Two crucial aspects of the data processing of the recently proposed DIGISTAR II and DIGISTAR. III multiple field-of-view (FOV) star trackers, which use one/two ...
Star Pattern Recognition and Mirror Assembly Misalignment for DIGISTAR II and III Multiple FOVs Star Sensors Daniele Mortari†, and Michela Angelucci‡ Two crucial aspects of the data processing of the recently proposed DIGISTAR II and DIGISTAR III multiple field-of-view (FOV) star trackers, which use one/two mirrors deflecting the sensor FOV to two/three orthogonal directions, respectively, are here analyzed. These aspects are the star identification process and the estimation of the mirror misalignment. In particular it is shown how to adapt the existing Search-Less Algorithm (SLA), which does not require searching phases, to these sensors and, in order to keep high the predicted gain in the attitude accuracy, two different methods to compute the mirror misalignment are provided. The first method, which evaluates the misalignment by a least-square approach, is used if the misalignment can be considered small enough, that is, such that the star-identification process can still be accomplished. The second method, which is used for any value of the mirror misalignment, implies, however, that the star identification process can be performed, separately, for each sub-FOV. Results and tests for the SLA adaptation as well as for the two methods to evaluate the mirror misalignment, are given.
Introduction Presently, Charge-Coupled Device (CCD) star trackers are the sensors providing the most accurate directions thanks to the improvement obtained by the star light defocusing technique, which has greatly increased the precision of the starlight direction identified by statistical methods. The recently proposed1 DIGISTAR II and DIGISTAR III multiple field-of-view (FOV) star trackers, which use one/two mirrors deflecting the sensor FOV to two/three orthogonal directions, are demonstrated to provide the attitude estimation algorithm with the optimal condition data set of nearly orthogonal observed directions. Figure 1 illustrates the FOV geometry for both DIGISTAR II and III sensors, and Fig. 2 shows the mirror displacement for DIGISTAR II. These new star sensors, which belong to a new type of small stars trackers2, allow a substantial improvement in the obtainable attitude accuracy1 (up to 28 times more than that of an equivalent-FOV sensor) as well as in the instrument's operating time. Even if the problem of identifying stars observed by star trackers has been tackled several times and in a variety of different manners for the last two decades3-15, at the present time, however, the star pattern recognition technique, when a multiple FOV star sensor is used, has still not developed. Just recently, however, a promising new star identification approach, 29 ad-hoc devised for DIGISTAR II †
Assistant Professor, Università degli Studi "La Sapienza" di Roma, Via Salaria 851, 00138 Roma, Italy. AIAA and AAS member, Tel.: +39-6-812-0529, FAX: +39 (06) 810-5241. http://crarisc1.psm.uniroma1.it/~daniele/ ‡ Graduate student, Università degli Studi "La Sapienza" di Roma, Via Salaria 851, 00138 Roma, Italy. Tel.: +39 (06) 812-0529, FAX: +39 (06) 810-5241. Paper AAS 99-182 of the 9th AAS/AIAA Space Flight Mechanics Meeting, Breckenridge, CA, February 7-10, 1999.
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and III sensors, is emerging, even though further studies to validate it are needed. In particular, this algorithm transforms the star identification problem into the problem of finding which are the admissible stars with a given direction, which seems to be a simpler problem.
Fig. 1 DIGISTAR II and III FOVs
Fig. 2 Conceptual DIGISTAR II Camera
This paper provides an adaptation to DIGISTAR II and III of the SLA15 star pattern identification (which is based on the construction of a proper vector of indices called the K-vector) and, in order to keep high the predicted gain in the attitude accuracy, it presents two different approaches to quantify the mirror assembly misalignment. Results and tests of the proposed methods are given. A DIGISTAR III star tracker with its optical axes pointing toward the x, y and z orthogonal directions, will be used for the test. In particular, let y and z be the deflected directions obtained by the mirror assembly, which is built as a rigid piece (block) containing the two mirrors. This means that, even though DIGISTAR III uses two deflective mirrors, it has only one misalignment problem, as DIGISTAR II. The purpose of the following two sections, therefore, is to estimate the misalignment of the mirror assembly for both the DIGISTAR II and DIGISTAR III star sensors.
Small Mirror Misalignment This section provides a procedure to compute the mirror misalignment matrix, which can be used when the misalignment can be considered small. The term small means that the mirror misalignment is such that the star identification process can still be accomplished (at least once). In the next section the large misalignment case, will be analyzed. Due to mirror assembly misalignment, the observed stars seen in the y and z directions all appear rotated by the misalignment matrix M. Let us consider a mirror misalignment small enough that the misalignment matrix can be well approximated by the linear misalignment matrix ML where ~ M L ≅ I 3×3 − ϑ (1) ~ and I3×3 and ϑ are the 3×3 unit matrix and the skew-symmetric matrix performing the vector cross product, respectively. Let V ( x ) ≡ [v1( x ) v 2( x ) " v n( xx ) ] , V ( y ) ≡ [v1( y ) v 2( y ) " v n( yy ) ] and V ( z ) ≡ [v1( z ) v 2( z ) " v n( zz ) ] (2) be the matrices containing the three inertial sets of stars, and S ( x ) ≡ [ s1( x ) s 2( x ) " s n( xx ) ] , S m( y ) ≡ [ s1( y ) s 2( y ) " s n( yy ) ] and S m( z ) ≡ [ s1( z ) s 2( z ) " s n( zz ) ]
(3)
be the corresponding matrices formed with the observed star directions of each sub-FOVs. The subscript m indicates that the stars associated with the corresponding sub-FOVs are affected by the mirror misalignment. 2
Let S ( y ) = M L S m( y ) = M L [ s1( y ) s 2( y ) " s n( yy ) ] and S ( z ) = M L S m( z ) = M L [ s1( z ) s 2( z ) " s n( zz ) ]
(4)
be the matrices containing the true observed deflected stars. This implies the following identities V ( x ) TV ( y ) = S ( x ) T S ( y ) = S ( x ) T M L S m( y ) (5) ( x)T ( z ) V V = S ( x ) T S ( z ) = S ( x ) T M L S m( z ) which, setting V ( m ) ≡ [V ( y ) V ( z ) ], S ( m ) ≡ [ S ( y ) S ( z ) ], and S m( m ) ≡ [ S m( y ) S m( z ) ] , can be written in the more compact form V ( x ) TV ( m ) = S ( x ) T S ( m ) = S ( x ) T M L S m( m ) (6) Using the linearized expression for the matrix misalignment of Eq (1), it is possible to write ~ V ( x ) TV ( m ) = S ( x ) T M L S m( m ) ≅ S ( x ) T S m( m ) + S ( x ) T S m( m ) ϑ where ~ ~ ~ S m( m ) = [ S m( y ) S m( z ) ]
~ S m( y ) = [~s1( y ) ~ (z) ~ ( z) S m = [ s1
and
~ s2( y ) " ~ sn(yy ) ] ~ s ( z) " ~ s ( z) ] 2
(7)
(8)
nz
Equation (7) leads to the least square solution ~ ~ ~ ϑ = {[ S ( x ) T S m( m ) ]T [ S ( x ) T S m( m ) ]}-1[ S ( x ) T S m( m ) ]T [V ( x ) TV ( m ) − S ( x ) T S m( m ) ] (9) The misalignment matrix ML computed using Eq. (1) is not orthogonal. However, it is known that Wahba's problem16, that is, the problem to find the optimal attitude matrix A from an attitude data set described by the attitude matrix B, and the problem to find the orthogonal matrix that is closest to a given one, in the sense of the Frobenius norm (also known as the Euclidean, or Schur, or Hilbert-Schmidt norm), are equivalent. This means that the existing procedures devised to solve Wahba's problem17-27, can all be used to find the orthogonal matrix MC closest to ML. This fact allows us to use one of the existing methods solving Wahba's problem. In particular, we have chosen the Energy Approach Algorithm (EAA) method,25 hereafter summarized. In this case the EAA method provides the following four equivalent solutions = ( M L M LT ) 1 / 2 M L-T = UΛU T M L-T T −1 / 2 M L = UΛ −1U T M L = ( M L M L ) MC = (10) T −1 / 2 = M L DΛ −1 D T = M L ( M L M L ) = M -T ( M T M ) 1 / 2 = M -T DΛD T L L L L where matrices U and D are the orthogonal eigenvector matrices of the symmetric matrices M L M LT and M LT M L , respectively ( M L M LT )U = UΛ 2 (11) T ( M L M L ) D = DΛ 2 and Λ is the diagonal matrix containing the eigenvalues of ML. We underline the fact that, in the case of all observed directions provided with the same precision (which occurs in the case under examination) the optimal attitude satisfying Wahba's optimality criterion and that computed with a least square method, are identical. Finally, it is to be noted that this method can be iterated by updating Eq. (8) by the MC computed with Eq. (10). Numerical tests The procedure to estimate the misalignment matrix, above described, is numerically tested using MATLAB28 software. Figures 3 and 4 show the numerical results of the proposed technique. These 3
Figures give the average results, obtained in N=1000 random tests, of the expected error E{MC,M} between the true M and the computed MC misalignment matrices π E{M C , M }= cos −1 [ trace( M CT M ) − 1] / 2 (12) 4
{
}
Fig. 3 Small misalignment, DIGISTAR II
Fig. 4 Small misalignment, DIGISTAR III
The expected error E{MC,M} gives us the information on the average distance between the misalignment matrix M and the computed misalignment matrix MC. In the performed tests the misalignment matrix M is built with random principal axes and principal angle Φ=0.4/π deg. (which implies E{M, I3×3}=Φπ/4=0.1 deg.).
Large Mirror Misalignment In the ideal case (star tracker observing with absolute precision) it is possible to write S ( x ) = AV ( x ) and S m( m ) = AmV ( m ) (13) where, due to mirror misalignment, the attitude matrix A differs from the attitude matrix Am. These two matrices will differ by the misalignment matrix M. In fact if ( x) ( x) S = AV then A = MS m( m )V ( m ) −1 = MAm (14) ( m) (m) MS m = AV and, therefore, M = AAmT (15) demonstrating the statement. In the real case, which implies a star tracker with finite precision, the Eqs. (13-14) are mathematically incorrect. However, they can still be considered correct but in the least square meaning, which is the case of the optimal attitude estimation problem. This implies that it is still possible the use of Eq. (15) by computing, separately, the A and Am matrices, which is the way proposed to tackle the large mirror misalignment problem in this section. Therefore, in the case of a large value for the mirror misalignment, the technique consists in performing the two star identification processes for S(x) and S m( m ) ≡ [ S m( y ) S m( z ) ] , separately, that is in identifying the matrices V(x) and V(m), then computing the attitude matrices A and Am and, finally, evaluating the misalignment using Eq. (15). This misalignment matrix should then be refined using the technique described in the "Small Mirror Misalignment" section. 4
Numerical tests Figures 5 and 6, similarly to Figs. 3-4, show the numerical results provided by the above described method to evaluate a large misalignment which is here considered with E{M,I3×3}=Φπ/4=1 degree. These Figures are produced using MATLAB28 software and they use the same assumptions adopted for Figs. 3-4. They give the average results, obtained in N=1000 tests, of the expected error E{MC,M} between a random misalignment M (built with a random principal axis and a principal angle = 4/π deg.) and the computed misalignment MC, given in Eq. (12).
Fig. 5 Large misalignment, DIGISTAR II
Fig. 6 Large misalignment, DIGISTAR III
Star Pattern Recognition A recent approach for star identification15, named the "Search-Less Algorithm" (SLA), is here demonstrated to be adaptable to the multiple FOVs DIGISTAR II and III star trackers. SLA is made of two subsequent identification processes: The "K-vector" Star Pairs Identification Technique This technique, which is summarized in this section, presents the indubitable advantage of avoiding the searching phases which characterize almost all of the existing alternative techniques. This is achieved thanks to a suitable n-long integer vector K, which provides, in a straightforward manner, the indices of the admissible star-pairs range which match with the observed ones. The method is general, and can be applied to any sorted n-long real vector Y, in which we want to identify all of the elements falling between (y–δ) and (y+δ). The vector K is built as follows. The straight line connecting the two extremes [1, Y(1)] and [n, Y(n)], has, on average, one element Y(i) for each D=[Y(n)– Y(1)]/(n–1) step. Let us consider the slightly steeper line which connects the two points [1, Y(1)–D/2] and [n, Y(n)+D/2]. This line, which simplifies the code by avoiding many index checks, assures K(1)=0 and K(n)=n, as it will be clear later. The equation of this line is a = nD /(n − 1) y = a1i + a0 where 1 (16) a 0 = Y (1) − a1 − D / 2 where i=1-n. Starting with K(1)=0, an integer vector K is then built as follows K (i ) = j where Y ( j ) ≤ a1i + a 0 < Y ( j + 1) (17) 5
From a practical point of view, the ith element of the K vector represents the number of elements Y(i) below the value Y(i)=a1i+a0. Once this vector is built, the evaluation of the two indices identifying, in the Y vector, the range of all the possible catalog star-pairs matching with the observed one, becomes an easy and fast task. The indices associated with these values in the Y vector are simply provided as jbot = bot{[ x − δ − a 0 ] / a1} (18) jtop = top{[ x + δ − a 0 ] / a1 } where the function top{z} is defined as the larger integer number next to z, and bot{z} is the integer number immediately below z. Once jbot and jtop are evaluated, it is possible to compute k start = K ( jbot ) + 1 and k end = K ( jtop ) (19) kstart and kend represent the extremes of the searched indices set k start ≤ k ≤ k end (20) The K-vector can easily be applied to the star pairs identification. The Y vector, therefore, contains the sorted cosines of all the observable star pair angles, and the (y–δ) and (y+δ) range limits represent cos(ϑ+2γ) and cos(ϑ–2γ), respectively, where ϑ is the angle between two observed stars and γ is the sensor precision. Ref. 16 contains a more detailed explanation of the K-vector technique. The Star Matching Identification Technique As a result of the K-vector technique application to a multiple-FOV star sensor, two different kinds of index vectors, that is, the set of [IA, JA, KA ] and the set of [IB, JB, KB], are provided. The first star pair kind is that associated with the case that both stars belong to the same FOV (same optical axis; for instance they are both directed along the "x-axis" FOV) whereas the second star pair kind is associated with the case when both stars belong to different FOVs (different optical axes; for instance one star is observed along the "x-axis" FOV while the other belongs to the "y-axis" FOV). The star pattern recognition philosophy here proposed is mainly an adaptation of the original SLA identification method. However, the used procedure differs from SLA in the star matching identification process. While SLA16 performs this task using a "reference star" and the star angular separations (with the other observed stars) our method uses angular separations among subsets of observed star triads. A proper cleaning phase has also been added. The main steps of the proposed star matching identification (for the si, sj, and sk observed star set) can be summarized as follows: 1. The K-vector method can be applied to all of the three different combinations: [si - sj], [si - sk], and [sj - sk] with the appropriate K-vector (vector KA when both stars belong to the same FOV and KB when the stars belong to different FOVs). For each observed star pair, the K-vector outputs the range [kstart - kend] of the indices as described by Eqs. (19,20). The kstart and kend indices allow then a fast identification of the admissible star pair set, using the I and J vectors of indices. The i-j star pair has the indices given by I[K(kij)] and J[K(kij)], where kij indicates the [kstart - kend] range. 2. All of the indices related to all of the three different combinations ([si - sj], [si - sk], and [sj - sk]), can be arranged in a m×2 matrix H of indices I [ K (k ij )] J [ K (k ij )] H = I [ K (k ik )] J [ K (k ik )] (21) jk jk I [ K (k )] J [ K (k )] 3. Now, since the true index of each observed star must appear twice in the H matrix (because it matches the other two) all of the rows, corresponding to all of the indices occurring only once in the H matrix, can be deleted. This new cleaning procedure allows us to proceed with a reduced H matrix which is, in almost all of the cases, consisting of only three rows: the right ones. 6
4. If the number of the H matrix rows after the cleaning procedure is three, then the index occurring twice in the rows 1 and 2 identifies the ith star, the index occurring twice in the rows 1 and 3 identifies the jth star, and the index occurring twice in the rows 2 and 3 identifies the kth star. 5. When the three selected stars are such that one (or two) of them is (are) already identified, then such information is used to identify the other two (or one) stars. 6. When all of the combinations of three stars have been considered, then the identification process ends and, using the identified stars, it will be possible to evaluate the spacecraft attitude. The above described identification method is reliable, fast and easy to code. Numerical tests Figures 7, 8 and 9 show some of the main characteristics of the above star identification procedure applied to the multiple FOVs DIGISTAR III star sensor. These figures, which have been produced using MATLAB28, show the average values obtained with N=1000 random tests. The DIGISTAR III under consideration has three orthogonal FOVs with aperture of 9.2176 deg, that is, it covers the same observed sky portion of an equivalent star sensor having a single FOV of 16 deg. The CCD of such an instrument has a uniform error distribution with maximum error of 10 arcsec. The magnitude threshold is such that the instrument observes stars down to magnitude 3.2. Figure 7 provides an idea on the instrument operability by plotting the histogram of how the observed stars are distributed in the three available FOVs. Three cases may occur: 1. when all the observed stars fall within the same FOV (Single operating mode). In this case the instrument works as a standard single FOV star sensor, 2. when the stars fall into two FOVs (Dual operating mode), in which case the DIGISTAR III simply works as a DIGISTAR II sensor, and 3. when the stars are observed by all the three FOVs (Full operating mode)
Fig. 7 Operability test results, DIGISTAR III 7
Fig. 8 Speed test results, DIGISTAR III Figure 8 shows the average values (continuous line) of the overall cumulative number of floating point operations needed to identify stars, as a function of the number of observed stars.
Fig. 9 Accuracy test results, DIGISTAR III 8
Finally, Figure 9 shows the average accuracy (continuous line), expressed in terms of expected direction error E{A,T} between the true attitude T and the attitude A computed with EAA (which fully complies with Wahba's optimality criterion) as a function of the observed star number. π E{ A, T }= cos −1 [ trace( A TT ) − 1] / 2 (22) 4
{
}
Conclusions This paper presents two algorithms for the mirror misalignment estimation process and an adaptation of the SLA star identification for the recently proposed multiple field-of-view (FOV) star sensors DIGISTAR II and DIGISTAR III, which observe stars in two or three orthogonal FOVs, respectively. The observation of almost orthogonal stars implies an attitude data set close to the best condition (the worst condition consists of parallel observed directions, or thereabout). This leads to a substantial improvement in the attitude estimation accuracy, while the availability of multiple FOVs improves the operability time with respect to a standard equivalent single FOV star sensor. The mirror assembly alignment precision is an important aspect that might become crucial to assure the predicted gain in the attitude determination accuracy. This is why it is important to be able to estimate in-flight the mirror misalignment matrix mainly because it might change with time. This paper provides two different solutions to this problem, the use of which depends on whether the misalignment is such that the star pattern identification process can be accomplished (small misalignment) or not (large misalignment). The star pattern recognition proposed is derived from the SLA identification technique, a very fast method that does not require any searching phases to identify stars within a large star catalog. The original technique, which is here adapted to the multiple FOVs DIGISTAR II and III, is reliable because it is able to identify and discard spikes, and particularly suitable for the lost-in-space case, that is, when no attitude information is available. The proposed method identifies star triads at once. The fact that DIGISTAR II and III sensors observe almost orthogonal stars yields the development of a new promising star pattern recognition29 algorithm, presently under study, which changes the star identification problem into the simpler problem of finding the admissible stars with a given direction. All of the proposed techniques are tested by simulation software and some main characteristics are shown by means of meaningful plots.
Acknowledgments We would warmly like to thank Bruno Bernabei, Carlo Arduini, Daniela Ianni, Antonio de Leo and Chiara Valente for support and advice provided, and we feel indebted to John Junkins, Tom Pollock, and to all of the DIGISTAR project people at Texas A&M University.
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