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

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 ( ) R O , i , j , k is an Earth centered equatorial inertial frame, with: J2000 J2000 J2000 J2000 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 J2000 01/01/2000; positive direction is towards the vernal equinox. k � orthogonal to the mean equatorial plane, at the date of 01/01/2000; positive direction is J2000 towards the north. j � completes the reference frame. J2000 1.1.2 Spacecraft Reference Frame – SC ( ) ( ) R O , i , j , k R O , i , j , k : it is assumed coincident with the gradiometer refence SC SC SC SC SC GR GR GR GR GR frame from DFACS point of view: = O O � at the intersection of the nominal gradiometer axes. SC GR i � SC along the launch vehicle axis; positive direction is towards the launch vehicle nose. k � orthogonal to the satellite earth face; positive direction is towards nadir. SC j � completes the reference frame. SC 1.1.3 Gradiometer RF – GR ( ) R i j k O , , , is a local non-inertial satellite reference frame, with: GR GR GR GR GR � O at the intersection of the nominal gradiometer axes. GR i i � nominally parallel to SC . GR k k � nominally parallel to . GR SC � j completes the reference frame. GR Star Tracker_BB.doc data creazione 08/ 05/ 2001 8.29.00 Pagina 1 di 6 data ultima revisione: 20/ 10/ 2010 18.02.00

1.1.4 Spacecraft Alignment Reference Frame – AL ( ) R O , i , j , k is the RF for alignment measurements of all equipments. The alignment RF is AL AL AL AL AL embodied by a master reference cube on the satellite: O � on the master reference cube. AL i , j , k i , j , k � parallel to SC . AL AL AL SC SC 1.1.5 Star-Tracker Alignment Reference Frame – STAL ( ) R O , i , j , k is a local satellite non-inertial frame STAL STAL STAL STAL STAL 1.1.6 Star-Tracker Measurement Reference Frame – STME ( ) R O , i , j , k is a local satellite non-inertial frame, defined by optical system and STME STME STME STME STME k focal plane of Star-Tracker: is aligned with the optical axis. SC 1.2 Quaternions The attitude parameters used in this context are the quaternions. Quaternions q are defined as an ordered quadruple of real numbers ( ) ( ) q � q , q , q , q = q , q (1.1) 1 2 3 0 v r ( ) q = q , q , q q = q where is called the “vectorial part” and is called the “real part”. v 1 2 3 r 0 The relations between the elements of a rotation matrix R and its quaternion q are given by: 1 q = ± 1 + r + r + r 0 11 22 33 2 1 ( ) q = r − r 1 32 23 4 q 0 (1.2) 1 ( ) q = r − r 2 13 31 4 q 0 1 ( ) q = r − r 3 21 12 4 q 0 r is the ( i,j ) element of R . A quaternion is also related to the Euler parameters by the following where ij relations: α α α α = = = = q cos , q u sin , q u sin , q u sin (1.3) 0 2 1 1 2 2 2 2 3 3 2 u ′ ⎡ ⎤ = ⎢ is the spatial rotation versor and α is the rotation angle. u u u where ⎥ ⎣ ⎦ 1 2 3 3 ∑ 2 = q � q 1 A quaternion is said to be unitary if . k k = 0 Star Tracker_BB.doc data creazione 08/ 05/ 2001 8.29.00 Pagina 2 di 6 data ultima revisione: 20/ 10/ 2010 18.02.00

Given a unitary quaternion q , the corresponding rotation matrix R can be computed as: ⎡ ( ) ( ) ⎤ 2 − 2 − 2 + 2 − + q q q q 2 q q q q 2 q q q q ⎢ ⎥ 1 2 3 0 1 2 3 0 1 3 2 0 ⎢ ⎥ ( ) ( ) ( ) = + − 2 + 2 − 2 + 2 − R q 2 q q q q q q q q 2 q q q q ⎢ ⎥ (1.4) 1 2 3 0 1 2 3 0 2 3 1 0 ⎢ ⎥ ( ) ( ) 2 2 2 2 2 q q − q q 2 q q + q q − q − q + q + q ⎢ ⎥ ⎣ ⎦ 1 3 2 0 2 3 1 0 1 2 3 0 R R R and their quaternions h h h , the product matrix , , � , , � Given n rotation matrices 1 2 n 1 2 n = R R R � R (1.5) 1 2 n is associated to the product quaternion h = h h � h (1.6) 1 2 n where ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ h − h − h − h g g − g − g − g h ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 1 2 3 0 0 1 2 3 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ h h − h h g g g g − g h ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ( ) ( ) = 1 0 3 2 1 = = 1 0 3 2 1 = hg F h g F g h ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1.7) h h h − h g g − g g g h 1 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 3 0 1 2 2 3 0 1 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ h − h h h g g g − g g h ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 3 2 1 0 3 3 2 1 0 3 1.3 Star-Tracker Model � R R J2000 R The instrument gives the attitude measurement of with respect to , i.e. , defined as STME J2000 STME � J2000 = J2000 R R R (1.8) STME STME E R is a rotation error matrix that takes into account systematic and random noises at STME level. where E = + R I d R , where For small errors E E ⎡ ⎤ − ψ θ 0 ⎢ ⎥ E E ⎢ ⎥ = ψ − ϕ d R 0 ⎢ ⎥ (1.9) E E E ⎢ ⎥ − θ ϕ 0 ⎢ ⎥ ⎣ ⎦ E E The rotation matrix from J2000 to STME is the following R J2000 = R J2000 R SC R AL R STAL (1.10) STME SC AL STAL STME According to (1.7), eqn. (1.8) is translated into quaternion representation as: ( ) J2000 = Ω J2000 � q q q (1.11) STME E STME where, due to small angle errors: Star Tracker_BB.doc data creazione 08/ 05/ 2001 8.29.00 Pagina 3 di 6 data ultima revisione: 20/ 10/ 2010 18.02.00

⎡ ⎤ ψ θ ϕ ⎢ ⎥ − 1 E E E ⎢ ⎥ 2 2 2 ⎢ ⎥ ψ ϕ ϑ ⎢ ⎥ − ⎢ E 1 E E ⎥ ⎢ ⎥ 2 2 2 Ω = ⎢ ( ) q (1.12) ⎥ θ ϕ ψ ERR ⎢ ⎥ − E E 1 E ⎢ ⎥ 2 2 2 ⎢ ⎥ ϕ ϑ ψ ⎢ ⎥ ⎢ − − − ⎥ E E E 1 ⎢ ⎥ ⎣ 2 2 2 ⎦ Each components of eqn. (1.10) is now expressed in terms of its (small) angular errors: ⎡ ⎤ − ψ SC θ SC 0 ⎢ ⎥ AL AL ( ) ⎢ ⎥ SC SC SC SC SC SC R = R I + d R ; d R = ψ 0 − ϕ ⎢ ⎥ (1.13) AL AL AL AL AL AL ⎢ ⎥ − θ SC ϕ SC ⎢ 0 ⎥ ⎣ ⎦ AL AL ⎡ ⎤ − ψ AL θ AL 0 ⎢ ⎥ STAL STAL ( ) ⎢ ⎥ AL AL AL AL AL AL R = R I + d R ; d R = ψ 0 − ϕ ⎢ ⎥ (1.14) STAL STAL STAL STAL STAL STAL ⎢ ⎥ − θ AL ϕ AL ⎢ 0 ⎥ ⎣ ⎦ STAL STAL ⎡ ⎤ − ψ STAL θ STAL 0 ⎢ ⎥ STME STME ( ) ⎢ ⎥ STAL STAL STAL STAL STAL STAL R = R I + d R ; d R = ψ 0 − ϕ ⎢ ⎥ (1.15) STME STME STME STME STME STME ⎢ ⎥ − θ STAL ϕ STAL ⎢ 0 ⎥ ⎣ ⎦ STME STME ⎡ ⎤ A A A R R ϕ θ ψ For DFACS purposes, the generic vector of angular errors from to , indicated by ⎦ , ⎢ ⎥ ⎣ A B B B B can be assumed as a random vector, with zero mean, uniformly distributed and uncorrelated. The half A e . amplitude of the uniform distribution is indicated as B T , tha attitude measured by the Star-Tracker at the k -th control cycle is: Assuming a sampling period S ( ) ( ) � J2000 = J2000 − SC AL STAL R R kT T R R R R kT (1.16) STME SC S D AL STAL STME E S T is the total delay time due to Star-Tracker measurement and elaboration. where D In quaternion form, taking into account relations (1.5), (1.6) and (1.7), eqn. (1.16) becomes: ( ) ( ) ( ) ( ) ( ) ( ) ( ) q � J2000 = Ω q Ω q STAL Ω q AL Ω q SC q J2000 − kT kT kT T (1.17) STME S E S STME STAL AL SC S D q J2000 q are the generic nominal rotation values, expressed as A where is the simulated satellite attitude, SC B quaternions, and: Star Tracker_BB.doc data creazione 08/ 05/ 2001 8.29.00 Pagina 4 di 6 data ultima revisione: 20/ 10/ 2010 18.02.00