Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 950675 8 pageshttpdxdoiorg1011552013950675

Research ArticleA New Celestial Navigation Method for Spacecraft ona Gravity Assist Trajectory

Ning Xiaolin Huang Panpan and Fang Jiancheng

School of Instrumentation Science amp Opto-electronics Engineering Beihang University (BUAA) Beijing 100191 China

Correspondence should be addressed to Ning Xiaolin ningxiaolinbuaaeducn

Received 5 February 2013 Revised 8 April 2013 Accepted 23 April 2013

Academic Editor Antonio F Bertachini A Prado

Copyright copy 2013 Ning Xiaolin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A practical and reliable capability for autonomous navigation needs to reduce operation cost to improve operational efficiencyand to increase mission safety Celestial navigation is a very attractive autonomous navigation solution for deep space spacecraftThere are mainly two kinds of celestial navigation methods the direct calculation method and the filter method The accuracy ofthe direct calculation method is low and very sensitive to the measurement noiseThe filter method can provide a better navigationperformance if a high accuracy dynamical model is available However the main practical problem existing in the autonomouscelestial navigation of spacecraft on a gravity assist trajectory is that the accuracy of trajectory model is not enough to be used inthe real navigation sometimes which may introduce large estimation error and even cause filter divergence To solve this problema new celestial navigation method is proposed in this paper which effectively combines the direct calculation method and the filtermethod using an interacting multiple model unscented Kalman filter (IMMUKF) The ground experimental results demonstratethat this method can provide better navigation performance and higher reliability than the traditional direct calculation methodand filter method

1 Introduction

The navigation of spacecrafts is performed primarily by theground tracking whose navigation accuracy can reach afew kilometers by the Doppler tracking very long base-line interferometry (VLBI) and delta very long baselineinterferometry (DVLBI) of deep space network (DSN) [12] However this earthbound tracking suffers from manylimitations such as huge cost long time delay discontinuityand easy disturbance Autonomous navigation has becomea key requirement for reliability and safety A variety ofautonomous navigationmethods for near earth satellites havebeen proposed and explored including the magnetometer-based method [3 4] global position system (GPS) [5 6]intersatellite link-based method and celestial navigationmethod but for deep space explorers celestial navigation isthe only feasible way

Generally celestial navigation methods for spacecraftcan be classified into two categories the direct calculationmethod and the filter methodThe direct calculation methoduses celestial measurements to directly calculate the position

of the spacecraft according to the geometric relations betweenthe spacecraft and celestial bodies [7] This method is verysimple and is independent of the system model but itsnavigation accuracy is relatively low and very sensitive tothe measurement error The filter method uses an optimalfilter combined with the celestial measurements and a systemmodel to estimate the position of the spacecraft Because thismethod can deal with measurement noise the navigationaccuracy is higher than that of the direct calculation but ahighly accurate dynamical model is needed [8ndash11]

In deep space missions a gravity assist trajectory is oftenused which uses the gravity of a planet (or other celestialbody) to alter the path and speed of a spacecraft This tech-nique allows to reach destinations which would not be acces-sible with current technology or to reach targets with signifi-cantly reduced propulsion requirements or in a reduced traveltime Many spacecrafts such as Voyager Galileo and Cassiniuse the gravity assist technique to achieve their targets Thetwo Voyager spacecrafts provide a classic example Voyager2 launched in August 1977 took one boost from Jupiter onefrom Saturn later fromUranus and then climbed all the way

2 Mathematical Problems in Engineering

to Neptune and beyond Voyager 1 launched the followingmonth obtained assists from Jupiter and Saturn respectivelyGalileo took one kick from Venus and two from Earth whileorbiting the Sun en route to its destination Jupiter Cassinipassed by Venus twice then Earth and finally Jupiter on theway to Saturn [12ndash14]

In a gravity assist trajectory angular momentum is trans-ferred from the orbiting planet to a spacecraft approachingfrom behind the planet in its progress about the sun Thevalue of spacecraftrsquos speed relative to planet is not changedduring a gravity assist flyby but its direction is changedHowever both value and direction of spacecraftrsquos speedrelative to the sun are changed during a gravity assistflyby Planetrsquos sun-relative orbital velocity is added to thespacecraftrsquos velocity and the spacecraft does not lose thiscomponent on its way out Instead the planet itself loses theenergy The massive planetrsquos loss is too small to be measuredbut the tiny spacecraftrsquos gain can be very great That is to saythat the orbit of spacecraft has changed before and after agravity assist flyby

When the spacecraft is on such a swing-by trajectory itsorbit changes very rapidly which will make it very difficultto build an accurate dynamical model If the inaccuratedynamicalmodel is used in the filtermethod it will introducelarge estimation error and even cause the filter divergenceSo this paper presents a new celestial navigation methodcombining the direct calculation method and the filtermethod for spacecraft on a gravity assist trajectory When anaccurate dynamical model is available the filter method isused to reduce the estimation error caused by measurementerror When such an accurate dynamical model is notavailable the result of direct calculation method is used tocorrect the estimation error caused by inaccurate dynamicalmodel and prevent the filter from divergence An interactingmultiple model unscented Kalman filter (IMMUKF) is usedto deal with the data association problem Compared to thetraditional celestial methods this newmethod can efficientlyimprove the accuracy and reliability in this case The feasi-bility of this new method is validated by ground experimentsusing a spacecraft on the Mars gravity assist trajectory

This paper is systematized in five sections After thisintroduction the principle of the direct calculationmethod isoutlined in Section 2 Then the principle of the filter methodis described followed by the state and measurement modelsin Section 3 Simulation results in Section 4 demonstratethe performance improvement Conclusions are drawn inSection 5

2 Direct Calculation Method

Celestial navigation method is based on the fact that theposition of celestial body (Sun Earth Mars and stars) ininertial frame at certain time is known and their positionobserved on spacecraft in the spacecraft body frame is afunction of spacecraftrsquos positionThe angle between the vectorpointing towardMars and the vector pointing toward the starfrom the spacecraft determines a cone The apex of the coneis located at the position ofMars the symmetry axis points to

Sun

119885

119874 119883

Mars

119884

The c

one o

f posit

ion 1

The cone of position 2

The cone of position 3

1198601

1198602

1198603

Spacecraft

Star 1

Star 2

Star 3

1198711

s1

s2

s3rR1

Figure 1 The principle of direct calculation method

OSun

PhobosMars

Spacecraft

119868119910

119868119911

119868119909

R1

R2

r

1205881 middot L1

1205882 middot L2

Figure 2 The geometry relationship of the spacecraft Mars andPhobos

the star and the opening angle is twice the measured angleThe spacecraft must be on the surface of the cone Obser-vations of two other stars can determine two other conesThe spacecraft must be on the line (L

1) where all three cones

intersect as illustrated in Figure 1 L1can be determined by

solving the following

L1sdot s1= cos119860

1

L1sdot s2= cos119860

2

L1sdot s3= cos119860

3

(1)

From an observation of the Phobos a second line ofposition (L

2) can be determined The position of spacecraft

can be specified uniquely on the point where the two linesintersect as shown in Figure 2

The geometry relationship of the spacecraft Mars andPhobos can be expressed as follows

r = R1+ 120588

1L1= R2+ 120588

2L2 (2)

120588

1L1minus 120588

2L2= R2minus R1 (3)

Mathematical Problems in Engineering 3

where r is the position vector of the spacecraft R1and R

2

are the position vectors of Mars and Phobos respectively 1205881

and 120588

2are the distances of spacecraft to Mars and Phobos

respectively Solving 120588

1and 120588

2from (3) the position of

spacecraft can be determined by (2)The limitation of the direct calculation method is that the

measurement error has a great impact on the calculation errorand it can not provide the velocity information

3 Filter Method

The filter method exploits the dynamics of the spacecraft toremove the effects of the noise and obtain a good estimationof the position and velocity of the spacecraft at the presenttime using the differences between the measured and esti-mated celestial data In the filter method the state (dynamic)and measurement models are needed The dynamical modeldescribes how the state variables change over time and themeasurement model describes the relationship between themeasurement variables and the state variables

The patched-conic technique [15 16] is a useful toolfor designing a fuel-saving gravity assist trajectory In thistechnique a two-body orbit about the Sun is first kine-matically patched onto a two-body hyperbolic orbit aboutthe planet Then the spacecraft proceeds along the planet-centered hyperbolic orbit during the flyby and finally a newtwo-body orbit about the Sun is patched onto the postflybyhyperbolic orbit For a spacecraft on Mars gravity assisttrajectory its trajectory can be broadly divided into two parts

When the spacecraft is outside Mars sphere of influenceits orbital motion can be described as a multibody problemwith the sun as the central body Its dynamical model isexpressed in the sun-centered inertial frame (J20000) asfollows

= V119909+ 119908

119909

119910 = V119910+ 119908

119910

= V119911+ 119908

119911

V119909= minus

120583

119904119909

119903

3

psminus 120583

119898[

(119909 minus 119909

1)

119903

3

pmminus

119909

1

119903

3

sm]

minus 120583

119890[

(119909 minus 119909

2)

119903

3

peminus

119909

2

119903

3

se] + 119908V

119909

V119910= minus

120583

119904119910

119903

3

psminus 120583

119898[

(119910 minus 119910

1)

119903

3

pmminus

119910

1

119903

3

sm]

minus 120583

119890[

(119910 minus 119910

2)

119903

3

peminus

119910

2

119903

3

se] + 119908V

119910

V119911= minus

120583

119904119911

119903

3

psminus 120583

119898[

(119911 minus 119911

1)

119903

3

pmminus

119911

1

119903

3

sm]

minus 120583

119890[

(119911 minus 119911

2)

119903

3

peminus

119911

2

119903

3

se] + 119908V

119911

(4)

where (119909

1 119910

1 119911

1) (1199092 119910

2 119911

2) and (119909 119910 119911) are the position

of Mars the Earth and the spacecraft in the sun-centeredinertial frame rse and rsm are the position vectors of theEarth and Mars relative to the Sun rps rpe and rpm arethe position vectors of the spacecraft relative to the Sunthe Earth and Mars respectively 120583

119904 120583

119898 and 120583

119890are the

gravitational constants of the Sun the Earth and Marsrespectively 119882 = [119908

119909 119908

119910 119908

119911 119908V119909

119908V119910

119908V119911

]

119879 is the systemnoise

When the spacecraft is in Mars sphere of influenceits orbital motion can be described as a perturbed two-body problem with Mars as the central body The dynamicalmodel in Mars-centered inertial frame (J20000) is written asfollows

1015840= V1015840

119909+ 119908

1015840

119909

119910

1015840= V1015840

119910+ 119908

1015840

119910

1015840= V1015840

119911+ 119908

1015840

119911

V1015840

119909= minus 120583

119898

119909

1015840

119903

10158403

pmminus 120583

119904[

119909

1015840minus 119909

1015840

1

119903

10158403

ps+

119909

1015840

1

119903

10158403

ms]

minus 120583

119890[

119909

1015840minus 119909

1015840

2

119903

10158403

pe+

119909

1015840

2

119903

10158403

me] + 119908

1015840

V119909

V1015840

119910= minus 120583

119898

119910

1015840

119903

10158403

pmminus 120583

119904[

119910

1015840minus 119910

1015840

1

119903

10158403

ps+

119910

1015840

1

119903

10158403

ms]

minus 120583

119890[

119910

1015840minus 119910

1015840

2

119903

10158403

pe+

119910

1015840

2

119903

10158403

me] + 119908

1015840

V119910

V1015840

119911= minus 120583

119898

119911

1015840

119903

10158403

pmminus 120583

119904[

119911

1015840minus 119911

1015840

1

119903

10158403

ps+

119911

1015840

1

119903

10158403

ms]

minus 120583

119890[

119911

1015840minus 119911

1015840

2

119903

10158403

pe+

119911

1015840

2

119903

10158403

me] + 119908

1015840

V119911

119903

1015840=radic119909

10158402+ 119910

10158402+ 119911

10158402

(5)

All parameters used in (5) have the same meaning asthose in (4) except that these parameters are expressed inMars-centered inertial frame instead of sun-centered inertialframe

Because these twomodels are in different frames the finalresults have to be transformed to the sun-centered inertialframe

The usually used celestial measurements include (1) thedirections of near celestial bodies (the sun and the planet) (2)the planet-planet angles and (3) the star-planet angles Here

4 Mathematical Problems in Engineering

we take the last one as an example and the correspondingmeasurement equation is given by the following

120572

119904= arccos(minus

rps sdot s119903ps

) + V120572119904

120572

119890= arccos(minus

rpe sdot s119903pe

) + V120572119890

120572

119898= arccos(minus

rpm sdot s119903pm

) + V120572119898

(6)

where s is the position vector of the navigation star in thesun-centered inertial frame which is obtained from the starcatalog by star identification Assume measurement 119885

1=

[120572

119904 120572

119890 120572

119898]

119879andmeasurement noise1198811= [V120572119904 V120572119890 V120572119898

]

119879 themeasurement model can be presented as follows

119885

1(119905) = 119867

1 [119883 (119905) 119905] + 119881

1(119905) (7)

which measurement noise covariance matrix 119877

1is presented

as

119877

1= 119864 [119881

1(119905) 119881

1(119905)

119879] = diag 1205902

119886119904 120590

2

119886119890 120590

2

119886119898 (8)

When the spacecraft flies far from the planets the star-planet angles change very slowlyThe star-planet angles alonecannot supply enough information for position estimationwithout the help of an accurate dynamical model in thefilter method However when the spacecraft is on a gravityassist trajectory its orbit changes very fast and an accuratedynamical model is not available which may introducelarge estimation error and even cause the filter diver-gence

4 New Method Based on IMMUKF

For solving problems of above two methods a new celestialnavigation method is presented for the flyby spacecraft Inthis new method the direct calculation method is used toget a preliminary position estimation result and then thispreliminary result is used as a supplementary measurementin the filter The difficulty existing in the new methodis that the measurement noise covariance matrix is notconstant because the position estimation error of the directcalculation method changes over time due to the movementof spacecraft An IMMUKF method is used to deal withthis problem and to enhance the adaptive capability of thefilter

41 System Model As mentioned above the orbital motionof the spacecraft can be described by two dynamical modelswhich are defined by (5) and (6) They can be writtenin general as

119883(119905) = 119891

1(119883(119905) 119905) + 119882

1(119905) and

119883(119905) =

119891

2(119883(119905) 119905) + 119882

2(119905) When the spacecraft is outside Marsrsquo

sphere of influence (119903pm gt 5773 times 10

6) Model 1 is usedWhen the spacecraft is in Marsrsquo influence sphere (119903pm le

5773 times 10

6) Model 2 is used The system noise covariance

Interaction

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Filtering Filtering119885(119896) 119885(119896)

Λ 1(119896) Λ119873(119896)

120583(119896 minus 1)

Modelprobabilityupdate

120583(119896)

Combination

X1 (k minus 1)P1 (k minus 1)

XN (k minus 1)PN (k minus 1)

X1 (k k minus 1)

P1 (k k minus 1)

XN (k k minus 1)

PN (k k minus 1)

X1(k)P1(k) PN(k)

XN(k)

X(k)P(k)

Figure 3 The IMMUKF algorithm

matrices are 119864[119882

1(119905)119882

1(119905)

119879] = 119876

1 119864[119882

2(119905)119882

2(119905)

119879] = 119876

2

respectivelyIn this new method the position estimation result

obtained by the direct calculation method is used as asupplementary measurement besides the star-planet anglemeasurement defined by (7) The measurement used in thisnew method can be described as follows

119885

2= [

119909

119899

119896

119910

119899

119896

119911

119899

119896

120572

119904 120572

119890 120572

119898]

119879

(9)

where [119909

119899 119910

119899 119911

119899]

119879 is the position estimation result obtainedby the direct calculation method 119896 is a scale factor which isused to make the magnitude of position measurement matchthat of star-planet angle measurement The measurementmodel can be presented as follows

119885

2(119905) = 119867

2 [119883 (119905) 119905] + 119881

2(119905) (10)

Because the measurement noise covariance matrix of thismodel is unknown and time varying a set of differentmatrices is used to approximate it Then 119873 measurementsubmodels are presented as 119885

2119894(119905) = 119867

2[119883(119905) 119905] + 119881

2119894(119905)

which measurement noise covariance matrixes 119877

2119894are as

follows

119877

2119894= 119864 [119881

2119894(119905) 119881

2119894(119905)

119879] = [119877119888119894

119877

1]

119877

119888119894=

119877

119888

119894

(119894 = 1 2 119873)

(11)

where 119877119888is the experimental measurement noise covariance

matrix of the direct calculation method In this paper 119877119888is

set as diag 46 times 10

719 times 10

784 times 10

5

42 IMMUKF Method The interacting multiple model(IMM) algorithm is a very effective estimation algorithm for

Mathematical Problems in Engineering 5

systems which can be described by a mixing of continuousstates and discrete modes [17 18] The switching betweensubmodels is governed by a Markov chain The IMMUKFmethod includes interaction filtering model probabilityupdate and combination of four steps [19ndash21] as shown inFigure 3

421 Interaction Suppose the Markov model transition pro-bability matrix 119875 is as follows

119875 =

[

[

[

[

[

[

[

119901

11119901

12sdot sdot sdot 119901

1119873

119901

21

119901

22sdot sdot sdot

119901

2119873

119901

1198731119901

1198732sdot sdot sdot 119901

119873119873

]

]

]

]

]

]

]

119873

sum

119894

119901

119894119895=

119873

sum

119895

119901

119894119895= 1

(12)

where 119901119894119895is the a priori probability of transition from model

119894 to model 119895 The initial value of 119875 in this paper is set as 119901119894119895

=

091 with 119894 = 119895 and 119901

119894119895= 001 with 119894 = 119895

The mixing probabilities at time 119896 minus 1 are computed asfollows

120583

119894119895(119896 119896 minus 1) =

1

119888

119895

sdot 119901

119894119895sdot 120583

119894(119896 minus 1) (13)

where 120583

119894(119896 minus 1) is the predicted model probability 119888

119895=

sum

119873

119894119901

119894119895sdot 120583

119894(119896 minus 1) is normalization constant The mixed initial

conditions for each submodel 119895 are as follows

119883

119895(119896 119896 minus 1) =

119873

sum

119894

119883

119894(119896 minus 1) sdot 120583

119894119895(119896 119896 minus 1)

119875

119895(119896 119896 minus 1) =

119873

sum

119894

119875

119894(119896 minus 1)

+ [119883

119894(119896 minus 1) minus 119883

119895(119896 119896 minus 1)]

sdot [119883

119894(119896 minus 1) minus 119883

119895(119896 119896 minus 1)]

119879

times 120583

119894119895(119896 119896 minus 1)

(14)

422 Filtering Because the state model and measurementmodel described above are nonlinear the optimal state andcovariance estimates 119883

119895(119896) and 119875

119895(119896) for each sub-model 119895

are calculated using UKF [22 23]

423 Model Probability Update Consider the following

119903

119894(119896) = 119885 (119896) minus 119885

119894(119896 119896 minus 1)

119878

119894(119896) = 119864 [119903

119894(119896) 119903

119894(119896)

119879]

Λ

119894(119896) = 119873 (119903

119894(119896) 0 119878

119894(119896))

=

exp [minus (12) 119903

119894(119896)

119879119878

119894(119896) 119903

119894(119896)]

radic2120587

1003816

1003816

1003816

1003816

119878

119894(119896)

1003816

1003816

1003816

1003816

119888 =

119873

sum

119894

Λ

119894(119896) 119888

119894

120583

119894(119896) =

1

119888

Λ

119894(119896)

119873

sum

119894

119901

119894119895sdot 120583

119894(119896 minus 1) =

1

119888

Λ

119894(119896) 119888

119894

(15)

where 119903119894(119896) and 119878

119894(119896) are innovation vector and its covariance

respectively Λ

119894(119896) is Model likelihood 119873(119909) is Gaussian

distribution

424 Combination Combined state119883(119896) and its covariance119875(119896) are calculated from the weighted state estimates 119883

119894(119896)

and covariances 119875119894(119896) as follows

119883 (119896) =

119873

sum

119894

119883

119894(119896) sdot 120583

119894(119896) (16)

119875 (119896) =

119873

sum

119894

119875

119894(119896) + [119883

119894(119896) minus 119883 (119896)]

sdot[119883

119894(119896) minus 119883 (119896)]

119879

sdot 120583

119894(119896)

(17)

5 Simulation Results

This section presents simulation results to show the perfor-mance improvement of this new method compared with thetraditional direct calculation method and the filter methodThe process of simulation is shown in Figure 4

The ideal trajectory of spacecraft is created by the STK(Satellite Tool Kit) whose initial value of semimajor axis is193216365381 km eccentricity 0236386 inclination 23455∘right Ascension of the ascending node 0258∘ and the argu-ment of periapsis 71347∘ The near bodies used in the directcalculation method are Mars and Phobos All measurementsare created by the celestial navigation simulation platform[24] using ideal trajectory of STK ephemeris of DE405 andTycho-2 catalogue The ideal trajectory of the spacecraft isshown in Figure 5 which has been checked according toTisserandrsquos criterion described in (17) [18 20]

119879 =

119903mars119886

+ 2

radic

119886 (1 minus 119890

2)

119903marscos 119894 (18)

The Tisserand parameter 119879 is conserved around minus02686

before and after the close encounter with Mars

6 Mathematical Problems in Engineering

Sensorsimulator

Stars

Earth SunPhobos

and MarsHigh dynamic celestial simulation system

Sensor

Star-planet angles

Direct calculationmethod

Spacecraftposition

measurement

Orbit and attitude

Spacecraft motion simulationsystem based on STK

Orbit dynamicssimulation

Attitude dynamicssimulation

IMMUKF

Navigation computer

Positionand velocity

Demo system

Figure 4 The process of simulation

1332119864 + 007

6659119864 + 006

0

minus6659119864 + 006

minus1332119864 + 007

minus1333119864 + 007 minus6663119864 + 006 6663119864 + 006 1333119864 + 0070

Figure 5 The ideal trajectory of the spacecraft

The accuracy of the Sun sensor Phobosrsquo sensor the Earthsensor and Mars sensor are selected to be 510158401015840 510158401015840 1510158401015840 and0005∘ respectively The accuracy of the star sensor is 310158401015840

Figure 6 shows the performance of the direct calculationmethod which is obtained with a 1-minute sampling intervalduring the 20-day periodwhen the spacecraft is nearMars Asit can be seen from Figure 6 the navigation accuracy of thismethod mainly depends on the accuracy of measurement soit is very sensitive to the measurement noise The geometryrelationship of the spacecraft Mars and Phobos also hasan impact on the navigation performance The shorter thedistance between spacecraft andMars (or Phobos) the higherthe navigation accuracy is

Figure 7 shows the performance comparison betweenthis new method and the traditional filter method which

0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

1400

1600

1800

2000Position estimation error

Time (day)

Posit

ion

erro

r (km

)

Figure 6 The position estimation error of the direct calculationmethod

only uses the star-planet angles as measurement under thesame simulation conditions As demonstrated in Figure 7the traditional filter method can provide a good navigationperformance when the spacecraft is far from Mars and ahighly accurate model is available but its estimation errorincreases greatly when the spacecraft is close to Mars Thisis because the star-planet angles alone cannot supply enough

Mathematical Problems in Engineering 7

Table 1 Performance comparison of different methods

Method Position estimation error(RMS) (km)

Velocity estimation error(RMS) (ms)

Position estimation error(Max) (km)

Velocity estimation error(Max) (ms)

Direct calculation method 19250 mdash 18162 mdash

Filter method 71725 01507 15468 04937

New method 62082 01118 16787 04486

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160Position estimation error

Time (day)

Posit

ion

estim

atio

n er

ror (

km)

UKFIMMUKF

0 2 4 6 8 10 12 14 16 18 20Time (day)

UKFIMMUKF

0

1

2

3

4

5

6Innovation sequence

Inno

vatio

n se

quen

ce (r

ad)

minus1

times10minus7

Figure 7 Results comparison between IMMUKF and UKF

information to correct the large estimation error introducedby the inaccurate dynamical model which even cause thedivergence of the filter after then The innovation sequenceof the two methods which has subtracted the measurementnoise is shown in Figure 7 The figure of the innovationsequence also demonstrates the divergence of the traditionalfilter method However the new method presented in thispaper shows a stable and satisfying navigation performanceThe details are given in Table 1

From these results we see that this IMMUKF-basednew celestial navigation method overcomes the shortcom-ings of the direct calculation method and the traditionalfilter method On one hand when an accurate dynamicalmodel is available the impact of measurement error onthe navigation performance is reduced by the dynamicalmodel On the other hand when an accurate dynamicalmodel is not available the impact of system error on thenavigation performance is reduced by the result of directcalculation method The navigation accuracy and reliabilityare greatly improved by the effective combination of thedirect calculationmethod and the filtermethodThese resultsdemonstrate that this new method is a particularly suitablecelestial navigation method of spacecrafts on a gravity assisttrajectory

6 Conclusions

A new autonomous celestial navigation method for space-crafts on a gravity assist trajectory is studied which effectivelycombines the direct calculationmethod and the filtermethodbased on IMMUKF The navigation performance of this newmethod is tested by the ground experiments and a positionestimation error within 20 km is obtained Compared withthe traditional methods this method can achieve a higheraccuracy and stability These results verify that this methodis a promising and attractive scheme of autonomous navi-gation for the spacecraft using gravity assist A limitation ofthis study is the significant increase in computational costThis method should have a broad application potential inautonomous navigation of deep space probes especially incases when the accurate dynamical model is not availableHow to enhance the adaptive capability and to reduce thecomputational cost is under further investigation

Acknowledgments

The research presented in this paper has been supportedby the National Natural Science Foundation of China(61233005) Program for New Century Excellent Talents in

8 Mathematical Problems in Engineering

University (NCET) (NCET-11-0771) and Aerospace Scienceand Technology Innovation Fund (10300002012117003) Theauthors wish to express their gratitude to all members of theScience and Technology on Inertial Laboratory and Funda-mental Science on Novel Inertial Instrument amp NavigationSystem Technology Laboratory for their valuable comments

References

[1] P W Kinman ldquoDoppler tracking of planetary spacecraftrdquo IEEETransactions on Microwave Theory and Techniques vol 40 no6 pp 1199ndash1204 1992

[2] B M Portock L A DrsquoAmario and R J Haw ldquoMars explorationrover orbit determination using ΔVLBI datardquo in Proceedings ofthe AIAAAAS Astrodynamics Specialist Conference MontereyCalif USA August 2002

[3] M L Psiaki L Huang and S M Fox ldquoGround tests of magne-tometer-based autonomous navigation (MAGNAV) for low-earth-orbiting spacecraftrdquo Journal of Guidance Control andDynamics vol 16 no 1 pp 206ndash214 1993

[4] H Jung and M L Psiaki ldquoTests of magnetometersun-sensororbit determination using flight datardquo Journal of GuidanceControl and Dynamics vol 25 no 3 pp 582ndash590 2002

[5] A Jaggi U Hugentobler H Bock and G Beutler ldquoPrecise orbitdetermination for GRACE using undifferenced or doubly dif-ferenced GPS datardquo Advances in Space Research vol 39 no 10pp 1612ndash1619 2007

[6] H T Son C H Roger C S Wendy and W Terri ldquoHigh accu-racy autonomous navigation using the Global Positioning Sys-tem (GPS)rdquo in Proceeding of the 12th International Symposiumon Space Flight Dynamics pp 73ndash78 Darmstadt Germany June1997

[7] R H Battin An Introduction to the Mathematics and Methodsof Astrodynamics American Institute of Aeronautics and Astro-nautics New York NY USA 1999

[8] B Kaufman R Campion B Vorder andW Richard ldquoAn over-view of the astrodynamics for the Deep Space Program ScienceExperiment mission (DSPSE)rdquo in Proceedings of the 44th Inter-national Astronautical Congress (IAF rsquo93) pp 141ndash155 1993

[9] R R Dasenbrock ldquoDSPSE autonomous position estimationexperimentrdquo in Proceedings of the International Symposium onAdvances in the Astronautical Sciences pp 203ndash217 April 1993

[10] A Long D Leung D Folta and C Gramling ldquoAutonomousnavigation of high-earth satellites using celestial objects anddoppler measurementsrdquo in Proceedings of the AIAAAAS Astro-dynamis Speialist Conferene 2000

[11] J R Yim J L Grassidis and J L Junkins ldquoAutonomousorbit navigation of interplanetary spacecraftrdquo in The AmericanInstitute of Aeronautics and Astronautics Guidance Navigationand Control Conference pp 53ndash61 Denver Colo USA August2000

[12] C R H Solorzano A A Sukhanov and A F B A Prado ldquoAstudy of trajectories to theNeptune systemusing gravity assistsrdquoAdvances in Space Research vol 40 no 1 pp 125ndash133 2007

[13] J D Anderson J K Campbell and M M Nieto ldquoThe energytransfer process in planetary flybysrdquoNewAstronomy vol 12 no5 pp 383ndash397 2007

[14] R P Russell and C A Ocampo ldquoGeometric analysis of free-return trajectories following a gravity-assisted flybyrdquo Journal ofSpacecraft and Rockets vol 42 no 1 pp 138ndash151 2005

[15] W Hohmann ldquoThe availability of heavenly bodiesrdquo NASAReport NASA-TT-F-44 1960

[16] W E Wiesel Spaceflight Dynamics McGraw-Hill New YorkNY USA 1989

[17] H A P Blom and Y Bar-Shalom ldquoInteracting multiple modelalgorithm for systems with Markovian switching coefficientsrdquoIEEE Transactions on Automatic Control vol 33 no 8 pp 780ndash783 1988

[18] Y Bar-Shalom K C Chang and H A P Blom ldquoTracking amaneuvering target using input estimation versus the interact-ing multiple model algorithmrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 25 no 2 pp 296ndash300 1992

[19] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[20] Y Boers and J N Driessen ldquoInteractingmultiplemodel particlefilterrdquo IEE Proceedings Radar Sonar and Navigation vol 150no 5 pp 344ndash349 2003

[21] R Toledo-Moreo M A Zamora-Izquierdo B Ubeda-Minarroand A F Gomez-Skarmeta ldquoHigh-integrity IMM-EKF-basedroad vehicle navigation with low-cost GPSSBASINSrdquo IEEETransactions on Intelligent Transportation Systems vol 8 no 3pp 491ndash511 2007

[22] O Payne and A Marrs ldquoAn unscented particle filter for GMTItrackingrdquo in Proceedings of the IEEE Aerospace ConferenceProceedings vol 3 pp 1869ndash1875 March 2004

[23] R Merwe A Doucet N Freitas and E Wan ldquoThe unscentedparticle filterrdquo Tech Rep CUEDF-INFENGTR 380 Cam-bridge University Engineering Department 2000

[24] W Quan J Fang F Xu andW Sheng ldquoHybrid simulation sys-tem study of SINSCNS integrated navigationrdquo IEEE Aerospaceand Electronic Systems Magazine vol 23 no 2 pp 17ndash24 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

to Neptune and beyond Voyager 1 launched the followingmonth obtained assists from Jupiter and Saturn respectivelyGalileo took one kick from Venus and two from Earth whileorbiting the Sun en route to its destination Jupiter Cassinipassed by Venus twice then Earth and finally Jupiter on theway to Saturn [12ndash14]

In a gravity assist trajectory angular momentum is trans-ferred from the orbiting planet to a spacecraft approachingfrom behind the planet in its progress about the sun Thevalue of spacecraftrsquos speed relative to planet is not changedduring a gravity assist flyby but its direction is changedHowever both value and direction of spacecraftrsquos speedrelative to the sun are changed during a gravity assistflyby Planetrsquos sun-relative orbital velocity is added to thespacecraftrsquos velocity and the spacecraft does not lose thiscomponent on its way out Instead the planet itself loses theenergy The massive planetrsquos loss is too small to be measuredbut the tiny spacecraftrsquos gain can be very great That is to saythat the orbit of spacecraft has changed before and after agravity assist flyby

When the spacecraft is on such a swing-by trajectory itsorbit changes very rapidly which will make it very difficultto build an accurate dynamical model If the inaccuratedynamicalmodel is used in the filtermethod it will introducelarge estimation error and even cause the filter divergenceSo this paper presents a new celestial navigation methodcombining the direct calculation method and the filtermethod for spacecraft on a gravity assist trajectory When anaccurate dynamical model is available the filter method isused to reduce the estimation error caused by measurementerror When such an accurate dynamical model is notavailable the result of direct calculation method is used tocorrect the estimation error caused by inaccurate dynamicalmodel and prevent the filter from divergence An interactingmultiple model unscented Kalman filter (IMMUKF) is usedto deal with the data association problem Compared to thetraditional celestial methods this newmethod can efficientlyimprove the accuracy and reliability in this case The feasi-bility of this new method is validated by ground experimentsusing a spacecraft on the Mars gravity assist trajectory

This paper is systematized in five sections After thisintroduction the principle of the direct calculationmethod isoutlined in Section 2 Then the principle of the filter methodis described followed by the state and measurement modelsin Section 3 Simulation results in Section 4 demonstratethe performance improvement Conclusions are drawn inSection 5

2 Direct Calculation Method

Celestial navigation method is based on the fact that theposition of celestial body (Sun Earth Mars and stars) ininertial frame at certain time is known and their positionobserved on spacecraft in the spacecraft body frame is afunction of spacecraftrsquos positionThe angle between the vectorpointing towardMars and the vector pointing toward the starfrom the spacecraft determines a cone The apex of the coneis located at the position ofMars the symmetry axis points to

Sun

119885

119874 119883

Mars

119884

The c

one o

f posit

ion 1

The cone of position 2

The cone of position 3

1198601

1198602

1198603

Spacecraft

Star 1

Star 2

Star 3

1198711

s1

s2

s3rR1

Figure 1 The principle of direct calculation method

OSun

PhobosMars

Spacecraft

119868119910

119868119911

119868119909

R1

R2

r

1205881 middot L1

1205882 middot L2

Figure 2 The geometry relationship of the spacecraft Mars andPhobos

the star and the opening angle is twice the measured angleThe spacecraft must be on the surface of the cone Obser-vations of two other stars can determine two other conesThe spacecraft must be on the line (L

1) where all three cones

intersect as illustrated in Figure 1 L1can be determined by

solving the following

L1sdot s1= cos119860

1

L1sdot s2= cos119860

2

L1sdot s3= cos119860

3

(1)

From an observation of the Phobos a second line ofposition (L

2) can be determined The position of spacecraft

can be specified uniquely on the point where the two linesintersect as shown in Figure 2

The geometry relationship of the spacecraft Mars andPhobos can be expressed as follows

r = R1+ 120588

1L1= R2+ 120588

2L2 (2)

120588

1L1minus 120588

2L2= R2minus R1 (3)

Mathematical Problems in Engineering 3

where r is the position vector of the spacecraft R1and R

2

are the position vectors of Mars and Phobos respectively 1205881

and 120588

2are the distances of spacecraft to Mars and Phobos

respectively Solving 120588

1and 120588

2from (3) the position of

spacecraft can be determined by (2)The limitation of the direct calculation method is that the

measurement error has a great impact on the calculation errorand it can not provide the velocity information

3 Filter Method

The filter method exploits the dynamics of the spacecraft toremove the effects of the noise and obtain a good estimationof the position and velocity of the spacecraft at the presenttime using the differences between the measured and esti-mated celestial data In the filter method the state (dynamic)and measurement models are needed The dynamical modeldescribes how the state variables change over time and themeasurement model describes the relationship between themeasurement variables and the state variables

The patched-conic technique [15 16] is a useful toolfor designing a fuel-saving gravity assist trajectory In thistechnique a two-body orbit about the Sun is first kine-matically patched onto a two-body hyperbolic orbit aboutthe planet Then the spacecraft proceeds along the planet-centered hyperbolic orbit during the flyby and finally a newtwo-body orbit about the Sun is patched onto the postflybyhyperbolic orbit For a spacecraft on Mars gravity assisttrajectory its trajectory can be broadly divided into two parts

When the spacecraft is outside Mars sphere of influenceits orbital motion can be described as a multibody problemwith the sun as the central body Its dynamical model isexpressed in the sun-centered inertial frame (J20000) asfollows

= V119909+ 119908

119909

119910 = V119910+ 119908

119910

= V119911+ 119908

119911

V119909= minus

120583

119904119909

119903

3

psminus 120583

119898[

(119909 minus 119909

1)

119903

3

pmminus

119909

1

119903

3

sm]

minus 120583

119890[

(119909 minus 119909

2)

119903

3

peminus

119909

2

119903

3

se] + 119908V

119909

V119910= minus

120583

119904119910

119903

3

psminus 120583

119898[

(119910 minus 119910

1)

119903

3

pmminus

119910

1

119903

3

sm]

minus 120583

119890[

(119910 minus 119910

2)

119903

3

peminus

119910

2

119903

3

se] + 119908V

119910

V119911= minus

120583

119904119911

119903

3

psminus 120583

119898[

(119911 minus 119911

1)

119903

3

pmminus

119911

1

119903

3

sm]

minus 120583

119890[

(119911 minus 119911

2)

119903

3

peminus

119911

2

119903

3

se] + 119908V

119911

(4)

where (119909

1 119910

1 119911

1) (1199092 119910

2 119911

2) and (119909 119910 119911) are the position

of Mars the Earth and the spacecraft in the sun-centeredinertial frame rse and rsm are the position vectors of theEarth and Mars relative to the Sun rps rpe and rpm arethe position vectors of the spacecraft relative to the Sunthe Earth and Mars respectively 120583

119904 120583

119898 and 120583

119890are the

gravitational constants of the Sun the Earth and Marsrespectively 119882 = [119908

119909 119908

119910 119908

119911 119908V119909

119908V119910

119908V119911

]

119879 is the systemnoise

When the spacecraft is in Mars sphere of influenceits orbital motion can be described as a perturbed two-body problem with Mars as the central body The dynamicalmodel in Mars-centered inertial frame (J20000) is written asfollows

1015840= V1015840

119909+ 119908

1015840

119909

119910

1015840= V1015840

119910+ 119908

1015840

119910

1015840= V1015840

119911+ 119908

1015840

119911

V1015840

119909= minus 120583

119898

119909

1015840

119903

10158403

pmminus 120583

119904[

119909

1015840minus 119909

1015840

1

119903

10158403

ps+

119909

1015840

1

119903

10158403

ms]

minus 120583

119890[

119909

1015840minus 119909

1015840

2

119903

10158403

pe+

119909

1015840

2

119903

10158403

me] + 119908

1015840

V119909

V1015840

119910= minus 120583

119898

119910

1015840

119903

10158403

pmminus 120583

119904[

119910

1015840minus 119910

1015840

1

119903

10158403

ps+

119910

1015840

1

119903

10158403

ms]

minus 120583

119890[

119910

1015840minus 119910

1015840

2

119903

10158403

pe+

119910

1015840

2

119903

10158403

me] + 119908

1015840

V119910

V1015840

119911= minus 120583

119898

119911

1015840

119903

10158403

pmminus 120583

119904[

119911

1015840minus 119911

1015840

1

119903

10158403

ps+

119911

1015840

1

119903

10158403

ms]

minus 120583

119890[

119911

1015840minus 119911

1015840

2

119903

10158403

pe+

119911

1015840

2

119903

10158403

me] + 119908

1015840

V119911

119903

1015840=radic119909

10158402+ 119910

10158402+ 119911

10158402

(5)

All parameters used in (5) have the same meaning asthose in (4) except that these parameters are expressed inMars-centered inertial frame instead of sun-centered inertialframe

Because these twomodels are in different frames the finalresults have to be transformed to the sun-centered inertialframe

The usually used celestial measurements include (1) thedirections of near celestial bodies (the sun and the planet) (2)the planet-planet angles and (3) the star-planet angles Here

4 Mathematical Problems in Engineering

we take the last one as an example and the correspondingmeasurement equation is given by the following

120572

119904= arccos(minus

rps sdot s119903ps

) + V120572119904

120572

119890= arccos(minus

rpe sdot s119903pe

) + V120572119890

120572

119898= arccos(minus

rpm sdot s119903pm

) + V120572119898

(6)

where s is the position vector of the navigation star in thesun-centered inertial frame which is obtained from the starcatalog by star identification Assume measurement 119885

1=

[120572

119904 120572

119890 120572

119898]

119879andmeasurement noise1198811= [V120572119904 V120572119890 V120572119898

]

119879 themeasurement model can be presented as follows

119885

1(119905) = 119867

1 [119883 (119905) 119905] + 119881

1(119905) (7)

which measurement noise covariance matrix 119877

1is presented

as

119877

1= 119864 [119881

1(119905) 119881

1(119905)

119879] = diag 1205902

119886119904 120590

2

119886119890 120590

2

119886119898 (8)

When the spacecraft flies far from the planets the star-planet angles change very slowlyThe star-planet angles alonecannot supply enough information for position estimationwithout the help of an accurate dynamical model in thefilter method However when the spacecraft is on a gravityassist trajectory its orbit changes very fast and an accuratedynamical model is not available which may introducelarge estimation error and even cause the filter diver-gence

4 New Method Based on IMMUKF

For solving problems of above two methods a new celestialnavigation method is presented for the flyby spacecraft Inthis new method the direct calculation method is used toget a preliminary position estimation result and then thispreliminary result is used as a supplementary measurementin the filter The difficulty existing in the new methodis that the measurement noise covariance matrix is notconstant because the position estimation error of the directcalculation method changes over time due to the movementof spacecraft An IMMUKF method is used to deal withthis problem and to enhance the adaptive capability of thefilter

41 System Model As mentioned above the orbital motionof the spacecraft can be described by two dynamical modelswhich are defined by (5) and (6) They can be writtenin general as

119883(119905) = 119891

1(119883(119905) 119905) + 119882

1(119905) and

119883(119905) =

119891

2(119883(119905) 119905) + 119882

2(119905) When the spacecraft is outside Marsrsquo

sphere of influence (119903pm gt 5773 times 10

6) Model 1 is usedWhen the spacecraft is in Marsrsquo influence sphere (119903pm le

5773 times 10

6) Model 2 is used The system noise covariance

Interaction

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Filtering Filtering119885(119896) 119885(119896)

Λ 1(119896) Λ119873(119896)

120583(119896 minus 1)

Modelprobabilityupdate

120583(119896)

Combination

X1 (k minus 1)P1 (k minus 1)

XN (k minus 1)PN (k minus 1)

X1 (k k minus 1)

P1 (k k minus 1)

XN (k k minus 1)

PN (k k minus 1)

X1(k)P1(k) PN(k)

XN(k)

X(k)P(k)

Figure 3 The IMMUKF algorithm

matrices are 119864[119882

1(119905)119882

1(119905)

119879] = 119876

1 119864[119882

2(119905)119882

2(119905)

119879] = 119876

2

respectivelyIn this new method the position estimation result

obtained by the direct calculation method is used as asupplementary measurement besides the star-planet anglemeasurement defined by (7) The measurement used in thisnew method can be described as follows

119885

2= [

119909

119899

119896

119910

119899

119896

119911

119899

119896

120572

119904 120572

119890 120572

119898]

119879

(9)

where [119909

119899 119910

119899 119911

119899]

119879 is the position estimation result obtainedby the direct calculation method 119896 is a scale factor which isused to make the magnitude of position measurement matchthat of star-planet angle measurement The measurementmodel can be presented as follows

119885

2(119905) = 119867

2 [119883 (119905) 119905] + 119881

2(119905) (10)

Because the measurement noise covariance matrix of thismodel is unknown and time varying a set of differentmatrices is used to approximate it Then 119873 measurementsubmodels are presented as 119885

2119894(119905) = 119867

2[119883(119905) 119905] + 119881

2119894(119905)

which measurement noise covariance matrixes 119877

2119894are as

follows

119877

2119894= 119864 [119881

2119894(119905) 119881

2119894(119905)

119879] = [119877119888119894

119877

1]

119877

119888119894=

119877

119888

119894

(119894 = 1 2 119873)

(11)

where 119877119888is the experimental measurement noise covariance

matrix of the direct calculation method In this paper 119877119888is

set as diag 46 times 10

719 times 10

784 times 10

5

42 IMMUKF Method The interacting multiple model(IMM) algorithm is a very effective estimation algorithm for

Mathematical Problems in Engineering 5

systems which can be described by a mixing of continuousstates and discrete modes [17 18] The switching betweensubmodels is governed by a Markov chain The IMMUKFmethod includes interaction filtering model probabilityupdate and combination of four steps [19ndash21] as shown inFigure 3

421 Interaction Suppose the Markov model transition pro-bability matrix 119875 is as follows

119875 =

[

[

[

[

[

[

[

119901

11119901

12sdot sdot sdot 119901

1119873

119901

21

119901

22sdot sdot sdot

119901

2119873

119901

1198731119901

1198732sdot sdot sdot 119901

119873119873

]

]

]

]

]

]

]

119873

sum

119894

119901

119894119895=

119873

sum

119895

119901

119894119895= 1

(12)

where 119901119894119895is the a priori probability of transition from model

119894 to model 119895 The initial value of 119875 in this paper is set as 119901119894119895

=

091 with 119894 = 119895 and 119901

119894119895= 001 with 119894 = 119895

The mixing probabilities at time 119896 minus 1 are computed asfollows

120583

119894119895(119896 119896 minus 1) =

1

119888

119895

sdot 119901

119894119895sdot 120583

119894(119896 minus 1) (13)

where 120583

119894(119896 minus 1) is the predicted model probability 119888

119895=

sum

119873

119894119901

119894119895sdot 120583

119894(119896 minus 1) is normalization constant The mixed initial

conditions for each submodel 119895 are as follows

119883

119895(119896 119896 minus 1) =

119873

sum

119894

119883

119894(119896 minus 1) sdot 120583

119894119895(119896 119896 minus 1)

119875

119895(119896 119896 minus 1) =

119873

sum

119894

119875

119894(119896 minus 1)

+ [119883

119894(119896 minus 1) minus 119883

119895(119896 119896 minus 1)]

sdot [119883

119894(119896 minus 1) minus 119883

119895(119896 119896 minus 1)]

119879

times 120583

119894119895(119896 119896 minus 1)

(14)

422 Filtering Because the state model and measurementmodel described above are nonlinear the optimal state andcovariance estimates 119883

119895(119896) and 119875

119895(119896) for each sub-model 119895

are calculated using UKF [22 23]

423 Model Probability Update Consider the following

119903

119894(119896) = 119885 (119896) minus 119885

119894(119896 119896 minus 1)

119878

119894(119896) = 119864 [119903

119894(119896) 119903

119894(119896)

119879]

Λ

119894(119896) = 119873 (119903

119894(119896) 0 119878

119894(119896))

=

exp [minus (12) 119903

119894(119896)

119879119878

119894(119896) 119903

119894(119896)]

radic2120587

1003816

1003816

1003816

1003816

119878

119894(119896)

1003816

1003816

1003816

1003816

119888 =

119873

sum

119894

Λ

119894(119896) 119888

119894

120583

119894(119896) =

1

119888

Λ

119894(119896)

119873

sum

119894

119901

119894119895sdot 120583

119894(119896 minus 1) =

1

119888

Λ

119894(119896) 119888

119894

(15)

where 119903119894(119896) and 119878

119894(119896) are innovation vector and its covariance

respectively Λ

119894(119896) is Model likelihood 119873(119909) is Gaussian

distribution

424 Combination Combined state119883(119896) and its covariance119875(119896) are calculated from the weighted state estimates 119883

119894(119896)

and covariances 119875119894(119896) as follows

119883 (119896) =

119873

sum

119894

119883

119894(119896) sdot 120583

119894(119896) (16)

119875 (119896) =

119873

sum

119894

119875

119894(119896) + [119883

119894(119896) minus 119883 (119896)]

sdot[119883

119894(119896) minus 119883 (119896)]

119879

sdot 120583

119894(119896)

(17)

5 Simulation Results

This section presents simulation results to show the perfor-mance improvement of this new method compared with thetraditional direct calculation method and the filter methodThe process of simulation is shown in Figure 4

The ideal trajectory of spacecraft is created by the STK(Satellite Tool Kit) whose initial value of semimajor axis is193216365381 km eccentricity 0236386 inclination 23455∘right Ascension of the ascending node 0258∘ and the argu-ment of periapsis 71347∘ The near bodies used in the directcalculation method are Mars and Phobos All measurementsare created by the celestial navigation simulation platform[24] using ideal trajectory of STK ephemeris of DE405 andTycho-2 catalogue The ideal trajectory of the spacecraft isshown in Figure 5 which has been checked according toTisserandrsquos criterion described in (17) [18 20]

119879 =

119903mars119886

+ 2

radic

119886 (1 minus 119890

2)

119903marscos 119894 (18)

The Tisserand parameter 119879 is conserved around minus02686

before and after the close encounter with Mars

6 Mathematical Problems in Engineering

Sensorsimulator

Stars

Earth SunPhobos

and MarsHigh dynamic celestial simulation system

Sensor

Star-planet angles

Direct calculationmethod

Spacecraftposition

measurement

Orbit and attitude

Spacecraft motion simulationsystem based on STK

Orbit dynamicssimulation

Attitude dynamicssimulation

IMMUKF

Navigation computer

Positionand velocity

Demo system

Figure 4 The process of simulation

1332119864 + 007

6659119864 + 006

0

minus6659119864 + 006

minus1332119864 + 007

minus1333119864 + 007 minus6663119864 + 006 6663119864 + 006 1333119864 + 0070

Figure 5 The ideal trajectory of the spacecraft

The accuracy of the Sun sensor Phobosrsquo sensor the Earthsensor and Mars sensor are selected to be 510158401015840 510158401015840 1510158401015840 and0005∘ respectively The accuracy of the star sensor is 310158401015840

Figure 6 shows the performance of the direct calculationmethod which is obtained with a 1-minute sampling intervalduring the 20-day periodwhen the spacecraft is nearMars Asit can be seen from Figure 6 the navigation accuracy of thismethod mainly depends on the accuracy of measurement soit is very sensitive to the measurement noise The geometryrelationship of the spacecraft Mars and Phobos also hasan impact on the navigation performance The shorter thedistance between spacecraft andMars (or Phobos) the higherthe navigation accuracy is

Figure 7 shows the performance comparison betweenthis new method and the traditional filter method which

0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

1400

1600

1800

2000Position estimation error

Time (day)

Posit

ion

erro

r (km

)

Figure 6 The position estimation error of the direct calculationmethod

only uses the star-planet angles as measurement under thesame simulation conditions As demonstrated in Figure 7the traditional filter method can provide a good navigationperformance when the spacecraft is far from Mars and ahighly accurate model is available but its estimation errorincreases greatly when the spacecraft is close to Mars Thisis because the star-planet angles alone cannot supply enough

Mathematical Problems in Engineering 7

Table 1 Performance comparison of different methods

Method Position estimation error(RMS) (km)

Velocity estimation error(RMS) (ms)

Position estimation error(Max) (km)

Velocity estimation error(Max) (ms)

Direct calculation method 19250 mdash 18162 mdash

Filter method 71725 01507 15468 04937

New method 62082 01118 16787 04486

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160Position estimation error

Time (day)

Posit

ion

estim

atio

n er

ror (

km)

UKFIMMUKF

0 2 4 6 8 10 12 14 16 18 20Time (day)

UKFIMMUKF

0

1

2

3

4

5

6Innovation sequence

Inno

vatio

n se

quen

ce (r

ad)

minus1

times10minus7

Figure 7 Results comparison between IMMUKF and UKF

information to correct the large estimation error introducedby the inaccurate dynamical model which even cause thedivergence of the filter after then The innovation sequenceof the two methods which has subtracted the measurementnoise is shown in Figure 7 The figure of the innovationsequence also demonstrates the divergence of the traditionalfilter method However the new method presented in thispaper shows a stable and satisfying navigation performanceThe details are given in Table 1

From these results we see that this IMMUKF-basednew celestial navigation method overcomes the shortcom-ings of the direct calculation method and the traditionalfilter method On one hand when an accurate dynamicalmodel is available the impact of measurement error onthe navigation performance is reduced by the dynamicalmodel On the other hand when an accurate dynamicalmodel is not available the impact of system error on thenavigation performance is reduced by the result of directcalculation method The navigation accuracy and reliabilityare greatly improved by the effective combination of thedirect calculationmethod and the filtermethodThese resultsdemonstrate that this new method is a particularly suitablecelestial navigation method of spacecrafts on a gravity assisttrajectory

6 Conclusions

A new autonomous celestial navigation method for space-crafts on a gravity assist trajectory is studied which effectivelycombines the direct calculationmethod and the filtermethodbased on IMMUKF The navigation performance of this newmethod is tested by the ground experiments and a positionestimation error within 20 km is obtained Compared withthe traditional methods this method can achieve a higheraccuracy and stability These results verify that this methodis a promising and attractive scheme of autonomous navi-gation for the spacecraft using gravity assist A limitation ofthis study is the significant increase in computational costThis method should have a broad application potential inautonomous navigation of deep space probes especially incases when the accurate dynamical model is not availableHow to enhance the adaptive capability and to reduce thecomputational cost is under further investigation

Acknowledgments

The research presented in this paper has been supportedby the National Natural Science Foundation of China(61233005) Program for New Century Excellent Talents in

8 Mathematical Problems in Engineering

University (NCET) (NCET-11-0771) and Aerospace Scienceand Technology Innovation Fund (10300002012117003) Theauthors wish to express their gratitude to all members of theScience and Technology on Inertial Laboratory and Funda-mental Science on Novel Inertial Instrument amp NavigationSystem Technology Laboratory for their valuable comments

References

[1] P W Kinman ldquoDoppler tracking of planetary spacecraftrdquo IEEETransactions on Microwave Theory and Techniques vol 40 no6 pp 1199ndash1204 1992

[2] B M Portock L A DrsquoAmario and R J Haw ldquoMars explorationrover orbit determination using ΔVLBI datardquo in Proceedings ofthe AIAAAAS Astrodynamics Specialist Conference MontereyCalif USA August 2002

[3] M L Psiaki L Huang and S M Fox ldquoGround tests of magne-tometer-based autonomous navigation (MAGNAV) for low-earth-orbiting spacecraftrdquo Journal of Guidance Control andDynamics vol 16 no 1 pp 206ndash214 1993

[4] H Jung and M L Psiaki ldquoTests of magnetometersun-sensororbit determination using flight datardquo Journal of GuidanceControl and Dynamics vol 25 no 3 pp 582ndash590 2002

[5] A Jaggi U Hugentobler H Bock and G Beutler ldquoPrecise orbitdetermination for GRACE using undifferenced or doubly dif-ferenced GPS datardquo Advances in Space Research vol 39 no 10pp 1612ndash1619 2007

[6] H T Son C H Roger C S Wendy and W Terri ldquoHigh accu-racy autonomous navigation using the Global Positioning Sys-tem (GPS)rdquo in Proceeding of the 12th International Symposiumon Space Flight Dynamics pp 73ndash78 Darmstadt Germany June1997

[7] R H Battin An Introduction to the Mathematics and Methodsof Astrodynamics American Institute of Aeronautics and Astro-nautics New York NY USA 1999

[8] B Kaufman R Campion B Vorder andW Richard ldquoAn over-view of the astrodynamics for the Deep Space Program ScienceExperiment mission (DSPSE)rdquo in Proceedings of the 44th Inter-national Astronautical Congress (IAF rsquo93) pp 141ndash155 1993

[9] R R Dasenbrock ldquoDSPSE autonomous position estimationexperimentrdquo in Proceedings of the International Symposium onAdvances in the Astronautical Sciences pp 203ndash217 April 1993

[10] A Long D Leung D Folta and C Gramling ldquoAutonomousnavigation of high-earth satellites using celestial objects anddoppler measurementsrdquo in Proceedings of the AIAAAAS Astro-dynamis Speialist Conferene 2000

[11] J R Yim J L Grassidis and J L Junkins ldquoAutonomousorbit navigation of interplanetary spacecraftrdquo in The AmericanInstitute of Aeronautics and Astronautics Guidance Navigationand Control Conference pp 53ndash61 Denver Colo USA August2000

[12] C R H Solorzano A A Sukhanov and A F B A Prado ldquoAstudy of trajectories to theNeptune systemusing gravity assistsrdquoAdvances in Space Research vol 40 no 1 pp 125ndash133 2007

[13] J D Anderson J K Campbell and M M Nieto ldquoThe energytransfer process in planetary flybysrdquoNewAstronomy vol 12 no5 pp 383ndash397 2007

[14] R P Russell and C A Ocampo ldquoGeometric analysis of free-return trajectories following a gravity-assisted flybyrdquo Journal ofSpacecraft and Rockets vol 42 no 1 pp 138ndash151 2005

[15] W Hohmann ldquoThe availability of heavenly bodiesrdquo NASAReport NASA-TT-F-44 1960

[16] W E Wiesel Spaceflight Dynamics McGraw-Hill New YorkNY USA 1989

[17] H A P Blom and Y Bar-Shalom ldquoInteracting multiple modelalgorithm for systems with Markovian switching coefficientsrdquoIEEE Transactions on Automatic Control vol 33 no 8 pp 780ndash783 1988

[18] Y Bar-Shalom K C Chang and H A P Blom ldquoTracking amaneuvering target using input estimation versus the interact-ing multiple model algorithmrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 25 no 2 pp 296ndash300 1992

[19] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[20] Y Boers and J N Driessen ldquoInteractingmultiplemodel particlefilterrdquo IEE Proceedings Radar Sonar and Navigation vol 150no 5 pp 344ndash349 2003

[21] R Toledo-Moreo M A Zamora-Izquierdo B Ubeda-Minarroand A F Gomez-Skarmeta ldquoHigh-integrity IMM-EKF-basedroad vehicle navigation with low-cost GPSSBASINSrdquo IEEETransactions on Intelligent Transportation Systems vol 8 no 3pp 491ndash511 2007

[22] O Payne and A Marrs ldquoAn unscented particle filter for GMTItrackingrdquo in Proceedings of the IEEE Aerospace ConferenceProceedings vol 3 pp 1869ndash1875 March 2004

[23] R Merwe A Doucet N Freitas and E Wan ldquoThe unscentedparticle filterrdquo Tech Rep CUEDF-INFENGTR 380 Cam-bridge University Engineering Department 2000

[24] W Quan J Fang F Xu andW Sheng ldquoHybrid simulation sys-tem study of SINSCNS integrated navigationrdquo IEEE Aerospaceand Electronic Systems Magazine vol 23 no 2 pp 17ndash24 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

where r is the position vector of the spacecraft R1and R

2

are the position vectors of Mars and Phobos respectively 1205881

and 120588

2are the distances of spacecraft to Mars and Phobos

respectively Solving 120588

1and 120588

2from (3) the position of

spacecraft can be determined by (2)The limitation of the direct calculation method is that the

measurement error has a great impact on the calculation errorand it can not provide the velocity information

3 Filter Method

The filter method exploits the dynamics of the spacecraft toremove the effects of the noise and obtain a good estimationof the position and velocity of the spacecraft at the presenttime using the differences between the measured and esti-mated celestial data In the filter method the state (dynamic)and measurement models are needed The dynamical modeldescribes how the state variables change over time and themeasurement model describes the relationship between themeasurement variables and the state variables

The patched-conic technique [15 16] is a useful toolfor designing a fuel-saving gravity assist trajectory In thistechnique a two-body orbit about the Sun is first kine-matically patched onto a two-body hyperbolic orbit aboutthe planet Then the spacecraft proceeds along the planet-centered hyperbolic orbit during the flyby and finally a newtwo-body orbit about the Sun is patched onto the postflybyhyperbolic orbit For a spacecraft on Mars gravity assisttrajectory its trajectory can be broadly divided into two parts

When the spacecraft is outside Mars sphere of influenceits orbital motion can be described as a multibody problemwith the sun as the central body Its dynamical model isexpressed in the sun-centered inertial frame (J20000) asfollows

= V119909+ 119908

119909

119910 = V119910+ 119908

119910

= V119911+ 119908

119911

V119909= minus

120583

119904119909

119903

3

psminus 120583

119898[

(119909 minus 119909

1)

119903

3

pmminus

119909

1

119903

3

sm]

minus 120583

119890[

(119909 minus 119909

2)

119903

3

peminus

119909

2

119903

3

se] + 119908V

119909

V119910= minus

120583

119904119910

119903

3

psminus 120583

119898[

(119910 minus 119910

1)

119903

3

pmminus

119910

1

119903

3

sm]

minus 120583

119890[

(119910 minus 119910

2)

119903

3

peminus

119910

2

119903

3

se] + 119908V

119910

V119911= minus

120583

119904119911

119903

3

psminus 120583

119898[

(119911 minus 119911

1)

119903

3

pmminus

119911

1

119903

3

sm]

minus 120583

119890[

(119911 minus 119911

2)

119903

3

peminus

119911

2

119903

3

se] + 119908V

119911

(4)

where (119909

1 119910

1 119911

1) (1199092 119910

2 119911

2) and (119909 119910 119911) are the position

of Mars the Earth and the spacecraft in the sun-centeredinertial frame rse and rsm are the position vectors of theEarth and Mars relative to the Sun rps rpe and rpm arethe position vectors of the spacecraft relative to the Sunthe Earth and Mars respectively 120583

119904 120583

119898 and 120583

119890are the

gravitational constants of the Sun the Earth and Marsrespectively 119882 = [119908

119909 119908

119910 119908

119911 119908V119909

119908V119910

119908V119911

]

119879 is the systemnoise

When the spacecraft is in Mars sphere of influenceits orbital motion can be described as a perturbed two-body problem with Mars as the central body The dynamicalmodel in Mars-centered inertial frame (J20000) is written asfollows

1015840= V1015840

119909+ 119908

1015840

119909

119910

1015840= V1015840

119910+ 119908

1015840

119910

1015840= V1015840

119911+ 119908

1015840

119911

V1015840

119909= minus 120583

119898

119909

1015840

119903

10158403

pmminus 120583

119904[

119909

1015840minus 119909

1015840

1

119903

10158403

ps+

119909

1015840

1

119903

10158403

ms]

minus 120583

119890[

119909

1015840minus 119909

1015840

2

119903

10158403

pe+

119909

1015840

2

119903

10158403

me] + 119908

1015840

V119909

V1015840

119910= minus 120583

119898

119910

1015840

119903

10158403

pmminus 120583

119904[

119910

1015840minus 119910

1015840

1

119903

10158403

ps+

119910

1015840

1

119903

10158403

ms]

minus 120583

119890[

119910

1015840minus 119910

1015840

2

119903

10158403

pe+

119910

1015840

2

119903

10158403

me] + 119908

1015840

V119910

V1015840

119911= minus 120583

119898

119911

1015840

119903

10158403

pmminus 120583

119904[

119911

1015840minus 119911

1015840

1

119903

10158403

ps+

119911

1015840

1

119903

10158403

ms]

minus 120583

119890[

119911

1015840minus 119911

1015840

2

119903

10158403

pe+

119911

1015840

2

119903

10158403

me] + 119908

1015840

V119911

119903

1015840=radic119909

10158402+ 119910

10158402+ 119911

10158402

(5)

All parameters used in (5) have the same meaning asthose in (4) except that these parameters are expressed inMars-centered inertial frame instead of sun-centered inertialframe

Because these twomodels are in different frames the finalresults have to be transformed to the sun-centered inertialframe

The usually used celestial measurements include (1) thedirections of near celestial bodies (the sun and the planet) (2)the planet-planet angles and (3) the star-planet angles Here

4 Mathematical Problems in Engineering

we take the last one as an example and the correspondingmeasurement equation is given by the following

120572

119904= arccos(minus

rps sdot s119903ps

) + V120572119904

120572

119890= arccos(minus

rpe sdot s119903pe

) + V120572119890

120572

119898= arccos(minus

rpm sdot s119903pm

) + V120572119898

(6)

where s is the position vector of the navigation star in thesun-centered inertial frame which is obtained from the starcatalog by star identification Assume measurement 119885

1=

[120572

119904 120572

119890 120572

119898]

119879andmeasurement noise1198811= [V120572119904 V120572119890 V120572119898

]

119879 themeasurement model can be presented as follows

119885

1(119905) = 119867

1 [119883 (119905) 119905] + 119881

1(119905) (7)

which measurement noise covariance matrix 119877

1is presented

as

119877

1= 119864 [119881

1(119905) 119881

1(119905)

119879] = diag 1205902

119886119904 120590

2

119886119890 120590

2

119886119898 (8)

When the spacecraft flies far from the planets the star-planet angles change very slowlyThe star-planet angles alonecannot supply enough information for position estimationwithout the help of an accurate dynamical model in thefilter method However when the spacecraft is on a gravityassist trajectory its orbit changes very fast and an accuratedynamical model is not available which may introducelarge estimation error and even cause the filter diver-gence

4 New Method Based on IMMUKF

For solving problems of above two methods a new celestialnavigation method is presented for the flyby spacecraft Inthis new method the direct calculation method is used toget a preliminary position estimation result and then thispreliminary result is used as a supplementary measurementin the filter The difficulty existing in the new methodis that the measurement noise covariance matrix is notconstant because the position estimation error of the directcalculation method changes over time due to the movementof spacecraft An IMMUKF method is used to deal withthis problem and to enhance the adaptive capability of thefilter

41 System Model As mentioned above the orbital motionof the spacecraft can be described by two dynamical modelswhich are defined by (5) and (6) They can be writtenin general as

119883(119905) = 119891

1(119883(119905) 119905) + 119882

1(119905) and

119883(119905) =

119891

2(119883(119905) 119905) + 119882

2(119905) When the spacecraft is outside Marsrsquo

sphere of influence (119903pm gt 5773 times 10

6) Model 1 is usedWhen the spacecraft is in Marsrsquo influence sphere (119903pm le

5773 times 10

6) Model 2 is used The system noise covariance

Interaction

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Filtering Filtering119885(119896) 119885(119896)

Λ 1(119896) Λ119873(119896)

120583(119896 minus 1)

Modelprobabilityupdate

120583(119896)

Combination

X1 (k minus 1)P1 (k minus 1)

XN (k minus 1)PN (k minus 1)

X1 (k k minus 1)

P1 (k k minus 1)

XN (k k minus 1)

PN (k k minus 1)

X1(k)P1(k) PN(k)

XN(k)

X(k)P(k)

Figure 3 The IMMUKF algorithm

matrices are 119864[119882

1(119905)119882

1(119905)

119879] = 119876

1 119864[119882

2(119905)119882

2(119905)

119879] = 119876

2

respectivelyIn this new method the position estimation result

obtained by the direct calculation method is used as asupplementary measurement besides the star-planet anglemeasurement defined by (7) The measurement used in thisnew method can be described as follows

119885

2= [

119909

119899

119896

119910

119899

119896

119911

119899

119896

120572

119904 120572

119890 120572

119898]

119879

(9)

where [119909

119899 119910

119899 119911

119899]

119879 is the position estimation result obtainedby the direct calculation method 119896 is a scale factor which isused to make the magnitude of position measurement matchthat of star-planet angle measurement The measurementmodel can be presented as follows

119885

2(119905) = 119867

2 [119883 (119905) 119905] + 119881

2(119905) (10)

Because the measurement noise covariance matrix of thismodel is unknown and time varying a set of differentmatrices is used to approximate it Then 119873 measurementsubmodels are presented as 119885

2119894(119905) = 119867

2[119883(119905) 119905] + 119881

2119894(119905)

which measurement noise covariance matrixes 119877

2119894are as

follows

119877

2119894= 119864 [119881

2119894(119905) 119881

2119894(119905)

119879] = [119877119888119894

119877

1]

119877

119888119894=

119877

119888

119894

(119894 = 1 2 119873)

(11)

where 119877119888is the experimental measurement noise covariance

matrix of the direct calculation method In this paper 119877119888is

set as diag 46 times 10

719 times 10

784 times 10

5

42 IMMUKF Method The interacting multiple model(IMM) algorithm is a very effective estimation algorithm for

Mathematical Problems in Engineering 5

systems which can be described by a mixing of continuousstates and discrete modes [17 18] The switching betweensubmodels is governed by a Markov chain The IMMUKFmethod includes interaction filtering model probabilityupdate and combination of four steps [19ndash21] as shown inFigure 3

421 Interaction Suppose the Markov model transition pro-bability matrix 119875 is as follows

119875 =

[

[

[

[

[

[

[

119901

11119901

12sdot sdot sdot 119901

1119873

119901

21

119901

22sdot sdot sdot

119901

2119873

119901

1198731119901

1198732sdot sdot sdot 119901

119873119873

]

]

]

]

]

]

]

119873

sum

119894

119901

119894119895=

119873

sum

119895

119901

119894119895= 1

(12)

where 119901119894119895is the a priori probability of transition from model

119894 to model 119895 The initial value of 119875 in this paper is set as 119901119894119895

=

091 with 119894 = 119895 and 119901

119894119895= 001 with 119894 = 119895

The mixing probabilities at time 119896 minus 1 are computed asfollows

120583

119894119895(119896 119896 minus 1) =

1

119888

119895

sdot 119901

119894119895sdot 120583

119894(119896 minus 1) (13)

where 120583

119894(119896 minus 1) is the predicted model probability 119888

119895=

sum

119873

119894119901

119894119895sdot 120583

119894(119896 minus 1) is normalization constant The mixed initial

conditions for each submodel 119895 are as follows

119883

119895(119896 119896 minus 1) =

119873

sum

119894

119883

119894(119896 minus 1) sdot 120583

119894119895(119896 119896 minus 1)

119875

119895(119896 119896 minus 1) =

119873

sum

119894

119875

119894(119896 minus 1)

+ [119883

119894(119896 minus 1) minus 119883

119895(119896 119896 minus 1)]

sdot [119883

119894(119896 minus 1) minus 119883

119895(119896 119896 minus 1)]

119879

times 120583

119894119895(119896 119896 minus 1)

(14)

422 Filtering Because the state model and measurementmodel described above are nonlinear the optimal state andcovariance estimates 119883

119895(119896) and 119875

119895(119896) for each sub-model 119895

are calculated using UKF [22 23]

423 Model Probability Update Consider the following

119903

119894(119896) = 119885 (119896) minus 119885

119894(119896 119896 minus 1)

119878

119894(119896) = 119864 [119903

119894(119896) 119903

119894(119896)

119879]

Λ

119894(119896) = 119873 (119903

119894(119896) 0 119878

119894(119896))

=

exp [minus (12) 119903

119894(119896)

119879119878

119894(119896) 119903

119894(119896)]

radic2120587

1003816

1003816

1003816

1003816

119878

119894(119896)

1003816

1003816

1003816

1003816

119888 =

119873

sum

119894

Λ

119894(119896) 119888

119894

120583

119894(119896) =

1

119888

Λ

119894(119896)

119873

sum

119894

119901

119894119895sdot 120583

119894(119896 minus 1) =

1

119888

Λ

119894(119896) 119888

119894

(15)

where 119903119894(119896) and 119878

119894(119896) are innovation vector and its covariance

respectively Λ

119894(119896) is Model likelihood 119873(119909) is Gaussian

distribution

424 Combination Combined state119883(119896) and its covariance119875(119896) are calculated from the weighted state estimates 119883

119894(119896)

and covariances 119875119894(119896) as follows

119883 (119896) =

119873

sum

119894

119883

119894(119896) sdot 120583

119894(119896) (16)

119875 (119896) =

119873

sum

119894

119875

119894(119896) + [119883

119894(119896) minus 119883 (119896)]

sdot[119883

119894(119896) minus 119883 (119896)]

119879

sdot 120583

119894(119896)

(17)

5 Simulation Results

This section presents simulation results to show the perfor-mance improvement of this new method compared with thetraditional direct calculation method and the filter methodThe process of simulation is shown in Figure 4

The ideal trajectory of spacecraft is created by the STK(Satellite Tool Kit) whose initial value of semimajor axis is193216365381 km eccentricity 0236386 inclination 23455∘right Ascension of the ascending node 0258∘ and the argu-ment of periapsis 71347∘ The near bodies used in the directcalculation method are Mars and Phobos All measurementsare created by the celestial navigation simulation platform[24] using ideal trajectory of STK ephemeris of DE405 andTycho-2 catalogue The ideal trajectory of the spacecraft isshown in Figure 5 which has been checked according toTisserandrsquos criterion described in (17) [18 20]

119879 =

119903mars119886

+ 2

radic

119886 (1 minus 119890

2)

119903marscos 119894 (18)

The Tisserand parameter 119879 is conserved around minus02686

before and after the close encounter with Mars

6 Mathematical Problems in Engineering

Sensorsimulator

Stars

Earth SunPhobos

and MarsHigh dynamic celestial simulation system

Sensor

Star-planet angles

Direct calculationmethod

Spacecraftposition

measurement

Orbit and attitude

Spacecraft motion simulationsystem based on STK

Orbit dynamicssimulation

Attitude dynamicssimulation

IMMUKF

Navigation computer

Positionand velocity

Demo system

Figure 4 The process of simulation

1332119864 + 007

6659119864 + 006

0

minus6659119864 + 006

minus1332119864 + 007

minus1333119864 + 007 minus6663119864 + 006 6663119864 + 006 1333119864 + 0070

Figure 5 The ideal trajectory of the spacecraft

The accuracy of the Sun sensor Phobosrsquo sensor the Earthsensor and Mars sensor are selected to be 510158401015840 510158401015840 1510158401015840 and0005∘ respectively The accuracy of the star sensor is 310158401015840

Figure 6 shows the performance of the direct calculationmethod which is obtained with a 1-minute sampling intervalduring the 20-day periodwhen the spacecraft is nearMars Asit can be seen from Figure 6 the navigation accuracy of thismethod mainly depends on the accuracy of measurement soit is very sensitive to the measurement noise The geometryrelationship of the spacecraft Mars and Phobos also hasan impact on the navigation performance The shorter thedistance between spacecraft andMars (or Phobos) the higherthe navigation accuracy is

Figure 7 shows the performance comparison betweenthis new method and the traditional filter method which

0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

1400

1600

1800

2000Position estimation error

Time (day)

Posit

ion

erro

r (km

)

Figure 6 The position estimation error of the direct calculationmethod

only uses the star-planet angles as measurement under thesame simulation conditions As demonstrated in Figure 7the traditional filter method can provide a good navigationperformance when the spacecraft is far from Mars and ahighly accurate model is available but its estimation errorincreases greatly when the spacecraft is close to Mars Thisis because the star-planet angles alone cannot supply enough

Mathematical Problems in Engineering 7

Table 1 Performance comparison of different methods

Method Position estimation error(RMS) (km)

Velocity estimation error(RMS) (ms)

Position estimation error(Max) (km)

Velocity estimation error(Max) (ms)

Direct calculation method 19250 mdash 18162 mdash

Filter method 71725 01507 15468 04937

New method 62082 01118 16787 04486

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160Position estimation error

Time (day)

Posit

ion

estim

atio

n er

ror (

km)

UKFIMMUKF

0 2 4 6 8 10 12 14 16 18 20Time (day)

UKFIMMUKF

0

1

2

3

4

5

6Innovation sequence

Inno

vatio

n se

quen

ce (r

ad)

minus1

times10minus7

Figure 7 Results comparison between IMMUKF and UKF

information to correct the large estimation error introducedby the inaccurate dynamical model which even cause thedivergence of the filter after then The innovation sequenceof the two methods which has subtracted the measurementnoise is shown in Figure 7 The figure of the innovationsequence also demonstrates the divergence of the traditionalfilter method However the new method presented in thispaper shows a stable and satisfying navigation performanceThe details are given in Table 1

From these results we see that this IMMUKF-basednew celestial navigation method overcomes the shortcom-ings of the direct calculation method and the traditionalfilter method On one hand when an accurate dynamicalmodel is available the impact of measurement error onthe navigation performance is reduced by the dynamicalmodel On the other hand when an accurate dynamicalmodel is not available the impact of system error on thenavigation performance is reduced by the result of directcalculation method The navigation accuracy and reliabilityare greatly improved by the effective combination of thedirect calculationmethod and the filtermethodThese resultsdemonstrate that this new method is a particularly suitablecelestial navigation method of spacecrafts on a gravity assisttrajectory

6 Conclusions

A new autonomous celestial navigation method for space-crafts on a gravity assist trajectory is studied which effectivelycombines the direct calculationmethod and the filtermethodbased on IMMUKF The navigation performance of this newmethod is tested by the ground experiments and a positionestimation error within 20 km is obtained Compared withthe traditional methods this method can achieve a higheraccuracy and stability These results verify that this methodis a promising and attractive scheme of autonomous navi-gation for the spacecraft using gravity assist A limitation ofthis study is the significant increase in computational costThis method should have a broad application potential inautonomous navigation of deep space probes especially incases when the accurate dynamical model is not availableHow to enhance the adaptive capability and to reduce thecomputational cost is under further investigation

Acknowledgments

The research presented in this paper has been supportedby the National Natural Science Foundation of China(61233005) Program for New Century Excellent Talents in

8 Mathematical Problems in Engineering

University (NCET) (NCET-11-0771) and Aerospace Scienceand Technology Innovation Fund (10300002012117003) Theauthors wish to express their gratitude to all members of theScience and Technology on Inertial Laboratory and Funda-mental Science on Novel Inertial Instrument amp NavigationSystem Technology Laboratory for their valuable comments

References

[1] P W Kinman ldquoDoppler tracking of planetary spacecraftrdquo IEEETransactions on Microwave Theory and Techniques vol 40 no6 pp 1199ndash1204 1992

[2] B M Portock L A DrsquoAmario and R J Haw ldquoMars explorationrover orbit determination using ΔVLBI datardquo in Proceedings ofthe AIAAAAS Astrodynamics Specialist Conference MontereyCalif USA August 2002

[3] M L Psiaki L Huang and S M Fox ldquoGround tests of magne-tometer-based autonomous navigation (MAGNAV) for low-earth-orbiting spacecraftrdquo Journal of Guidance Control andDynamics vol 16 no 1 pp 206ndash214 1993

[4] H Jung and M L Psiaki ldquoTests of magnetometersun-sensororbit determination using flight datardquo Journal of GuidanceControl and Dynamics vol 25 no 3 pp 582ndash590 2002

[5] A Jaggi U Hugentobler H Bock and G Beutler ldquoPrecise orbitdetermination for GRACE using undifferenced or doubly dif-ferenced GPS datardquo Advances in Space Research vol 39 no 10pp 1612ndash1619 2007

[6] H T Son C H Roger C S Wendy and W Terri ldquoHigh accu-racy autonomous navigation using the Global Positioning Sys-tem (GPS)rdquo in Proceeding of the 12th International Symposiumon Space Flight Dynamics pp 73ndash78 Darmstadt Germany June1997

[7] R H Battin An Introduction to the Mathematics and Methodsof Astrodynamics American Institute of Aeronautics and Astro-nautics New York NY USA 1999

[8] B Kaufman R Campion B Vorder andW Richard ldquoAn over-view of the astrodynamics for the Deep Space Program ScienceExperiment mission (DSPSE)rdquo in Proceedings of the 44th Inter-national Astronautical Congress (IAF rsquo93) pp 141ndash155 1993

[9] R R Dasenbrock ldquoDSPSE autonomous position estimationexperimentrdquo in Proceedings of the International Symposium onAdvances in the Astronautical Sciences pp 203ndash217 April 1993

[10] A Long D Leung D Folta and C Gramling ldquoAutonomousnavigation of high-earth satellites using celestial objects anddoppler measurementsrdquo in Proceedings of the AIAAAAS Astro-dynamis Speialist Conferene 2000

[11] J R Yim J L Grassidis and J L Junkins ldquoAutonomousorbit navigation of interplanetary spacecraftrdquo in The AmericanInstitute of Aeronautics and Astronautics Guidance Navigationand Control Conference pp 53ndash61 Denver Colo USA August2000

[12] C R H Solorzano A A Sukhanov and A F B A Prado ldquoAstudy of trajectories to theNeptune systemusing gravity assistsrdquoAdvances in Space Research vol 40 no 1 pp 125ndash133 2007

[13] J D Anderson J K Campbell and M M Nieto ldquoThe energytransfer process in planetary flybysrdquoNewAstronomy vol 12 no5 pp 383ndash397 2007

[14] R P Russell and C A Ocampo ldquoGeometric analysis of free-return trajectories following a gravity-assisted flybyrdquo Journal ofSpacecraft and Rockets vol 42 no 1 pp 138ndash151 2005

[15] W Hohmann ldquoThe availability of heavenly bodiesrdquo NASAReport NASA-TT-F-44 1960

[16] W E Wiesel Spaceflight Dynamics McGraw-Hill New YorkNY USA 1989

[17] H A P Blom and Y Bar-Shalom ldquoInteracting multiple modelalgorithm for systems with Markovian switching coefficientsrdquoIEEE Transactions on Automatic Control vol 33 no 8 pp 780ndash783 1988

[18] Y Bar-Shalom K C Chang and H A P Blom ldquoTracking amaneuvering target using input estimation versus the interact-ing multiple model algorithmrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 25 no 2 pp 296ndash300 1992

[19] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[20] Y Boers and J N Driessen ldquoInteractingmultiplemodel particlefilterrdquo IEE Proceedings Radar Sonar and Navigation vol 150no 5 pp 344ndash349 2003

[21] R Toledo-Moreo M A Zamora-Izquierdo B Ubeda-Minarroand A F Gomez-Skarmeta ldquoHigh-integrity IMM-EKF-basedroad vehicle navigation with low-cost GPSSBASINSrdquo IEEETransactions on Intelligent Transportation Systems vol 8 no 3pp 491ndash511 2007

[22] O Payne and A Marrs ldquoAn unscented particle filter for GMTItrackingrdquo in Proceedings of the IEEE Aerospace ConferenceProceedings vol 3 pp 1869ndash1875 March 2004

[23] R Merwe A Doucet N Freitas and E Wan ldquoThe unscentedparticle filterrdquo Tech Rep CUEDF-INFENGTR 380 Cam-bridge University Engineering Department 2000

[24] W Quan J Fang F Xu andW Sheng ldquoHybrid simulation sys-tem study of SINSCNS integrated navigationrdquo IEEE Aerospaceand Electronic Systems Magazine vol 23 no 2 pp 17ndash24 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

we take the last one as an example and the correspondingmeasurement equation is given by the following

120572

119904= arccos(minus

rps sdot s119903ps

) + V120572119904

120572

119890= arccos(minus

rpe sdot s119903pe

) + V120572119890

120572

119898= arccos(minus

rpm sdot s119903pm

) + V120572119898

(6)

where s is the position vector of the navigation star in thesun-centered inertial frame which is obtained from the starcatalog by star identification Assume measurement 119885

1=

[120572

119904 120572

119890 120572

119898]

119879andmeasurement noise1198811= [V120572119904 V120572119890 V120572119898

]

119879 themeasurement model can be presented as follows

119885

1(119905) = 119867

1 [119883 (119905) 119905] + 119881

1(119905) (7)

which measurement noise covariance matrix 119877

1is presented

as

119877

1= 119864 [119881

1(119905) 119881

1(119905)

119879] = diag 1205902

119886119904 120590

2

119886119890 120590

2

119886119898 (8)

When the spacecraft flies far from the planets the star-planet angles change very slowlyThe star-planet angles alonecannot supply enough information for position estimationwithout the help of an accurate dynamical model in thefilter method However when the spacecraft is on a gravityassist trajectory its orbit changes very fast and an accuratedynamical model is not available which may introducelarge estimation error and even cause the filter diver-gence

4 New Method Based on IMMUKF

For solving problems of above two methods a new celestialnavigation method is presented for the flyby spacecraft Inthis new method the direct calculation method is used toget a preliminary position estimation result and then thispreliminary result is used as a supplementary measurementin the filter The difficulty existing in the new methodis that the measurement noise covariance matrix is notconstant because the position estimation error of the directcalculation method changes over time due to the movementof spacecraft An IMMUKF method is used to deal withthis problem and to enhance the adaptive capability of thefilter

41 System Model As mentioned above the orbital motionof the spacecraft can be described by two dynamical modelswhich are defined by (5) and (6) They can be writtenin general as

119883(119905) = 119891

1(119883(119905) 119905) + 119882

1(119905) and

119883(119905) =

119891

2(119883(119905) 119905) + 119882

2(119905) When the spacecraft is outside Marsrsquo

sphere of influence (119903pm gt 5773 times 10

6) Model 1 is usedWhen the spacecraft is in Marsrsquo influence sphere (119903pm le

5773 times 10

6) Model 2 is used The system noise covariance

Interaction

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Filtering Filtering119885(119896) 119885(119896)

Λ 1(119896) Λ119873(119896)

120583(119896 minus 1)

Modelprobabilityupdate

120583(119896)

Combination

X1 (k minus 1)P1 (k minus 1)

XN (k minus 1)PN (k minus 1)

X1 (k k minus 1)

P1 (k k minus 1)

XN (k k minus 1)

PN (k k minus 1)

X1(k)P1(k) PN(k)

XN(k)

X(k)P(k)

Figure 3 The IMMUKF algorithm

matrices are 119864[119882

1(119905)119882

1(119905)

119879] = 119876

1 119864[119882

2(119905)119882

2(119905)

119879] = 119876

2

respectivelyIn this new method the position estimation result

obtained by the direct calculation method is used as asupplementary measurement besides the star-planet anglemeasurement defined by (7) The measurement used in thisnew method can be described as follows

119885

2= [

119909

119899

119896

119910

119899

119896

119911

119899

119896

120572

119904 120572

119890 120572

119898]

119879

(9)

where [119909

119899 119910

119899 119911

119899]

119879 is the position estimation result obtainedby the direct calculation method 119896 is a scale factor which isused to make the magnitude of position measurement matchthat of star-planet angle measurement The measurementmodel can be presented as follows

119885

2(119905) = 119867

2 [119883 (119905) 119905] + 119881

2(119905) (10)

Because the measurement noise covariance matrix of thismodel is unknown and time varying a set of differentmatrices is used to approximate it Then 119873 measurementsubmodels are presented as 119885

2119894(119905) = 119867

2[119883(119905) 119905] + 119881

2119894(119905)

which measurement noise covariance matrixes 119877

2119894are as

follows

119877

2119894= 119864 [119881

2119894(119905) 119881

2119894(119905)

119879] = [119877119888119894

119877

1]

119877

119888119894=

119877

119888

119894

(119894 = 1 2 119873)

(11)

where 119877119888is the experimental measurement noise covariance

matrix of the direct calculation method In this paper 119877119888is

set as diag 46 times 10

719 times 10

784 times 10

5

42 IMMUKF Method The interacting multiple model(IMM) algorithm is a very effective estimation algorithm for

Mathematical Problems in Engineering 5

systems which can be described by a mixing of continuousstates and discrete modes [17 18] The switching betweensubmodels is governed by a Markov chain The IMMUKFmethod includes interaction filtering model probabilityupdate and combination of four steps [19ndash21] as shown inFigure 3

421 Interaction Suppose the Markov model transition pro-bability matrix 119875 is as follows

119875 =

[

[

[

[

[

[

[

119901

11119901

12sdot sdot sdot 119901

1119873

119901

21

119901

22sdot sdot sdot

119901

2119873

119901

1198731119901

1198732sdot sdot sdot 119901

119873119873

]

]

]

]

]

]

]

119873

sum

119894

119901

119894119895=

119873

sum

119895

119901

119894119895= 1

(12)

where 119901119894119895is the a priori probability of transition from model

119894 to model 119895 The initial value of 119875 in this paper is set as 119901119894119895

=

091 with 119894 = 119895 and 119901

119894119895= 001 with 119894 = 119895

The mixing probabilities at time 119896 minus 1 are computed asfollows

120583

119894119895(119896 119896 minus 1) =

1

119888

119895

sdot 119901

119894119895sdot 120583

119894(119896 minus 1) (13)

where 120583

119894(119896 minus 1) is the predicted model probability 119888

119895=

sum

119873

119894119901

119894119895sdot 120583

119894(119896 minus 1) is normalization constant The mixed initial

conditions for each submodel 119895 are as follows

119883

119895(119896 119896 minus 1) =

119873

sum

119894

119883

119894(119896 minus 1) sdot 120583

119894119895(119896 119896 minus 1)

119875

119895(119896 119896 minus 1) =

119873

sum

119894

119875

119894(119896 minus 1)

+ [119883

119894(119896 minus 1) minus 119883

119895(119896 119896 minus 1)]

sdot [119883

119894(119896 minus 1) minus 119883

119895(119896 119896 minus 1)]

119879

times 120583

119894119895(119896 119896 minus 1)

(14)

422 Filtering Because the state model and measurementmodel described above are nonlinear the optimal state andcovariance estimates 119883

119895(119896) and 119875

119895(119896) for each sub-model 119895

are calculated using UKF [22 23]

423 Model Probability Update Consider the following

119903

119894(119896) = 119885 (119896) minus 119885

119894(119896 119896 minus 1)

119878

119894(119896) = 119864 [119903

119894(119896) 119903

119894(119896)

119879]

Λ

119894(119896) = 119873 (119903

119894(119896) 0 119878

119894(119896))

=

exp [minus (12) 119903

119894(119896)

119879119878

119894(119896) 119903

119894(119896)]

radic2120587

1003816

1003816

1003816

1003816

119878

119894(119896)

1003816

1003816

1003816

1003816

119888 =

119873

sum

119894

Λ

119894(119896) 119888

119894

120583

119894(119896) =

1

119888

Λ

119894(119896)

119873

sum

119894

119901

119894119895sdot 120583

119894(119896 minus 1) =

1

119888

Λ

119894(119896) 119888

119894

(15)

where 119903119894(119896) and 119878

119894(119896) are innovation vector and its covariance

respectively Λ

119894(119896) is Model likelihood 119873(119909) is Gaussian

distribution

424 Combination Combined state119883(119896) and its covariance119875(119896) are calculated from the weighted state estimates 119883

119894(119896)

and covariances 119875119894(119896) as follows

119883 (119896) =

119873

sum

119894

119883

119894(119896) sdot 120583

119894(119896) (16)

119875 (119896) =

119873

sum

119894

119875

119894(119896) + [119883

119894(119896) minus 119883 (119896)]

sdot[119883

119894(119896) minus 119883 (119896)]

119879

sdot 120583

119894(119896)

(17)

5 Simulation Results

This section presents simulation results to show the perfor-mance improvement of this new method compared with thetraditional direct calculation method and the filter methodThe process of simulation is shown in Figure 4

The ideal trajectory of spacecraft is created by the STK(Satellite Tool Kit) whose initial value of semimajor axis is193216365381 km eccentricity 0236386 inclination 23455∘right Ascension of the ascending node 0258∘ and the argu-ment of periapsis 71347∘ The near bodies used in the directcalculation method are Mars and Phobos All measurementsare created by the celestial navigation simulation platform[24] using ideal trajectory of STK ephemeris of DE405 andTycho-2 catalogue The ideal trajectory of the spacecraft isshown in Figure 5 which has been checked according toTisserandrsquos criterion described in (17) [18 20]

119879 =

119903mars119886

+ 2

radic

119886 (1 minus 119890

2)

119903marscos 119894 (18)

The Tisserand parameter 119879 is conserved around minus02686

before and after the close encounter with Mars

6 Mathematical Problems in Engineering

Sensorsimulator

Stars

Earth SunPhobos

and MarsHigh dynamic celestial simulation system

Sensor

Star-planet angles

Direct calculationmethod

Spacecraftposition

measurement

Orbit and attitude

Spacecraft motion simulationsystem based on STK

Orbit dynamicssimulation

Attitude dynamicssimulation

IMMUKF

Navigation computer

Positionand velocity

Demo system

Figure 4 The process of simulation

1332119864 + 007

6659119864 + 006

0

minus6659119864 + 006

minus1332119864 + 007

minus1333119864 + 007 minus6663119864 + 006 6663119864 + 006 1333119864 + 0070

Figure 5 The ideal trajectory of the spacecraft

The accuracy of the Sun sensor Phobosrsquo sensor the Earthsensor and Mars sensor are selected to be 510158401015840 510158401015840 1510158401015840 and0005∘ respectively The accuracy of the star sensor is 310158401015840

Figure 6 shows the performance of the direct calculationmethod which is obtained with a 1-minute sampling intervalduring the 20-day periodwhen the spacecraft is nearMars Asit can be seen from Figure 6 the navigation accuracy of thismethod mainly depends on the accuracy of measurement soit is very sensitive to the measurement noise The geometryrelationship of the spacecraft Mars and Phobos also hasan impact on the navigation performance The shorter thedistance between spacecraft andMars (or Phobos) the higherthe navigation accuracy is

Figure 7 shows the performance comparison betweenthis new method and the traditional filter method which

0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

1400

1600

1800

2000Position estimation error

Time (day)

Posit

ion

erro

r (km

)

Figure 6 The position estimation error of the direct calculationmethod

only uses the star-planet angles as measurement under thesame simulation conditions As demonstrated in Figure 7the traditional filter method can provide a good navigationperformance when the spacecraft is far from Mars and ahighly accurate model is available but its estimation errorincreases greatly when the spacecraft is close to Mars Thisis because the star-planet angles alone cannot supply enough

Mathematical Problems in Engineering 7

Table 1 Performance comparison of different methods

Method Position estimation error(RMS) (km)

Velocity estimation error(RMS) (ms)

Position estimation error(Max) (km)

Velocity estimation error(Max) (ms)

Direct calculation method 19250 mdash 18162 mdash

Filter method 71725 01507 15468 04937

New method 62082 01118 16787 04486

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160Position estimation error

Time (day)

Posit

ion

estim

atio

n er

ror (

km)

UKFIMMUKF

0 2 4 6 8 10 12 14 16 18 20Time (day)

UKFIMMUKF

0

1

2

3

4

5

6Innovation sequence

Inno

vatio

n se

quen

ce (r

ad)

minus1

times10minus7

Figure 7 Results comparison between IMMUKF and UKF

information to correct the large estimation error introducedby the inaccurate dynamical model which even cause thedivergence of the filter after then The innovation sequenceof the two methods which has subtracted the measurementnoise is shown in Figure 7 The figure of the innovationsequence also demonstrates the divergence of the traditionalfilter method However the new method presented in thispaper shows a stable and satisfying navigation performanceThe details are given in Table 1

From these results we see that this IMMUKF-basednew celestial navigation method overcomes the shortcom-ings of the direct calculation method and the traditionalfilter method On one hand when an accurate dynamicalmodel is available the impact of measurement error onthe navigation performance is reduced by the dynamicalmodel On the other hand when an accurate dynamicalmodel is not available the impact of system error on thenavigation performance is reduced by the result of directcalculation method The navigation accuracy and reliabilityare greatly improved by the effective combination of thedirect calculationmethod and the filtermethodThese resultsdemonstrate that this new method is a particularly suitablecelestial navigation method of spacecrafts on a gravity assisttrajectory

6 Conclusions

A new autonomous celestial navigation method for space-crafts on a gravity assist trajectory is studied which effectivelycombines the direct calculationmethod and the filtermethodbased on IMMUKF The navigation performance of this newmethod is tested by the ground experiments and a positionestimation error within 20 km is obtained Compared withthe traditional methods this method can achieve a higheraccuracy and stability These results verify that this methodis a promising and attractive scheme of autonomous navi-gation for the spacecraft using gravity assist A limitation ofthis study is the significant increase in computational costThis method should have a broad application potential inautonomous navigation of deep space probes especially incases when the accurate dynamical model is not availableHow to enhance the adaptive capability and to reduce thecomputational cost is under further investigation

Acknowledgments

The research presented in this paper has been supportedby the National Natural Science Foundation of China(61233005) Program for New Century Excellent Talents in

8 Mathematical Problems in Engineering

University (NCET) (NCET-11-0771) and Aerospace Scienceand Technology Innovation Fund (10300002012117003) Theauthors wish to express their gratitude to all members of theScience and Technology on Inertial Laboratory and Funda-mental Science on Novel Inertial Instrument amp NavigationSystem Technology Laboratory for their valuable comments

References

[1] P W Kinman ldquoDoppler tracking of planetary spacecraftrdquo IEEETransactions on Microwave Theory and Techniques vol 40 no6 pp 1199ndash1204 1992

[2] B M Portock L A DrsquoAmario and R J Haw ldquoMars explorationrover orbit determination using ΔVLBI datardquo in Proceedings ofthe AIAAAAS Astrodynamics Specialist Conference MontereyCalif USA August 2002

[3] M L Psiaki L Huang and S M Fox ldquoGround tests of magne-tometer-based autonomous navigation (MAGNAV) for low-earth-orbiting spacecraftrdquo Journal of Guidance Control andDynamics vol 16 no 1 pp 206ndash214 1993

[4] H Jung and M L Psiaki ldquoTests of magnetometersun-sensororbit determination using flight datardquo Journal of GuidanceControl and Dynamics vol 25 no 3 pp 582ndash590 2002

[5] A Jaggi U Hugentobler H Bock and G Beutler ldquoPrecise orbitdetermination for GRACE using undifferenced or doubly dif-ferenced GPS datardquo Advances in Space Research vol 39 no 10pp 1612ndash1619 2007

[6] H T Son C H Roger C S Wendy and W Terri ldquoHigh accu-racy autonomous navigation using the Global Positioning Sys-tem (GPS)rdquo in Proceeding of the 12th International Symposiumon Space Flight Dynamics pp 73ndash78 Darmstadt Germany June1997

[7] R H Battin An Introduction to the Mathematics and Methodsof Astrodynamics American Institute of Aeronautics and Astro-nautics New York NY USA 1999

[8] B Kaufman R Campion B Vorder andW Richard ldquoAn over-view of the astrodynamics for the Deep Space Program ScienceExperiment mission (DSPSE)rdquo in Proceedings of the 44th Inter-national Astronautical Congress (IAF rsquo93) pp 141ndash155 1993

[9] R R Dasenbrock ldquoDSPSE autonomous position estimationexperimentrdquo in Proceedings of the International Symposium onAdvances in the Astronautical Sciences pp 203ndash217 April 1993

[10] A Long D Leung D Folta and C Gramling ldquoAutonomousnavigation of high-earth satellites using celestial objects anddoppler measurementsrdquo in Proceedings of the AIAAAAS Astro-dynamis Speialist Conferene 2000

[11] J R Yim J L Grassidis and J L Junkins ldquoAutonomousorbit navigation of interplanetary spacecraftrdquo in The AmericanInstitute of Aeronautics and Astronautics Guidance Navigationand Control Conference pp 53ndash61 Denver Colo USA August2000

[12] C R H Solorzano A A Sukhanov and A F B A Prado ldquoAstudy of trajectories to theNeptune systemusing gravity assistsrdquoAdvances in Space Research vol 40 no 1 pp 125ndash133 2007

[13] J D Anderson J K Campbell and M M Nieto ldquoThe energytransfer process in planetary flybysrdquoNewAstronomy vol 12 no5 pp 383ndash397 2007

[14] R P Russell and C A Ocampo ldquoGeometric analysis of free-return trajectories following a gravity-assisted flybyrdquo Journal ofSpacecraft and Rockets vol 42 no 1 pp 138ndash151 2005

[15] W Hohmann ldquoThe availability of heavenly bodiesrdquo NASAReport NASA-TT-F-44 1960

[16] W E Wiesel Spaceflight Dynamics McGraw-Hill New YorkNY USA 1989

[17] H A P Blom and Y Bar-Shalom ldquoInteracting multiple modelalgorithm for systems with Markovian switching coefficientsrdquoIEEE Transactions on Automatic Control vol 33 no 8 pp 780ndash783 1988

[18] Y Bar-Shalom K C Chang and H A P Blom ldquoTracking amaneuvering target using input estimation versus the interact-ing multiple model algorithmrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 25 no 2 pp 296ndash300 1992

[19] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[20] Y Boers and J N Driessen ldquoInteractingmultiplemodel particlefilterrdquo IEE Proceedings Radar Sonar and Navigation vol 150no 5 pp 344ndash349 2003

[21] R Toledo-Moreo M A Zamora-Izquierdo B Ubeda-Minarroand A F Gomez-Skarmeta ldquoHigh-integrity IMM-EKF-basedroad vehicle navigation with low-cost GPSSBASINSrdquo IEEETransactions on Intelligent Transportation Systems vol 8 no 3pp 491ndash511 2007

[22] O Payne and A Marrs ldquoAn unscented particle filter for GMTItrackingrdquo in Proceedings of the IEEE Aerospace ConferenceProceedings vol 3 pp 1869ndash1875 March 2004

[23] R Merwe A Doucet N Freitas and E Wan ldquoThe unscentedparticle filterrdquo Tech Rep CUEDF-INFENGTR 380 Cam-bridge University Engineering Department 2000

[24] W Quan J Fang F Xu andW Sheng ldquoHybrid simulation sys-tem study of SINSCNS integrated navigationrdquo IEEE Aerospaceand Electronic Systems Magazine vol 23 no 2 pp 17ndash24 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

systems which can be described by a mixing of continuousstates and discrete modes [17 18] The switching betweensubmodels is governed by a Markov chain The IMMUKFmethod includes interaction filtering model probabilityupdate and combination of four steps [19ndash21] as shown inFigure 3

421 Interaction Suppose the Markov model transition pro-bability matrix 119875 is as follows

119875 =

[

[

[

[

[

[

[

119901

11119901

12sdot sdot sdot 119901

1119873

119901

21

119901

22sdot sdot sdot

119901

2119873

119901

1198731119901

1198732sdot sdot sdot 119901

119873119873

]

]

]

]

]

]

]

119873

sum

119894

119901

119894119895=

119873

sum

119895

119901

119894119895= 1

(12)

where 119901119894119895is the a priori probability of transition from model

119894 to model 119895 The initial value of 119875 in this paper is set as 119901119894119895

=

091 with 119894 = 119895 and 119901

119894119895= 001 with 119894 = 119895

The mixing probabilities at time 119896 minus 1 are computed asfollows

120583

119894119895(119896 119896 minus 1) =

1

119888

119895

sdot 119901

119894119895sdot 120583

119894(119896 minus 1) (13)

where 120583

119894(119896 minus 1) is the predicted model probability 119888

119895=

sum

119873

119894119901

119894119895sdot 120583

119894(119896 minus 1) is normalization constant The mixed initial

conditions for each submodel 119895 are as follows

119883

119895(119896 119896 minus 1) =

119873

sum

119894

119883

119894(119896 minus 1) sdot 120583

119894119895(119896 119896 minus 1)

119875

119895(119896 119896 minus 1) =

119873

sum

119894

119875

119894(119896 minus 1)

+ [119883

119894(119896 minus 1) minus 119883

119895(119896 119896 minus 1)]

sdot [119883

119894(119896 minus 1) minus 119883

119895(119896 119896 minus 1)]

119879

times 120583

119894119895(119896 119896 minus 1)

(14)

422 Filtering Because the state model and measurementmodel described above are nonlinear the optimal state andcovariance estimates 119883

119895(119896) and 119875

119895(119896) for each sub-model 119895

are calculated using UKF [22 23]

423 Model Probability Update Consider the following

119903

119894(119896) = 119885 (119896) minus 119885

119894(119896 119896 minus 1)

119878

119894(119896) = 119864 [119903

119894(119896) 119903

119894(119896)

119879]

Λ

119894(119896) = 119873 (119903

119894(119896) 0 119878

119894(119896))

=

exp [minus (12) 119903

119894(119896)

119879119878

119894(119896) 119903

119894(119896)]

radic2120587

1003816

1003816

1003816

1003816

119878

119894(119896)

1003816

1003816

1003816

1003816

119888 =

119873

sum

119894

Λ

119894(119896) 119888

119894

120583

119894(119896) =

1

119888

Λ

119894(119896)

119873

sum

119894

119901

119894119895sdot 120583

119894(119896 minus 1) =

1

119888

Λ

119894(119896) 119888

119894

(15)

where 119903119894(119896) and 119878

119894(119896) are innovation vector and its covariance

respectively Λ

119894(119896) is Model likelihood 119873(119909) is Gaussian

distribution

424 Combination Combined state119883(119896) and its covariance119875(119896) are calculated from the weighted state estimates 119883

119894(119896)

and covariances 119875119894(119896) as follows

119883 (119896) =

119873

sum

119894

119883

119894(119896) sdot 120583

119894(119896) (16)

119875 (119896) =

119873

sum

119894

119875

119894(119896) + [119883

119894(119896) minus 119883 (119896)]

sdot[119883

119894(119896) minus 119883 (119896)]

119879

sdot 120583

119894(119896)

(17)

5 Simulation Results

This section presents simulation results to show the perfor-mance improvement of this new method compared with thetraditional direct calculation method and the filter methodThe process of simulation is shown in Figure 4

The ideal trajectory of spacecraft is created by the STK(Satellite Tool Kit) whose initial value of semimajor axis is193216365381 km eccentricity 0236386 inclination 23455∘right Ascension of the ascending node 0258∘ and the argu-ment of periapsis 71347∘ The near bodies used in the directcalculation method are Mars and Phobos All measurementsare created by the celestial navigation simulation platform[24] using ideal trajectory of STK ephemeris of DE405 andTycho-2 catalogue The ideal trajectory of the spacecraft isshown in Figure 5 which has been checked according toTisserandrsquos criterion described in (17) [18 20]

119879 =

119903mars119886

+ 2

radic

119886 (1 minus 119890

2)

119903marscos 119894 (18)

The Tisserand parameter 119879 is conserved around minus02686

before and after the close encounter with Mars

6 Mathematical Problems in Engineering

Sensorsimulator

Stars

Earth SunPhobos

and MarsHigh dynamic celestial simulation system

Sensor

Star-planet angles

Direct calculationmethod

Spacecraftposition

measurement

Orbit and attitude

Spacecraft motion simulationsystem based on STK

Orbit dynamicssimulation

Attitude dynamicssimulation

IMMUKF

Navigation computer

Positionand velocity

Demo system

Figure 4 The process of simulation

1332119864 + 007

6659119864 + 006

0

minus6659119864 + 006

minus1332119864 + 007

minus1333119864 + 007 minus6663119864 + 006 6663119864 + 006 1333119864 + 0070

Figure 5 The ideal trajectory of the spacecraft

The accuracy of the Sun sensor Phobosrsquo sensor the Earthsensor and Mars sensor are selected to be 510158401015840 510158401015840 1510158401015840 and0005∘ respectively The accuracy of the star sensor is 310158401015840

Figure 6 shows the performance of the direct calculationmethod which is obtained with a 1-minute sampling intervalduring the 20-day periodwhen the spacecraft is nearMars Asit can be seen from Figure 6 the navigation accuracy of thismethod mainly depends on the accuracy of measurement soit is very sensitive to the measurement noise The geometryrelationship of the spacecraft Mars and Phobos also hasan impact on the navigation performance The shorter thedistance between spacecraft andMars (or Phobos) the higherthe navigation accuracy is

Figure 7 shows the performance comparison betweenthis new method and the traditional filter method which

0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

1400

1600

1800

2000Position estimation error

Time (day)

Posit

ion

erro

r (km

)

Figure 6 The position estimation error of the direct calculationmethod

only uses the star-planet angles as measurement under thesame simulation conditions As demonstrated in Figure 7the traditional filter method can provide a good navigationperformance when the spacecraft is far from Mars and ahighly accurate model is available but its estimation errorincreases greatly when the spacecraft is close to Mars Thisis because the star-planet angles alone cannot supply enough

Mathematical Problems in Engineering 7

Table 1 Performance comparison of different methods

Method Position estimation error(RMS) (km)

Velocity estimation error(RMS) (ms)

Position estimation error(Max) (km)

Velocity estimation error(Max) (ms)

Direct calculation method 19250 mdash 18162 mdash

Filter method 71725 01507 15468 04937

New method 62082 01118 16787 04486

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160Position estimation error

Time (day)

Posit

ion

estim

atio

n er

ror (

km)

UKFIMMUKF

0 2 4 6 8 10 12 14 16 18 20Time (day)

UKFIMMUKF

0

1

2

3

4

5

6Innovation sequence

Inno

vatio

n se

quen

ce (r

ad)

minus1

times10minus7

Figure 7 Results comparison between IMMUKF and UKF

information to correct the large estimation error introducedby the inaccurate dynamical model which even cause thedivergence of the filter after then The innovation sequenceof the two methods which has subtracted the measurementnoise is shown in Figure 7 The figure of the innovationsequence also demonstrates the divergence of the traditionalfilter method However the new method presented in thispaper shows a stable and satisfying navigation performanceThe details are given in Table 1

From these results we see that this IMMUKF-basednew celestial navigation method overcomes the shortcom-ings of the direct calculation method and the traditionalfilter method On one hand when an accurate dynamicalmodel is available the impact of measurement error onthe navigation performance is reduced by the dynamicalmodel On the other hand when an accurate dynamicalmodel is not available the impact of system error on thenavigation performance is reduced by the result of directcalculation method The navigation accuracy and reliabilityare greatly improved by the effective combination of thedirect calculationmethod and the filtermethodThese resultsdemonstrate that this new method is a particularly suitablecelestial navigation method of spacecrafts on a gravity assisttrajectory

6 Conclusions

A new autonomous celestial navigation method for space-crafts on a gravity assist trajectory is studied which effectivelycombines the direct calculationmethod and the filtermethodbased on IMMUKF The navigation performance of this newmethod is tested by the ground experiments and a positionestimation error within 20 km is obtained Compared withthe traditional methods this method can achieve a higheraccuracy and stability These results verify that this methodis a promising and attractive scheme of autonomous navi-gation for the spacecraft using gravity assist A limitation ofthis study is the significant increase in computational costThis method should have a broad application potential inautonomous navigation of deep space probes especially incases when the accurate dynamical model is not availableHow to enhance the adaptive capability and to reduce thecomputational cost is under further investigation

Acknowledgments

The research presented in this paper has been supportedby the National Natural Science Foundation of China(61233005) Program for New Century Excellent Talents in

8 Mathematical Problems in Engineering

University (NCET) (NCET-11-0771) and Aerospace Scienceand Technology Innovation Fund (10300002012117003) Theauthors wish to express their gratitude to all members of theScience and Technology on Inertial Laboratory and Funda-mental Science on Novel Inertial Instrument amp NavigationSystem Technology Laboratory for their valuable comments

References

[1] P W Kinman ldquoDoppler tracking of planetary spacecraftrdquo IEEETransactions on Microwave Theory and Techniques vol 40 no6 pp 1199ndash1204 1992

[2] B M Portock L A DrsquoAmario and R J Haw ldquoMars explorationrover orbit determination using ΔVLBI datardquo in Proceedings ofthe AIAAAAS Astrodynamics Specialist Conference MontereyCalif USA August 2002

[3] M L Psiaki L Huang and S M Fox ldquoGround tests of magne-tometer-based autonomous navigation (MAGNAV) for low-earth-orbiting spacecraftrdquo Journal of Guidance Control andDynamics vol 16 no 1 pp 206ndash214 1993

[4] H Jung and M L Psiaki ldquoTests of magnetometersun-sensororbit determination using flight datardquo Journal of GuidanceControl and Dynamics vol 25 no 3 pp 582ndash590 2002

[5] A Jaggi U Hugentobler H Bock and G Beutler ldquoPrecise orbitdetermination for GRACE using undifferenced or doubly dif-ferenced GPS datardquo Advances in Space Research vol 39 no 10pp 1612ndash1619 2007

[6] H T Son C H Roger C S Wendy and W Terri ldquoHigh accu-racy autonomous navigation using the Global Positioning Sys-tem (GPS)rdquo in Proceeding of the 12th International Symposiumon Space Flight Dynamics pp 73ndash78 Darmstadt Germany June1997

[7] R H Battin An Introduction to the Mathematics and Methodsof Astrodynamics American Institute of Aeronautics and Astro-nautics New York NY USA 1999

[8] B Kaufman R Campion B Vorder andW Richard ldquoAn over-view of the astrodynamics for the Deep Space Program ScienceExperiment mission (DSPSE)rdquo in Proceedings of the 44th Inter-national Astronautical Congress (IAF rsquo93) pp 141ndash155 1993

[9] R R Dasenbrock ldquoDSPSE autonomous position estimationexperimentrdquo in Proceedings of the International Symposium onAdvances in the Astronautical Sciences pp 203ndash217 April 1993

[10] A Long D Leung D Folta and C Gramling ldquoAutonomousnavigation of high-earth satellites using celestial objects anddoppler measurementsrdquo in Proceedings of the AIAAAAS Astro-dynamis Speialist Conferene 2000

[11] J R Yim J L Grassidis and J L Junkins ldquoAutonomousorbit navigation of interplanetary spacecraftrdquo in The AmericanInstitute of Aeronautics and Astronautics Guidance Navigationand Control Conference pp 53ndash61 Denver Colo USA August2000

[12] C R H Solorzano A A Sukhanov and A F B A Prado ldquoAstudy of trajectories to theNeptune systemusing gravity assistsrdquoAdvances in Space Research vol 40 no 1 pp 125ndash133 2007

[13] J D Anderson J K Campbell and M M Nieto ldquoThe energytransfer process in planetary flybysrdquoNewAstronomy vol 12 no5 pp 383ndash397 2007

[14] R P Russell and C A Ocampo ldquoGeometric analysis of free-return trajectories following a gravity-assisted flybyrdquo Journal ofSpacecraft and Rockets vol 42 no 1 pp 138ndash151 2005

[15] W Hohmann ldquoThe availability of heavenly bodiesrdquo NASAReport NASA-TT-F-44 1960

[16] W E Wiesel Spaceflight Dynamics McGraw-Hill New YorkNY USA 1989

[17] H A P Blom and Y Bar-Shalom ldquoInteracting multiple modelalgorithm for systems with Markovian switching coefficientsrdquoIEEE Transactions on Automatic Control vol 33 no 8 pp 780ndash783 1988

[18] Y Bar-Shalom K C Chang and H A P Blom ldquoTracking amaneuvering target using input estimation versus the interact-ing multiple model algorithmrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 25 no 2 pp 296ndash300 1992

[19] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[20] Y Boers and J N Driessen ldquoInteractingmultiplemodel particlefilterrdquo IEE Proceedings Radar Sonar and Navigation vol 150no 5 pp 344ndash349 2003

[21] R Toledo-Moreo M A Zamora-Izquierdo B Ubeda-Minarroand A F Gomez-Skarmeta ldquoHigh-integrity IMM-EKF-basedroad vehicle navigation with low-cost GPSSBASINSrdquo IEEETransactions on Intelligent Transportation Systems vol 8 no 3pp 491ndash511 2007

[22] O Payne and A Marrs ldquoAn unscented particle filter for GMTItrackingrdquo in Proceedings of the IEEE Aerospace ConferenceProceedings vol 3 pp 1869ndash1875 March 2004

[23] R Merwe A Doucet N Freitas and E Wan ldquoThe unscentedparticle filterrdquo Tech Rep CUEDF-INFENGTR 380 Cam-bridge University Engineering Department 2000

[24] W Quan J Fang F Xu andW Sheng ldquoHybrid simulation sys-tem study of SINSCNS integrated navigationrdquo IEEE Aerospaceand Electronic Systems Magazine vol 23 no 2 pp 17ndash24 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Sensorsimulator

Stars

Earth SunPhobos

and MarsHigh dynamic celestial simulation system

Sensor

Star-planet angles

Direct calculationmethod

Spacecraftposition

measurement

Orbit and attitude

Spacecraft motion simulationsystem based on STK

Orbit dynamicssimulation

Attitude dynamicssimulation

IMMUKF

Navigation computer

Positionand velocity

Demo system

Figure 4 The process of simulation

1332119864 + 007

6659119864 + 006

0

minus6659119864 + 006

minus1332119864 + 007

minus1333119864 + 007 minus6663119864 + 006 6663119864 + 006 1333119864 + 0070

Figure 5 The ideal trajectory of the spacecraft

The accuracy of the Sun sensor Phobosrsquo sensor the Earthsensor and Mars sensor are selected to be 510158401015840 510158401015840 1510158401015840 and0005∘ respectively The accuracy of the star sensor is 310158401015840

Figure 6 shows the performance of the direct calculationmethod which is obtained with a 1-minute sampling intervalduring the 20-day periodwhen the spacecraft is nearMars Asit can be seen from Figure 6 the navigation accuracy of thismethod mainly depends on the accuracy of measurement soit is very sensitive to the measurement noise The geometryrelationship of the spacecraft Mars and Phobos also hasan impact on the navigation performance The shorter thedistance between spacecraft andMars (or Phobos) the higherthe navigation accuracy is

Figure 7 shows the performance comparison betweenthis new method and the traditional filter method which

0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

1400

1600

1800

2000Position estimation error

Time (day)

Posit

ion

erro

r (km

)

Figure 6 The position estimation error of the direct calculationmethod

only uses the star-planet angles as measurement under thesame simulation conditions As demonstrated in Figure 7the traditional filter method can provide a good navigationperformance when the spacecraft is far from Mars and ahighly accurate model is available but its estimation errorincreases greatly when the spacecraft is close to Mars Thisis because the star-planet angles alone cannot supply enough

Mathematical Problems in Engineering 7

Table 1 Performance comparison of different methods

Method Position estimation error(RMS) (km)

Velocity estimation error(RMS) (ms)

Position estimation error(Max) (km)

Velocity estimation error(Max) (ms)

Direct calculation method 19250 mdash 18162 mdash

Filter method 71725 01507 15468 04937

New method 62082 01118 16787 04486

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160Position estimation error

Time (day)

Posit

ion

estim

atio

n er

ror (

km)

UKFIMMUKF

0 2 4 6 8 10 12 14 16 18 20Time (day)

UKFIMMUKF

0

1

2

3

4

5

6Innovation sequence

Inno

vatio

n se

quen

ce (r

ad)

minus1

times10minus7

Figure 7 Results comparison between IMMUKF and UKF

information to correct the large estimation error introducedby the inaccurate dynamical model which even cause thedivergence of the filter after then The innovation sequenceof the two methods which has subtracted the measurementnoise is shown in Figure 7 The figure of the innovationsequence also demonstrates the divergence of the traditionalfilter method However the new method presented in thispaper shows a stable and satisfying navigation performanceThe details are given in Table 1

From these results we see that this IMMUKF-basednew celestial navigation method overcomes the shortcom-ings of the direct calculation method and the traditionalfilter method On one hand when an accurate dynamicalmodel is available the impact of measurement error onthe navigation performance is reduced by the dynamicalmodel On the other hand when an accurate dynamicalmodel is not available the impact of system error on thenavigation performance is reduced by the result of directcalculation method The navigation accuracy and reliabilityare greatly improved by the effective combination of thedirect calculationmethod and the filtermethodThese resultsdemonstrate that this new method is a particularly suitablecelestial navigation method of spacecrafts on a gravity assisttrajectory

6 Conclusions

A new autonomous celestial navigation method for space-crafts on a gravity assist trajectory is studied which effectivelycombines the direct calculationmethod and the filtermethodbased on IMMUKF The navigation performance of this newmethod is tested by the ground experiments and a positionestimation error within 20 km is obtained Compared withthe traditional methods this method can achieve a higheraccuracy and stability These results verify that this methodis a promising and attractive scheme of autonomous navi-gation for the spacecraft using gravity assist A limitation ofthis study is the significant increase in computational costThis method should have a broad application potential inautonomous navigation of deep space probes especially incases when the accurate dynamical model is not availableHow to enhance the adaptive capability and to reduce thecomputational cost is under further investigation

Acknowledgments

The research presented in this paper has been supportedby the National Natural Science Foundation of China(61233005) Program for New Century Excellent Talents in

8 Mathematical Problems in Engineering

University (NCET) (NCET-11-0771) and Aerospace Scienceand Technology Innovation Fund (10300002012117003) Theauthors wish to express their gratitude to all members of theScience and Technology on Inertial Laboratory and Funda-mental Science on Novel Inertial Instrument amp NavigationSystem Technology Laboratory for their valuable comments

References

[1] P W Kinman ldquoDoppler tracking of planetary spacecraftrdquo IEEETransactions on Microwave Theory and Techniques vol 40 no6 pp 1199ndash1204 1992

[2] B M Portock L A DrsquoAmario and R J Haw ldquoMars explorationrover orbit determination using ΔVLBI datardquo in Proceedings ofthe AIAAAAS Astrodynamics Specialist Conference MontereyCalif USA August 2002

[3] M L Psiaki L Huang and S M Fox ldquoGround tests of magne-tometer-based autonomous navigation (MAGNAV) for low-earth-orbiting spacecraftrdquo Journal of Guidance Control andDynamics vol 16 no 1 pp 206ndash214 1993

[4] H Jung and M L Psiaki ldquoTests of magnetometersun-sensororbit determination using flight datardquo Journal of GuidanceControl and Dynamics vol 25 no 3 pp 582ndash590 2002

[5] A Jaggi U Hugentobler H Bock and G Beutler ldquoPrecise orbitdetermination for GRACE using undifferenced or doubly dif-ferenced GPS datardquo Advances in Space Research vol 39 no 10pp 1612ndash1619 2007

[6] H T Son C H Roger C S Wendy and W Terri ldquoHigh accu-racy autonomous navigation using the Global Positioning Sys-tem (GPS)rdquo in Proceeding of the 12th International Symposiumon Space Flight Dynamics pp 73ndash78 Darmstadt Germany June1997

[7] R H Battin An Introduction to the Mathematics and Methodsof Astrodynamics American Institute of Aeronautics and Astro-nautics New York NY USA 1999

[8] B Kaufman R Campion B Vorder andW Richard ldquoAn over-view of the astrodynamics for the Deep Space Program ScienceExperiment mission (DSPSE)rdquo in Proceedings of the 44th Inter-national Astronautical Congress (IAF rsquo93) pp 141ndash155 1993

[9] R R Dasenbrock ldquoDSPSE autonomous position estimationexperimentrdquo in Proceedings of the International Symposium onAdvances in the Astronautical Sciences pp 203ndash217 April 1993

[10] A Long D Leung D Folta and C Gramling ldquoAutonomousnavigation of high-earth satellites using celestial objects anddoppler measurementsrdquo in Proceedings of the AIAAAAS Astro-dynamis Speialist Conferene 2000

[11] J R Yim J L Grassidis and J L Junkins ldquoAutonomousorbit navigation of interplanetary spacecraftrdquo in The AmericanInstitute of Aeronautics and Astronautics Guidance Navigationand Control Conference pp 53ndash61 Denver Colo USA August2000

[12] C R H Solorzano A A Sukhanov and A F B A Prado ldquoAstudy of trajectories to theNeptune systemusing gravity assistsrdquoAdvances in Space Research vol 40 no 1 pp 125ndash133 2007

[13] J D Anderson J K Campbell and M M Nieto ldquoThe energytransfer process in planetary flybysrdquoNewAstronomy vol 12 no5 pp 383ndash397 2007

[14] R P Russell and C A Ocampo ldquoGeometric analysis of free-return trajectories following a gravity-assisted flybyrdquo Journal ofSpacecraft and Rockets vol 42 no 1 pp 138ndash151 2005

[15] W Hohmann ldquoThe availability of heavenly bodiesrdquo NASAReport NASA-TT-F-44 1960

[16] W E Wiesel Spaceflight Dynamics McGraw-Hill New YorkNY USA 1989

[17] H A P Blom and Y Bar-Shalom ldquoInteracting multiple modelalgorithm for systems with Markovian switching coefficientsrdquoIEEE Transactions on Automatic Control vol 33 no 8 pp 780ndash783 1988

[18] Y Bar-Shalom K C Chang and H A P Blom ldquoTracking amaneuvering target using input estimation versus the interact-ing multiple model algorithmrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 25 no 2 pp 296ndash300 1992

[19] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[20] Y Boers and J N Driessen ldquoInteractingmultiplemodel particlefilterrdquo IEE Proceedings Radar Sonar and Navigation vol 150no 5 pp 344ndash349 2003

[21] R Toledo-Moreo M A Zamora-Izquierdo B Ubeda-Minarroand A F Gomez-Skarmeta ldquoHigh-integrity IMM-EKF-basedroad vehicle navigation with low-cost GPSSBASINSrdquo IEEETransactions on Intelligent Transportation Systems vol 8 no 3pp 491ndash511 2007

[22] O Payne and A Marrs ldquoAn unscented particle filter for GMTItrackingrdquo in Proceedings of the IEEE Aerospace ConferenceProceedings vol 3 pp 1869ndash1875 March 2004

[23] R Merwe A Doucet N Freitas and E Wan ldquoThe unscentedparticle filterrdquo Tech Rep CUEDF-INFENGTR 380 Cam-bridge University Engineering Department 2000

[24] W Quan J Fang F Xu andW Sheng ldquoHybrid simulation sys-tem study of SINSCNS integrated navigationrdquo IEEE Aerospaceand Electronic Systems Magazine vol 23 no 2 pp 17ndash24 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 Performance comparison of different methods

Method Position estimation error(RMS) (km)

Velocity estimation error(RMS) (ms)

Position estimation error(Max) (km)

Velocity estimation error(Max) (ms)

Direct calculation method 19250 mdash 18162 mdash

Filter method 71725 01507 15468 04937

New method 62082 01118 16787 04486

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160Position estimation error

Time (day)

Posit

ion

estim

atio

n er

ror (

km)

UKFIMMUKF

0 2 4 6 8 10 12 14 16 18 20Time (day)

UKFIMMUKF

0

1

2

3

4

5

6Innovation sequence

Inno

vatio

n se

quen

ce (r

ad)

minus1

times10minus7

Figure 7 Results comparison between IMMUKF and UKF

information to correct the large estimation error introducedby the inaccurate dynamical model which even cause thedivergence of the filter after then The innovation sequenceof the two methods which has subtracted the measurementnoise is shown in Figure 7 The figure of the innovationsequence also demonstrates the divergence of the traditionalfilter method However the new method presented in thispaper shows a stable and satisfying navigation performanceThe details are given in Table 1

From these results we see that this IMMUKF-basednew celestial navigation method overcomes the shortcom-ings of the direct calculation method and the traditionalfilter method On one hand when an accurate dynamicalmodel is available the impact of measurement error onthe navigation performance is reduced by the dynamicalmodel On the other hand when an accurate dynamicalmodel is not available the impact of system error on thenavigation performance is reduced by the result of directcalculation method The navigation accuracy and reliabilityare greatly improved by the effective combination of thedirect calculationmethod and the filtermethodThese resultsdemonstrate that this new method is a particularly suitablecelestial navigation method of spacecrafts on a gravity assisttrajectory

6 Conclusions

A new autonomous celestial navigation method for space-crafts on a gravity assist trajectory is studied which effectivelycombines the direct calculationmethod and the filtermethodbased on IMMUKF The navigation performance of this newmethod is tested by the ground experiments and a positionestimation error within 20 km is obtained Compared withthe traditional methods this method can achieve a higheraccuracy and stability These results verify that this methodis a promising and attractive scheme of autonomous navi-gation for the spacecraft using gravity assist A limitation ofthis study is the significant increase in computational costThis method should have a broad application potential inautonomous navigation of deep space probes especially incases when the accurate dynamical model is not availableHow to enhance the adaptive capability and to reduce thecomputational cost is under further investigation

Acknowledgments

The research presented in this paper has been supportedby the National Natural Science Foundation of China(61233005) Program for New Century Excellent Talents in

8 Mathematical Problems in Engineering

University (NCET) (NCET-11-0771) and Aerospace Scienceand Technology Innovation Fund (10300002012117003) Theauthors wish to express their gratitude to all members of theScience and Technology on Inertial Laboratory and Funda-mental Science on Novel Inertial Instrument amp NavigationSystem Technology Laboratory for their valuable comments

References

[1] P W Kinman ldquoDoppler tracking of planetary spacecraftrdquo IEEETransactions on Microwave Theory and Techniques vol 40 no6 pp 1199ndash1204 1992

[2] B M Portock L A DrsquoAmario and R J Haw ldquoMars explorationrover orbit determination using ΔVLBI datardquo in Proceedings ofthe AIAAAAS Astrodynamics Specialist Conference MontereyCalif USA August 2002

[3] M L Psiaki L Huang and S M Fox ldquoGround tests of magne-tometer-based autonomous navigation (MAGNAV) for low-earth-orbiting spacecraftrdquo Journal of Guidance Control andDynamics vol 16 no 1 pp 206ndash214 1993

[4] H Jung and M L Psiaki ldquoTests of magnetometersun-sensororbit determination using flight datardquo Journal of GuidanceControl and Dynamics vol 25 no 3 pp 582ndash590 2002

[5] A Jaggi U Hugentobler H Bock and G Beutler ldquoPrecise orbitdetermination for GRACE using undifferenced or doubly dif-ferenced GPS datardquo Advances in Space Research vol 39 no 10pp 1612ndash1619 2007

[6] H T Son C H Roger C S Wendy and W Terri ldquoHigh accu-racy autonomous navigation using the Global Positioning Sys-tem (GPS)rdquo in Proceeding of the 12th International Symposiumon Space Flight Dynamics pp 73ndash78 Darmstadt Germany June1997

[7] R H Battin An Introduction to the Mathematics and Methodsof Astrodynamics American Institute of Aeronautics and Astro-nautics New York NY USA 1999

[8] B Kaufman R Campion B Vorder andW Richard ldquoAn over-view of the astrodynamics for the Deep Space Program ScienceExperiment mission (DSPSE)rdquo in Proceedings of the 44th Inter-national Astronautical Congress (IAF rsquo93) pp 141ndash155 1993

[9] R R Dasenbrock ldquoDSPSE autonomous position estimationexperimentrdquo in Proceedings of the International Symposium onAdvances in the Astronautical Sciences pp 203ndash217 April 1993

[10] A Long D Leung D Folta and C Gramling ldquoAutonomousnavigation of high-earth satellites using celestial objects anddoppler measurementsrdquo in Proceedings of the AIAAAAS Astro-dynamis Speialist Conferene 2000

[11] J R Yim J L Grassidis and J L Junkins ldquoAutonomousorbit navigation of interplanetary spacecraftrdquo in The AmericanInstitute of Aeronautics and Astronautics Guidance Navigationand Control Conference pp 53ndash61 Denver Colo USA August2000

[12] C R H Solorzano A A Sukhanov and A F B A Prado ldquoAstudy of trajectories to theNeptune systemusing gravity assistsrdquoAdvances in Space Research vol 40 no 1 pp 125ndash133 2007

[13] J D Anderson J K Campbell and M M Nieto ldquoThe energytransfer process in planetary flybysrdquoNewAstronomy vol 12 no5 pp 383ndash397 2007

[14] R P Russell and C A Ocampo ldquoGeometric analysis of free-return trajectories following a gravity-assisted flybyrdquo Journal ofSpacecraft and Rockets vol 42 no 1 pp 138ndash151 2005

[15] W Hohmann ldquoThe availability of heavenly bodiesrdquo NASAReport NASA-TT-F-44 1960

[16] W E Wiesel Spaceflight Dynamics McGraw-Hill New YorkNY USA 1989

[17] H A P Blom and Y Bar-Shalom ldquoInteracting multiple modelalgorithm for systems with Markovian switching coefficientsrdquoIEEE Transactions on Automatic Control vol 33 no 8 pp 780ndash783 1988

[18] Y Bar-Shalom K C Chang and H A P Blom ldquoTracking amaneuvering target using input estimation versus the interact-ing multiple model algorithmrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 25 no 2 pp 296ndash300 1992

[19] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[20] Y Boers and J N Driessen ldquoInteractingmultiplemodel particlefilterrdquo IEE Proceedings Radar Sonar and Navigation vol 150no 5 pp 344ndash349 2003

[21] R Toledo-Moreo M A Zamora-Izquierdo B Ubeda-Minarroand A F Gomez-Skarmeta ldquoHigh-integrity IMM-EKF-basedroad vehicle navigation with low-cost GPSSBASINSrdquo IEEETransactions on Intelligent Transportation Systems vol 8 no 3pp 491ndash511 2007

[22] O Payne and A Marrs ldquoAn unscented particle filter for GMTItrackingrdquo in Proceedings of the IEEE Aerospace ConferenceProceedings vol 3 pp 1869ndash1875 March 2004

[23] R Merwe A Doucet N Freitas and E Wan ldquoThe unscentedparticle filterrdquo Tech Rep CUEDF-INFENGTR 380 Cam-bridge University Engineering Department 2000

[24] W Quan J Fang F Xu andW Sheng ldquoHybrid simulation sys-tem study of SINSCNS integrated navigationrdquo IEEE Aerospaceand Electronic Systems Magazine vol 23 no 2 pp 17ndash24 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

University (NCET) (NCET-11-0771) and Aerospace Scienceand Technology Innovation Fund (10300002012117003) Theauthors wish to express their gratitude to all members of theScience and Technology on Inertial Laboratory and Funda-mental Science on Novel Inertial Instrument amp NavigationSystem Technology Laboratory for their valuable comments

References

[1] P W Kinman ldquoDoppler tracking of planetary spacecraftrdquo IEEETransactions on Microwave Theory and Techniques vol 40 no6 pp 1199ndash1204 1992

[2] B M Portock L A DrsquoAmario and R J Haw ldquoMars explorationrover orbit determination using ΔVLBI datardquo in Proceedings ofthe AIAAAAS Astrodynamics Specialist Conference MontereyCalif USA August 2002

[3] M L Psiaki L Huang and S M Fox ldquoGround tests of magne-tometer-based autonomous navigation (MAGNAV) for low-earth-orbiting spacecraftrdquo Journal of Guidance Control andDynamics vol 16 no 1 pp 206ndash214 1993

[4] H Jung and M L Psiaki ldquoTests of magnetometersun-sensororbit determination using flight datardquo Journal of GuidanceControl and Dynamics vol 25 no 3 pp 582ndash590 2002

[5] A Jaggi U Hugentobler H Bock and G Beutler ldquoPrecise orbitdetermination for GRACE using undifferenced or doubly dif-ferenced GPS datardquo Advances in Space Research vol 39 no 10pp 1612ndash1619 2007

[6] H T Son C H Roger C S Wendy and W Terri ldquoHigh accu-racy autonomous navigation using the Global Positioning Sys-tem (GPS)rdquo in Proceeding of the 12th International Symposiumon Space Flight Dynamics pp 73ndash78 Darmstadt Germany June1997

[7] R H Battin An Introduction to the Mathematics and Methodsof Astrodynamics American Institute of Aeronautics and Astro-nautics New York NY USA 1999

[8] B Kaufman R Campion B Vorder andW Richard ldquoAn over-view of the astrodynamics for the Deep Space Program ScienceExperiment mission (DSPSE)rdquo in Proceedings of the 44th Inter-national Astronautical Congress (IAF rsquo93) pp 141ndash155 1993

[9] R R Dasenbrock ldquoDSPSE autonomous position estimationexperimentrdquo in Proceedings of the International Symposium onAdvances in the Astronautical Sciences pp 203ndash217 April 1993

[10] A Long D Leung D Folta and C Gramling ldquoAutonomousnavigation of high-earth satellites using celestial objects anddoppler measurementsrdquo in Proceedings of the AIAAAAS Astro-dynamis Speialist Conferene 2000

[11] J R Yim J L Grassidis and J L Junkins ldquoAutonomousorbit navigation of interplanetary spacecraftrdquo in The AmericanInstitute of Aeronautics and Astronautics Guidance Navigationand Control Conference pp 53ndash61 Denver Colo USA August2000

[12] C R H Solorzano A A Sukhanov and A F B A Prado ldquoAstudy of trajectories to theNeptune systemusing gravity assistsrdquoAdvances in Space Research vol 40 no 1 pp 125ndash133 2007

[13] J D Anderson J K Campbell and M M Nieto ldquoThe energytransfer process in planetary flybysrdquoNewAstronomy vol 12 no5 pp 383ndash397 2007

[14] R P Russell and C A Ocampo ldquoGeometric analysis of free-return trajectories following a gravity-assisted flybyrdquo Journal ofSpacecraft and Rockets vol 42 no 1 pp 138ndash151 2005

[15] W Hohmann ldquoThe availability of heavenly bodiesrdquo NASAReport NASA-TT-F-44 1960

[16] W E Wiesel Spaceflight Dynamics McGraw-Hill New YorkNY USA 1989

[17] H A P Blom and Y Bar-Shalom ldquoInteracting multiple modelalgorithm for systems with Markovian switching coefficientsrdquoIEEE Transactions on Automatic Control vol 33 no 8 pp 780ndash783 1988

[18] Y Bar-Shalom K C Chang and H A P Blom ldquoTracking amaneuvering target using input estimation versus the interact-ing multiple model algorithmrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 25 no 2 pp 296ndash300 1992

[19] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[20] Y Boers and J N Driessen ldquoInteractingmultiplemodel particlefilterrdquo IEE Proceedings Radar Sonar and Navigation vol 150no 5 pp 344ndash349 2003

[21] R Toledo-Moreo M A Zamora-Izquierdo B Ubeda-Minarroand A F Gomez-Skarmeta ldquoHigh-integrity IMM-EKF-basedroad vehicle navigation with low-cost GPSSBASINSrdquo IEEETransactions on Intelligent Transportation Systems vol 8 no 3pp 491ndash511 2007

[22] O Payne and A Marrs ldquoAn unscented particle filter for GMTItrackingrdquo in Proceedings of the IEEE Aerospace ConferenceProceedings vol 3 pp 1869ndash1875 March 2004

[23] R Merwe A Doucet N Freitas and E Wan ldquoThe unscentedparticle filterrdquo Tech Rep CUEDF-INFENGTR 380 Cam-bridge University Engineering Department 2000

[24] W Quan J Fang F Xu andW Sheng ldquoHybrid simulation sys-tem study of SINSCNS integrated navigationrdquo IEEE Aerospaceand Electronic Systems Magazine vol 23 no 2 pp 17ndash24 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of