Star-Tracker Attitude Measurement Model

Transcription

1 Star-Tracker Star-Tracker Attitude Measurement Model Basilio BONA, nrico CANUTO Dipartimento di Automatica e Informatica, Politecnico di Torino Corso Duca degli Abruzzi 4 9 Torino, Italy tel , fax bona@polito.it 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.. Inertial Reference Frame R i j k is an arth centered equatorial inertial frame, with: O at the centre of the arth. i along the intersection of the mean ecliptic plane with the mean equatorial plane, at the date of //; positive direction is towards the vernal equinox. k orthogonal to the mean equatorial plane, at the date of //; positive direction is towards the north. j completes the reference frame... Spacecraft Reference Frame R ( O, i, j, k ) : it is assumed coincident with the gradiometer ( O,,, ) frame from DFACS point of view: R i j k refence O = O at the intersection of the nominal gradiometer axes. i 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. j completes the reference frame...3 Gradiometer RF R i j k is a local non-inertial satellite reference frame, with: O at the intersection of the nominal gradiometer axes. i nominally parallel to i. k nominally parallel to k. j completes the reference frame. Star Tracker_BB.doc data creazione 8/5/ 8.9. Pagina di 6 data ultima revisione: // 8..

2 ..4 Spacecraft Alignment Reference Frame AL R i j k is the RF for alignment measurements of all equipments. The alignment RF is AL embodied by a master reference cube on the satellite: O on the master reference cube. AL i, j, k parallel to i AL, j, k...5 Star-Tracker Alignment Reference Frame STAL R i j k is a local satellite non-inertial frame STAL..6 Star-Tracker Measurement Reference Frame STM R i j k is a local satellite non-inertial frame, defined by optical system and STM STM STM STM STM focal plane of Star-Tracker: k is aligned with the optical axis.. Quaternions The attitude parameters used in this context are the quaternions. Quaternions q are defined as an ordered quadruple of real numbers where = ( q, q, q v 3) ( q, q, q, q 3 ) = (, q v r) q q (.) q is called the vectorial part and q = q is called the real part. r The relations between the elements of a rotation matrix R and its quaternion q are given by: q =± + r + r + r q = ( r r 3 3) 4q q = ( r r 3 3) 4q q = ( r r ) 4q 33 3 (.) where r is the (i,j) element of R. A quaternion is also related to the uler parameters by the following ij relations: q = cos α, q = u sin α, q = u sin α, q = u sin α (.3) 3 3 where u = u u u 3 is the spatial rotation versor and α is the rotation angle. A quaternion is said to be unitary if 3 q q =. k= k Star Tracker_BB.doc data creazione 8/5/ 8.9. Pagina di 6 data ultima revisione: // 8..

3 Given a unitary quaternion q, the corresponding rotation matrix R can be computed as: ( ) ( ) ( qq q q ) q q q q 3 3 ( q q qq 3 ) ( ) ( + ) + + q q q q qq q q qq q q R( q ) = (.4) qq qq qq qq q q q q Given n rotation matrices R, R, R and their quaternions h, h, h, the product matrix n n is associated to the product quaternion R = R R R (.5) n h = h h h (.6) n where h h h h g g g g g h 3 3 h h h h g g g g g h 3 3 hg = ( ) ( ) h h h h g = F h g = g g g g h = F g h (.7) 3 3 h h h h g g g g g h Star-Tracker Model The instrument gives the attitude measurement of R with respect to R STM, i.e. R STM, defined as R = R R (.8) STM STM where R is a rotation error matrix that takes into account systematic and random noises at STM level. For small errors R = I+ dr, where The rotation matrix from to STM is the following ψ θ dr = ψ ϕ (.9) θ ϕ R = R R R R (.) AL STAL STM AL STAL STM According to (.7), eqn. (.8) is translated into quaternion representation as: where, due to small angle errors: STM ( ) q =Ω q q (.) STM Star Tracker_BB.doc data creazione 8/5/ 8.9. Pagina 3 di 6 data ultima revisione: // 8..

4 ψ θ ϕ ψ ϕ ϑ Ω ( q ) = RR θ ϕ ψ (.) ϕ ϑ ψ ach components of eqn. (.) is now expressed in terms of its (small) angular errors: ψ θ R = R ( I+ d RAL) ; dr = ψ ϕ AL θ ϕ (.3) ψ θ AL R = R ( I+ d RSTAL) ; dr = ψ ϕ STAL θ ϕ (.4) ψ θ STM STM R = R STM STM ( I+ d RSTM) ; dr = ψ ϕ STM STM STM θ ϕ STM STM (.5) For DFACS purposes, the generic vector of angular errors from R to R, indicated by A A A ϕ θ ψ A B B B B, can be assumed as a random vector, with zero mean, uniformly distributed and uncorrelated. The half amplitude of the uniform distribution is indicated as e. Assuming a sampling period T, tha attitude measured by the Star-Tracker at the k-th control cycle is: S A B ( kt T ) ( kt ) R = R R R R R (.6) AL STAL STM S D AL STAL STM S where T is the total delay time due to Star-Tracker measurement and elaboration. D In quaternion form, taking into account relations (.5), (.6) and (.7), eqn. (.6) becomes: where quaternions, and: ( kt ) =Ω( ( kt )) Ω( ) Ω S S ( ) Ω( ) ( kt T S D) q q q q q q (.7) ST STM STM ST q is the simulated satellite attitude, A q are the generic nominal rotation values, expressed as B Star Tracker_BB.doc data creazione 8/5/ 8.9. Pagina 4 di 6 data ultima revisione: // 8..

5 The generic error ε ( kt ) terms: where: i ( kt ) ( kt ) ( kt ) ε ε ε z S y S x S ε ( kt ) ε ( kt ) ε ( kt ) z S x S y S ( ( )) Ω q kt = (.8) S εy( kts) εx ( kts) εz ( kts) εx ( kts) εy( kts) εz ( kts) i S, where i denotes the generic x, y or z component, can be written as a sum of three ( kt ) ( kt ) ( kt ) ε = ε + ε + ε (.9) i S i ic S iw S ε is a random gaussian zero mean variable, with standard deviation σ STAL ( e ) ( e AL STAL) ( estm) σ = + + (.) i ε ic ( kts ) is a gaussian stochastic process, with PSD equal to σ C ( f; ) ic i i ω, where f is the focal length and ω is the S/C angular velocity. A discrete approximation is given by the following expression: ( kt ) = p (( k ) T ) + p (( k ) T ) n (( k ) T ) ε ε σ ( K T v C S ) p = e ic S ic S ic S ic S (.) where n is a random white noise with zero mean and standard deviation σ =, K is a ic n C parameter that takes into account the relation between the image speed on focal plane and the v = v ω. correlation time, v is the mean star image velocity on the focal plane ( ) ε ( kt ) is a white gaussian stochastic process with power equal to C T v( ) iw S Assuming T constant, a first approximation is given by: exp iw i ( ; exp ) σ ω. 3 σ = σ a v a v a v i i (.) Furthermore it is assumed that processes ε ic ( k ) and ( k iw ) processes ε xc ( k ), ε yc ( k ) and ( k zc ) processes ε ( k ), ε ( k ) and ( k ) xw yw Disregarding ε effects, v is related to i ε are not correlated; ε are not correlated; ε are not correlated. zw ω by: Star Tracker_BB.doc data creazione 8/5/ 8.9. Pagina 5 di 6 data ultima revisione: // 8..

6 v = ω + ω (.3) x,stm Y,STM where ω ω x,stm x, ω ω y,stm = AL STAL STM y, AL STAL ( ) R R R (.4) ω ω z,stm z, Star Tracker_BB.doc data creazione 8/5/ 8.9. Pagina 6 di 6 data ultima revisione: // 8..