AUTONOMOUS RELATIVE NAVIGATION FOR ... - CiteSeerX

AUTONOMOUS RELATIVE NAVIGATION FOR FORMATION-FLYING SATELLITES USING GPS

Cheryl

GRAMLING

and J. Russell

NASA Goddard Greenbelt,

Anne

LONG,

Space

Flight

Maryland

David

CARPENTER Center

USA 20771

KELBEL,

and Taesul

LEE

Computer Sciences Corporation Lanham-Seabrook, Maryland USA 20706

ABSTRACT-

The

advanced

spacecraft

formation

flyers.

Goddard systems This

Space

Flight

to provide

paper

Center

autonomous

discusses

is

currently

navigation

autonomous

developing and

control

relative

of

navigation

performance for a formation of four eccentric, medium-altitude Earth-orbiting satellites using Global Positioning System (GPS) Standard Positioning Service (SPS) and "GPS-like" intersatellite measurements. The performance of several candidate relative navigation approaches is evaluated. These analyses indicate that an autonomous relative navigation position accuracy of 1 meter root-meansquare can be achieved by differencing measurements from common GPS space estimated solutions.

high-accuracy vehicles are

filtered solutions if only used in the independently-

1 - INTRODUCTION Formation-flying missions science Center

techniques

and data.

and

satellite

autonomy

enable many small, inexpensive The Guidance, Navigation, and

(GSFC)

has

successfully

developed

this

experience

navigation

and control

to

develop

of formation

To support this effort, achievable for proposed

advanced

Laboratory,

Telephone

Published

flight

determination for rendezvous navigation maintaining This

class

mission

Electronics,

data results performance and docking

performance a relatively of missions to study

the

and

spacecraft

Earth

science

satellite

navigation

systems

that

systems

communications the GNCC has

provide

autonomous

and relative navigation intersatellite measurements.

accuracy Several

transceivers that support this tracking concept for these include Johns Hopkins Applied Physics

Telegraph,

and Stanford

([Braz

and

space and ground 94, Hart 97]. Recently,

the GNCC is assessing the absolute formations using GPS and "GPS-like"

International

space

flyers.

are developing GPS Research Laboratory;

Cincinnati

autonomous

Administration's (GPS) [Gram

universities and corporations NASA and the Air Force Laboratory,

revolutionize

to fly in formation and gather concurrent Center (GNCC) at Goddard Space Flight

high-accuracy

using the National Aeronautics and Space systems and the Global Positioning System leveraged

will

satellites Control

Honeywell,

University

96], [Schi 98], [More

[Baue

Motorola,

Jet

Propulsion

99].

98], [Kama 99])

have

shown

relative

orbit

at the meter- to decameter-level using relative GPS pseudorange data scenarios in low Earth orbits. This paper addresses the level of relative

achievable

for a different

tight formation, is represented Earth's

aurora.

eccentric Earth orbits of approximately 10 to 30 kilometers. Maneuvers would

class

in a relatively by

a tetrahedral

This

formation

of missions,

ecentric

500 by 7000 kilometer be performed monthly

than

two

vehicles

orbit.

formation consists

i.e. more

designed of

four

to support satellites

a proposed maintained

in

altitudes, with initial separations to maintain this configuration.

of To

supportautonomousplanning of thesemaneuvers,the total relativeposition and velocity accuracy must be about 100 metersand 2 centimetersper second,respectively.Later in the mission, the separationwould be reducedto about500 meters,which would reducethe total relative position accuracyrequirementsto about 5 meters.This paper quantifiesthe relative navigation accuracy achievablefor this formationby differencingsatellitestatevectorsthat are independentlyestimated using eithera geometricpoint solutionmethodor a high-accuracyextendedKalman filter. 2 -RELATIVE The most

NAVIGATION

straightforward

CONCEPTS

relative

differencing

the absolute

differencing

method

navigation

position

approach

vectors

can be used

of each

to support

control strategies. Fig. 1 illustrates decentralized control of a distributed

computes

satellite

the satellite

relative

in the formation

decentralized,

centralized,

[Zyla

positions 93].

by

The state

or hierarchical

formation

a possible concept for using this approach to support satellite formation. In this case, each satellite independently

computes its absolute state vector using GPS and possibly "GPS-like" intersatellite measurements and transfers it via an intersatellite communications link to every other satellite in the formation. Each satellite computes its relative position to the other satellites by state uses this relative state to plan and execute formation maintenance maneuvers position

within

the formation.

control

strategies.

[Carp 00] discusses

• GPS/Intersatellite Tracking • Absolute State Vector Estimation • Absolute State Vector Differencing for Relative Navigation

a recent

investigation

Via Intersatellite Link State Vector N Via Intersatellite Link

Differencing

Configuration

with Decentralized

The absolute state vector computation can be performed method or a real-time filtered algorithm. The typical computes

the

real-time

solving a set of simultaneous of four GPS space vehicles solutions.

The

ionospheric ephemerides of the time) position

major

delays

the

of

error

Selective

on the order

filtered

Control

either an instantaneous Standard Positioning

spacecraft

position

in the

GPS

Availability

SPS

(SA)

and

measurements

corruption

point solution Service (SPS)

receiver

algorithm,

of 150 meters

arise

applied

to limit geometric solutions to approximately 100 meters when SA is enabled. Typically, GPS receiver vendors

accuracy

A real-time

three-dimensional

using GPS

Formation

time

bias

by

equations constructed using pseudorange measurements to a minimum (SVs). These products are often referred to as geometric or point

sources

and

formation

Satellite N

Satellite 1

receiver

of decentralized

• GPS/Intersatellite Tracking • Absolute State Vector Estimation • Absolute State Vector Differencing for Relative Navigation • Formation Maintenance

State Vector 1

J • Formation Maintenance

Fig. 1 State Vector

vector differencing and to maintain its desired

from

uncorrected

to the GPS

signals

and

(two-dimensional, 95 percent advertise three-dimensional

(lcr).

such as that implemented

in the GPS

Enhanced

Orbit

Determination

(GEODE) software [Godd 00], reduces the impact of the measurement errors by using an extended Kalman filter in conjunction with a high-fidelity orbital dynamics model. In addition to the simple state

vector

increasingly •

•

more

complex

approach,

the

relative

navigation

Simultaneous

estimation

measurements

between

Simultaneous

estimation

using •

differencing

GPS

measurements

Simultaneous estimation satellites and "GPS-like"

of the the local

real-time

local

filtered

can

support

the

following

approaches and

remote

and remote

of the local

approach

satellites

using

singly-differenced

GPS

SV measurement

biases

satellites

and remote

satellites

and GPS

from all satellites of the local measurements

and remote satellites using GPS measurements between the local and remote satellites 2

to all

It is anticipatedthat the more complex approacheswill provide more accuraterelative navigation solutions(for example,see[Axel 86], [Gald93], [Zyla 93], [Binn 97], [Cora98]). 3 - PERFORMANCE SIMULATION PROCEDURE The formation studied consists of four satellites maintained in eccentric Earth orbits at an inclination of 80 degrees with altitudes of approximately 500 kilometers at perigee by 7000kilometers at apogee.All satelliteshavenearly identicalsurfaceareasof 0.6613meters2 and massesof 200 kilograms.Eachsatelliteis offset from the othersatellitesby a total of approximately 10to 30 kilometers,in and/orout of the orbit plane,forming a tetrahedron.To quantify the level of absoluteand relative navigationperformancethat is achievable,GPSand"GPS-like" intersatellite measurementswere simulatedfor the two satellitesthat are in the sameorbital planeand usedto estimatetheir absoluteandrelativepositionsandvelocities. Realistic GPS pseudorangeand "GPS-like" intersatellitemeasurementswere simulated for each satellitebasedon truth ephemeridescomputedusing the GoddardTrajectoryDeterminationSystem (GTDS), which is the primary orbit determinationprogramusedfor operationalsatellite supportat GSFC. The truth ephemerideswere computedusing a high-fidelity force model that includeda 70 by 70 Joint GoddardModel (JGM)-2 for nonsphericalgravity forces,Jet Propulsion Laboratory Definitive Ephemeris200 for solar and lunar gravitational forces, a Harris-Priesteratmospheric density model, and solar radiation pressureforces. Table 1 lists the measurementsimulation options. Table 1.MeasurementSimulationParameters Parameter Measurement

.....

data rate

_Valu-e

............................

ii _ __

'_:_-_:i_ _-_-

-_i._-_i -_ ..... -

GPS: Every 1 minute from all visible GPS SVs lntersatellite: Every 3 minutes from each transmitting satellite

GPS SV ephemerides

Broadcast ephemerides

GPS signal characteristics: SA errors Transmitting antenna pattern

for July 19-22, 1998

25 meter (1-sigma) GPS L-band pattern, modeled from 0 to 90 degrees down from boresight

Transmitted power User antenna models:

28 dB-Watts in maximum gain direction

Visibility constraints

• •

Earth blockage with 50 km altitude tropospheric mask GPS SV transmitting antenna main beam and receiving antenna horizon masks

•

No constraints on intersatellite link

•

9 channels for GPS

•

3 channels for intersatellite link

•

35 dB-Herlz receiver acquisition threshold

Hemispherical

Receiver characteristics

GPS antenna: anti-nadir pointing Maximum gain : 4.9dBic Horizon mask: 85 degrees from boresight

Receiver clock bias white noise spectral density

9.616 x 10.2osecondsz per second

Receiver clock drift rate white noise spectral density Random measurement errors

1.043

Ionospheric

21.3 meters at 500 kilometer height 3.4 meters at 1000 kilometer height

The GPS

and the GPS Block

signal

I! signal

configuration strength

antenna

secondsz per seconds3

GPS pseudorange: 2 meters (1-sigma) Intersatellite pseudorange: 3 meters (1-sigma)

delays

constellation

x 10 27

was based

at the GPS receiver's

pattern

(including

on the GPS location

both the main

broadcast

was modeled

attenuation

model

that was

used

provides

realistic

for the

assuming

and side lobes).

hemispherical GPS antenna, with boresight anti-nadir pointing. threshold was set at 35 dB-hertz, consistent with the performance SV signal

messages Each

user

epoch

date

the nominal satellite

GPS

had

one

The GPS receiver's acquisition of most space receivers. The GPS

signal

acquisition

predictions

[More

99]. The numberof simultaneousmeasurements waslimited to nine GPS(selectedbasedon highest signal-to-noiseratio) and three intersatellite,consistentwith the use of a twelve-channel GPS transreceiver.No constraintswereplacedon acquisitionof the intersatellitesignal. Fig. 2 showsthe numberof GPSSVsvisible asa function of time andaltitude.The satellites'single hemisphericalantennaconsiderablylimits GPS visibility at high altitudes.The periods of lowest visibility (4 or fewer) occur when the satellitesare at altitudesabove 5500 kilometers,where the visibility is highly dependenton the exactposition of the GPS SVs within eachorbit plane. The periodsof best visibility with 6 or morevisible GPSSVs occurwhen the satellitesarewithin the main lobe of the GPSsignal(i.e. below approximately3000kilometers). 14

12

....

-4

_10 g)

! i,

! ;

0 0.2

0.4

0.8

0.8

80OO

i-s°°°................. ?000

I-

t-I I

I

p

_,ooo -

i ....................

--

t_

I ....... I

g

30OO 2O0O

........

_ ........

1000

,--_. .......

_

V

0 0.2

0.4

0.6

0.8

Elapsed Days

Fig. 2. GPS Selective

Availability

level, using the model was used errors

when

function

(SA)

errors

were

as a Function

applied

Lear4 autoregressive integrated with a 2-meter (1-sigma) error

SA is disabled,

of the

SV Visibility

height

which

of the signal

and Altitude

measurements

at a 25-meter

(1-sigma)

moving average time series model [JSC 93]. This level to model the residual SPS ephemeris and clock

above

computed for the test orbit using the Bent noise was simulated assuming a highly-stable of 0.16(109). A twice-integrated simulate the clock bias and clock

to the GPS

is expected path

Time

by 2005. the Earth,

Ionospheric which

ionospheric model crystal oscillator

random walk model, drift noise contributions

was

delays based

were

modeled

on ionospheric

as a delays

available in GTDS. Receiver clock with a l-second root Allan variance

which is based on [Brow to the GPS measurement

97], was errors.

used

to

To minimize potentially large biases in the intersatellite measurements, each transmitting satellite would estimate its clock offset from GPS time and frequency offset from nominal based on GPS measurements and steer its clock to be synchronized to within 100 nanoseconds (30 m) with GPS time. whole

Each

transmitting

millisecond,

Alternately,

satellite

synchronized

the transmitting

part of its navigation expected intersatellite estimates were not

would

transmit

with GPS

satellite

could

a pseudorandom

Time

to within

provide

message and these biases pseudorange measurement included in the preliminary

its clock

noise

the accuracy offset

(PN)

code

of its clock

and frequency

starting offset

bias

could be included in the measurement biases in the transmitter and receiver analysis reported in this paper.

at the estimate.

estimates

as

model. The clock offset

The GPSpseudorangemeasurementsetswereprocessedusing both the point solution methodand the real-time filtered algorithm that is availablein the GEODE flight software.GEODE was also used to process measurementsets that included intersateUitemeasurements.The absolute navigationerrorswerecomputedby differencingthe truth andestimatedabsolutestatevectors.The estimatedrelative statevectorswerecomputedby differencingthe estimatedabsolutestatevectors for the two satellites.The relative navigationerrorswerecomputedby differencingthe true relative statevectorsandthe estimatedrelative statevectors. 4 - RELATIVE

NAVIGATION

PERFORMANCE

BY DIFFERENCING

POINT

SOLUTIONS This

section

method

presents

the absolute

for computation

and relative

of the

absolute

navigation

state

results

vectors.

obtained

The point

absolute position vector and receiver clock bias for each satellite which a minimum of four GPS measurements is available. This pseudorange The

measurements

left-hand

side

from

of Fig. 3 shows

errors of the point solutions The left-hand side of Fig. representative antenna, there

up to a maximum the

is five or fewer

absolute

and the geometrical

solution

provides

the

at every measurement time for method uses all available GPS

root-mean-square

(RMS)

with SA enabled. every orbit when

distribution

and

maximum

position

six or more GPS SVs are visible. position error versus time for a Using a single anti-nadir pointing fewer than four measurements are

available and point solutions cannot be computed. With or without errors of more than 1 kilometer occur for solutions near apogee, satellites

the point

method

of nine GPS SVs.

with and without SA enabled, when 4 shows the absolute point solution

set of individual point solutions are periods of about 70 minutes

using

solution

is poor.

SA enabled, where the During

the peak absolute number of visible

periods

of good

visibility

(i.e. six or more visible GPS SVs), the absolute radial, in-track, and cross-track (RIC) RMS position errors are 47 meters, 20 meters and 17 meters, respectively. Elimination of the SA-induced GPS ephemeris meters, This

and clock 5.4 meters

point

velocity

significantly

method

were

vectors,

approximately

significantly

and 4.5 meters,

solution

vectors

position

errors

10 larger

computed

meters errors

the absolute

not provide

absolute

by differentiating at

per when

RIC RMS

position

errors

to about

19

respectively.

does

computed

reduces

a

1-second

second fewer

during

velocity

a quadratic

solutions

directly.

polynomial

spacing.

This

produced

periods

with

6 or more

The

fit to the RMS

velocity

visible

GPS

absolute

four

nearest

errors SVs,

of with

than 6 GPS SVs are visible.

Absolute Position Error

Relative Position Error

700 600 500

II'-] RMS

(meters)

I

Maximum

(meters)]

400 300 200 100 0

r

>6 SVs With SA

26 SVs Without

SA

99.8% With SA

94% With SA

Percent of GPS

Fig. 3. Absolute

and Relative Satellite With Six or More

55% With SA

99.8% Without

SVs in Common

Position Errors Using Visible GPS SVs

Point

Solutions

SA

...............

250

-T-°t--al-A-bs°iu_te-P.-°-s-l-t!.°--n-En'--°r-I_ete_-

2OO o

•

:_

150

_ _

u

• •

•*

•

100 -'; .... [..o •

• ,

i

•

,

o

80

i

•

•

•

_ ;

•

Total

Relative

Position

o_

'*°

•

so ...................

I

•

i

o;

° ]

•

s :

",

"11,_

•

f'_

_

I

?.

,',

o

_

°.

,p 1.5

0.5

Elapsed

Fig 4. Absolute The right-hand

o

Satellite

for each common

solution

versus

error

navigation

error

_

time

is significantly

relative

navigation

Position

Errors

With

the relative

When

the

absolute

correlated is enabled

when

the percentage

solutions primarily

smaller

would

solutions solution

but

significant.

still

to the relative

are used

in the computation

5 - RELATIVE

equal

obtained

absolute

identical

•

..

errors

position

the

without relative error

of the absolute

NAVIGATION

errors

absolute

error.

g

e

•

2

2.5

Days

Using

obtained

GPS

Point

Solutions

by differencing

SVs is high

When

the

(99.8%

with

of common

from measurement error cancellation

would occur if the absolute point from common GPS SVs. In this of the individual

SA are differenced, position

with

the percentage

error contribution are differenced,

due to ionospheric

RMS

11;

visible GPS SVs, varying the percentage side of Fig. 4 shows the relative point

of common

the root sum square

The

.

are differenced, the solution error contributions from to SA and ionospheric delay) cancel and the relative

than

(primarily

.

1.5

SA Enabled

solution

largest relative navigation error measurement sets that are not

errors

•

1

GPS SVs used by each satellite decreases, less of the solution error is correlated. In these cases when the absolute solutions significantly less. The were computed using

.

0.5

3

satellite with six or more GPS SVs. The right-hand

the absolute errors (due

i •

0

2.5

of Fig. 3 summarizes

SA). In this case when correlated measurement

_,

Elapsed

absolute point solutions of measurements from position

"_i

**.

i

•

Days

and Relative

side

•

2

.

20 •

0

i

•

o

0

(mete_)

"

-:-v ;- "_)o:-

•

Error

t

$

• •

0o ! i

o_"_ ._ *_ .... __ : ..... •

• i

......................

RMS

error

errors)

without

(7 meters),

solutions case, the

position

cancellation

delay

SA enabled

absolute

of the

when

SA (6 meters) only

errors.

remaining

is less than

when

is

SA

is nearly

common

SVs

state vectors.

PERFORMANCE

BY DIFFERENCING

FILTERED

SOLUTIONS This section algorithm available sets,

as

GEODE in the offset

presents

well

as

processing

simulated

and relative

navigation

parameters.

sets

that

Atmospheric

included

intersatellite

pseudorange

left hand

obtained

using

a real-time

measurements.

drag and solar radiation

pressure

Table forces

measurements and clock

that were

measurement

side of Fig. 5 compares

performed. Ensemble solutions obtained by

created

by the varying

filtering

filter algorithm measurement 2 lists

were

using atmospheric drag and solar radiation pressure coefficients and 3 percent respectively from the values used in the truth

A Monte Carlo error analysis was for absolute and relative navigation

for the SA, random,

results

of the absolute state vectors. The extended Kalman flight software was used to process the GPS pseudorange

measurement

state propagation by 10 percent

generation. accumulated

The

the absolute

for computation in the GEODE

the

included that were ephemeris

error statistics were processing 50 sets of

the random

number

seeds

errors.

the steady-state

time-wise

ensemble•

true RMS

and maximum

errors of the RMS/maximum

absolute position estimates, error is the RMS/maximum

with and without SA enabled. The ensemble of the true error (difference between the estimated

true and

the true state) wise ensemble

at each time computed across all 50 Monte Carlo solutions. The steady-state timetrue RMS/maximum error is the RMS/maximum along the time axis of the ensemble

true errors,

omitting

RMS

of the absolute

error

the initial

and with SA disabled,

convergence

position

period

(l day).

and velocity

estimates

2. GEODE

Processing

Figures

6 and 7 show

for representative

with

Value

Earth Gravity model

30x30 Joint Goddard Model (JGM)-2

Solar and lunar ephemeris

Low-precision analytical ephemeris 100 meters

Initial position error in each component Initial velocity error in each component

0.1 meter per second

drag coefficient error

0.22 (10 percent)

Solar radiation pressure coefficient error Initial receiver time bias error

0.042 (3 percent) 100 meters

Initial receiver time bias rate error

1 meter per second

Estimated state

= •

GPS SV ephemerides Intersatellite transmitter ephemerides

Broadcast GPS ephemerides for July 19-22, 1998 Real-time filtered solution for same period

Ionospheric

500 kilometer minimum height of signal path

editing

Measurement

processing rate

User position and velocity GPS receiver time bias and time bias drift

GPS: All available pseudoranges

every 3 minutes

Intersatellite: All available pseudoranges

30

Absolute

SA enabled

Parameters

Parameter

Atmospheric

cases

true

respectively. Table

Nonspherical

the ensemble

Position

Error

Relative

Position

every 3 minutes

Error

25 mRMS meters) m Maximum (meters)]

2O 15 10 5 0 With

SA

Without

SA

99.8%

with

SA

94% with

SA

55% with

SA

99.8%

without

55% without

SA

SA

Percent of GPS SVs in Common

Fig. 5. Absolute

Total

and Relative Steady-State Using Filtered Solutions

RMS Position

Error

Time-Wise Ensemble True with GPS Measurements Total

(meters)

,4

__-_U_-2_-_-_I-TU_L--_TgT-T]UL----.----IT-U21T2_L'In

,_

12

....................................................

12

,o

I

Error

(millimeters

Errors

per second)

' ............ F-A-.---I.......... --i ,o

'I 6

4 t_i

RMS Velocity

Position

6

................................................

4

.,

0.5

1

1.5 Elapsed

Fig 6. Absolute

2

_Relatlvs 0.5

2.5

Error 1

,

, 1.5

Elapsed

Days

2

and Relative Ensemble True RMS Position and Velocity Errors Using Filtered Solutions with 99.8% GPS SVs in Common 7

2.5

Days

With

SA Enabled

3

Total Total RMS Position ................................... -

• 14

!

12

Absolute

Error

(mete=rs_! ............... -

Error

"

= i

14

i

12

Error

(millimeters

per second} J

'

_

RMS Velocity

.....

-

i

i

lo

i

8

8

6

6

4

4

-_i -t_" -If 0

,_ ___E.,_r-._...,'--tit

0.5

1

1.5 Elapsed

Fig 7. Absolute

With

2 2

2.5

2.5

1.5 Elapsed

Days

the absolute

filtered

point solution position errors shown meters, and 2.0 meters, respectively.

entire

1

position

errors

are

significantly

3

Days

and Relative Ensemble True RMS Position and Velocity Errors Using Filtered Solutions with 99.8% GPS SVs in Common

SA enabled,

velocity second,

O.S

3

With

smaller

SA Disabled

than

the

absolute

in Fig. 2, with RIC RMS position errors of 2.6 meters, 5.9 The filtered solutions also provide reliable, high-accuracy

estimates with RIC RMS velocity errors of 3.6 millimeters per second, 1.4 millimeters and 1.2 millimeters per second, respectively. The level of accuracy is consistent over orbit,

insensitive

SA-induced

GPS

approximately

to the number

ephemeris

2.8 meters,

and

of visible

GPS

errors

reduces

clock

4.5 meters

and

1.3 meters,

SVs at any the

specific

absolute

time.

RIC

RMS

filter

compute

the absolute

processing

was

synchronized

filter solutions

so that a high percentage

were

relative position errors of 0.17 meters, errors of 0.36 millimeter per second,

from

of the

position

errors

to

respectively.

The right hand side of Fig. 5 summarizes the relative solution errors obtained absolute filtered solutions for each satellite. The lowest relative errors were absolute

Elimination

per the

common

of the measurements

SVs (99.8%

0.56 meters, and 0.17 meters, 0.09 millimeter per second,

by differencing obtained when

with SA),

yielding

the the

used RMS

to

RIC

and RMS RIC relative velocity and 0.I millimeter per second,

respectively. Fig. 6 also shows the relative ensemble true RMS position and velocity errors versus time when the percentage of common GPS SVs is high (99.8% with SA). In this case, when the absolute solutions are differenced, the error contributions from correlated measurement and dynamic errors cancel and the relative navigation errors. Since the satellites are in nearly the same and

a large

percentage

of the

dynamic

error

accuracy is significantly better than the absolute orbits, the dynamic errors are highly correlated,

contribution

cancels

in all of these

cases.

When

the

percentage of common GPS SVs used by each satellite decreases, less of the measurement error is correlated. Consequently, the absolute error cancellation in the differenced solutions is less and relative errors increase as the percentage of common GPS SVs decreases. With SA disabled (99.8% without

SA),

complete

the RMS

satellite

A preliminary measurements state vector pseudorange

relative

synchronization

position

errors

(99.8%

are comparable

to the

SA-enabled

from the other

with

nearly

with SA).

assessment was made of the impact of augmenting with "GPS-like" intersatellite measurements. In the approach of the local satellite was estimated independently using measurements

case

satellites

in the formation.

the GPS evaluated, GPS and

This simulation

pseudorange the absolute intersatellite assumed

that

a navigation message containing the absolute state vector of each of the transmitting satellites would be provided to the receiving satellite using the intersatellite communications link and used to model the intersatellite measurements and to compute the relative state vector by the state vector differencing by estimation absolute and measurements.

method.

When

the ephemeris

provided

using only GPS measurements relative solution errors are When

the

true

ephemeris

for each transmitting

(with RMS comparable

of each

transmitting

satellite

was

that obtained

position errors of about 8 meters), the to those obtained using only GPS satellite

was

provided,

the

absolute

solutionerror for the local receivingsatellitedecreased to below2 metersbut the relative navigation increaseddueto the reductionin the cancellationof the solutionerror contributionsfrom correlated dynamic and measurementerrors. Thesepreliminary results indicate that high accuracyrelative navigation solutions using cross-link measurements will requirethat cross-link measurementsbe includedin the solutionsfor all the satellitesin formation, in orderto maximizethe cancellationof the absoluteerrorcontributions. 6 - CONCLUSIONS AND FUTURE DIRECTIONS This study assesses the feasibility of using statevector differencingfor relative navigation.When SA is enabled,relativenavigationaccuraciesof betterthan 10metersRMS areachievablewhen six or more GPSSVs aremutually visible by differencingpoint solutions,if only measurementsfrom GPS SVs commonto both satellitesare usedin the absolutesolutions.However, the use of point solutionsis not suitablefor continuousreal-timenavigationin configurationswhere fewer than six GPS SVs are visible during significant portions of the orbit. For the 500 by 7000 kilometer formationusedin this study,reliable pointsolutionswould beavailableonly about50 percentof the time using a single hemisphericalantenna,improving to about90 percentof the time using an omnidirectional antennaconfiguration.Becausepoint solutions do not provide accuratevelocity estimates,they arenot suitablefor applicationsin which statevector informationmust be predicted aheadin time, e.g.,to supportautonomousmaneuverplanning. If the absolutestate vectors are computedusing a high-accuracyfilter, the relative navigation accuracyimprovesto betterthan 1 meterand0.5 millimeter per secondRMS if only measurements from GPS SVs common-to both satellites are used in the absolute solutions. This level of performanceis maintainedover the entireorbit, insensitiveto the numberof visible GPSSVs at any specific time. This relative accuracy satisfies the most stringent navigation requirementsfor autonomousmaneuverplanning of the formation studied.The relative accuracydegradesas the percentageof measurementsfrom commonGPSSVsthatareusedin the point solutionsdecreases. In formationsfor which the separationdistancesarelessthan 100kilometers,suchasthe formation studied in this report, GPSSV visibility will be nearly identicalfor all usersatellites.In this case, restrictingthe measurements processedto thosefrom commonGPSSVs is not unreasonable. Future directions will initially focus on a detailedinvestigation of improvementsto be realized throughthe simultaneousestimationof the local andremotesatellitestatevectors.Expectationsare that the relative navigation performancewill be improved through the processingof singledifferenced measurementsor the estimation of GPS-SV-relatedbiases and the processing of intersatellitemeasurements to simultaneouslyestimatethe locationof all satellitesin the formation. The GEODE flight software, which has recently been enhancedto support multiple satellite estimationandadditionalmeasurement types, will be used in these studies. The relative navigation version of GEODE is being integrated GSFC GNCC. This formation-flying in GNCC's

formation-flying

into a low cost GPS satellite receiver receiver will be used to demonstrate

being developed by the end-to-end performance

test bed.

REFERENCES [Axel

86]

P. Axelrad and J. Kelley, "Near using GPS," Proceedings of Conference,

[Baue

99]

F. Bauer Enhanced

November

Earth Orbit Determination the 1986 1EEE Position

Navigation Navigation

1986, pp. 184-191.

et al., "Enabling Spacecraft Autonomy Technology,"

September

and Rendezvous Location and

Formation Proceedings

Flying Through Spacebourne GPS and of the ION GPS-99, Nashville, TN,

1999

[Binn

97]

P. Binning, Proceedings

"Satellite to Satellite Relative Navigation of the 1997 ION National Technical Meeting,

[Braz

96]

J. Brazzel, et al. "Flight Test Results from STS-69," Space Flight Mechanics 1996, AAS 9

Real-Time Publications

Using January Relative Office,

GPS Pseudoranges," 1997, pp.407-415. GPS Experiment San Diego, CA

on

¢

1

[Brow

97]

R. G. Brown Filtering,

[Carp

00]

J. R.

and P. Y. C. Hwang,

Third

Carpenter,

Formations,"

[Gald

98]

93]

"A

March

T. Corrazini,

John Wiley

264

20-22,

et al.,

at

Signals

and Applied

Kalman

1997

Invetigation

presented

of Decentralized

the

IEEE

Control

Aerospace

for

Satellite

Conference,

Big

Sky

2000

"Experimental

Formation

Flying

Spacecraft,"

J. Galdos,

et al.,

"A GPS

Capture,"

to Random

and Sons,

Preliminary

Paper

Montana, [Cora

Edition,

Intro_htction

Proceedings

Demonstration

Navigation, Relative

of the

of GPS

45(3),

ION

Sensor

for

pp. 195-207.

Navigation

1993

as a Relative

Filter

National

for Automatic

Technical

Rendezvous

Meeting,

and

January

1993,

pp.83-94. [Godd

00]

Goddard

Space Enhanced

Version

5, T. Lee

February [Gram

94]

C.

J.

[Kama

[More

99]

98]

the

99]

Global

Space Charles

Kamano,

Flight

et

al.

1999

Moreau

Moreau

and

Space [Zyla

93]

H.

et

al.,

GPS

Flight

System

Specifications,

Sciences

Corporation,

L. Zyla Vehicle

"GPS

Evaluation

Receiver

1998,

of

Autonomous AIAA

of STS-80

AAS Publications

and M. Montez, "Use of Two Orbital Rendezvous," Proceedings

301-312

10

Filters,

William

Docking

Conference,

Portland,

Analysis

Nashville, and

Eccentric

Relative

Mechanics

Rendezvous

GN&C

Post-flight

Massachusetts,

Flight

Spacecraft

19-2 l, 1997, pp. 123-133

GPS Navigation

Architecture

in Highly

Theory

1994, pp.253-267

of the

May

for

Flight

1993

of the 1999 "RGPS

17-19,

of the SSTI/Lewis

3345,

of the ION GPS-98,

Cambridge

et al. "Results

June

(TONS)

MechanicsEstimation May

Proceedings

Bias Models

Proceedings

System

Navigation

(GPS),

Laboratory,

Marcille,

Flight 3265,

Publication

Range

Navigation

Mechanics

of the

System

Conference

and

Navigation

Publication

"Autonomous

Proceedings

ION Meeting,

E. Schiesser,

Proceedings

"Result

OR, August

Autonomous

98]

Positioning

Mathematical

by Computer

Onboard

Conference

Draper

of ETS-VII,"

M.

prepared

"TDRSS

Center,

Stark

Experiment G.

Global

(GEODE)

(CSC),

Positioning

1997, NASA

M. Lear,

Annual [Schi

al.,

and T. Lee,

Demonstrations," [More

A. Long

NASA

A. Long,

Johnson

I.

CSC-96-968-01ROUD1,

Determination

Experiment,"

Symposium [JSC 93]

et

1994,

R. Hart, Using

and

Gramling

Symposium 97]

Center,

Orbit

2000

Qualification

[Hart

Flight

(GPS)

Expected

Orbits,"

June 28-30, GPS Office,

of

ARP-K

TN, September

Flight 1998

Performance

Proceedings

of the

for 55th

1999

Navigation San Diego,

Flight

Experiment,"

CA

GPS Receivers in Order to Perform Space of the ION-GPS-93, September 1993, pp.