JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICSVol. 18, No. 6, November-December 1995

Roll/Pitch Determination with Scanning Horizon Sensors:Oblateness and Altitude Corrections

Hari B. Hablani*Rockwell International Corporation, Downey, California 90241-7009

Analytical models are developed to relate horizon sensor locations, measurements, and their mounting geometrywith roll/pitch attitude of Earth-pointing spacecraft, negotiating circular orbits around an oblate Earth. Twoarrangements of a pair of horizon sensors are considered: left and right and aft and forward of the velocity vector,tilted in the pitch-yaw or roll-yaw plane, respectively. The corresponding roll/pitch oblateness corrections areformulated that involve 1) noncircularity of the Earth disk seen from space, 2) slight changes in azimuth anglesof the horizon crossing points, and 3) scan angles of the sensors to travel the Earth disk. These corrections arecompared with the closed-form roll/pitch components of the angle between geodetic and geocentric normals atthe instantaneous location of the spacecraft—an alternate approach, also developed in the paper, to determine thecorrections. The two approaches yield nearly the same result if a spacecraft is equipped with a pair of sensors (orfour diagonal sensors). In that instance, the use onboard of the closed-form expressions of oblateness corrections,independent of the sensor geometry and location, suggests itself. But if due to cost, weight, or configurationconstraints, only one Earth sensor is used, the corrections will then depend on the sensor geometry and locationand may be determined for a given orbit before flight and stored numerically to minimize the processing load onthe flight computer. The altitude corrections in the roll/pitch measurements, corresponding to the use of one ortwo sensors and arising from the spacecraft's altitude variations, are also formulated and illustrated in the paper.

I. Introduction

HORIZON sensors have been used for nearly three decadesto measure roll and pitch attitude of Earth-pointing and spin-

stabilized satellites; see, for example, Ref. 1 for TIROS satellites andRef. 2 for more recent satellites wherein a horizon sensor is pairedwith a fixed head star tracker, fine sun sensor, digital sun sensor, ora three-axis magnetometer, depending on the attitude determinationaccuracy desired. In this long span, great strides have been madein horizon sensing techniques, resulting in significantly improvedaccuracy in the attitude determination.2"7 The accuracy of the mea-surements is known to depend on random instrumental errors andon the correction of such deterministic errors as 1) seasonal varia-tion in Earth's radiance (not including mesoscale weather patternsor sudden polar stratospheric warmings5), 2) Earth's oblateness andspacecraft altitude variations, 3) ambient temperature, 4) spin pe-riod change, and 5) misalignment and biases; see, for instance, Alexand Shrivastava6 and Space Sciences7 for the contribution of eachsource to the attitude error budget (0.454 deg rms, total) and how thesum total is reduced to nearly one-eighth amplitude (0.06 deg rms)by onboard corrections. Of the five deterministic error sources justmentioned, this paper focuses on the attitude errors caused by oblate-ness and spacecraft altitude variations and onboard corrections toeliminate these errors. The paper is limited to the scanning sensorson Earth-pointing satellites; static Earth sensors and spin-stabilizedsatellites are not considered.

In the literature, Earth's oblateness has been accounted for in twoways: 1) by determining the crossing points of the sensor scanpathon the oblate Earth disk (Liu,8 Ohtakay and Havens,9 and Collins10)and 2) by determining the deviation of the geodetic normal fromthe geocentric normal (Lebsock and Eterno11). The first approach isindispensable when only one scanning sensor is used for the attitudedetermination. The second approach, on the other hand, is an out-growth of the classical Earth's oblateness modeling technique usedfor inertial navigation of aircraft (Britting,12 Sees. 3.4,4.1, and 4.2).It is likely that both approaches have been modeled, analyzed, andcompared in company reports to determine accurately the roll and

Received Oct. 9,1993; revision received Jan. 13,1995; accepted for publi-cation Jan. 16,1995. Copyright © 1995 by Hari B. Hablani. Published by theAmerican Institute of Aeronautics and Astronautics, Inc., with permission.

* Senior Engineering Specialist, Advanced Programs, Avionics and Soft-ware Group, Space Systems Division, 12214 Lakewood Boulevard. Asso-ciate Fellow AIAA.

pitch attitudes, but the treatment in the open literature is fragmentaryand scattered. Dissatisfied with thready details available, we presentin this paper the models of roll/pitch attitude determination usingscanning horizon sensors—solo or paired in the pitch-yaw or theroll-yaw plane. The contents of the paper are as follows. SectionII deals with roll and pitch attitude determination with a pair ofscanning horizon sensors on an Earth-pointing satellite, orbitinga spherical Earth. Basic relationships are established between themeasurements from the horizon sensors left and right of the forwardvelocity, their tilt and cone angle, and the roll/pitch attitude of thespacecraft. Expressions of the azimuth angles of the four crossingpoints on a circular Earth disk are also formulated. The oblate Earthis considered in Sec. III. A simplified expression of noncircularity ofthe Earth disk, valid up to the first order of the flatness factor, is de-veloped to arrive at the roll/pitch oblateness corrections. Deviationof the geodetic normal from the geocentric normal along the space-craft orbit and its roll/pitch components are formulated in Sec. IVand compared numerically with the roll/pitch corrections developedin Sec. III. Section V presents the roll/pitch attitude determinationand oblateness corrections for the sensors aft and forward of the ve-locity vector. The roll/pitch corrections arising from the deviationof the spacecraft from its nominal constant altitude for both pairsof sensors are formulated in Sec. VI. Finally, the paper is summedup in Sec. VII. To bring more cohesion in the technical literature,the literal relationships and the numerical results in this paper arecompared with those openly available.

II. Sensors in Pitch-Yaw Plane: Spherical EarthThroughout the paper, we are concerned with small roll and pitch

angles of Earth-pointing spacecraft in a circular orbit, and thereforewe shall treat their determination separately. Figure 1 depicts thegeometry of a scanning horizon sensor on the right of the spacecraft'sforward motion. For a circular orbit, a local vertical local horizontalframe Tc is customarily defined to consist of a unit vector c\ alongthe velocity vector, c3 along the nadir from the spacecraft masscenter to Earth's mass center, and a unit vector c-i opposite to theorbit normal and equal to c^ x c\. The spacecraft rotates clockwise(cw) once per orbit about the axis c2 with an angular rate of &>0(&>o > 0). When the roll, pitch, and yaw attitude angles are all zero,the body-fixed unit vector triad b\,bi,b^ is aligned with the triadCi, C2, 03. The spin axis $2 of the scanning Earth sensor is tiltedby an angle £ (10-25 deg, generally) from the pitch axis c2 in the

1355

1356 HABLANI

SPACECRAFTMASS CENTER

ZERO ROLL/PITCH:C 2 C 3 ANDS 2 S 3

IN THE PITCH/YAWPLANE

Fig. 1 Geometry and scan path: right tilted sensor.

pitch-yaw plane, as shown. When there is a nonzero roll or yawangle, the angle £ will be measured from the body-fixed axis b2, notc2. The optic axis O of the sensor makes a semi-cone angle 8C (30-60 deg) with the spin axis and forms a cone in space, intersectingEarth. Usually, the sensor scans counterclockwise (ccw) about theaxis s2 and the associated angular momentum is negligible (say,10~4 N • m • s) to preclude its influence on the spacecraft dynamics.However, for bias momentum satellites, the sensor could be mountedon a momentum wheel, forming a Scanwheel® (manufactured byIthaco Space Systems). The associated angular momentum about theaxis s2 is then dynamically significant, and it is usually clockwise soas to add to the satellite's angular momentum arising from the once-per-orbit clockwise rotation about the axis c2 and render a morestable satellite. (In that circumstance, there must be a left sensor,also tilted by an angle £, so that the net bias momentum is alongthe pitch axis.) The roll and pitch measurements and the oblatenesscorrections do not depend on the sense of rotation of the sensor;however, the space-to-Earth crossing point 00/? and the Earth-to-space crossing point O\R, shown in Fig. 1 for counterclockwise scanmotion, will interchange for the clockwise scan motion, and the formof the roll and pitch equations and oblate corrections will changecorrespondingly. These changes will be illustrated subsequently.Continuing with Fig. 1 for now, the Earth disk visible from thesatellite at an altitude h is of angular radius pe defined by

sinpe = (1)

where the subscript 0 denotes a quantity pertaining to a sphericalEarth of radius jRe. The scan angle width (OQR to OIR) of the spher-ical Earth about the axis s2, for zero roll and pitch angles, is denoted29 WQ . The tangency of the optic axis O at the transit points OQR andOIR yields

where O is the unit vector along the optic axis O. Expressing O atthe transit points in the frame Fc, the semiscan angle 9W® for thespherical Earth can be shown to be governed by

-,ce«9 (3)for sin(-) and c - for cos(-)where the common practice of writing s -

is adopted.

Roll/Pitch Determination: Right SensorFor a positive roll angle ct\, the scanwidth 29 w (from OQR to

OIR) of the Earth disk, for the right sensor, is larger than 29W®,corresponding to the zero roll and pitch attitude of the spacecraft.For small roll and pitch angles, the roll angle a i is in the plane c2c3,and therefore, for nonzero a\, Eq. (3) modifies to

+ oil) (4)

valid for both crossing points OQR and OIR. For OL\ <£ 1, the semis-can angle 9W will be only slightly different from 9W&, and thereforewe can write

where |AJ <^C 9W$. For small a\ and Aw, Eq. (4) then leads to thelinear relationship

where the nominal slope K is governed by the nominal parameters§, 8C, and ̂ ® (and therefore the altitude h) as follows:

K = sm9w®/(cot8c - tan f cos 9W®) (7)

cos p® = (2)For zero pitch angle, when the reference mark on the scanning sen-sor aligns itself momentarily with the axis 53 in the pitch-yaw plane

HABLANI 1357

(Fig. 1), the scanpath OQRO\R (equal to 20W) is divided equally.However, for nonzero pitch angle the scanpath from OQR to thereference mark and from the reference mark to O\R will be, respec-tively, 0^0* and 9wlR (9wOR > 0^ for a positive pitch angle). For anonzero pitch angle, then, Eq. (6) changes to

+ 9W\R — 20u,®) (8)

where the scan angles 9W^R and 9W\R are furnished by the sensor and0m® is known. The scanwidth (9w()R + 9W\R) is the so-called chord.

For determining the pitch angle a2 about the unit vector c2> theunit vector 6 at the transit point O0# or O\R is first expressed in theframe Si $2*3 (Fig-1). It is then transformed to the spacecraft-attachedframe bib2b3 through the angle f and then to the local vertical localhorizontal frame Tc through the pitch angle a2. Applying Eq. (2)then yields, at O\R,

= c8cst= (9)

The equation for the in-crossing point O0/? is obtained fromEq. (9) by replacing 9wiR with -9wQR. Invoking the small-angleapproximation similar to Eq. (5) and ignoring the second-orderproducts, for the ccw scan, Eq. (9) and its companion for O^Rlead to

(10)

where, as stated earlier, 9WQR > 9W\R for a2 > 0- If the sensor wereto scan Earth clockwise (as in the case of a Scan wheel), the transitpoints OQR and O\R in Fig. 1 will interchange and Eq. (10) wouldthen transform, for the cw scan, to

) COS £ (ID

For a2 > 0, we would now have 9W\R > 9w()R. The roll and pitchangles thus remain the same regardless of the scan direction of thesensor.

In the literature, the pitch angle ct2 is often expressed in terms of aphase angle. Since the chord 29w = 9wOR + 9wiR, Eqs. (10) and (11)can be rewritten in the form

<*2 = (9woR - 9W) cos f = (9W - 9wlR) cos £ (12a)

a2 = (9W - 9wQR) cos £ = (0^* - 9W) cos f (12b)

for ccw and cw scanning, respectively, where 9wQR — 9W, Ow — OwiR,etc., are the phase angles.

The pitch angle c*2 is not sensitive to the altitude variations of thespacecraft because it (the pitch) depends on the measured angles9wOR and 9W\R and the tilt angle f. The accuracy of the roll angle[Eq. (8)], on the other hand, is influenced by the altitude variations,for both the nominal angle 0^® and the slope K are prestored for anominal altitude. If the true altitude is known, the parameters K and9W($ can be updated in flight as the altitude varies; otherwise, a well-known alternative is to use another sensor on the left of the velocityvector (the — c2 side). This configuration is considered next.

Roll/Pitch Determination: Left SensorReferring to Fig. 2, the scan cone axis — s2 of the left sensor is

tilted by the angle f from the axis — c2 (the negative pitch axis, orthe orbit normal). The sensor scans with the same counterclockwiseangular velocity cos about the axis s2 as before, except that the s2axes of the two sensors are not parallel. Instead, they are arrangedsymmetrically about the pitch axis b2 (or c2 if the roll/pitch/yawangles are all zero) in the pitch-yaw plane such that the net angularmomentum is about the pitch axis b2. In the case of a Scan wheel,which has a significant angular momentum by design, the angularvelocity cos about the axis s2 is usually clockwise; but Fig. 2 per-tains to a scenario in which the angular momentum of the sensor isnegligible and it spins counterclockwise about the axis s2. The cor-responding transit points space-to-Earth, O()L, and Earth-to-space,OIL, and the scanpath on the Earth disk are also shown in Fig. 2.The scan angles from O()L to the reference mark aligned with s3and from the reference mark to OIL are 0^01 and 9W\L, respectively.

S 2 S 3 ANDC 2 C 3

IN THE PITCH/YAWPLANE

CENTER

Fig. 2 Geometry and scan path: left tilted sensor.

1358 HABLANI

As a side comment, note that, by a still different arrangement, thesensors could scan Earth in opposite directions; see Sec. V.

Equation (3), derived for the right sensor for zero roll/pitch/yawangles, is valid even for the left sensor. But as the roll angle a\ isnow opposite to the tilt angle f in Fig. 2, the angle a\ in Eq. (4) isreplaced by —a\. Therefore,

cpe = c8cs(% - ai) + s8cc(t; - ai)cOWtiL i = 0, 1 (13a)

valid for both in-crossing and out-crossing points. On the other hand,the tangency equation (2) applied to the left out-crossing point, withthe spacecraft at a pitch angle «2, is, at On,

- a2s8cs9wiL (I3b)

formally the same as Eq. (9) for the right out-crossing point, becauseif the right sensor's optic axis were located at — f and (—180 — 6C)deg, it would occupy the left sensor's optic axis position. Substi-tuting these angles in place of £ and 8C and Ow\i in place of 0W\Rin Eq. (9), Eq. (13b) follows. For positive roll angle, the scanwidth(OwQL +0iwiz,) diminishes (see Fig. 2, Ref. 6). Following the previouslinear analysis for the right sensor and using Eqs. (13a), (13b), andits counterpart for the in-crossing point, we arrive at these roll andpitch measurement equations:

ot\ = — ^ H- 9wn —

and

oi2 = 3 cos £(0u,oi-0,1,11)

for counterclockwise scanning and

(14)

(15a)

(15b)

for clockwise scanning. The comments made earlier for the rightsensor regarding the corrections for altitude variations are valid forthe left sensor as well.

Roll/Pitch Determination: Both SensorsTo eliminate the dependence of the roll angle on the altitude

knowledge in Eqs. (8) and (14), these two measurements fromthe two sensors are combined and averaged. Likewise, averagingthe pitch angles from the previous equations, the following roll andpitch equations are found:

-9wlL) (16)

(17a)

- 9wOL) (lib)

and

for counterclockwise scanning and

oil = \cos^(9wiR -

for clockwise scanning. The dependence of the roll angle on thealtitude is not completely eliminated, though, since the slope K inEq. (16) is a function of the semiscan angle 9W$9 which in turndepends on the altitude. But as we shall see later, that is a compar-atively weak dependence. Furthermore, not surprisingly, Eqs. (16)and (17b) match with the roll and pitch measurement equation (3)by Ohtakay and Havens9 for the SEASAT-A bias momentum satel-lite, establishing thereby the literal values of their constants kr andkp\ kr = 4/K and kp = 4/cos£. And finally, Eqs. (16) and (17)can be interpreted in terms of phase and chord associated with eachscanpath.

ORBIT~z ' NORMAL

Fig. 3 Four horizon crossing points for spherical Earth and positiveroll angle: sensors scanning counterclockwise about pitch axis.

Points of Horizon CrossingsAnticipating our later needs while calculating the oblateness cor-

rections, we now determine the azimuths of the four crossing pointson the circular Earth disk (see Fig. 3). The azimuth angle, denoted byT/r, is measured from the local east E about the local vertical ur at theinstantaneous location of the spacecraft in the orbit or, equivalently(for a spherical Earth), at the subsatellite point. For this purpose,let i be the orbit inclination angle and coQt be the spacecraft's trueanomaly measured from the ascending node line, which is at an an-gle £2an from the vernal equinox. Then, the longitude 0 measuredfrom the vernal equinox and latitude A of the subsatellite point, bothangles in the geocentric inertial frame, and the heading angle y ofthe spacecraft's velocity (the unit vector cO, measured ccw from thelocal east (see Fig. 3), are all given by the following11:

Longitude

Latitude

Heading angle

tan"1 (cos i

A = sin"1 (sin/ sino>oO

y = tan"1 (tan i cos coQt)

(18a)

(18b)

(18c)

Now, the azimuth angle V" is determined most conveniently interms of the angle ty' = ty — y, measured from the unit vector ci.Specifically, the unit vector 6 along the optic axis O has the compo-nents O • ci and O • c2 along the unit vectors c\ and c2, respectively,and therefore (Fig. 3)

tarn// = -O-c2/d-Ci (19)

For zero roll and zero pitch angle and counterclockwise scan, ex-pressing the unit vector O at the out-crossing point of the right sensorin the frame Fc\ c\c2c^ and, using Eq. (19), we arrive at

tan ty(R = (-c (20a)

where, as before, the subscript 1 R denotes the out-crossing point.The expression for the in-crossing azimuth ty'QR f°r the right sensoris obtained from Eq. (20a) by replacing the semiscan angle 9W\Rwith -9wQR, yielding

tan ^QR = (-c (20b)

The expressions for the in-crossing and out-crossing azimuth anglesfor the left sensor, ^'QL and T/^L, derived likewise, are as follows:

tan i^;L = (c8cctj -

tan VQL = (c8ccl; - s

(20c)

(20d)

HABLANI 1359

For a spherical Earth and zero roll and zero pitch angles, the foursemi scan angles are all equal:

"w()R — (21)

and therefore, using Eqs. (20) and the angles <5C, £, and 9we, theinstantaneous nominal azimuth angles of the four crossing pointscan be determined.

III. Sensors in Pitch-Yaw Plane: Oblate EarthNoncircularity of the Horizon Caused by Oblateness

In this study, Earth's atmosphere and its radiance are not con-sidered and therefore the oblate Earth's radius given by Eq. (4.14)(Liu8) is simplified to

R = (22)

where Req is the equatorial radius, 6378.14 km; R is the radius at lat-itudeA.;s2- = sin2 • ; and /is the ellipticity factor,/ = 0.00335281,of the reference spheroid. It is noted that R in Eq. (22) is indepen-dent of the longitude </>. Next, the angular radius p of the spheroid'shorizon given by Eq. (4.24) (Liu8) at any subsatellite point is exceed-ingly complex, so it is simplified below by using R [Eq. (22)], byinvoking binomial expansion in /, and then by ignoring all higherorder terms of /, leading to

p = cor1 [cotp® + /(cotp®c2A,s2vI/ + tan p®s2X + s2Xsi4>)](23)

where c1- = cos2 • and s2"k — sin2X. Because / is very small,the angular radius p would be only slightly different from p® andtherefore one may write13

p±, (24)

where \sp\ <£ p®. Comparing Eq. (24) with Eq. (23) and using\sp\ <$; 1 rad in trigonometric expansions yield

2A tan p®s (25)

which shows that the noncircularity of the horizon is of the sameorder as /.

Another measure of oblateness is the slight difference betweenthe semi scan angle 0«,® for a spherical Earth, with zero roll andzero pitch angles, and the four quasisemiscan angles 9Wtij (i = 0, 1;j = L, R) for an oblate Earth (quasi because OWQR and 9w\Rand 9wn may not be all equal but each is almost equal toDenoting this slight difference as s w j j , we have

— ft J_ c— uw® i t-w 1 rad (26)

where the small angle ewjj defined for an oblate Earth and zeroroll and pitch angles should not be confused with the angle A^introduced earlier in Eq. (5) for nonzero roll and pitch angles. Theangle ew>ij is related to sptij (ep at the crossing point ij) in a wayestablished by generalizing Eq. (3) for an oblate Earth:

+ s8cct;c9W i = 0, 1; j = L, R (27)

where p/y equals Earth's angular radius at the crossing point i j .Differentiation of Eq. (27) furnishes

i =0,1; 7= L, /? (28)

where the first right-hand side (RHS) is exact and the second isbased on the fact that sWtij and SPJJ are both of the order 0.01 rad,so they both can be calculated using the spherical Earth parameterspe and 0we instead of the oblate Earth quantities p,7- and 0W ,,-_/. Thequantities SPJJ for the four crossing points are calculated using Eq.(25) where the azimuth vl> corresponding to the crossing points areobtained from Eqs. (20), valid for a spherical Earth and nominalattitude of the spacecraft.

Roll/Pitch Determination: Right SensorFor the oblate Earth, Eq. (4) corresponding to the roll measure-

ment is generalized by substituting piR (i = 0, 1) for p® and —9wQRor OW\R for 9W9 to consider the in-crossing and out-crossing tran-sit points. To linearize, we use the expansion (5), appending thesubscripts OR or 1R, and, additionally, use the expansion

i = 0, 1 (29)

where |Ap//j | <3C p®. Comparing the expansions (24) and (29), weobserve that, strictly speaking, Ap t i j corresponding to the nonzeroroll/pitch angles will be different from sp}ij corresponding to zeroroll/pitch angles; but for small roll and pitch angles, this differ-ence is negligible because ep varies slowly with the latitude A andazimuth ^r. Therefore, this difference will be ignored henceforth.Substitution of the expansions (5) and (29) in the generalized formsof Eq. (4) for the in-crossing and out-crossing points of the rightsensor and linearization lead to

(30)

where the roll angle a\mtR measured by the right sensor is formallyidentical to Eq. (8):

&wlR — 20,,,®) = ^ (31)

and the roll oblateness correction angle 8a\R, after using the rela-tionship (28), is

(32)

Since the true roll angle a\ is defined relative to the geocentricvertical, Eqs. (30-32) indicate that a\ = 0 when AwiR = swiR(i = 0, 1). The measured roll angle a\mtR is then clearly nonzero.On the other hand, ct\mtR = 0 whenever the geocentric roll angle ofthe spacecraft equals 8otiR. Thus, one infers that the measured rollangle a\mtR must be a geodetic roll angle and

0 <& (XimtR = ~8ofiR

/? =0 <3>ai =8aiR

(33a)

(33b)

for geocentric and geodetic Earth-pointing orientations, respec-tively.

Consider the pitch angle a2 next. Equation (9) for the out-crossingpoint and the companion equation for the in-crossing point are gen-eralized for an oblate Earth, as before. The linear analysis thenleads to

(34)

where the measured pitch angle oi2m,R using the right sensor, scan-ning counterclockwise, is

= \ (35)

analogous to Eq. (10), and the corresponding pitch oblateness cor-rection 8a2R is

- swQR) (36)

Arguing as before for the roll angle, a2m,R is a geodetic pitch anglemeasured by the right sensor and

oil — 0 & ct2m,R = -8a2R

a2mR = 0 <^ «2

(37a)

(37b)

for geocentric and geodetic Earth-pointing orientations, respec-tively.

1360 HABLANI

Roll/Pitch Determination: Left SensorWith the spacecraft at a roll angle a\9 the tangency equation at

the in-crossing and out-crossing points for the left sensor, scanningthe oblate Earth, is

cpiL = c8cs(t; - ai) + s8cc(t; - i = 0, 1 (38)

analogous to Eq. (13a) for the right sensor. Substituting the expan-sions of piL and 0Wtii, similar to the expansions (29) and (5), inEq. (38) and performing a linear analysis again lead to

(39)

where the measured roll angle ot\m,i and the roll oblateness correc-tion 5o?iL are

~ 26U) = -^ ) (40a)

(40b)

<*im,L being formally the same as Eq. (14), except that ot\m,L is ageodetic roll angle similar to a\mtR [Eq. (31)].

Regarding the pitch angle, we generalize Eq. (13b) and its com-panion for the left in-crossing point. The first-order expansion ofthe two equations then leads to

+ 8(X2L (41)

where the measured geodetic pitch angle and the associated oblate-ness correction are, for counterclockwise scanning,

8oi2L = 3 - ewOL)

(42a)

(42b)

The right-hand sides of Eqs. (15a) and (42a) both for ccw scan areformally the same, although 9wOL and 9W\L in Eq. (42a) pertain toscanning an oblate Earth and those in Eq. (15a) pertain to scanninga spherical Earth.

Roll/Pitch Determination: Both SensorsWhen a spacecraft is fitted with both right and left sensors, the

average geocentric roll angle is obtained by adding Eqs. (30) and(39), leading to

ai = otlm + 8ai (43)

where the measured geodetic roll angle ot\m is formally the same asEq. (16) and the roll oblateness correction 8a\ is

regardless of the scan direction. The pitch angle is likewise obtainedby adding and averaging Eqs. (34) and (41), arriving at

where ct2m is formally the same as the right-hand sides of Eqs. (17)and the pitch oblateness correction 8a2 is

8a2 = \ cos %(SWIR ~ ewOR + sw\L - swOL) (46a)

for counterclockwise scanning and

8a2 = \ COS%(SWQR - swiR + swOL - swlL) (46b)

for clockwise scanning. While the pitch oblateness corrections (36)and (42b) resemble Eq. (9) of Ref. 9 (wherein the sensor scansclockwise), the roll oblateness corrections above cannot be relatedso easily with Eq. (5) of Ref. 9 because the partial derivatives thereinare not worked out.

IllustrationThe sensor tilt angle f shown in Fig. 1 is introduced so as to

increase the scan width 29 w of Earth and the slope K, increasing inturn the measurability of the roll and pitch angles. For a spacecraftin a circular orbit, Fig. 4 illustrates the semiscan angle 9W& for aspherical Earth vs the tilt angle f for three semicone angles 8C. Asexpected, 9W$ increases with both 8C and f , although, as 8C increases,the influence of £ on 9W@ weakens. The corresponding variation inthe slope K is illustrated in Fig. 5. A tempting conclusion fromFigs. 4 and 5 is to select 8C = 60 deg and £ = 20 deg, but furtherparametric studies are desired to arrive at an optimum selection ofthese angles.

The roll oblateness corrections associated with the right and theleft sensors, separately and combined [Eqs. (32), (40b), and (44)],are illustrated in Fig. 6 starting from the ascending node. The correc-tion —8ctiR, instead of 8a\R, is shown to aid its visual comparisonwith 8ctiL. Over one orbital period, the magnitudes of the correc-tions associated with the right and the left sensors interchange. Sur-prisingly, the maximum roll correction for one sensor for a 100-minnear-polar orbit can be as much as 0.43 deg, but if the two sensors areused together, this amplitude reduces to 0.09 deg. These results arein accord with those in Fig. 7 of Ref. 6, except that, apparently, Alexand Shrivastava6 have plotted negative of the correction. The averageroll correction 8oti is zero at the equator (a)Qt = 0, n) and maximumnear poles (a^t = ±7r/2), but instead of varying at twice the orbitrate, as stated in Ref. 11, it varies sinusoidally at the orbit rate. Thesignificance of the quantity -rj cos y in Fig. 6 is explained in Sec. IV.

The pitch oblateness corrections for the same parameters as thosein Fig. 6 are illustrated in Fig. 7. Unlike the roll corrections, the

75.000-

68.125 -

61.250-

54.375-

47.500-

40.625-

33.750-

26.875-

20.0000 8 10 12 14 16 18 20

Fig. 4 Semiscan width 9W® vs sensor tilt angle £ for different values ofsemicone angle 6C; R$ = 6378.14 km, h = 833.4 km, and i = 98.7 deg.

2.1000-|

1.8375-

1.5750-

1.3125-

1.0500-

0.7875-

0.5250-

0.2625-

0.0000

6C = 60 deg

45 deg

30 deg

0—1——I——I——I———I——I

10 12 14 16 18 20

Fig. 5 Slope K vs sensor tilt angle £ for different values of semiconeangle £c; fle = 6378.14 km, h = 833.4 km, and i = 98.7 deg.

HABLANI 1361

0.100 -

0.025 -

-0.050 -

-0.125-

-0.275-

-0.350-

-0.425 -

-0.5000 45 90 135 180 225 270 315 360

o)0t(deg)

Fig. 6 Roll oblateness corrections Sotn and SOL\R for left and rightsensors and their average 8oL\; comparison with closed-form roll cor-rection —77 cos 7; #0 = 6378.14 km, h = 797 km, i = 108 deg, 6C = 45 deg,and £ = 0 deg.

0 45 90 135 180 225 270 315 360co0t (deg)

Fig. 7 Average and individual pitch oblateness corrections for left andright scanning sensors; comparison with closed-form pitch correction-77 sin 7; #e = 6378.14 km, h = 797 km, i = 108 deg, 8C - 45 deg, and£ = 0 deg.

pitch corrections vary sinusoidally at twice the orbit frequency with~0.13 deg amplitude, although neither <5a2L nor ^IR exhibits thesame peak over one orbit period. The significance of the quantity—77 sin y is explained in Sec. IV. Also, these corrections match withthose in Fig. 7 of Ref. 6, although, as for the roll corrections, theauthors have apparently plotted the negative oblateness corrections.

IV. Roll/Pitch Components of Deviation of GeodeticNormal from Geocentric Normal

Because of Earth's oblateness, the geodetic normal n at any pointon the reference spheroid is, in general, not aligned with the geo-centric radius vector passing through that point or through a pointabove it (see Fig. 8). It seems therefore that this deviation must berelated to the roll and pitch corrections formulated above. It is thisrelationship that we attempt to formulate below.

Deviation of Geodetic NormalIn a geocentric inertial frame, any point on Earth's surface or

above can be located by its radial distance r, longitude 0, and latitudeX. The axis of rotation of the longitude 0, ccw positive, passesthrough the celestial north pole, whereas the unit vector E along thelocal celestial east is the axis of rotation of the latitude A, cw positive.Liu8 has furnished the outward, geodetic normal n at any point onthe oblate Earth in a Cartesian coordinate system [Eq. (4.17), Ref. 8].However, the deviation of n from the geocentric radial vector appearsto be more readily calculable in a spherical coordinate system. Thespherical coordinate trio r, 0, A is not right handed because of cw

E NORMAL TO THE~ PLANE OF THE

PAPER

(GEODETICP SUBSATELLITE

POINT)

Fig. 8 Formulation of deviation of geodetic vertical from geocentricvertical.

positive X, so we introduce the right-handed spherical coordinatesr, kc, 0 (in this sequence), where X6. is colatitude: A,c. = 90 deg —A.The associated dextral unit vector triad is ur, u\c, M0 where ur is aradial unit vector, uXc is the unit vector in the direction of increasingAc, and u^ is the unit vector along the local east (11$ = E). Theoutward normal n is then given by13

n = ur - f sin 2Xc (47)

where /2 and higher order terms have been ignored. Clearly, n is inthe plane formed by ur and uXc (the plane of Fig. 8), and therefore,the deviation angle 77 between n and ur is about the unit vectoru(f)( = E) axis. Up to the first order of /, Refs. 12 and 13 showthat the angle 77 between the geodetic normal n passing through thespacecraft mass center at a height h and the geocentric unit vectorur = —c3 at the spacecraft mass center is

(48)

about the east unit vector at the spacecraft location and positive, asshown in Fig. 8.

Roll/Pitch Components of DeviationWe note from Fig. 3 that

E = cos yc\ + sin yc-i (49)

Hence, the small-angle vector r\E has the components rjcy and 17 syabout the c\ and c2 vectors, respectively. To render explicit the de-pendence of these components on the orbit angle coot, define

i?™ = fReq/(Rcq + h) (50)Reference 1 3 then shows that the components of r\E are

77 cos y = r]mx sin 2i sin coot 77 sin y = r\mx sin2 / sin 2a)Qt

and the angle 77 varies with the orbit angle as follows:j_

77 = 2r]mx sin i sin & > o / l — sin2 / sin2 cotf) 2

(51)

(52)

To relate the roll and pitch components [Eqs. (51)] with the oblate-ness corrections 8a\ and <5a2, recall that the geodetic roll and pitchangles are zero (alm = 0 = a2m) when the spacecraft's yaw axis#3 passes through the geodetic subsatellite point p, that is, when&s = — n. This implies that, to align #3 with the geodetic nadir —n,

1362 HABLANI

the spacecraft must turn by a roll angle ot\ and a pitch angle ot2equal to

= —r] cos y = — r\ sin y (53)

the negative signs stemming from c3 X b3 = -rjE = a\c\ + a2c2.These quantities were illustrated in Figs. 6 and 7, wherein we nowobserve that they agree very well with the oblateness correctionsformulated in Sec. Ill for a pair of horizon sensors. Therefore, whenthere are two sensors, the oblateness corrections are

8a\ & —f] cos y 8a2 & —r] sin Y (54)

which is a handy conclusion, for the corrections then need not becalculated using the analysis of Sec. III. Indeed, substituting Sa\and 8a2 from Eq. (54) in Eqs. (43) and (45) and recalling Eqs. (48)and (50), we arrive at

= a\m - Vmx cos y sin2A

= a2m - VmX sin y sin 2X

(55a)

(55b)

which are the same as those stated without proof by Lebsock andEterno11 (1987 version). Perhaps more importantly, we also con-clude that if only one sensor is onboard the spacecraft, Eqs. (54)and (55) become invalid and the corrections then must be calculatedas shown in Sec. III.

V. Aft and Forward Horizon Sensors inRoll-Yaw Plane

While packaging various instruments in a spacecraft bus, it issometimes not feasible to install the scanning horizon sensors in thepitch-yaw plane; for example, the solar arrays might be occludingthe sensor's field of view. In that event, the Earth sensors may bemounted on the aft and/or forward face of the spacecraft. Moreover,if both aft and forward sensors are used, they may scan Earth inthe same direction or in the opposite direction. In the latter case,the net spin angular momentum of the two sensors is virtually zero,although the angular momentum of a scanning sensor is usually verysmall to begin with. In any event, Fig. 9 depicts the scanpaths of aftand forward sensors, tilted in the roll-yaw plane and each spinninganticlockwise about their own scan axes. Below we develop theroll and pitch angles as measured by these sensors, individually andtogether, and the associated oblateness corrections.

Aft SensorEquation (3), which determines the semiscan angle 9W$ of the

sensor for a spherical Earth with the spacecraft in the zero roll/pitchorientation, still applies. Developing the math model and performing

EARTH DISC

FORWARDO., p OUT-CROSSING

POINT (\|/1F)

AFTOUT-CROSSINGPOINT (\|f1A)

FORWARDOOF IN-CROSSING

POINT (\J/op)

Fig. 9 In-crossing and out-crossing points of aft and forward scanning

the linear analysis in the same fashion as in Sees. II and III, wearrive at the geocentric roll angle equation similar to Eq. (30), withthe subscript R replaced by the subscript A (for aft). The geodeticmeasured roll angle aim,A and the associated oblateness correctionBa\A in terms of the measured scan angles are now

)

SalA = l

(56a)

(56b)

Equation (56a) is clearly right because, for a positive roll angle aboutCi, the in-crossing scan angle 9WQA up to the reference mark will besmaller than the out-crossing scan angle 6W\A. However, notice thatthe form of Eqs. (56) is similar to the measured pitch angle a2,n,/?[Eq. (35)] and the associated oblateness correction 8a2R [Eq. (36)],because, earlier, the scan axis was in the pitch-yaw plane and nowit is in the roll-yaw plane. The geocentric pitch angle, on the otherhand, is given by an equation similar to Eq. (34), with the subscriptR replaced by the subscript A. The measured geodetic pitch anglect2m,A and the associated oblateness correction 8ot2A are defined by

8oi2A = — 2^(£wOA + GWIA) (57b)

which are analogous to, as expected, the roll equations (31) and (32).

Forward SensorThis sensor spins counterclockwise about an axis tilted by an

angle £ from the unit vector c\ in the roll-yaw plane. The corre-sponding scanpath on the Earth disk is shown in Fig. 9. The roll andpitch angles are given by equations similar to those for the aft sen-sor, with the subscript A replaced by F (for forward). The measuredangles and the oblateness corrections are the following:

<*lm,F= j

&*I,F = \

8a2<F =

(58a)

(58b)(59a)

(59b)

Because the aft sensor and the forward sensor are spinning oppo-sitely, Eqs. (58) and (59) appear negative of Eqs. (56) and (57). Also,as in the case of aft sensor, the roll equations (58) resemble the pitchequations (35) and (36) and the pitch equations (59) resemble theroll equations (31) and (32).

Aft and Forward SensorsWhen the roll and pitch measurements from the two sensors

and the associated oblateness corrections are added and averaged,Eqs. (43) and (45) follow where, now,

- 9wQA 0u;lF)

8a2 = \

(60a)

(60b)(61a)

(61b)

The evaluation of the oblateness corrections Ba\ and 8a2 requiresdetermination of eWtij (i = 0, 1; j = A, F), which are governedby Eqs. (25) and (28). The horizon crossing points ^v appearing inepjj are determined as follows.

Horizon Crossing PointsCalling upon the definition (19) of the crossing point ̂ (i —

0, 1; j = A, F) and determining the components along Ci and c2 ofthe four optic vectors, for zero roll/pitch attitude, we arrive at thefollowing:

Aft out-crossing

tan vl/' = -s8cs9wiA/(-c8cct; + s8cs^c9wlA) (62a)

HABLANI 1363

Aft in-crossing

tan V'QA = s8cs9wOA/(-c8cct; + s8cs

Forward in-crossing

tan vJ/(')F = -s8cs0wQF/(c8cct; - sS

Forward out-crossing

tan V[F = s8cs0wlF/(c8cct; - s8cs^

where ̂ = Vu - y (i = 0, 1; j = A, F).

(62b)

(62c)

(62d)

IllustrationFigures 10 and 1 1 illustrate the roll and pitch oblateness correc-

tions associated with the aft and forward sensors for the parameterslisted therein. To determine these corrections the crossing azimuthsare determined first using Eqs. (62) for a spherical Earth. These arethen substituted along with other quantities in Eqs. (25) and (28) todetermine sWiij (i — 0, 1; j = A, F). The crossing azimuths for anoblate Earth could be calculated next using 9Wtij instead of Ow® inEqs. (62), but these need not be used to obtain more accurate sWtijbecause the improved accuracy is negligible. The roll correctionsSctiA and 8aiF for the aft and forward sensors and the average correc-tion 5«i are illustrated in Fig. 10 and compared with the closed-formcorrection -77 cos y . In contrast with Fig. 6 where the sensors arealong the pitch axis, the four roll corrections in Fig. 10 stay verynear each other. The peaks of the average correction 8a{ and of theclosed-form correction —77 cos y are only slightly different, and theindividual sensor corrections have slight lead or lag in their phases.

8C = 60 deg£ = 13.64 degi = 98.7 degh = 833.4km

0 45 90 135 180 225 270 315 360co0t (deg)

Fig. 10 Roll oblateness corrections for aft and forward sensors andtheir average; comparison with closed-form roll correction —77 cos 7.

45 90 135 180 225 270 315 360a)0t (deg)

Fig. 11 Pitch oblateness corrections for aft and forward sensorsand their average; comparison with closed-form oblateness correction—77 sin 7.

This general agreement in roll, however, does not carry over to thepitch axis, as we see in Fig. 11, wherein the individual pitch correc-tions are nearly four times larger than the average pitch correction,which in.turn is nearly the same as the closed-form pitch correction(-77 sin y). Figures 10 and 11, just as Figs. 6 and 7, suggest that ifboth aft and forward sensors are employed, the closed-form expres-sions —77 cos y and —77 sin y can be used for oblateness correction;if not, the corrections may be computed on the ground for the se-lected solo sensor and stored in the flight computer in a table lookupform before flight.

VI. Roll/Pitch Altitude CorrectionsSimple, linear, attitude determination equations of the preceding

allow for the evaluation of roll/pitch corrections to compensate forthe spacecraft altitude variations caused by eccentricity of the orbit.Because both the oblateness corrections and the altitude correctionsare small, they can be treated separately and so, in this section, wefocus on the altitude corrections.

The first quantity affected by the spacecraft altitude variation 8his the angular radius pe of the Earth disk. Differentiating Eq. (1),we obtain

(63)

where 6pe is the variation in pe caused by 8h. This, then, changesthe scanwidth 20W(B of Earth; quantitatively, Eq. (3) yields

80W® = -sin pe tan pe<$/z/[(/?e + h) sin Sc cos £ sin 9w9] (64)

Examining Eqs. (6) and (7) we observe that when a single sensor isused for attitude determination, the variation 89W® affects the atti-tude measurement accuracy directly as well as through the change inthe slope K. The change 8K is determined by differentiating Eq. (7):

8K = -tan£)/(cotSc - (65)

The altitude corrections for individual sensors can be now formu-lated as shown below. However, we note that if the instantaneousaltitude of the spacecraft is known and used in flight to update theslope K and the semiscan width 9W®, no additional altitude correc-tion will be required because, then, the measured scan angle 29 w , K ,and 9W® all pertain to the same altitude. However, the following per-turbation analysis helps in the preflight estimation of the roll/pitchcorrections for a possible range of the altitude variation 8h.

Left and Right SensorsInspecting the roll equation (6) and the pitch equation (10), we

observe that the pitch angle requires no altitude correction becausethe tilt angle £ is fixed regardless of the altitude; the roll angle,however, requires this correction, for the slope K and the semiscanangle 9W(S) depend on the altitude. The semiscan width 9W is a mea-surement whereas K and 9W® pertain to a nominal altitude. If 9W ismeasured at a nonnominal altitude but K and 9W^ remain nominal,the true roll attitude will be obtained by using Eq. (30) where theroll altitude correction 8ctiR will be

8alR = (8K/K)alm,R - K86W (66)

the variations 8K and 89W® being given by Eqs. (64) and (65).Thus, the altitude correction consists of two parts. The bias partK89W®, however, can be eliminated if a left sensor is used as well.Specifically, differentiating Eq. (14), we arrive at

(67)

Comparing Eqs. (66) and (67) we observe that the bias correctionsfor the left and right sensors are mutually opposite, so if the twosensors are used together, the residual roll correction will be

tei - (SK/K)ctlm

with aim equal to the right-hand side of Eq. (16).

(68)

1364 HABLANI

1.00-

0.75-

0.50-

0.25-

o.oo-

-0.25-

-0.50-

-0.75-

_ i nn _

5C = 45 deg _ K 6e /^^ = Odeg w©^i = 98.7 deg J?'h = 833'4km /<(5K/K)a2r

/" - K 89W@

(8K/K) a2mjA /'

/XX

/"-50 -40 -30 -20 -10 0 10 20 30 40 50

5h (km)

Fig. 12 Pitch altitude corrections for aft sensor vs altitude variationwith measured pitch angle equal to 2 deg.

Aft and Forward SensorsExamining the roll and pitch measurement equations (56a) and

(57a) or (58a) and (59a), we infer that, unlike the case of the left andright sensors, the roll measurements now need no altitude correctionbut the pitch measurements do. Differentiating the pitch measure-ment equations (57a) and (59a) corresponding to the aft and forwardsensors, the altitude corrections associated with each sensor sepa-rately are found to be

8a2A = (8K/K)ct2m,A -

8ot2F = (8K/K)a2m,F +

(69a)

(69b)

When both sensors are used together, the combined altitude correc-tion is

8ct2 = (8K/K)ct2m (70)

where the measured pitch angle a2m is given by Eq. (62a).

IllustrationFigure 12 depicts the pitch corrections for an aft sensor when the

nominal altitude varies by ±50 km and the other nominal parametersare fixed as noted in the figure. The bias correction — K80W® predom-inates over the slope correction. For example, when 8h = 50 km,—K80W® « 0.95 deg. Compare this with the pitch correction(8K/K)oi2m,A caused by the slope change 8K, equal to -0.04 degwhen the measured pitch angle is 2 deg. If the true altitude is known,through navigation or ephemeris or radar altimeter, the pitch alti-tude correction will not be needed altogether; if not, a forward sensorcan be employed whose pitch measurements, combined with the aftsensor measurements, will eliminate the bias error K80W®. A smallresidual error (8K/K)a2m will still remain, but this may be withinthe error budget.

VII. Concluding RemarksRoll/pitch oblateness corrections to the scanning horizon sen-

sor data can be determined in two ways: 1) by the crossingpoint approach, wherein the azimuth angles of the space-to-Earthand Earth-to-space transit points of the horizon sensor's scan are

determined, and 2) by decomposing the angular deviation of thegeocentric normal from the geodetic normal into the roll and pitchcomponents. The two approaches yield essentially the same cor-rections if the Earth-pointing spacecraft is equipped with multiplescanning sensors—left and right along the pitch axis or aft andforward along the roll axis or, by the same token, four diagonalsensors. In that event, the second approach, which furnishes closed-form expressions for the roll/pitch oblateness corrections, can beused onboard and is highly recommended. However, if a spacecraftis outfitted with only one sensor, the corrections about one axis—roll if the sensor is along the pitch axis and pitch if the sensor isalong the roll axis—is far different from that predicted by the geo-centric/geodetic angular deviation approach. The corrections thenmay be calculated on the ground for the desired sensor locationand stored in the flight computer using the crossing point approach.The paper treats two most common arrangements of the horizonsensors—a pair tilted in the pitch-yaw plane and one in the roll-yaw plane—and obtains, through linear perturbation analysis, thecorrections to compensate Earth's oblateness and the spacecraft'saltitude variations arising from the orbit eccentricity.

References^indorfer, W., and Muhlfelder, L., "Attitude and Spin Control for TIROS

Wheel," AIAA/JACC Guidance and Control Conference (Seattle, WA), 1966,pp. 448-462.

2Brasoveanu, D., and Hashmall, J., "Spacecraft Attitude Determina-tion Accuracy from Mission Experience," Proceedings of Flight Mechan-ics/Estimation Theory Symposium, NASA-CP-3265, Goddard Space FlightCenter, Greenbelt, MD, 1994, pp. 153-166. .

3 Anderson, R. H., Heiling, C. I., Okomoto, A. Y., and Zimmerman, H. E.,"Optimization of Earth Sensor Thresholding Techniques," Journal of Guid-ance and Control, Vol. 4, No. 2, 1981, pp. 136-140.

4Gontin, R. A., and Ward, K. A., "Horizon Sensor Accuracy ImprovementUsing Earth Horizon Profile Phenomenology," AIAA Guidance, Navigation,and Control Conference (Monterey, CA), AIAA, New York, 1987, pp. 1495-1502.

5Alex, T. K., and Seshamani, R., "Generation of Infrared Earth Radiancefor Attitude Determination," Journal of Guidance, Control, and Dynamics,Vol. 12, No. 2, 1989, pp. 275-277.

6Alex, T. K., and Shrivastava, S. K., "On-Board Corrections of SystematicErrors of Earth Sensors," IEEE Transactions on Aerospace and ElectronicSystems, Vol. 25, No. 3, 1989, pp. 373-379.

7Space Sciences, "Attitude Determination and Control Systems: Prod-ucts, Services," A AS Guidance and Control Conference, Keystone, CO, Feb.1992.

8Liu, K., "Earth Oblateness Modeling," Spacecraft Attitude Determina-tion and Control, edited by J. R. Wertz, Reidel, Boston, MA, 1984, Sec. 4.3.

9Ohtakay, H., and Havens, W. R, "Attitude Determination System forSEAS AT-A Mission," Proceedings of the AIAA Guidance and Control Con-ference (Hollywood, FL), AIAA, New York, 1977, pp. 6-17.

10Collins, S., "Geocentric Nadir and Range from Horizon Sensor Obser-vations of the Oblate Earth," AAS/AIAA Space Flight Mechanics Meeting,Colorado Springs, CO, Feb. 1992 (AAS 92-176).

HLebsock, K., and Eterno, J., "Design of a Maneuverable Momen-tum Bias Attitude Control System," AIAA/AAS Astrodynamics Confer-ence, Minneapolis, MN, 1988, pp. 704-713; also Paper AAS 87-511,AAS/AIAA Astrodynamics Specialists Conference, Kalispell, MT, Aug.1987.

12Britting, K. R., Inertial Navigation Systems Analysis, Wiley-Inter-science, New York, 1971.

13Hablani, H. B., "Modeling of Roll/Pitch Determination with Hori-zon Sensors: Oblate Earth," Proceedings of AIAA Guidance, Control, andNavigation Conference (Monterey, CA), AIAA, Washington, DC, 1993,pp. 1133-1147.