Full text of DOI:10.2514/2.4846

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS
–
Vol. 24, No. 6, November December 2001
Fast Algorithm for Prediction of Satellite Imaging
and Communication Opportunities
Yan Mai¤ and Philip Palmer†
University of Surrey, Guildford, England GU2 7XH, United Kingdom
A new equation for nadir tracking estimation, taking into account secular perturbations from the Earth’s
gravitationalŽ eld and the in uence of drag, is presented. The advantageof this proposed approach is that it needs
(
to be solved just twice per day with respect to a speciŽ c ground target there are usually 14 orbital periods per day
)
for sun synchronous orbit . The resultant approximate near nadir angle time, whose root of mean square error is
(
)
(
around a few seconds i.e., » 30-km groundtrack error , is then further reŽ ned using a newly developed controlling
)
equation, reducing the maximum error to about 0.4 s » 2:8 km groundtrack for over 20 days prediction period.
This is acceptable when the imagingŽ eld of view of 10 km is considered for high resolution small satellite cameras.
The prediction period of 60 dayscan be used for a typicalsmall satellite camera whose Ž eld of view is about100 km.
This method can also be expanded to solve the rise-and-set time problem. Because of the low complexity of the
proposed method, it is very suitable for implementation on the onboard processor whose computationalresources
are generally limited. The new computational process is described and simulation results are presented.
)
Nomenclature
satellite’s orbital semimajor axis
elevation angle
http://www.dbsindustries.com/investor.html#esat , which consists
of a 6 enhanced microsatellite constellationtargeted at the gas and
a
h
i
=
=
=
=
(
)
electricutilityfor GlobalEnergyMeteringService GEMS ; Global
Altimeter Network Designed to Evaluate Risk,1 which operates a
constellation of 16 small satellites that will orbit the Earth in rapid
succession, observing sea surface winds and waves so that ships at
sea can be constantly updated on the sea conditions around them;
inclination
N
integer representingthe number of satellite
passages
n
r
=
=
mean motion
radial distance of the satellite from the center
of the Earth
2;3
and a proposed 7-microsatellite constellation to deliver global
frequent disaster monitoring. Accordingly, the number of users
of Earth images will increase rapidly. Among all sorts of space-
crafts carrying imaging instruments, low-cost small satellites will
become a main source because of their cost effectiveness and fast
reaction.
T
t0
=
=
orbital period
time when the satellite Ž rst crosses over a given
latitude line on the ascending pass
angle of camera Ž eld of view
maximum longitudinaloffset angle for target
to be in Ž eld of view of camera
longitudinalangle margin for ground target
to be visible to satellite
°
1Àmax
=
=
With the increasing capability of modern micro- and minisatel-
lites to undertake autonomous imaging of targets on the surface of
the Earth by employing sophisticated onboard computers and data
processing techniques, one requirement is to be able to determine
the time of closest approachto nadir of a satellite in low Earth orbit
for a speciŽc ground location. Because the satellite may not  y di-
µÀ
¸
=
=
argument of latitude measured on orbital
plane
(
)
rectlyoverthe desiredtargetarea for some period daysor weeks , it
is Ž rst necessaryto determine whether a given target will be within
the narrow Ž eld of view of the satellite cameras to capture an image
within a shortertime span. In the conventionalapproach,it has been
customary to solve this problem by letting the satellite run through
its ephemeris and then checking at each instant to see where the
subsatellite nadir point falls. An orbital simulation is advanced in
ÀS
ÀT
=
=
=
=
=
=
=
geodetic longitude of satellite footprint
geodetic longitude of ground target
geocentric latitude of ground target
geodetic latitude of ground target
right ascension of ascending node
argument of perigee
’
T
ÁT
Ä
!
!
Earth rotation rate
©
t
time by some small time increment 1 , and a possibility check is
performed at each step. This method is called trajectory checking.
However, this method requires Keplerian equations to be solved
hundreds of times per orbital period, and is, therefore, not suitable
for onboard processing because of the computationalload.
I. Introduction
UTURE generations of Earth observation satellites must be
considerably more operable and autonomous to enable con-
F
tinuing military and civil applications such as resources investiga-
tion, fast-changingphenomenonmonitoring, and geographicinfor-
mation generation. The need for greater operability and autonomy
stems from shrinkinggovernmentalsupport for imaging space mis-
sions and commercialinterestin deployinglow-cost small satellites
for Earth observation. Greatly decreased operations costs will not
only enable more such missions, but will also enable systems to be
Acloselyrelatedproblemistherise-and-settimeproblemofwhen
a satellite is visible from a given point on the Earth. An important
application of this is to Ž nd when the satellite is visible over a
ground station for data transmission.The simplest way to solve this
problem is by using a numerical method such as described in the
preceding paragraph. Escobal4 proposed a faster method to solve
this problem by developinga closed-formsolution for the visibility
periods.He introduceda singletranscendentalequationas a function
of the eccentric anomaly of the satellite orbit, which he called the
controllingequation.Numericalmethods5 were thenusedto Ž nd the
rise-and-settimes. The advantageof this equation is that it is solved
only once per orbital period, in contrast with the hundreds of times
the Keplerian equation must be solved with the standard step-by-
step technique of hill climbing. The controlling equation, however,
is only valid for two-body motion, and our main aim is to solve
(
scalable to meet commercial goals. These include ESAT URL:
Received 13 June 2000; revision received 29 August 2000; accepted for
°c
publication 4 December 2000. Copyright
2001 by Yan Mai and Philip
Palmer. Published by the American Institute of Aeronautics and Astronau-
tics, Inc., with permission.
¤Ph.D. Student, Surrey Space Centre.
†Senior Lecturer, Surrey Space Centre.
1118

MAI AND PALMER
1119
D
¡
the satellitenadir trackingproblem, which means in applicationthis
equation needs further development.
1À ÀS ÀT . After each orbital period the satellite revisits the
TLL and the Earth rotates under it, bringing the target closer to
the satellite’s longitudinal position. The satellite will see the tar-
Besides the controllingequationmethod,Lawton6 has developed
–
–
T n
get approaching by an amount !© or !©2¼= , where !© is the
Earth’s rotation rate. The target closest satellite passage TCSP oc-
another method to solve for satellite satellite and satellite ground
station visibility periods for vehicles in circular or near circular
(
)
t
d
n
orbits by approximating the visibility function Ã. / by a Fourier
curs when the longitudinal difference À is smaller than !©2¼= .
series. More recently, Alfano and Negron7 see also Ref. 8 further
Therefore, we obtain the following fundamental equation:
(
)
t
developedthe Ã. / function to suit all orbitaltypes. However, these
( )
D N
n C d
1À
!©2¼=
À
1
methods are only valid for solving satellite rise-and-set time prob-
lems and are not suitable for satellite nadir tracking and still have
their own limitations.
d
where À is the longitudinal difference between the subsatellite
point and the target at TCSP.
In this paper, a new method is presentedthat can be used not only
for two-bodymotion, but also accommodatessecularperturbations,
short and long periodic perturbations, and the in uence of drag in
a straightforward manner, providing a simple equation for nadir
tracking and solving the satellite visibility problem. The proposed
formulation only needs to be solved twice per day, which makes
it very suitable for implementation onboard a small satellite. In
Sec. II, we describetheŽ rstphaseofthe newmethod,whichiscalled
coarse search. It works in two-body, secular perturbations arising
from the Earth’s oblateness and atmospheric drag perturbations.In
Sec. III,we introducethesecondphaseofthemethod,whichiscalled
reŽ nement. ReŽ nement improves the accuracy of the new method.
Simulationresultsare presentedinSec. IV, aswellasthecomparison
of CPU processing time between the conventionalmethod and this
new method. Finally, in Sec. V, we set out our conclusions.
Hence,
( )
2
N D
n
[.1À=2¼/. =!©/]
where the square brackets implie the integer part.
In other words, the closestapproachto the target will occur when
· d
n
À < !©2¼= . Therefore, as long as we know the initial pas-
0
t0
T
sage time of the TLL and the satellite’s orbital period , we
can derive the possible closest approach time over long intervals of
time. We name the procedure of TCSP estimation as coarse search
because the satellite maximum elevation time over the target does
not necessarily occur at TLL crossing.
Even in thisTCSP case,however,the satellitemaximumelevation
angle of this pass may not be suitable for imaging because the on-
board camera Ž eld of view FOV is very small. Therefore,we need
(
)
to Ž nd the maximum longitudinaloffset angle 1Àmax for the target
d
·
to be in the FOV of the camera and to check whether
À
1Àmax.
II. Coarse Search for Satellite Passes
A. Fundamental Algorithm: Two-Body Analysis
We can easily estimate the satellite closest approach time by
checking the satellite ascending and descending passage once, re-
spectively, per day. Set the orbital period of the satellite to be
The calculation of 1Àmax will be described in Sec. II.A.2.
To determine the rise-and-set times of the satellite over a given
ground station, some modiŽ cation to the description given has to
be made. Instead of 1Àmax , we need to set another angle margin
µÀ within which the ground target is visible to the satellite Fig. 2 .
Vallado and McClain8 deŽ ne this as the ground-range angle. The
calculationof µÀ will be describedin Sec. II.A.2. Figure3 shows the
(
)
) ( )
n
where is the mean motion and the time when
T D
n
t
.
2¼=
0
the satellite Ž rst crosses over a given latitude line on the ascending
(
)
pass Fig. 1 . We call the circleof constantlatitudethatrunsthrough
(
)
the target location the target latitude line TLL . The key point of
our approachis to use the fact that, for two-body motion, a satellite
will revisit exactly the same point in an inertial coordinate system
(
)
Fig. 1 . This means that the satellite
T
after each orbital period
(
)
.
t0 C T
will make another ascending pass over the TLL at time
To simplify the discussion,we shall ignorethe descendingpassages
over the TLL and include them again only at the end. Note that in
this method satellitepositionis expressedby the redundantepicycle
(
)
r
I
r
coordinates , ¸, , and Ä Ref. 9 , where is the radial distance
of the satellite from the center of the Earth, ¸ is the argument of
i
latitude measured on the orbital plane, is the inclination,and Ä is
the right ascension of the ascending node.
If the location of a target on the Earth is ÀT ; ÁT , where ÀT and
(
)
ÁT are the geodetic longitude and latitude, respectively, then the
(
)
,
t C NT
t C N
n
2¼=
satellite will pass over the TLL every
or
0
0
N
where is an integer representingthe number of satellitepassages.
t
At time 0, the satellite is over the TLL and the initial longitu-
Fig. 2 Within longitude angle µÀ satellite S is visible to ground
target T.
dinal difference between the satellite footprint ÀS and target ÀT is
Fig. 3 Visibility of satellite passes.
Fig. 1 Satellite orbiting around Earth showing crossings of TLL.

1120
MAI AND PALMER
Fig. 5 Location of target and satellite at closest approach; O is center
of Earth, where offset angle µ is deŽ ned; ° is FOV of camera.
Fig. 4 Geometry of ¸0 in Earth-centered inertial coordinate.
basic idea of our new method for satellite rise-and-settimes. When
the satellite is visible, its longitude ÀS must satisfy the following
condition:
( )
3
¡
·
·
C
ÀT µÀ ÀS ÀT µÀ
’
¡
To test whetherthe passesare visible,we startfromÀS ÀT µÀ .
If this is a visible pass, we add it to our coarse search list. When
¡
C
ÀS < ÀT µÀ , we add2¼ toÀS .WhenÀS > ÀT µÀ , we cancompute
D
¡
C
the difference in longitude 1ÀÀ ÀS .ÀT µÀ / that will bring it
to within the visibility of the ground station. Therefore, we get the
following formula for satellite visible estimation:
( )
4
N D
n
C
[.1ÀÀ =2¼/. =!©/]
1
Fig. 6 Computation of maximum longitudinal offset angle margin
( )
D
Àmax, given ground track of satellite l and spherical offset angle µc.
1. Finding Initialization Argument of Latitude ¸0
In the preceding section, we pointed out that we need to know
For typical Earth observation orbits, which are near circular and
polar, the angle µ. / in Eq. 6 can be approximatedby the constant
t
the initial passage time of the TLL. In our approach, we only
0
( )
r
need to calculate the correspondinginitial ÀS0. Therefore, we need
angle
(
)
¸
the initial argument of latitude for TLL. This is found from the
0
T
sphericaltriangle shown in Fig. 4, where is the ground target and
is the geocentric latitude of the ground target:
¡1
( )
7
D
a a
¡
µc sin [. = e/ sin.° =2/] ° =2
’
T
a
where is the satellite’s orbital semimajor axis. From the angle
µc, we can compute the longitudinal offset angle margin 1Àmax
Figure 6 shows how 1Àmax is related to the satellite inclination
( )
D
i
sin ¸0 sin’T =sin
5
.
2. LongitudinalOffset Angle Margin
i
angle , the upper limit of the spherical offset angle µc, and ground
target latitude ’T . The arc represents the satellite’s ground track.
To Ž nd 1Àmax , we start by Ž nding . Then the longitudinal offset
angle 1Àmax can be solved by following the equation for a small
a. Anglemarginforsatellitemaximumelevationangleestimation.
For nadirtracking,sometimesthesatellitemaximumelevationangle
to a speciŽ c target may not be high enough for the payloadto image
the ground target. Therefore,we need to Ž nd the longitudinaloffset
angle,whichdetermineswhethera givenpassageis a suitableclosest
approachpassage.The range of an image is dependenton the phys-
l
p
11
(
)
circle Werts, page 727, Eq. A-3 :
2
p ¡
cos
sin ’T
cos2 ’T
( )
8
D
cos 1Àmax
(
)
ical dimension of the charge-coupled device CCD array and the
focal length of the lens. For the narrow angle camera of PoSAT-1,10
11
(
p
To Ž nd we use the law of cosines for sides Werts, page 732,
Eq. A-26 :
(
£
which has an interlacedCCD matrix of dimension576 560 about
8:4 6:5 mm and focal length of 50 mm, the ground cover is
150 100km froman orbitalaltitudeof800km, which corresponds
to the camera’s half FOV of 3.6 deg. For nadir tracking, we need
to Ž nd the longitudinaloffset angle margin 1Àmax to check whether
the previousapproachestimationwouldbe acceptable,which means
)
)
£
£
( )
9
p D
l
C
l
cos
cos cos ´ sin sin ´ cos 0
l
where the three angles of , ´, and 0 are given in terms of ’T , µc,
and using the law of sines Werts, page 732, Eq. A-25 :
11
(
)
i
d
·
we need to check whether
À
1Àmax .
i
sin µc sin
We assume the shape of the Earth is spherical. Let the camera
FOV angle be ° . Then the upper limit of the spherical offset angle
(
)
D
tan 0
10
¡
i
sin ’T cos sin µc
(
)
r
µ. / is given by Fig. 5
sin µc
(
(
)
)
D
sin ´
11
12
¡1
sin0
( )
6
r D
r a
¡
µ. / sin [. = e/ sin.° =2/] °=2
sin ’T
l D
sin
r
a
where is the geocentric range of the satellite and is the Earth’s
e
i
sin
equatorial radius. Vallada and McClain8 describe in detail the ge-
ometry of surveillanceand reconnaissanceoperations.
This completes the approach estimation procedure.

MAI AND PALMER
1121
b. Angle margin for rise-and-settimes.
The rise time of a satel-
The effect of drag on the argumentof latitudecan be incorporated
into the epicycle equationsas
lite shouldoccur when the satellite,at a given orbitalheight, crosses
the horizon plane. In this case, we set up another angle margin µÀ
as shown in Fig. 2 and simpliŽ ed the calculation for it.
2
( )
19
D
C
C
B
¸
®.1 ·/ 1:5 ®
S
a D R C H
.
R
If the orbital radius of the satellite is
, where is
B
where is the drag coefŽ cient.
)
H
the radius of the Earth and
is the satellite orbital height then
We start by Ž nding the change in the epicycle phase ® over one
(
)
see Ref. 8 :
)
D
nodalperiod.By setting¸ to be2¼, we Ž ndthesolutionfor®. 1®
(
)
from Eq. 19 :
( )
13
D R a
cos µÀ
=
£ ¯¡
¢¤
p
(
)
D
C
¢
C
C
B
12¼
1® [4¼=.1 ·/]
1
1
1
20
We, therefore, wish to estimate the times when the satellite
§
reaches the target longitude within µÀ . However, because this a
(
)
Using this in Eq. 17 , we obtain
simpliŽed calculation for satellite longitudinal angle margin, we
R
reduce by a Ž xed fraction to avoid missing some low passes.
(
)
N D f
n
¡
n g
.1À=1®/[ =.!© µ /]
21
This completes our discussion of the coarse search, where we
have included the secular perturbationsand atmospheric drag.
B. Adding Secular Perturbations
A satellite under the in uence of an inverse square gravitational
law has trulyconstantorbitalelements.In reality,however,thereis a
gradualchangein the orbitalelementsdue to the Earth’s oblateness.
The principaleffect of this is to introducea short-periodoscillation
ofthe orbitalelements,which we canignorein mostcases.Theargu-
ment of perigee! and longitudeof the ascendingnode Ä, however,
experience a secular drift that signiŽ cantly changes the long-term
prediction of maximum elevation angle. We can adopt the method
we have outlined in Sec. II.A to take proper account of all of these
secular variations. In the following description we will introduce
the formulas for the satellite maximum elevation angle prediction.
The procedure for satellite rise-and-settimes is similar to this.
First, we can easily add secularperturbationsto the coarsesearch
procedure for the effect on the argument of latitude ¸ that changes
the nodal period if the satellite comes back to the same TLL:
III. ReŽ ning the Estimates of Maximum Elevation and
Rise-and-Set Times
Having estimated the approachtime to the target at TLL, we now
need a procedure that will reŽ ne this estimate to an application set
tolerance. For this we extend Escobal’s4 approach to determine the
maximumelevationanglebyintroducinga newcontrollingequation
based on the epicycle equations.
In Fig. 7, we show the geometry of a satellite pass. The target
T
ground station is located on the surface of an oblate Earth, and
the vector zT is the local normal to the ground target surface. The
S
position of the satellite is . We have the position of both the target
13
(
)
and thesatelliteinEarth-centered,Earth-Ž xed ECEF coordinates
r I
, , Ä, ¸, and ® from the epicycle equations, from
expressed in
which we compute the slant vector P:
( )
14
D
C
¸
®.1 ·/
( )
22
D
¡
P
XS XT
where · is the coefŽ cient of seculardrifts in the epicycleequations9
D nt
This gives the positionof the satelliteas seen from the target.The
elevation angle is the angle measured from the horizon up to the
and epicycle phase ®
. Thus, there is a change in ® for each
)
D
C
TLL crossing of 1® 2¼=.1 · .
h
satellite. If this angle is , then
P
The second effect is the precession of the orbital plane Ä. This
moves the target away from the orbital plane, Ä > 0. We can incor-
P
(
(
)
)
¢
D P
h
sin
P ZT
23
24
porate this effect into the rotation rate of the Earth:
Therefore, we name a new controlling equation:
P
Ä
( )
15
D
¡
!
!
eff
©
F
D
h D
¢
P
.®/ sin
.P ZT /=
In the epicycle description of the orbit,9 the variation in Ä is
F
is a function of ® only through Xs , and ZT and XT are constant
expressed as
vectors in the ECEF coordinate system. Xs varies with ® both be-
cause the satellite moves along its orbits and through the Earth’s ro-
tation in the transformationfrom Earth-centeredinertial13 to ECEF
coordinates. It is obvious that the maximal and zero points of the
( )
16
D
C
Ä0 µ®
Ä
9
where µ is the secular coefŽcient of plane precession. Hence,
P
Ä
h
F
D n
µ .
elevationangle representthe maximal and zero of function .®/,
( )
( )
We can incorporate these results into Eqs. 1 and 2 for the
respectively. Therefore, to Ž nd the maximum elevation angle, we
F
D
coarse search to get
just need to Ž nd ®max such that d =d®.®max
/
0, in which case we
(
(
)
)
D
¡ n N
n
1À .!© µ / .1®= /
17
18
Therefore,
µ
¶
n
C
1À .1 ·/
N D
C. Accounting for Drag
¡
n
2¼ .!© µ /
Gravity is not the only force acting on the satellite. Another im-
portant effect comes from the Earth’s atmosphere, which still has a
signiŽ cant effect on orbits up to altitudes as high as 1000 km. Be-
causemost of our Earth observationsatellitesorbitat altitudeslower
than this, we need to consider the effects of atmosphericdrag. Drag
is very difŽ cult to model because of the many factors affecting the
Earth’s upper atmosphere and the satellite’s attitude, which affects
the cross-sectional area. In this paper, we only consider the effect
of drag on the satellite’s argument of lattitude for the coarse search
r
and include the effect on in the reŽ nement. To test our result, the
Fig. 7 Geometry of ground target T and satellite S in ECEF
coordinates.
SGP4 model12 has been used for drag modeling.

1122
MAI AND PALMER
(
)
F
D
C
C
take the derivative of .®/, and the solution should represent the
satellite maximum elevation time.
Ä
¸
Ä0 µ® 1Ä sin 2®
27
D
C
A a
¡
3 B
¯
2
C
¡
¡
¯
.2 = /[sin.® ®p/ sin ®p] 2Â.1 cos ¯/
The computationof satellitelocationXS is carriedout as follows.
The epicycle equationswhich express
time can be written as
( )
r I
, , Ä, ¸ as functionsof
2
(
(
)
)
C
C
1¸ sin 2¯
28
29
where we have included the effects of atmospheric drag,14 and
r D a
C
.1 ½/
¡ A
¡
C a
cos.® ®p/
 sin ¯
D
C
¯
.1 ·/®
(
(
)
)
C a
¡
B
1r cos 2¯ 2 ¯
25
26
where ½, ·, and µ are the coefŽ cients for secular perturbation; Â
representslong periodic perturbationcoefŽcients;and 1 represents
the short periodic terms.
I D I C
¡
1I .1 cos 2®/
0
(
) (
)
We deŽ ne the satellite position », ´ Fig. 8 on the orbital plane
usingCartesiancoordinateswiththe» axisalongthe ascendingnode
of the orbit. Hence,
D r
D r
»
cos ¸;
´
sin ¸
)
,
E
D X
.
Y
Z
XS can be expressedin ECEF coordinatesas XS
where
;
;
E
E
XE
D
¡
C
I
¡
» cos.º Ä/ ´ cos sin.º Ä/
YE D ¡
¡
C
I
¡
» sin.º Ä/ ´ cos cos.º Ä/
(
)
30
ZE
D
I
´ sin
(
)
Fig. 8 Earth-centered inertial coordinates X, Y, Z and orbital plane
coordinates », ´, J .
(
)
(
and º is the local ephemeris time the angle between the Ž rst point
)
axis in the ECEF frame . These equations
X
of Aries ° and the
E
F
together describe the dependenceof on ®.
IV. Test and Result
Results for two-body and secular perturbation expansion are as
follows.
For many practicalproblems,theapproximationof two-bodymo-
tion is sufŽ cient, especially if two closely neighboring points on a
trajectoryare under investigation.However, for the long-termsatel-
lite passes prediction,we cannot ignorethe cumulative effect of the
gradualvariationof elementsfrom their two-bodyvaluesto achieve
the required accuracy for satellite imaging and communication. In
Fig. 9, we show the prediction of our method compared with the
SGP4 model.12 The diamonds clearly indicate that after only a few
hours the timing error of our prediction based on two-body theory
is already up to 8 s, and within one day the timing error is around
Fig. 9 Diamondsshow timing error of two-body prediction when com-
pared with SGP4 model; squares show error when J2 is incorporated.
£
1 min. Images from PoSAT-1 cover an area 150 100 km on the
Fig. 10 Timing errors when short and long variations are included, when compared with SGP4.

MAI AND PALMER
1123
ground which means that the prediction timing errors should be at
least within 7 s 1 s corresponds to approximately 7 km ground-
ahead time. As Mai and Palmer16 pointed out, when atmospheric
dragcan be ignored,thedifferencebetweenour predictionandSGP4
arisesbecausetheaccuracyofSGP4 isonly10¡6, andthereisa small
drift of ¸ between the epicycleequationsand SGP4 that buildsup to
a signiŽ cant error. This demonstratesthat over a look-aheadtime of
a few days, when drag effects can be ignored, we have achievedthe
prediction timing accuracy required by the high-resolutioncamera
(
)
track to keep the target within the image. The two-body prediction
»
is, therefore, only adequate for image capture within 2 3 h.
To reduce the timing errors, we have included the secular ef-
fects into our coarse search. Unlike Escobal’s4 original controlling
F
equation, our function .®/ not only includes secular drift but also
(
)
has short- and long-periodic perturbations taken into account. We
present in Table 1 a comparison of the epicycle prediction with
an accurate propagator15 to look at the timing errors from the pre-
diction when atmospheric drag is ignored. Table 1 shows that the
timing errors are as small as 0.15 s for a look-ahead time of almost
300 days.
on UoSat-12 Ref. 17 .
We next consider the drag compensation that we introduced
in Sec. II. In Fig. 11, we show the timing errors compared with
SGP4, now with drag included in the model. Both predictions are
based on the same set of initial conditions taken for the same
(
)
North American Aerospace Defence Command NORAD Ž le
(
)
In Fig. 10 we show a comparison of our prediction with SGP4.
Withanexhaustivesearchapproachwe seethatthetimingdifference
between our method and SGP4 is less than 1 s for two months look-
URL:http://www.celestrak.com , and the predictions extend over
100 days. With a look-aheadtime of 100 days, the timing error has
now been reduced to about 2 s. Without drag compensation,for the
Fig. 11 Timing errors when atmospheric drag is included, when compared with SGP4.
Fig. 12 Prediction errors of single pass using our method, for look-ahead times of up to 40 days using NORAD data at different epochs.

1124
MAI AND PALMER
Table 1 Timing errors as a function
of look-ahead time, comparing
the predictions with an accurate
orbit propagator
Although we have shown that, particularly as we approach so-
lar maximum, the variability of atmospheric drag degrades perfor-
–
mance, it is still adequate to predict imaging times to within 1 2 s
overa timescaleof a month. Estimates can be automaticallyupdated
during this interval to monitor the stability of the image capture
time and, hence, remove the effects of the uncertainty in the drag
parameters.
Look-ahead
time, day
Timing
error, s
0.96
4.53
9.26
——
298.99
8:7e¡3
2:2e¡2
3:8e¡2
——
1:5e¡1
We have shown elsewhere16 how to translate NORAD elements,
which are freely available for all traded satellites over the Internet,
to epicycleelements.Hence, thismethod can be used by any system
that has access to these NORAD Ž les.
Acknowledgments
Table 2 Processing time on a Pentium II,
averaged over 10,000 experimentsa
The authors are grateful for the Ž nancial support of the Surrey
(
)
Space Centre SSC and Surrey Satellite Technology Limited. The
authorswish to acknowledgethe supportgiven by Martin Sweeting
of SSC.
Proposed
method
Estimation,
s
ReŽ ning,
s
Two-body
J2
1.71
1.86
2.86
3.71
References
1Da Silva Curiel, R. A., Sun, W., Sweeting, M., Jolly, G., and Stephens,
P., “The ‘GANDA’ Mission,” International Academy of Astronautics,
Paper IAA-00-IAA.11.4.05,Oct. 2000.
a
D
Current program, SGP4 786 s.
2Chu, V., Da Silva Curiel, R. A., Sun, W., and Sweeting, M., “Dis-
aster Monitering Constellation,” International Academy of Astronautics,
Paper IAA-00-IAA.11.4.03,Oct. 2000.
same accuracy level, the look-ahead time is only one week. Thus,
for PoSAT-1, with a typical small satellite remote sensing camera,
we can predict imaging opportunities for up to 100 days ahead.
For UoSat-12, which has a high-resolution camera onboard, we
can predict imaging opportunitiesfor up to 1 month with sufŽ cient
accuracy.
To remove the drift errors in SGP4, we performed one last ex-
periment where we compared the predictionsof our algorithm with
itself, using two different NORAD Ž les. The separation in time be-
tween the two NORAD Ž les was anything up to 40 days, and the
timing errors for the same pass are shown in Fig. 12. One of these
predictions was based on a NORAD data set from just before the
pass. The dates used for this experiment were from May to July of
1997. The variability in prediction time is due to the variability of
atmospheric drag.
3Sweeting,M., and Chen, F.Y., “Network ofLowCost Small Satellites for
Monitoringand MitigationofNatural Disasters,” InternationalAstronautical
Federation, Paper IAF-96-B.3.P215,Oct. 1996.
4Escobal, P. R., Methodsof Orbit Determination, Wiley, New York, 1976,
–
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–
lishing, New York, 1997, pp. 65 78.
6
–
Lawton, J. A., “Numerical Method for Rapidly Determining Satellite
Satellite and Satellite Ground Station In-View Periods,” Journal of Guid-
–
–
ance, Control, and Dynamics, Vol. 10, No. 1, 1987, pp. 32 36.
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–
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and Applications, McGraw Hill, New York, 1997, pp. 741 815.
9Hashida, Y., and Palmer, P., “Epicyclic Motion of Satellites About an
The algorithmis severalordersof magnitudefasterto run thanthe
exhaustivesearchusing SGP4 thatwe have employed.In Table 2 we
presentsome timings for the estimation on a Pentium II. These tim-
ings are sufŽ cientlyshort for this algorithmto be used on hand-held
receiversand are sufŽ cientlyaccurateto controlimaging deviceson
satellites.
(
Oblate Planet,” Journal of Guidance, Control, and Dynamics to be pub-
)
lished .
10Fouquet, M., Sweeting, M. N., “Earth Observation Using Low Cost
Micro/Minisatellites,” Acta Astronautica, Vol. 39, No. 9/12, 1996, pp. 823
–
826.
11Wertz, J. R., Spacecraft AttitudeDetermination and Control, D. Reidel,
Dordrecht, The Netherlands, 1978, pp. 727, 732.
V. Conclusions
12Hoots, F. R., and Roehrich, R. L., “Models for Propagation of NORAD
Element Sets,” Aerospace Defense Center, Spacetrack Rept. 3, Peterson,
We have introduced a new method to predict the passes of a
satellite’s closest approach to a speciŽ c target on the ground. This
is useful for satellite nadir tracking and solving the satellite vis-
ibility problem. We have Ž rst described a coarse search phase of
this method, including two-body motion, secular perturbation, and
atmosphericdrag. We have then describedthe second phase, reŽ ne-
–
AFB, CO, 1980, pp. 10 20.
13Chobotv, V. A., Orbital Mechanics, 2nd ed., AIAA, Reston, VA, 1996,
–
pp. 11 16.
14Hashida, Y., and Palmer, P., “Epicycle Motion of Satellites Under the
(
)
Atmospheric Drag Perturbation,” submitted for publication .
15Palmer, P. L., Aarseth, S. J., Mikkola, S., and Hashida, Y., “High Pre-
cision Integration Methods for Orbit Propagation,”Journal of Astronautical
F
D
ment, which usesa furtherdevelopedcontrollingequation .®/
0
based on the epicycle equations.We have shown that, ignoringdrag
effects,we can achievetiming accuraciesof 1 s for look-aheadtimes
of 60 days. When drag compensationis included, we provide sufŽ -
ciently accuratetiming estimates for over 100 days ahead. For most
imaging and communicationapplicationsusing small satellites,this
is sufŽ cient. For high-resolution imaging, look-ahead time is re-
duced to about 1 month.
–
Sciences, Vol. 46, No. 4, 1998, pp. 329 342.
16Mai, Y., and Palmer, P. L., “Conversion of North American Aerospace
Defence Command Elements to Epicycle Elements,” Journal of Guidance,
–
Control, and Dynamics, Vol. 24, No. 2, 2001, pp. 406 408.
17Fouquet, M., and Sweeting, M., “UoSAT-12 Minisatellite for High Per-
formance Earth Observationat Low Cost,”Acta Astronautica, Vol. 41, No. 3,
–
1997, p. 173 182.