China,100081

In order to achieve the requirements of high precision and update rate satellite attitude determination, attitude determination solution based on star-sensor and gyro filtering algorithm is proposed. Quaternion model is established for satellite attitude determination, and Extended Kalman Filter is used to complete the fusion of sensor information as well as the correction of attitude error and gyro drift error. For the gyro's update rate is much higher than the star sensor’s, two multi-step-length attitude determination methods which are direct integration method and P-matrix (estimate error covariance matrix) one-step prediction method are designed, during the star-sensor’s output interval, the first method directly uses the gyro’s output to estimate the attitude quaternion, the second method predicts P-matrix (estimate error covariance matrix) additionally based on the first method. Combined with mathematical simulation and analysis, it is concluded that both multi-step-length satellite attitude determination methods can provide higher precision and update rate satellite attitude than fixed-step-length method. Considering the small difference between the estimation precisions of two multi-step-length methods, direct integration method is more applicable than P-matrix one-step prediction methods.

Nomenclature

b = gyro constant drift boC = rotation matrix of orbit coordinate system relative to the satellite body coordinate system

d = gyro time correlative drift

( )F t = State Transition Matrix

H = observation matrix K = filter gain P = estimate error covariance matrix Q = system noise covariance matrix

q = attitude quaternion of the body coordinate system relative to the orbit coordinate system

R = measurement noise covariance matrix T = filtering period V = observation noise vector

、 、 = star-sensor’s measurement noises

W = system noise vector 1 Master student, Beijing institute of technology, 5 South, Zhongguancun Street, Haidian district, Beijing. gk0214@163.com 2 Professor, Beijing institute of technology, 5 South, Zhongguancun Street, Haidian district, Beijing.duxiaojing2633@163.com 3 Master student, Beijing institute of technology, 5 South, Zhongguancun Street, Haidian district, Beijing. inertial@bit.edu.cn

AIAA Guidance, Navigation, and Control Conference08 - 11 August 2011, Portland, Oregon

AIAA 2011-6645

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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X = system state matrix Z = system observation matrix

g = output of gyro

bib = the angular velocity vector of satellite body coordinate system relative to the inertial coordinate

system oio = the angular velocity vector of orbit coordinate system relative to inertial coordinate system

o = the orbital angular velocity bob = the angular velocity vector of satellite body coordinate system relative to orbit coordinate

system

s s s 、 、 = star-sensor’s measurement output

ˆˆ ˆ 、 、 = star-sensor’s estimate output

、 、 = star-sensor’s measurement bias

g = gyro measurement noise

d , b = gyro drift slope noises

= correlative time constant

I. Introduction YRO and star-sensor are commonly used in satellite attitude measurement, these two components have different update rates, gyro output’s frequency is high and star-sensor’s is lower. Attitude is commonly

determined by using the conventional Extended Kalman Filter, filter update rate is equal to the star-sensor’s update rate, and the attitude estimate value remains unchanged during the star-sensor output interval. This method can provide accurate attitude when the angular rate of satellite attitude is low, but if the angular rate is high, because the update period is long, attitude estimate precision will become lower. What is more, when the attitude update frequency required is higher than the update rate of star-sensor, because of the long system update period, attitude information required cannot be provided.1

Considering the gyro’s update rate is much higher than the star-sensor’s, two multi-step-length attitude determination methods which are direct integration method and P-matrix one-step prediction method are designed in this paper, during the star-sensor’s output interval, the first method directly uses the gyro’s output to estimate the attitude quaternion, the second method predicts P-matrix (estimate error covariance matrix) one step ahead based on the first method. Combined with mathematical simulation, it is concluded that both multi-step-length satellite attitude determination methods can provide higher precision and update rate satellite attitude, but considering the practical application, direct integration multi-step-length attitude determination method is more applicable.

This paper is structured as follows: First, we describe the precise satellite attitude determination system, including the error model of gyro and star-sensor measurement, the state equation and observation equation of system; Second, we introduce the filtering algorithm considering multi-step-length, including the basic filtering steps of fixed-step-length method and two multi-step-length methods; Third, we analyze the results of mathematical simulation using the three methods separately;Forth, we make a conclusion of the full paper.

II. Precise satellite attitude determination system

A. Error model of sensors 1) Error model of gyro measurement

The error model of gyro measurement used for designing the filtering algorithm of attitude determination is

bg ib gd b (1)

In the equation, g is the output of gyro; the satellite body coordinate system is rotating relative to the inertial

coordinate system, bib is the component of the rotate angular rate in satellite coordinate system; d is the gyro time

G

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correlative drift; b is the gyro constant drift; g is the gyro measurement noise which is white noise. In this equation,

d and b are defined as

1d

b

d d

b

(2)

In the equation, ,d b are gyro drift slope noises which are described by white noises, 2, 3 is the correlative

time constant. 2) Error model of star-sensor measurement

This paper uses three attitude angles handled by the star-sensor data processing unit as the star-sensor’s output, star-sensor’s measurement bias is the error between measurement output and estimate output.4

ˆ

ˆ

ˆ

s

s

s

(3)

In the equation, s s s 、 、 are star-sensor’s measurement outputs, 、 、 are star-sensor’s

measurement noises, which are described by white noises.

B. System state equation The attitude kinematics equation describes satellite body coordinate system’s rotating relative to the orbit

coordinate system, 5 its quaternion expression form is

1

2bobq q (4)

In the equation, q is the attitude quaternion of the body coordinate system relative to the orbit coordinate

system; bob is extended rotate angular velocity, [0 ( ) ]b b T T

ob ob ; bob is the component of angular

velocity in the satellite body coordinate system, which describes the rotation of the satellite body coordinate system relative to the inertial coordinate system. By its definition,

b b b oob ib o ioC (5)

In the equation, Tb b b b

ib ibx iby ibz is the component of angular velocity in the inertial coordinate

system, which describes the rotation of the satellite body coordinate system relative to the inertial coordinate system;

0 0To

io o is the angular velocity vector of orbit coordinate system relative to the inertial coordinate

system; o is the orbital angular velocity.

The error quaternion q between the real quaternion q and the estimate quaternion q̂ is defined as

0 1 2 3[ ]Tq q q q q , its incremental form is described as

ˆq q q (6) By substitution of Eq.(6) into Eq.(4), we can get:

1 1 1

ˆ ˆ2 2 2

b b bob ob obq q q q (7)

In the equation, bob is the angular velocity error. By the definition of quaternion square, we can get

01 1

ˆ ˆ2 2

b bob ob b

ob

q qq

(8)

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What is more, because error quaternion is a small quantity, we can get the approximation that

0 1 2 3[ ] [1 0 0 0]T Tq q q q q , so

1 1

ˆ ( )2 2

b b bob ob obq o q (9)

In the equation, ( )bobo q is infinitely small quantity of higher order. By substitution Eqs.(8) and(9)

into Eq.(7), and ignoring the small quantity of second order, we can get the linear representation of kinematics equation as follows:

1

13 2 13 13

3

0

1 1ˆ ˆ( )

2 2

0

b b b bob ob ob ob

q

q q q q

q

q

(10)

In the equation, symbol represents the anti-symmetric matrix:

ˆ ˆ0

ˆ ˆ ˆ( ) 0

ˆ ˆ 0

b bobz oby

b b bob obz obx

b boby obx

(11)

Because of gyro’s measurement model Eq.(1), Eq.(10) can be transformed as follows:

13 13

0

1 1 1ˆ( )

2 2 20

bob gq q d b

q

(12)

In the equation, d is the error of d and b is the error of b. So we can choose the vector part of error quaternion, estimate error of gyro’s time correlative drift and constant

drift as the system state variables:

9 1 13[ ]T T T TX q d b (13)

The error state equation can be expressed as follows:

( ) ( ) ( ) ( )X t F t X t W t (14) In the equation,

3 3 3 3

3 3 3 3

3 3 3 3 3 3 9 9

ˆ( ) 0.5 0.5

( ) 0 0

0 0 0

( ) [ 0.5 ]

bob

Tg d b

I I

F t D

W t

(15)

In the equation,1 1 1

[ ]x y z

D diag , is the correlative time constant.

For Extended Kalman Filter algorithm, we need to discretize Eq.(14), and then we can get:

, 1 , 1 1k k k k k kX X W (16)

In the equation, , 1k k is one-step transition matrix from time 1kt to kt .

If we make 1kt t , we can get

1

, 1k

k k XI FT I F T

(17)

In the equation, T is the filtering period; 1kW is system noise vector:

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{ } 0, { , }Tk k j k kjE W E W W Q (18)

In the equation, kQ is system noise covariance matrix. By Eq.(15) we can get:

2 2 23 3 3 3 3 3(0.25 )k g d bQ diag I I I (19)

C. System observation equation By the error model of star-sensor measurement said earlier, we choose the star-sensor’s measurement bias as

the system observation variables:

TZ (20)

The observation equation is expressed as follows: ( , ) ( )Z h X t V t (21) We can linearize and discretize Eq.(20):

k k k kZ H X V (22)

In the equation, kH is observation matrix:

/ 1

3 3 3 3 3 3

( )

[ ( ), ]2 0 0

( )k k k

k kk

k X t X

h X t tH I

X t

(23)

kV is observation noise vector,

{ } 0, { , }Tk k j k kjE V E V V R (24)

In the equation, kR is measurement noise covariance matrix.

Assuming that the star-sensor has the same measurement accuracy in three axis attitude angles which is(3 )k , we can get:

23 3kR k I (25)

III. Filtering algorithm considering multi update rate In common fixed-step-length attitude determination algorithm, output interval of the lowest update rate sensor,

which is star-sensor here, is usually employed as the filtering interval. The estimation value of attitude is keeping invariant during the star-sensor’s output interval. When the star-sensor outputs measurement information, the state variables are updated. The attitude estimate error and gyro drift estimate error are compensated by using the state variables. Then it will go into the next filtering cycle. Considering that the gyro’s update rate is much higher than the star-sensor’s, it will improve the estimation accuracy and update rate of attitude by using the gyro’s output during the star-sensor’s output interval. Two methods estimating attitude by using gyro’s output are employed, both of them are multi-step-length methods. During the star-sensor’s output interval, the direct integration method estimates the attitude quaternion by the transition matrix composed of gyro’s output, but besides using the direct integration method, P-matrix one-step prediction method makes one-step prediction of the P-matrix (estimate error covariance matrix) by using the state transition matrix composed of gyro’s output.

D. Direct integration multi-step-length method Based on the Extended Kalman Filter, considering the different update rate of sensors, it is assumed that the

star-sensor’s output interval T is M times as the gyro’s output interval t , the filtering process can be described as follows: 1) Forecast calculation

At the time 1kt i t (i=1,2,3,…M-1) during the star-sensor’s output interval, we do forecast calculation

using gyro’s output:

ˆˆ ˆ( ) ( , ) ( 1)bobq i t q i (26)

In the equation, ˆ( , )bob t is the quaternion transition matrix:

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4 4

4 4

ˆ0 ( )ˆ( , )

ˆ ˆ( )

b Tb obob b b

ob ob

t I t

(27)

2) Update calculation with observation information

At the time 1kt M t (it also is the time kt ), the star-sensor is outputting data, we can get the current

estimate quaternion by the estimate quaternion at ( 1)t M t :

| 1 ˆˆ ˆ( , ) ( ( 1) )bk k obq t q t M t (28)

The forecast calculation of estimate error covariance matrix is:

| 1 1 1( ) ( )Tk k k kP T P T Q (29)

In the equation, ( )M t is filtering state transition matrix:

( ) ( )T I F t T (30) We do measurement update calculation by using the star-sensor’s output, and the filter gain is:

1| 1 | 1( )T T

k k k k k k k k kK P H H P H R (31)

Update the estimate error covariance matrix:

| 1( ) ( )T Tk k k k k k k k k kP I K H P I K H K R K (32)

Update the state:

ˆk k kX K Z (33)

3) Update the attitude quaternion and gyro drift

After the optimal state variables 13ˆ ˆˆ ˆ[( ) ( ) ( ) ]T T T

k k k kX q d b are calculated, the gyro correlative

drift and constant drift can be corrected as follows:

| 1

| 1

ˆ ˆ ˆ

ˆ ˆ ˆk k k k

k k k k

d d d

b b b

(34)

The equation of attitude quaternion correction is expressed as follows:

| 1ˆ ˆ ˆ( ) (( ) ) ( )k k k kq q q (35)

Considering the constraint that the norm of quaternion is 1, we can get:

13 13

13

ˆ ˆ1 ( )ˆ( )ˆ

Tk k

k

k

q qq

q

(36)

After the correction of attitude and drifts, the state variables X̂ reset to zeros. This method estimates the attitude quaternion by integrating the gyro’s output directly during the star-sensor’s

output interval, and updates the state variables when the star-sensor outputs measurement information. Compared with the conventional fixed-step-length method, this method doesn’t increase too much computation.

E. P-matrix one-step prediction multi-step-length method The interval of star-sensor and gyro is defined as it in section 2.1, the filtering process can be described as

follows: 1) Forecast calculation

At the time 1kt i t (i=1,2,3,…M-1) during the star-sensor’s output interval, we use gyro’s output to predict

the attitude quaternion, because we don’t have observation information, we don’t update the state:

ˆ ˆ( ) ( 1)

ˆ ˆ( ) ( 1)

b i b i

d i d i

(37)

According to the gyro’s output, the P-matrix which is estimate error covariance matrix can be predicted as follows:

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1( ) ( ) ( 1) ( )TkP i t P i t Q (38)

In the equation, ( )t is the filtering state transition matrix:

( ) ( )t I F t t (39) 2) Update calculation with observation information

At the time 1kt M t (it also is the time kt ), the star-sensor is outputting data, so we can get the current

estimate quaternion and P-matrix by them at ( 1)t M t :

| 1

| 1 1

ˆˆ ˆ( , ) ( ( 1) )

( ) ( ( 1) ) ( )

bk k ob

Tk k k

q t q t M t

P t P t M t t Q

(40)

According to the Eqs.(31),(32) and(33), we can update the state by star-sensor’s output. 3) Update the attitude quaternion and gyro drift

After the optimal state variables 13ˆ ˆˆ ˆ[( ) ( ) ( ) ]T T T

k k k kX q d b are calculated, as the section 2.1

said, we can correct the attitude quaternion and gyro’s drifts by Eqs.(34),(35) and(36). After the correction of

attitude and drifts, the state variables X̂ reset to zeros. This method uses the gyro’s output to estimate the attitude quaternion and to predict the P-matrix during the

star-sensor’s output interval. Compared with the direct integration multi-step-length method, this method adds the process of P-matrix one-step prediction but doesn’t increase too much computation.

IV. Mathematical simulation and analysis

F. Comparison of the multi-step-length method and fixed-step-method

The simulation parameters of gyro are: gyro constant drift 3 /b h , gyro constant drift white noise mean

square deviation 30.03 /b h , time correlative drift initial value 0 0.1 /d h , time correlative drift white

noise mean square deviation 30.1 /d h , correlation time 1h , measurement white noise mean square

deviation 0.02 /g h , update rate 20gf Hz ; The simulation parameters of star-sensor are: measurement

white noise mean square deviation 3s , update rate 20sf Hz ; other simulation parameters are: orbital

angular velocity 31 10 /o rad s , initial state variables 0 9 1ˆ 0 0 0

TX

, initial error covariance matrix

60 9 9ˆ 1 10P I

, simulation time 600s , simulation step-length 0.2s .

The first simulation is at the low dynamic condition that the satellite attitude angular velocity 41 10 deg/b

ob s , the second simulation is at the high dynamic condition that the satellite attitude angular

velocity bob changes according to the sine rule, amplitude 0.005deg/A s ,frequency 0.1f Hz .

The simulation results of three methods in low dynamic condition are shown as follows, Fig.1 shows the real three-axis attitude angular, Fig.2 shows the three-axis attitude angular estimation errors of three determination methods; table1 shows the standard deviation of the estimation errors:

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Figure 1 Real three-axis attitude angular in low dynamic condition

Figure 2 Three-axis attitude angular estimation errors of three methods in low dynamic condition

0 100 200 300 400 500 6000

0.05

0.1

time,s

gam

a,de

g

0 100 200 300 400 500 6000

0.05

0.1

time,s

thet

a,de

g

0 100 200 300 400 500 6000

0.05

0.1

time,s

psi,d

eg

0 100 200 300 400 500 600-5

0

5

10

time,s

gam

a,"

0 100 200 300 400 500 600-10

0

10

time,s

thet

a,"

0 100 200 300 400 500 600-10

0

10

time,s

psi,"

Fixed-step-length method

Direct integration methodP-matrix one-step prediction method

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Table 1. Standard deviation of estimation errors in low dynamic condition（ '' ）

Standard Deviation( )

Fixed-step-length method

Direct integration multi-step-length method

P-matrix one-step prediction multi-step-length method

Roll 0.65 0.66 0.75 Pitch 1.10 1.06 1.22 Yaw 1.23 1.37 1.58

The simulation results of high dynamic condition are shown as follows, Fig.3 shows the real three-axis attitude

angular, Fig.4 shows the three-axis attitude angular estimation errors of three determination methods; table2 shows the standard deviation of the estimation errors:

Figure 3 Real three-axis attitude angular in high dynamic condition

0 100 200 300 400 500 600-0.02

0

0.02

time,s

gam

a,de

g

0 100 200 300 400 500 600-0.02

0

0.02

time,s

thet

a,de

g

0 100 200 300 400 500 600-0.02

0

0.02

time,s

psi,d

eg

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Figure 4 Three-axis attitude angular estimation errors of three methods in high dynamic condition

Table 2. Standard deviation of estimation errors in high dynamic condition（ '' ）

Standard Deviation( )

Fixed-step-length method

Direct integration multi-step-length method

P-matrix one-step prediction multi-step-length method

Roll 2.09 1.12 1.15 Pitch 2.33 1.32 1.36 Yaw 2.45 1.75 1.83

From the simulation results, we can make three conclusions as follows: First, all the three methods can provide high precision satellite attitude information when the satellite is in low

dynamic condition; Second, when the satellite is in high dynamic condition, the direct integration method has the same precision as

the P-matrix one-step prediction method, but the fixed-step-length method has lower precision than the two multi-step-length methods obviously. This is because during the star-sensor’s output interval, the attitude estimatation value of fixed-step-length method keeps invariant, but by using the gyro’s output which has direct

relationship with the attitude angular velocity bob , the two multi-step-length methods are more sensitive to the

dynamic changes of the satellite attitude; Third, no matter in low or high dynamic condition, the attitude update rate of fixed-step-length method is equal

to the observation variables’ update rate which is the star-sensor’s update rate 5Hz, but both the two

0 100 200 300 400 500 600-10

0

10

time,s

gam

a,"

0 100 200 300 400 500 600-10

0

10

time,s

thet

a,"

0 100 200 300 400 500 600-10

0

10

time,s

psi,"

Fixed-step-length method

Direct integration method

P-matrix one-step prediction method

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multi-step-length methods can provide satellite attitude information whose update rate is equal to the gyro’s update rate 20Hz.

In sum, we can conclude that both multi-step-length methods are more applicable to the high precision and high update rate satellite attitude determination than fixed-step-length method.

G. Comparison of the two multi-step-length methods The first simulation uses high precision gyro, the simulation parameters of gyro are: gyro constant drift

3 /b h , gyro constant drift white noise mean square deviation 30.03 /b h , time correlative drift initial

value 0 0.1 /d h , time correlative drift white noise mean square deviation 30.1 /d h , correlation time

1h , measurement white noise mean square deviation 0.02 /g h , update rate 20gf Hz ; the second

simulation uses medium precision gyro, except correlation time and update rate gf , the values of gyro’s

simulation parameters are 10 times large as the values of gyro’s parameters in the first simulation. Other parameters of the two simulations are same, including star-sensor’s measurement white noise mean

square deviation 3s , update rate 20sf Hz , orbital angular velocity 31 10 /o rad s , initial state

variables 0 9 1ˆ 0 0 0

TX

, initial error covariance matrix 6

0 9 9ˆ 1 10P I

, simulation time 600s ,

simulation step-length 0.2s The results of simulation using high precision gyro are shown as Fig.1, Fig.2 and table1 which are the results

of two multi-step-length methods in low dynamic condition. The results of simulation using medium precision gyro are shown as follows, Figs.5 shows the real three-axis

attitude angular, Fig.6 shows the three-axis attitude angular estimation errors of two methods; table3 shows the standard deviation of the estimation errors:

Figure 5 Real three-axis attitude angular using medium precision gyro

0 100 200 300 400 500 6000

0.5

1

time,s

gam

a,de

g

0 100 200 300 400 500 6000

0.5

1

time,s

thet

a,de

g

0 100 200 300 400 500 6000

0.5

1

time,s

psi,d

eg

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Figure 6 Three-axis attitude estimation errors of two methods using medium precision gyro

Table 3. Standard deviation of estimate errors of two methods using medium precision gyro（ '' ）

Standard Deviation( ) Direct integration method P-matrix one-step prediction method

Roll 1.11 1.31 Pitch 1.16 1.40 Yaw 1.22 1.49

From the simulation results, we can make two conclusions as follows: First, when the gyro’s precision is high, the direct integration method’s calculation precision is as high as the

P-matrix one-step prediction method’s, this is because during the star-sensor’s output interval, the difference of two methods’ estimation error brought by the gyro’s drift is very small;

Second, when the gyro’s precision is medium, the direct integration method’s calculation precision is a little higher than the P-matrix one-step prediction method’s, about 0.2 arcsecond(1 ) in three-axis. This is because after estimating the attitude quaternion, P-matrix one-step prediction method updates P-matrix additionally by using the gyro’s output which has measurement errors. The errors are big enough to make obvious difference between the estimation precisions of two multi-step-length methods.

In sum, considering that the estimation precision of direct integration method is a little higher than it of P-matrix one-step prediction method in any simulation condition, we hold that direct integration method is more applicable to the high precision and update rate satellite attitude determination.

0 100 200 300 400 500 600-5

0

5

time,s

gam

a,"

0 100 200 300 400 500 600-5

0

5

10

time,s

thet

a,"

0 100 200 300 400 500 600-10

0

10

time,s

psi,"

Direct integration method

P-matrix one-step prediction method

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V. Conclusion By analyzing the measurement characteristics of attitude sensors, this paper designs multi-step-length satellite

attitude determination methods considering the different update rates of sensors. During the star-sensor’s output interval, the gyro’s output is used to estimate satellite attitude and predict the P-matrix, the state variables updating and error correction are completed after the star-sensor outputs measurement information. The mathematical simulation shows that compared with fixed-step-length method, both two multi-step-length methods can provide high precision and update rate attitude information. Considering that the estimation precision of direct integration method is a little higher than it of P-matrix one-step prediction method in any simulation condition, we hold that direct integration multi-step-length method is more applicable to the high precision and update rate satellite attitude determination.

References

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3Friedland, Control System Design:An Introduction to State Space Methods, McGraw-Hill, New York, 1986. Periodicals

4Yang Feng, Zhou Zongxi, and Liu Shuguang, “Satellite Attitude Determination Algorithm Based on Star-sensor and FOG,” Control Engineering of China, Vol.13, No.4, 2006, pp. 374-376. (In Chinese) Books

5Zhang Renwei, Dynamics and Control of Satellite Orbit and Attitude, Press of Beijing University of Aeronautics and Astronautics, Beijing, 1998, Chaps 5,6. (In Chinese)