( ) ( ) R O , i , j , k R O , i , j , k : it is - - PDF document

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Star-Tracker Attitude Measurement Model

Basilio BONA, Enrico CANUTO Dipartimento di Automatica e Informatica, Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino, Italy

• tel. 011 564 7026, fax 011 564 7099

bona@polito.it

1 Star-Tracker

The Star-Tracker provides the measurements of S/C attitude angles to be compared with the estimated ones in order to produce the attitude error used in the control algorithm. The following reference frames are applicable

1.1.1 Inertial Reference Frame – J2000

( )

J2000 J2000 J2000 J2000 J2000

, , , O R i j k

is an Earth centered equatorial inertial frame, with:

• J2000

O

at the centre of the Earth.

• J2000

i

along the intersection of the mean ecliptic plane with the mean equatorial plane, at the date of 01/01/2000; positive direction is towards the vernal equinox.

• J2000

k

• rthogonal to the mean equatorial plane, at the date of 01/01/2000; positive direction is

towards the north.

• J2000

j

completes the reference frame.

1.1.2 Spacecraft Reference Frame – SC

( )

SC SC SC SC SC

, , , O R i j k

: it is assumed coincident with the gradiometer

( )

GR GR GR GR GR

, , , O R i j k

refence frame from DFACS point of view:

• SC

GR

O O =

at the intersection of the nominal gradiometer axes.

• SC

i

along the launch vehicle axis; positive direction is towards the launch vehicle nose.

• SC

k

• rthogonal to the satellite earth face; positive direction is towards nadir.
• SC

j

completes the reference frame.

1.1.3 Gradiometer RF – GR

( )

GR GR GR GR GR

, , , O R i j k

is a local non-inertial satellite reference frame, with:

• GR

O

at the intersection of the nominal gradiometer axes.

• GR

i

nominally parallel to SC

i

.

• GR

k

nominally parallel to

SC

k

.

• GR

j

completes the reference frame.

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1.1.4 Spacecraft Alignment Reference Frame – AL

( )

AL AL AL AL AL

, , , O R i j k

is the RF for alignment measurements of all equipments. The alignment RF is embodied by a master reference cube on the satellite:

• AL

O

• n the master reference cube.
• AL

AL AL

, , i j k

parallel to SC

SC SC

, , i j k

.

1.1.5 Star-Tracker Alignment Reference Frame – STAL

( )

STAL STAL STAL STAL STAL

, , , O R i j k

is a local satellite non-inertial frame

1.1.6 Star-Tracker Measurement Reference Frame – STME

( )

STME STME STME STME STME

, , , O R i j k

is a local satellite non-inertial frame, defined by optical system and focal plane of Star-Tracker:

SC

k

is aligned with the optical axis.

1.2 Quaternions

The attitude parameters used in this context are the quaternions. Quaternions q are defined as an ordered quadruple of real numbers

( ) ( )

1 2 3

, , , ,

v r

q q q q q =

• q

q

(1.1) where

( )

1 2 3

, ,

v

q q q = q

is called the “vectorial part” and

r

q q =

is called the “real part”. The relations between the elements of a rotation matrix R and its quaternion q are given by:

( ) ( ) ( )

11 22 33 1 32 23 2 13 31 3 21 12

1 1 2 1 4 1 4 1 4 q r r r q r r q q r r q q r r q = ± + + + = − = − = −

(1.2) where

ij

r is the (i,j) element of R . A quaternion is also related to the Euler parameters by the following

relations:

1 1 2 2 3 3

cos , sin , sin , sin 2 2 2 2 q q u q u q u α α α α = = = =

(1.3) where

1 2 3

u u u ′ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ u

is the spatial rotation versor and α is the rotation angle. A quaternion is said to be unitary if

3 2

1

k k

q

=

=

∑

• q

.

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Given a unitary quaternion q , the corresponding rotation matrix R can be computed as:

( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2 2 2 1 2 3 1 2 3 1 3 2 0 2 2 2 2 1 2 3 0 1 2 3 2 3 1 0 2 2 2 2 1 3 2 0 2 3 1 0 1 2 3

2 2 2 2 2 2 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ⎡ ⎤ − − + − + ⎢ ⎥ ⎢ ⎥ = + − + − + − ⎢ ⎥ ⎢ ⎥ − + − − + + ⎢ ⎥ ⎣ ⎦ R q

(1.4) Given n rotation matrices

1 2

, ,

n

• R

R R and their quaternions

1 2

, ,

n

• h

h h , the product matrix

1 2 n

=

• R

R R R

(1.5) is associated to the product quaternion

1 2 n

=

• h

h h h

(1.6) where

( ) ( )

1 2 3 1 2 3 1 3 2 1 1 3 2 1 1 2 2 3 1 2 2 3 1 2 3 2 1 3 3 2 1 3

h h h h g g g g g h h h h h g g g g g h h h h h g g g g g h h h h h g g g g g h ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ − − − − − − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = = = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ hg F h g F g h

(1.7)

1.3 Star-Tracker Model

The instrument gives the attitude measurement of

STME

R

with respect to

J2000

R

, i.e.

J2000 STME

• R

, defined as

J2000 J2000 STME STME E

=

• R

R R

(1.8) where

E

R is a rotation error matrix that takes into account systematic and random noises at STME level.

For small errors

d

E E

= + R I R , where d

E E E E E E E

ψ θ ψ ϕ θ ϕ ⎡ ⎤ − ⎢ ⎥ ⎢ ⎥ = − ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ R

(1.9) The rotation matrix from J2000 to STME is the following

J2000 J2000 SC AL STAL STME SC AL STAL STME

= R R R R R

(1.10) According to (1.7), eqn. (1.8) is translated into quaternion representation as:

( )

J2000 J2000 STME STME E

= Ω

• q

q q

(1.11) where, due to small angle errors:

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1 2 2 2 1 2 2 2 ( ) 1 2 2 2 1 2 2 2

E E E E E E ERR E E E E E E

ψ θ ϕ ψ ϕ ϑ θ ϕ ψ ϕ ϑ ψ ⎡ ⎤ ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ Ω = ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − − − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ q

(1.12) Each components of eqn. (1.10) is now expressed in terms of its (small) angular errors:

( )

SC SC AL AL SC SC SC SC SC SC AL AL AL AL AL AL SC SC AL AL

d ; d ψ θ ψ ϕ θ ϕ ⎡ ⎤ − ⎢ ⎥ ⎢ ⎥ = + = − ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ R R I R R

(1.13)

( )

AL AL STAL STAL AL AL AL AL AL AL STAL STAL STAL STAL STAL STAL AL AL STAL STAL

d ; d ψ θ ψ ϕ θ ϕ ⎡ ⎤ − ⎢ ⎥ ⎢ ⎥ = + = − ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ R R I R R

(1.14)

( )

STAL STAL STME STME STAL STAL STAL STAL STAL STAL STME STME STME STME STME STME STAL STAL STME STME

d ; d ψ θ ψ ϕ θ ϕ ⎡ ⎤ − ⎢ ⎥ ⎢ ⎥ = + = − ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ R R I R R

(1.15) For DFACS purposes, the generic vector of angular errors from

A

R

to

B

R

, indicated by

A A A B B B

ϕ θ ψ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ,

can be assumed as a random vector, with zero mean, uniformly distributed and uncorrelated. The half amplitude of the uniform distribution is indicated as

A B

e .

Assuming a sampling period

S

T , tha attitude measured by the Star-Tracker at the k-th control cycle is:

( ) ( )

J2000 J2000 SC AL STAL STME SC AL STAL STME S D E S

kT T kT = −

• R

R R R R R

(1.16) where

D

T is the total delay time due to Star-Tracker measurement and elaboration.

In quaternion form, taking into account relations (1.5), (1.6) and (1.7), eqn. (1.16) becomes:

( ) ( )

( ) ( ) ( ) ( )

( )

J2000 STAL AL SC J2000 STME STME STAL AL SC S E S S D

kT kT kT T = Ω Ω Ω Ω −

• q

q q q q q

(1.17) where

J2000 SC

q

is the simulated satellite attitude,

A B

q are the generic nominal rotation values, expressed as

quaternions, and:

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( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2

z S y S x S z S x S y S E S y S x S z S x S y S z S

kT kT kT kT kT kT kT kT kT kT kT kT kT ε ε ε ε ε ε ε ε ε ε ε ε ⎡ ⎤ ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ Ω = ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − − − ⎢ ⎥ ⎣ ⎦ q

(1.18) The generic error (

)

i S

kT ε

, where i denotes the generic x, y or z component, can be written as a sum of three terms:

( ) ( ) ( )

i S i iC S iW S

kT kT kT ε ε ε ε = + +

(1.19) where:

• i

ε is a random gaussian zero mean variable, with standard deviation

i

σ

( ) ( ) ( )

2 2 2 SC AL STAL AL STAL STME

3 3 3

i

e e e σ = + +

(1.20)

• (

)

iC S

kT ε

is a gaussian stochastic process, with PSD equal to

( )

2

;

iC i

C f σ ω

, where f is the focal length and ω is the S/C angular velocity. A discrete approximation is given by the following expression:

( ) ( )

( )

( )

( )

( )

( )

( )

2

1 1 1 1

C S

iC S iC S iC S iC S K T v

kT p k T p k T n k T p e ε ε σ

−

= − + − − − =

(1.21) where

iC

n

is a random white noise with zero mean and standard deviation

1

n

σ =

,

C

K is a

parameter that takes into account the relation between the image speed on focal plane and the correlation time, v is the mean star image velocity on the focal plane

( )

v v ω =

.

• (

)

iW S

kT ε

is a white gaussian stochastic process with power equal to

( )

( )

2 exp; iW i

C T v σ ω

. Assuming

exp

T

constant, a first approximation is given by:

2 3 1 2 3

1

i i

a v a v a v σ σ ⎛ ⎞ ⎟ ⎜ = + + + ⎟ ⎜ ⎟ ⎝ ⎠

(1.22) Furthermore it is assumed that

• processes

( )

iC k

ε

and

( )

iW k

ε

are not correlated;

• processes

( )

xC k

ε

,

( )

yC k

ε

and

( )

zC k

ε

are not correlated;

• processes

( )

xW k

ε

,

( )

yW k

ε

and

( )

zW k

ε

are not correlated. Disregarding

i

ε effects, v is related to

SC

ω

by:

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2 2 ,STME ,STME x Y

v ω ω = +

(1.23) where

( )

,STME ,SC SC AL STAL ,STME AL STAL STME ,SC ,STME ,SC x x y y z z

ω ω ω ω ω ω ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ′ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ R R R

(1.24)