Acta Astronautica 66 (2010) 508 -- 515

Contents lists available at ScienceDirect

Acta Astronautica

journal homepage: www.e lsev ier .com/ locate /ac taast ro

An analytical approach to star identification reliability

Mrinal Kumar∗, Daniele Mortari, John L. Junkins

Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA

A R T I C L E I N F O A B S T R A C T

Article history:Received 24 March 2009Received in revised form26 June 2009Accepted 11 July 2009Available online 6 August 2009

Keywords:Star identificationSpherical trigonometryAnalytical modelsMonte Carlo estimates

The problem of real-time, on-board star pattern identification is considered which mustprecede any spacecraft attitude estimation algorithm based on measured line of sight di-rections to stars. Following the use of the search-less k-vector method to access feasiblecandidate stars (detailed in a previous paper) a tiered logical structure is introduced inwhich inter-star angles for pairs, triples and general elementary star configurations areused to match measured patterns to corresponding patterns in a star catalog. Analyticalexpressions are developed for the expected frequency of matching an observed star patternwith an incorrect pattern in the star catalog due to measurement error. Such knowledgeof erroneous match frequencies can be used to rigorously terminate the star identificationprocess with a virtually certain match, while obviating the need for expensive Monte Carlosimulations. Developments shown are supported with a few simulation results.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

This paper provides the mathematical tools which estab-lish in a closed form, the expected frequency that a givenmeasured star pattern may be matched to an invalid set ofstars in the catalog due to measurement error. We considerthe case of no prior information, so all conceivable star pat-terns from the whole sky must be considered as candidatesfor mismatches. It is usually required to conduct expensiveMonte Carlo simulations to incorporatemeasurement uncer-tainty in the star-identification process. These simulationsbecome increasingly computationally expensive (and even-tually unfeasible) with increase in the number of stars inthe pattern. This manifestation of the curse of dimension-ality (the number of trials required for a converged resultgrows exponentially with every added star in the pattern) istreated in this paper by developing analytical estimates forthe frequency of incorrect matches for general star-patterns.Knowledge of such estimates is valuable because a negligibly

∗ Corresponding author.E-mail addresses: mrinal@neo.tamu.edu (M. Kumar),

mortari@aeromail.tamu.edu (D. Mortari), junkins@tamu.edu (J.L. Junkins)URL: http://people.tamu.edu/∼mrinal (M. Kumar).

0094-5765/$ - see front matter © 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.actaastro.2009.07.005

small estimated probability of mismatch allows the termina-tion of the star-identification process with virtual certainty.

Typical star-identification algorithms involve construc-tion of star-patterns using known stars followed by match-ing the pattern to available database to determine the errorprobability [1]. An alternate method is to match each indi-vidual star to a signature pattern in the viewing section ofthe sky using its sensor image [1,6]. In the current work, weare interested in the former approach based on matchingstar-patterns (typically spherical polygons) to the catalog. Atthe same time, we consider the case of no a priori informa-tion, so that the entire catalog is considered while countingincorrectly matched patterns. Kosik [2] described several al-gorithms in which star polygons are analyzed by looking atthe various star-pairs. The most popular technique probablyis to consider the case of star-triangles andmatch the patternto the onboard catalog [3,4,7]. The missing element from thebulk of the existing literature is adequate statistical analysis.The current work aims to derive a compact set of formulaethat estimate the frequency of matching an observed starpattern to an incorrect set of stars in the catalog. Such errorscan occur simply due to measurement error. We considerthe general problem of a star-polygon with n stars and keepthe option of incorporating partial or complete information

M. Kumar et al. / Acta Astronautica 66 (2010) 508 -- 515 509

celestial spherecelestial spherebi

j

i kl

vij vikvil

An Open Star Structure with Four Stars.

bi

j

i kl

vijvikvilvjlvkl

vjk

A Star Tetrahedron (Four Star ClosedPolygon)

Fig. 1. Open and closed star patterns. (a) An open star structure with four stars. (b) A star tetrahedron (four star closed polygon).

about the polygon. In the former case of partial information(see Fig. 1(a)), inter-star angles are considered only betweenthe central star paired individually with all other stars, lead-ing to an open-star structure. In the latter case (Fig. 1(b)),all possible inter-star angles in the closed star-pattern areconsidered before a mismatch frequency is estimated. In aprevious work, [10] the authors laid out the principles ofthe pyramid star-identification technique along with thepreliminaries of analytical mismatch frequency estimation.The current paper extends this work to compute analyticalexpressions for mismatch frequencies for star-polygonscomprising of 3, 4, and in general, n stars. Following thesearch-less k-vector method to access feasible candidatestars [5,9], a tiered logical structure is introduced in whichinter-star angles for pairs, triples and general elementarypolygons are used to match measured star patterns tocorresponding patterns in a star catalog.

A star polygon consisting of n stars is defined by the setof M =∑n−1

k=1 k= n(n− 1)/2 inter-star angles associated witha “spherical polygon”. Examples are pairs (n = 2), triangles(n = 3), pyramids (n = 4), or more stars (n>4). The actualmeasurements made during star-identification are the setof n line of sight vectors, {(bi, bj) : (i, j) ∈ {1, 2, . . . ,n}} whichpoint from the sensor toward the vertices of the star poly-gon on the celestial sphere. The various inter-star angles arecalculated using the expression �ij=�ji=cos−1(bTi bj). The ob-jective of star-identification then is to find correspondencebetween the measured and cataloged star indices. Once thiscorrespondence is achieved within a specified tolerance,the star-identification process may be terminated. Note thattermination of the identification process does not guaranteethe match of individual stars to those in the catalog, espe-cially since no a priori information has been assumed. Itis probable (with the calculated probability) that the mea-sured pattern was matched to an invalid set of stars thatmake the same pattern in the catalog. However, if computedprobability of such a match is extremely low (say 10−20), theidentification process can be successfully terminated. Hav-ing analytical expressions for calculating probabilities ofsuch order is useful because Monte Carlo simulations wouldrequire a prohibitive number of trials for convergence to

such low orders of magnitude. Computational statistical-inference approaches based on random sampling becomeinfeasible if one is pursuing frequencies on the order of10−7 or smaller, because of the necessity of performing avery large number (certainly >108) of trials to establish astatistically significant estimate for such rare events. Indeed,it would be desirable to reduce the expected frequency ofa random star identification to be significantly less thanunity over the lifetime of a given mission [i.e., 1/(# of staridentifications over the lifetime of a multi-year mission),which for the anticipated high frame rate active pixel cam-eras corresponds to a frequency less than 10−10], which ispossible using the current approach.

2. Expected match frequency of measured star patternto invalid star polygons in catalog

Let us consider the entire sky with a uniform star distri-bution. This implies that the star density � (which dependson the given magnitude threshold m) is simply given as

�(m) = N(m)4�

(1)

where N(m) is the overall number of stars with magnitudeless thanm. The relationship between themagnitude thresh-oldm and ln{N(m)} can be approximated [9] by the followinglinear expression:

m�0.8985 lnN − 2.0474 ± 0.2 (2)

The above is a least-squares straight line fit. A quadraticleast-squares fit provides a better approximation as

m�0.0126(lnN)2 + 0.7109 lnN − 1.3734 ± 0.1 (3)

Fig. 2 shows the residuals between the linear and quadraticbest fits together with the associated standard deviations.We conclude that Eq. (2) provides an adequate approxima-tion, especially forN>1000, because the camera determinedmagnitude of a star seldom matches to within 0.1 of the

510 M. Kumar et al. / Acta Astronautica 66 (2010) 508 -- 515

Fig. 2. Star magnitude fitting using linear and quadratic functions.

catalog value.1 It is well known that the actual star distri-bution is not uniform in practice. Studies of the actual stardensity indicate that variations in density by a factor of twoor more can occur when averaged over the typical size of astar camera field of view. We can take these variations intoaccount by conservatively interpreting the frequency resultsobtained. However, our studies indicate that the uniformdensity approximation is more than adequate for order ofmagnitude expected frequency analysis. We note that a fu-ture extension of this work can address the known depar-ture of the actual star position density distribution from theuniform distribution assumed herein, which ultimately willlead to increased accuracy in the further generalized analo-gies of the frequency formulas derived herein.

We now proceed to obtain the error frequency in match-ing star-patterns to the catalog. We begin with the simpleststar-pattern, namely the star-pair having an inter-star an-gle of �ij. The measurement error is modeled by the angulardistance K�, where �2 is the centroiding error variance ofthe instrument and K is a chosen amplification factor. It hasbeen suggested in Mortari et al. [10] that a value of K = 6.4is a good choice based on the analysis of data from the Star-Nav I experiment. Further details can be obtained from thereferred paper. It suffices to say here that the chosen ampli-fication factor works even for actual non-uniform star den-sities and provides conservative estimates for the mismatchfrequencies. We also note that while it is theoretically pos-sible that measurement error of the instrument depend onthe inter-star angle, the magnitude of such error is vanish-ingly small to be of consequence for the current application.

1 Note that the frequency appearing in these residuals depend onthe fact that the star catalog provides the magnitude information withprecision truncated to 0.1.

For the sake of establishing continuity, we briefly red-erive the mismatch frequency for the star-pair from Mor-tari et al. [10]. To this end, consider Fig. 3(a) which showsa star indexed i as the pivot. An infinitesimal area dS(�ij) isshown, which represents an infinitesimal region on the ce-lestial sphere containing stars that have an angular separa-tion from the pivot in the range [�ij,�ij + d�ij]. Clearly, thenumber of stars in the infinitesimal area dS(�ij) is

dn(�ij) = �∗ dS(�ij) = (N − 1)2

sin�ij d�ij (4)

where �∗ = (N − 1)/4� indicates a uniform star density ob-tained by excluding the pivot star, i. Next, we apply the con-dition that no a priori information about the location of thepivot star is available. It is therefore possible that any oneof the N stars in the catalog be the pivot. We are thus leadto the following expression for the infinitesimal expectedfrequency of mismatches for a star-pair with angular sepa-ration �ij:

df ij(�ij) = 12Ndn(�ij) = N(N − 1)

4sin�ij d�ij (5)

The factor of 0.5 in the above equation compensates for du-plicate counting, i.e. it ensures that the pairs ij and ji arecounted as one. Finally, integrating the infinitesimal areaover the range [�ij − K�,�ij + K�] we obtain the total fre-quency of possible mismatches in star-pairs with nominalangular separation �ij as the following:

f(�ij,K�

)= N(N − 1)4

∫ �ij+K�

�ij−K�sin�d�

= N(N − 1)2

sinK� sin�ij (6)

M. Kumar et al. / Acta Astronautica 66 (2010) 508 -- 515 511

Fig. 3. Infinitesimal areas in consideration of two and three star patterns. (a) Area cut out on the surface of a sphere by a cone of half angle �. (b) Startriangle geometry.

The above equation represents the expected frequency ofstar-pairs “i–j” in the catalog (with the specified inter-starangle �ij) that may be matched incorrectly to the observedstar-pair, to within the measurement precision specified bythe band K�.

3. Expected catalog match frequency for open starpatterns

3.1. Open star pattern with two legs

Consider the case of a three star pattern ijk forming atwo-leg open star structure similar to the one shown inFig. 1(a). We are concerned here with processing only partialinformation about this set of stars. We arbitrarily assign thestar i to be the central star and seek to match the measuredinter-star angles (�ij ± K�) and (�ik ± K�) to the catalog. Inother words, we “pivot” about the i-th measured star. Thus,using Eq. (4), it is easy to evaluate the number of stars, f̄ij,displaced from i by an angle varying between (�ij −K�) and(�ij + K�):

f̄ij =∫ �ij+K�

�ij−K�dn(�) = (N − 1) sinK� sin�ij (7)

Note that the above equation is different from Eq. (6) in thesense that in Eq. (7), the star “i” is fixed, whereas in Eq. (6), allpossible candidates for the i-th star have been considered. Inthe same manner, the expected number f̄ik of stars displacedfrom i by an angle which varies from (�ik −K�) to (�ik +K�)is f̄ik = (N − 1) sinK� sin�ik. Now we consider all possiblecandidates for the i-th star, which can be any one of theN cataloged stars. Doing so, the expected frequency that ameasured star matches two neighboring stars with both legs(star pairs ij and ik) is

fi−(j,k) = Nf̄ij f̄ik = N[(N − 1) sinK�]2 sin�ij sin�ik (8)

Some comments are in place here. Obviously, note that�ij ��ik, or else the two stars j and k would lie in the sameK� strip for star i. Specifically, the validity condition for

Eq. (8) is |�ij − �ik|>2K�. This holds for every star pair ap-pearing in all ensuing analysis. Also, note that Eq. (8) doesnot provide the frequency for invalid match of a star trian-gle. Instead, it provides the frequency of matching only twolegs with one end at star i. Moreover, no duplicate countingoccurs in this case as for star pairs, so no factor appears inEq. (8) as in Eq. (6).

3.2. Open star pattern with M legs

Eq. (8) can be easily generalized to the expected fre-quency that a star matches withM legs (that is, withM otherstars identified by), giving us:

fi−(j1,. . .,jM) = NjM∏j=j1

f̄ij

= N[(N − 1) sinK�]MjM∏j=j1

sin�ij (9)

that complete the searched solution.We reiterate that the above results, Eqs. (8) and (9), do

not match all the inter-star angles in the closed star poly-gon. Therefore these formulas may admit many more ex-pected spurious match events than necessary, because theyhave not used all of the measured information and are thusconservative. The frequency count for closed star patternsis addressed below, beginning with the most fundamentalcase of a three star measured pattern.

In what follows, we develop generalizations of Eq. (6) sothat the objects of interest are general star polygons con-taining n>2 stars.

4. Expected catalog match frequency for closed starpatterns

4.1. Case of three stars (star triangle)

Let us now consider the frequency of random occurrenceassociated with matching all three angles of a star triangle.With reference to Fig. 3(b), let us consider one of the fij star

512 M. Kumar et al. / Acta Astronautica 66 (2010) 508 -- 515

i

θik

j

R

θ i-θi+

θik+

θikθik-

R Sin θikKσk

θjk

Fig. 4. An approximation of the area on the celestial sphere.

pairs, where fij has the expression provided by Eq. (6), andlet us consider the intersection area �A of the two spheri-cal surfaces associated with the angles �ik (centered at thei-star) and �jk (centered at the j-star). Note for the givenmeasured angles, and associated uncertainties, the k-th starmust lie in one of the small areas �A. In order to get an es-timate of the area �A, consider Fig. 4. The area �A is the oneenclosed by the bold solid lines on the surface of the celes-tial sphere. We can write an approximate expression for thisarea as follows:

�Aij−k ≈ (�+i + �−

i )(�+j + �−

j ) sin�ik sin�jk (10)

The angles corresponding to star i have been shown inFig. 4. Angles for star j are defined analogously. Using spher-ical trigonometry, we have the following (exact) expressionsfor the various angles appearing in Eq. (10):

�+ik = cos−1(cosK� cos�ik + sinK� sin�ik cos�k) (11)

�−ik = cos−1(cosK� cos�ik − sinK� sin�ik cos�k) (12)

�+i = sin−1

(sinK� sin�k

sin�+ik

)(13)

�−i = sin−1

(sinK� sin�k

sin�−ik

)(14)

Similar expressions can be obtained for angles �+j and �−

j .Thus, the expected frequency of random occurrence that agiven star triangle is matchedwithinmeasurement precisionis given by

fij−k = fij�∗∗�Aij−k

= N(N − 1)2

sinK� sin�ij(N − 2)

4��Aij−k (15)

where �∗∗ = (N− 2)/(4�) is a modified star density that doesnot take into account the stars i and j that obviously can-not fall into the small �Aij−k cone intersection area. Using

L1L2L3L4

i

k

j

δA

Fig. 5. Evaluating the exact area of intersection.

Eq. (10) to approximate �Aij−k, we have that

fij−k = N(N − 1)(N − 2)8�

(�+i + �−

i )(�+j + �−

j )

× sinK� sin�ij sin�ik sin�jk (16)

We note again that Eq. (10) is an approximate estimate ofthe concerned area of intersection in which the k-th starlies. It is possible to obtain an exact expression in terms ofdifferences of spherical lenses as shown in Fig. 5. Clearly, thearea of concern �A is given by the following expression:

�Aexact = 12 [(L1 − L2) − (L3 − L4)] (17)

Finally, we note that when trying to identify a star triangle,there is a possibility of the existence of a mirror image tri-angle. This possibility can be eliminated by checking to seeif the cataloged triple (r1, r2, r3) is consistent with the associ-ated measured triple (b1, b2, b3). The two triangles are thenconsistent if the following condition

sign[rT1(r2 × r3)] = sign[bT1(b2 × b3)] (18)

is satisfied. Upon checking Eq. (18), only the triple starpattern identifications which passes this test are furtherconsidered.

4.2. Star polygons exceeding three stars

For star patterns exceeding three stars, the analytical ex-pressions are somewhat complicated. The basic idea for anyM star (M>2) structure is to obtain the frequency of find-ing incorrect star matches for the (M − 1) star structure,and then to multiply it with the frequency of obtaining anincorrect match for the M th star relative to the (M − 1)star structure. Note that the area on the celestial sphere inwhich the last (M th) star can be found gets smaller andsmaller as M increases and it becomes increasingly difficultto obtain its exact value. Without introducing any simplify-ing assumptions, the area in which the M th star lies is the

M. Kumar et al. / Acta Astronautica 66 (2010) 508 -- 515 513

intersection of the [−K�,K�] error bands with each of theother (M − 1) stars as pivots. A simple estimate of this areacan be obtained by taking the minimum of all possible areaintersections taking two stars at a time. This allows us toutilize the formula obtained in Eq. (10) and use theminimumover all star-pairs. Note that this is a conservative estimate,since

⋂i �Ai �mini �Ai. Mathematically, we have

fi1i2,. . .,iM−1−(iM) = fi1i2,. . .,iM−2−(iM−1)�M−1�AM (19)

where �M−1 is the star density obtained upon excluding thefirst M− 1 stars, which are known and no longer contributeto uncertainty. Consequently, we have �M−1 = [N− (M−1)]/4�. As described above, �AM is the area in which the M thstar resides, given the (M − 1) star structure. The “two-star”estimate of this area is given by

�AM = mini

{�A(i1, i2),�A(i1, i3), . . . ,�A(i1, iM−1),

. . . ,�A(iM−2, iM−1)} (20)

where each of �A(m,n) is computed using Eq. (10). InEq. (19), there is some subjectivity since we can form the(M − 1) star polygon by leaving out any one of the M stars,leading to identical formulae based on different inter-starangle combinations. However, since we use these equa-tions for order of magnitude analysis only, any of thesefrequency formulas is appropriate. Also, to avoid consid-eration of reflections, each cataloged and measured triplefrom the hypothesized match of p measured and catalogedstars should be tested using Eq. (18), all hypothesized iden-tification of triples that do not pass these tests should berejected.

4.2.1. Example: four-star polygonLet the stars be indexed by the letters i, j, k and l. The

number of possible incorrectly matched triangles is givenby fij−(k) found in the previous section. Clearly then, usingEq. (19), we get the following expression for the frequencyof obtaining incorrect matches for the 4-star polygon:

fijk−(l) = fij−(k)�3 min{�A(i, j),�A(i, k),�A(j, k)} (21)

In Eqs. (6), (16), (21), we see the inherent competitionbetween the size of the catalog (N), and the precision ofthe measurement (K�)—as a larger N tends to increasethe frequencies and smaller (K�) tends to decrease thefrequencies. Looking at the dependency of the frequencyrelationships on the error band K�, it is evident that we candrive the frequency of incorrect random matches arbitrarilysmall if we are able to consider a large number of measuredstars.

5. Expected catalog match frequency for the generalstar structure

For the most general case, the following two formulas:

• Eq. (9), which gives the closed form expression of the fre-quency of a star with M legs and

• Eq. (19), which provides the expression of an M-star poly-gon (of which the simplest case is the triangle),

allow us to build other more complicated cases. Let us giveby means of an example, the general idea on how to pro-ceed. Consider seven measured stars identified by the in-dices i, j, k, l, m, n, and r, and let us quantify the frequencyassociated with only a subset of all the possible consideredlegs. One possibility could be that we consider all the an-gular legs of the triangle (i–j–k), and all the angular legsbetween the stars (l–m–n–r) with the triangle (i–j–k), butnot the angles among the stars (l–m–n–r). Thus we have athree star polygon (i.e. triangle) and four three-leg star pat-terns centered at l, m, n and r respectively. The frequencyfor this selection is simply obtained by the simultaneous ap-plication of the above-mentioned formulae. In this case wewill have

f = fi,j−kfl−(i,j,k)fm−(i,j,k)fn−(i,j,k)fr−(i,j,k) (22)

Note that the above described case is an illustration of howmuch data we wish to consider in obtaining the error prob-ability. If the information left out is significant, the estimatewill be extremely conservative. The point however is thatwe would have the flexibility to consider any star structureconstructed out of a number of measured stars. Generallyspeaking, we should base accepting a hypothesis for staridentification as valid (based on a given pattern of measuredinter-star angles matching those from a cataloged pattern towithin measurement error), when the frequency of a ran-dom match between the measured and cataloged patternsis less than some tolerance. Note we may frequently sat-isfy this criterion with fewer than the maximum number ofinter-star angles among the observed stars.

6. Numerical example

To observe the implications of the above developments,consider the following typical values: N=5000, �=25�rad,k = 6.4. Let us consider five stars indexed by i, j, k, l andm. We consider nominal values for the various measuredinter-star angles, i.e. �ij, �ik, �il, �im, �jk, etc. Various starpolygon results are shown in Table 1 with comparisons toMonte Carlo results. Table 2 shows results for the open starstructure with two, three, and four legs alongside a compar-ison to the Monte Carlo method. We see that the frequen-cies match well with Monte Carlo analysis (when feasible).We mention that it is not feasible to obtain high-confidenceresults for polygons with more than three stars with theMonte Carlo approach because of the large number of sam-ple points required to obtain a converged result. It is clearfrom Table 1 that matching of four or five star patterns re-sults in a very low expected frequency of occurrence that anincorrect random pattern from the star catalog could havematched the measured star pattern. In another way, match-ing a measured star pattern with four or more stars to withinmeasurement precision, we approach engineering certaintythat we have the correct star identification, especially if starpatterns in successive frames have such matches and thereis some overlap in the pattern matched star-fields. We reit-erate that the above analytical results for star polygons ex-ceeding three stars are considerably conservative becauseof the “minimum”-approximation used in Eq. (20) to com-pute the area of intersection in which every additional star

514 M. Kumar et al. / Acta Astronautica 66 (2010) 508 -- 515

Table 1Numerical results for star polygons with two, three, four, and five stars.

� (�rad) Method fij fij−k fijk−l fikjl−m

25 Analytical 845 2.2 × 10−2 3.9 × 10−7 9.2 × 10−12

25 Monte Carlo 846 2.6 × 10−2 Not feasible Not feasible250 Analytical 8450 22 3.8 × 10−2 9.2 × 10−5

250 Monte Carlo 8448 28 4.1 × 10−2 Not feasible/attempted

N = 5000 stars in catalog.

Table 2Numerical results for the star-leg structure pattern match with two, three,and four legs, and star “i” at the vertex.

Method fi−(j,k) fi−(j,k,l) fi−(j,k,l,m)

Analytical 304 67 11Monte Carlo 308 69 11

N = 5000 stars in catalog and � = 25�rad.

introduced lies. It is possible to obtain order-of-magnitudeestimates of how adding stars to a closed pattern improvesmatching probability. This analysis has been performed inthe next section. To get a feel of such conservatism, we in-creased the error band to 2000�rad and reduced the num-ber of stars in the catalog to 2000 stars to perform a roughMonte Carlo counting analysis. We found that the expres-sions provided in this paper led to 32 incorrect matches forthe 4-star polygon while the Monte Carlo analysis (based on1 experiments) resulted in only two false matches. Note thatthe error band (�) will not assume such a high value in areal-life situation.

Also note from Table 2 that the open star leg structureshave high probability of incorrect matches because all inter-star angles are not considered. Furthermore, adding starsto open star-patterns does not lead to a significant reduc-tion in mismatch frequency. This point has been explainedelucidated in the next section on order-of-magnitude anal-ysis. The Monte Carlo results match extremely well in thiscase because there are no simplifying assumptions used inderiving the concerned formulae. Comparing Table 2 to thefirst row of Table 1, it is evident that including all inter-star angles in the star identification pattern match is vital toobtain high confidence identification with only four or fivestars.

7. Order of magnitude analysis

In this section, we perform order of magnitude analysisfor the analytical mismatch-frequency expressions derivedabove. This analysis illustrates the true power of the expres-sions derived above because such insight is not availablefrom a numerical Monte Carlo type simulation. It is possibleto get quick estimates of how many stars would be neededto obtain a virtually perfect (probabilistically speaking) star-identification. It also sheds light on how adding an additionalstar to the star-pattern reduces the estimated mismatch fre-quency and why closed-star patterns offer a better way toperform star identification. We first look at open-star pat-terns, beginning with the star-pair (Eq. (6)) for which the

following order-of-magnitude relation holds:

O[fij] =O

[N(N − 1)

2sinK� sin�ij

]

≈ O

[N2

2× K� × 1

]=O

[N2

2K�

](23)

In a similar fashion, we look at open star-patterns with two(Eq. (8)) and M (Eq. (9)) legs:

O[fi−(jk)] ≈ O[N3(K�)2] (24)

O[fi−(j1,j2,. . .,jM)] ≈ O[NM+1(K�)M] (25)

The above equations clearly illustrate that adding additionalstars to an open-star structure does not help reduce themismatch-frequency order-of-magnitude estimate. This isdue to the fact that adding an additional star (jM) increasesknowledge by only one inter-star angle, namely �ijM . Thisgrowth in information is offset by the increase in the pos-sible number of mismatches. In other words, the rate ofgrowth of information is balanced out by the rate at whichincorrect matches grow due to the addition of one star to anopen star pattern. This is evident from the results shown inTable 2. Although the mismatch frequency is seen to reducewith the addition of new stars, there is no clear order-of-magnitude advantage. The observed reduction in mismatchfrequency is due to the details skipped in the order analysis,e.g. O[N(N − 1)] ≈ O[N2].

Let us now consider the case of closed star-patterns. Wefirst consider the order of the common area �A encounteredwhile considering the star-triangle (Eq. (10)) and shown inFigs. 3(b) and 4. Clearly,

O[�A] ≈ O[(2K�)2] (26)

In obtaining the above equation, Eqs. (11)–(14) have alsobeen used (e.g. O[�+

i ] = O[K�]). Thus we obtain the orderof magnitude for the triangle mismatch frequency fromEq. (15) as follows:

O[fij−k] ≈ O

[N(N − 1)(N − 2)

8�× (K�) × (2K�)2

]

= O

[N3(K�)3

�

](27)

Note the difference in frequency orders between Eqs. (24)and (27). The former is for an open structure with three starswhile the latter is a closed structure with three stars. The in-clusion of the additional inter-star angles in the closed poly-gon causes information to grow faster than the number of

M. Kumar et al. / Acta Astronautica 66 (2010) 508 -- 515 515

unfavorable samples, thus leading to a significant improve-ment. Eq. (21) gives us an approximate order-of-magnitudeestimate for the four-star closed polygon:

O[fijk−l] =O[fij−k × �3 × min{�A(i, j),�A(i, k),�A(j, k)]

≈ O

[N3(K�)3

�× N − 3

4�× (2K�)2

]

≈ O

[N4(K�)5

�2

](28)

Finally, for the M-star closed polygon, we obtain the follow-ing general order-of-magnitude equation for the mismatchfrequency:

O[fi1,i2,. . .,iM−1−iM ] =O

[NM(K�)2M−3

�M−2

](29)

The above equation underlines the advantages of the cur-rent approach. It provides a quick estimate of the level ofaccuracy that can be expected from a star-identificationprocess without having to go through tedious Monte Carlosimulations. The above equation shows that with every ad-ditional star used in a closed polygon, the polygon mismatchfrequency goes down approximately by (at least) an orderof N(K�)2/�. This result is apparent in the results shownin Table 1. For � = 25�rad, we have O[N(K�)2/�] ≈ 10−5

while for � = 250�rad, O[N(K�)2/�] ≈ 10−3. Table 1 clearlyreflects these numbers. The Monte Carlo results (when fea-sible) corroborate these estimates. We thus conclude thatthe estimates obtained in this work are extremely use-ful for obtaining a virtually certain (probabilistically) star-identification.

8. Conclusions

In this paper, we have introduced a novel method toquantify the frequency of false matching in star patternidentification, based on the assumption of uniform star dis-tribution over the celestial sphere. We have introduced an

analytical means to compute the expected frequency of ran-dom occurrence that a cataloged polygon of stars could pos-sibly match, to within camera precision, the given measuredpolygon. This analytical means of computing the expectedfrequency is novel and important to eliminate the need forexpensive and slowly converging Monte Carlo estimates ofstar identification reliability.

In particular, we derive the expected random frequenciesassociated with matching inter-star angles from measuredstar polyhedra. Using recursively the frequency analysis formatching triangles, we show how to develop the expectedfrequency for four and five star patterns with associatedformulas derived for the expected frequency of randommatches to within measurement precision. These formulasare original contributions which permit, for the first time,a rigorous analytical basis for deciding upon the validity ofan identified star pattern with knowledge of the frequencythat an invalid match could occur.

References

[1] C. Padgett, K. Kruetz-Delgado, S. Udomkesmalee, Evaluation of staridentification techniques, Journal of Guidance Control and Dynamics20 (2) (1997) 259–267.

[2] J.C. Kosik, Star pattern identification aboard an inertially stabilizedspacecraft, Journal of Guidance Control and Dynamics 14 (2) (1991)230–235.

[3] E. Groth, A pattern-matching algorithm for two-dimensionalcoordinate lists, Astronomical Journal 91 (1986) 1244–1248.

[4] M.S. Scholl, Star-field identification for autonomous attitudedetermination, Journal of Guidance Control and Dynamics 18 (1)(1995) 61–65.

[5] D. Mortari, Search-less algorithm for star pattern recognition, Journalof the Astronautical Sciences 45 (2) (1997) 179–194.

[6] S. Udomkesmalee, J.W. Alexander, A.F. Tolivar, Stochastic staridentification, Journal of the Astronautical Sciences 17 (6) (1994)1283–1286.

[7] R.W.H. Van Bezooijen, A star pattern recognition algorithm forautonomous attitude determination, in: Proceedings of the EleventhIFAC Symposium on Automatic Control in Aerospace, Isubuka, Japan,1989, pp. 61–70.

[9] D. Mortari, A fast on-board autonomous attitude determinationsystem based on a new star-ID technique for a wide FOV star tracker,Advances in the Astronautical Sciences 93 (Pt. II) (1996) 893–903.

[10] D. Mortari, M.A. Samaan, C. Bruccoleri, J.L. Junkins, The pyramid staridentification technique, Navigation 51 (3) (2004) 171–183.